VandenBroeck and Keller, Equation (6)

Percentage Accurate: 76.9% → 99.0%

Time: 38.2s

Alternatives: 11

Speedup: 1.0×

• could not determine a ground truth

  1. F = -552346281143277300.0
  2. l = -576726006623873.5

Specification

Math

\[\begin{array}{l} \\ \pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))

C

double code(double F, double l) {
	return (((double) M_PI) * l) - ((1.0 / (F * F)) * tan((((double) M_PI) * l)));
}

Java

public static double code(double F, double l) {
	return (Math.PI * l) - ((1.0 / (F * F)) * Math.tan((Math.PI * l)));
}

Python

def code(F, l):
	return (math.pi * l) - ((1.0 / (F * F)) * math.tan((math.pi * l)))

Julia

function code(F, l)
	return Float64(Float64(pi * l) - Float64(Float64(1.0 / Float64(F * F)) * tan(Float64(pi * l))))
end

MATLAB

function tmp = code(F, l)
	tmp = (pi * l) - ((1.0 / (F * F)) * tan((pi * l)));
end

Wolfram

code[F_, l_] := N[(N[(Pi * l), $MachinePrecision] - N[(N[(1.0 / N[(F * F), $MachinePrecision]), $MachinePrecision] * N[Tan[N[(Pi * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 76.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))

C

double code(double F, double l) {
	return (((double) M_PI) * l) - ((1.0 / (F * F)) * tan((((double) M_PI) * l)));
}

Java

public static double code(double F, double l) {
	return (Math.PI * l) - ((1.0 / (F * F)) * Math.tan((Math.PI * l)));
}

Python

def code(F, l):
	return (math.pi * l) - ((1.0 / (F * F)) * math.tan((math.pi * l)))

Julia

function code(F, l)
	return Float64(Float64(pi * l) - Float64(Float64(1.0 / Float64(F * F)) * tan(Float64(pi * l))))
end

MATLAB

function tmp = code(F, l)
	tmp = (pi * l) - ((1.0 / (F * F)) * tan((pi * l)));
end

Wolfram

code[F_, l_] := N[(N[(Pi * l), $MachinePrecision] - N[(N[(1.0 / N[(F * F), $MachinePrecision]), $MachinePrecision] * N[Tan[N[(Pi * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 1000000000000:\\ \;\;\;\;\pi \cdot l\_m - \frac{\frac{\tan \left(\pi \cdot l\_m\right)}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m + \frac{\sin \left(\pi \cdot l\_m\right)}{F} \cdot \frac{\frac{-1}{\mathsf{fma}\left({l\_m}^{2}, \mathsf{fma}\left({l\_m}^{2}, \mathsf{fma}\left(-0.001388888888888889, {l\_m}^{2} \cdot \mathsf{log1p}\left(\mathsf{expm1}\left({\pi}^{6}\right)\right), 0.041666666666666664 \cdot {\pi}^{4}\right), -0.5 \cdot {\pi}^{2}\right), 1\right)}}{F}\\ \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= (* PI l_m) 1000000000000.0)
    (- (* PI l_m) (/ (/ (tan (* PI l_m)) F) F))
    (+
     (* PI l_m)
     (*
      (/ (sin (* PI l_m)) F)
      (/
       (/
        -1.0
        (fma
         (pow l_m 2.0)
         (fma
          (pow l_m 2.0)
          (fma
           -0.001388888888888889
           (* (pow l_m 2.0) (log1p (expm1 (pow PI 6.0))))
           (* 0.041666666666666664 (pow PI 4.0)))
          (* -0.5 (pow PI 2.0)))
         1.0))
       F))))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if ((((double) M_PI) * l_m) <= 1000000000000.0) {
		tmp = (((double) M_PI) * l_m) - ((tan((((double) M_PI) * l_m)) / F) / F);
	} else {
		tmp = (((double) M_PI) * l_m) + ((sin((((double) M_PI) * l_m)) / F) * ((-1.0 / fma(pow(l_m, 2.0), fma(pow(l_m, 2.0), fma(-0.001388888888888889, (pow(l_m, 2.0) * log1p(expm1(pow(((double) M_PI), 6.0)))), (0.041666666666666664 * pow(((double) M_PI), 4.0))), (-0.5 * pow(((double) M_PI), 2.0))), 1.0)) / F));
	}
	return l_s * tmp;
}

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	tmp = 0.0
	if (Float64(pi * l_m) <= 1000000000000.0)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(tan(Float64(pi * l_m)) / F) / F));
	else
		tmp = Float64(Float64(pi * l_m) + Float64(Float64(sin(Float64(pi * l_m)) / F) * Float64(Float64(-1.0 / fma((l_m ^ 2.0), fma((l_m ^ 2.0), fma(-0.001388888888888889, Float64((l_m ^ 2.0) * log1p(expm1((pi ^ 6.0)))), Float64(0.041666666666666664 * (pi ^ 4.0))), Float64(-0.5 * (pi ^ 2.0))), 1.0)) / F)));
	end
	return Float64(l_s * tmp)
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * If[LessEqual[N[(Pi * l$95$m), $MachinePrecision], 1000000000000.0], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(N[Tan[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * l$95$m), $MachinePrecision] + N[(N[(N[Sin[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision] * N[(N[(-1.0 / N[(N[Power[l$95$m, 2.0], $MachinePrecision] * N[(N[Power[l$95$m, 2.0], $MachinePrecision] * N[(-0.001388888888888889 * N[(N[Power[l$95$m, 2.0], $MachinePrecision] * N[Log[1 + N[(Exp[N[Power[Pi, 6.0], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(0.041666666666666664 * N[Power[Pi, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 1e12

   1. Initial program 78.7%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l/ 78.8%

         \[\leadsto \pi \cdot \ell - \color{blue}{\frac{1 \cdot \tan \left(\pi \cdot \ell\right)}{F \cdot F}} \]

      2. *-un-lft-identity 78.8%

         \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\tan \left(\pi \cdot \ell\right)}}{F \cdot F} \]

      3. associate-/r* 84.9%

         \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}} \]

   4. Applied egg-rr 84.9%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}} \]

   if 1e12 < (*.f64 (PI.f64) l)

   1. Initial program 50.9%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   2. Step-by-step derivation

      1. *-commutative 50.9%

         \[\leadsto \pi \cdot \ell - \color{blue}{\tan \left(\pi \cdot \ell\right) \cdot \frac{1}{F \cdot F}} \]

      2. sqr-neg 50.9%

         \[\leadsto \pi \cdot \ell - \tan \left(\pi \cdot \ell\right) \cdot \frac{1}{\color{blue}{\left(-F\right) \cdot \left(-F\right)}} \]

      3. associate-*r/ 50.9%

         \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\tan \left(\pi \cdot \ell\right) \cdot 1}{\left(-F\right) \cdot \left(-F\right)}} \]

      4. sqr-neg 50.9%

         \[\leadsto \pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right) \cdot 1}{\color{blue}{F \cdot F}} \]

      5. *-rgt-identity 50.9%

         \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\tan \left(\pi \cdot \ell\right)}}{F \cdot F} \]

   3. Simplified 50.9%

      \[\leadsto \color{blue}{\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. tan-quot 50.9%

         \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\frac{\sin \left(\pi \cdot \ell\right)}{\cos \left(\pi \cdot \ell\right)}}}{F \cdot F} \]

      2. div-inv 50.9%

         \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\sin \left(\pi \cdot \ell\right) \cdot \frac{1}{\cos \left(\pi \cdot \ell\right)}}}{F \cdot F} \]

   6. Applied egg-rr 50.9%

      \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\sin \left(\pi \cdot \ell\right) \cdot \frac{1}{\cos \left(\pi \cdot \ell\right)}}}{F \cdot F} \]

   7. Taylor expanded in l around 0 60.7%

      \[\leadsto \pi \cdot \ell - \frac{\sin \left(\pi \cdot \ell\right) \cdot \frac{1}{\color{blue}{1 + {\ell}^{2} \cdot \left(-0.5 \cdot {\pi}^{2} + {\ell}^{2} \cdot \left(-0.001388888888888889 \cdot \left({\ell}^{2} \cdot {\pi}^{6}\right) + 0.041666666666666664 \cdot {\pi}^{4}\right)\right)}}}{F \cdot F} \]
   8. Step-by-step derivation

      1. times-frac 88.2%

         \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\sin \left(\pi \cdot \ell\right)}{F} \cdot \frac{\frac{1}{1 + {\ell}^{2} \cdot \left(-0.5 \cdot {\pi}^{2} + {\ell}^{2} \cdot \left(-0.001388888888888889 \cdot \left({\ell}^{2} \cdot {\pi}^{6}\right) + 0.041666666666666664 \cdot {\pi}^{4}\right)\right)}}{F}} \]

   9. Applied egg-rr 88.2%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\sin \left(\pi \cdot \ell\right)}{F} \cdot \frac{\frac{1}{\mathsf{fma}\left({\ell}^{2}, \mathsf{fma}\left({\ell}^{2}, \mathsf{fma}\left(-0.001388888888888889, {\ell}^{2} \cdot {\pi}^{6}, 0.041666666666666664 \cdot {\pi}^{4}\right), -0.5 \cdot {\pi}^{2}\right), 1\right)}}{F}} \]
   10. Step-by-step derivation

       1. log1p-expm1-u 99.5%

          \[\leadsto \pi \cdot \ell - \frac{\sin \left(\pi \cdot \ell\right)}{F} \cdot \frac{\frac{1}{\mathsf{fma}\left({\ell}^{2}, \mathsf{fma}\left({\ell}^{2}, \mathsf{fma}\left(-0.001388888888888889, {\ell}^{2} \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left({\pi}^{6}\right)\right)}, 0.041666666666666664 \cdot {\pi}^{4}\right), -0.5 \cdot {\pi}^{2}\right), 1\right)}}{F} \]

   11. Applied egg-rr 99.5%

       \[\leadsto \pi \cdot \ell - \frac{\sin \left(\pi \cdot \ell\right)}{F} \cdot \frac{\frac{1}{\mathsf{fma}\left({\ell}^{2}, \mathsf{fma}\left({\ell}^{2}, \mathsf{fma}\left(-0.001388888888888889, {\ell}^{2} \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left({\pi}^{6}\right)\right)}, 0.041666666666666664 \cdot {\pi}^{4}\right), -0.5 \cdot {\pi}^{2}\right), 1\right)}}{F} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 1000000000000:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell + \frac{\sin \left(\pi \cdot \ell\right)}{F} \cdot \frac{\frac{-1}{\mathsf{fma}\left({\ell}^{2}, \mathsf{fma}\left({\ell}^{2}, \mathsf{fma}\left(-0.001388888888888889, {\ell}^{2} \cdot \mathsf{log1p}\left(\mathsf{expm1}\left({\pi}^{6}\right)\right), 0.041666666666666664 \cdot {\pi}^{4}\right), -0.5 \cdot {\pi}^{2}\right), 1\right)}}{F}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \left(\pi \cdot l\_m + \frac{\sin \left(\pi \cdot l\_m\right)}{F} \cdot \frac{\frac{-1}{\mathsf{fma}\left({l\_m}^{2}, \mathsf{fma}\left({l\_m}^{2}, \mathsf{fma}\left(-0.001388888888888889, {l\_m}^{2} \cdot {\pi}^{6}, 0.041666666666666664 \cdot {\pi}^{4}\right), -0.5 \cdot {\pi}^{2}\right), 1\right)}}{F}\right) \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (+
   (* PI l_m)
   (*
    (/ (sin (* PI l_m)) F)
    (/
     (/
      -1.0
      (fma
       (pow l_m 2.0)
       (fma
        (pow l_m 2.0)
        (fma
         -0.001388888888888889
         (* (pow l_m 2.0) (pow PI 6.0))
         (* 0.041666666666666664 (pow PI 4.0)))
        (* -0.5 (pow PI 2.0)))
       1.0))
     F)))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	return l_s * ((((double) M_PI) * l_m) + ((sin((((double) M_PI) * l_m)) / F) * ((-1.0 / fma(pow(l_m, 2.0), fma(pow(l_m, 2.0), fma(-0.001388888888888889, (pow(l_m, 2.0) * pow(((double) M_PI), 6.0)), (0.041666666666666664 * pow(((double) M_PI), 4.0))), (-0.5 * pow(((double) M_PI), 2.0))), 1.0)) / F)));
}

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	return Float64(l_s * Float64(Float64(pi * l_m) + Float64(Float64(sin(Float64(pi * l_m)) / F) * Float64(Float64(-1.0 / fma((l_m ^ 2.0), fma((l_m ^ 2.0), fma(-0.001388888888888889, Float64((l_m ^ 2.0) * (pi ^ 6.0)), Float64(0.041666666666666664 * (pi ^ 4.0))), Float64(-0.5 * (pi ^ 2.0))), 1.0)) / F))))
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * N[(N[(Pi * l$95$m), $MachinePrecision] + N[(N[(N[Sin[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision] * N[(N[(-1.0 / N[(N[Power[l$95$m, 2.0], $MachinePrecision] * N[(N[Power[l$95$m, 2.0], $MachinePrecision] * N[(-0.001388888888888889 * N[(N[Power[l$95$m, 2.0], $MachinePrecision] * N[Power[Pi, 6.0], $MachinePrecision]), $MachinePrecision] + N[(0.041666666666666664 * N[Power[Pi, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.4%

   \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
2. Step-by-step derivation

   1. *-commutative 72.4%

      \[\leadsto \pi \cdot \ell - \color{blue}{\tan \left(\pi \cdot \ell\right) \cdot \frac{1}{F \cdot F}} \]

   2. sqr-neg 72.4%

      \[\leadsto \pi \cdot \ell - \tan \left(\pi \cdot \ell\right) \cdot \frac{1}{\color{blue}{\left(-F\right) \cdot \left(-F\right)}} \]

   3. associate-*r/ 72.5%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\tan \left(\pi \cdot \ell\right) \cdot 1}{\left(-F\right) \cdot \left(-F\right)}} \]

   4. sqr-neg 72.5%

      \[\leadsto \pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right) \cdot 1}{\color{blue}{F \cdot F}} \]

   5. *-rgt-identity 72.5%

      \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\tan \left(\pi \cdot \ell\right)}}{F \cdot F} \]

3. Simplified 72.5%

   \[\leadsto \color{blue}{\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. tan-quot 72.5%

      \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\frac{\sin \left(\pi \cdot \ell\right)}{\cos \left(\pi \cdot \ell\right)}}}{F \cdot F} \]

   2. div-inv 72.5%

      \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\sin \left(\pi \cdot \ell\right) \cdot \frac{1}{\cos \left(\pi \cdot \ell\right)}}}{F \cdot F} \]

6. Applied egg-rr 72.5%

   \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\sin \left(\pi \cdot \ell\right) \cdot \frac{1}{\cos \left(\pi \cdot \ell\right)}}}{F \cdot F} \]

7. Taylor expanded in l around 0 77.3%

   \[\leadsto \pi \cdot \ell - \frac{\sin \left(\pi \cdot \ell\right) \cdot \frac{1}{\color{blue}{1 + {\ell}^{2} \cdot \left(-0.5 \cdot {\pi}^{2} + {\ell}^{2} \cdot \left(-0.001388888888888889 \cdot \left({\ell}^{2} \cdot {\pi}^{6}\right) + 0.041666666666666664 \cdot {\pi}^{4}\right)\right)}}}{F \cdot F} \]
8. Step-by-step derivation

   1. times-frac 94.1%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\sin \left(\pi \cdot \ell\right)}{F} \cdot \frac{\frac{1}{1 + {\ell}^{2} \cdot \left(-0.5 \cdot {\pi}^{2} + {\ell}^{2} \cdot \left(-0.001388888888888889 \cdot \left({\ell}^{2} \cdot {\pi}^{6}\right) + 0.041666666666666664 \cdot {\pi}^{4}\right)\right)}}{F}} \]

9. Applied egg-rr 94.1%

   \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\sin \left(\pi \cdot \ell\right)}{F} \cdot \frac{\frac{1}{\mathsf{fma}\left({\ell}^{2}, \mathsf{fma}\left({\ell}^{2}, \mathsf{fma}\left(-0.001388888888888889, {\ell}^{2} \cdot {\pi}^{6}, 0.041666666666666664 \cdot {\pi}^{4}\right), -0.5 \cdot {\pi}^{2}\right), 1\right)}}{F}} \]

10. Final simplification 94.1%

    \[\leadsto \pi \cdot \ell + \frac{\sin \left(\pi \cdot \ell\right)}{F} \cdot \frac{\frac{-1}{\mathsf{fma}\left({\ell}^{2}, \mathsf{fma}\left({\ell}^{2}, \mathsf{fma}\left(-0.001388888888888889, {\ell}^{2} \cdot {\pi}^{6}, 0.041666666666666664 \cdot {\pi}^{4}\right), -0.5 \cdot {\pi}^{2}\right), 1\right)}}{F} \]
11. Add Preprocessing

Alternative 3: 88.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 4 \cdot 10^{+33}:\\ \;\;\;\;\pi \cdot l\_m - \frac{\frac{\tan \left(\pi \cdot l\_m\right)}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m + \frac{\sin \left(\pi \cdot l\_m\right) \cdot \frac{-1}{1 + {l\_m}^{2} \cdot \left(-0.5 \cdot {\pi}^{2} + {l\_m}^{2} \cdot \left(0.041666666666666664 \cdot {\pi}^{4} + -0.001388888888888889 \cdot \left({l\_m}^{2} \cdot \mathsf{log1p}\left(\mathsf{expm1}\left({\pi}^{6}\right)\right)\right)\right)\right)}}{F \cdot F}\\ \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= (* PI l_m) 4e+33)
    (- (* PI l_m) (/ (/ (tan (* PI l_m)) F) F))
    (+
     (* PI l_m)
     (/
      (*
       (sin (* PI l_m))
       (/
        -1.0
        (+
         1.0
         (*
          (pow l_m 2.0)
          (+
           (* -0.5 (pow PI 2.0))
           (*
            (pow l_m 2.0)
            (+
             (* 0.041666666666666664 (pow PI 4.0))
             (*
              -0.001388888888888889
              (* (pow l_m 2.0) (log1p (expm1 (pow PI 6.0))))))))))))
      (* F F))))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if ((((double) M_PI) * l_m) <= 4e+33) {
		tmp = (((double) M_PI) * l_m) - ((tan((((double) M_PI) * l_m)) / F) / F);
	} else {
		tmp = (((double) M_PI) * l_m) + ((sin((((double) M_PI) * l_m)) * (-1.0 / (1.0 + (pow(l_m, 2.0) * ((-0.5 * pow(((double) M_PI), 2.0)) + (pow(l_m, 2.0) * ((0.041666666666666664 * pow(((double) M_PI), 4.0)) + (-0.001388888888888889 * (pow(l_m, 2.0) * log1p(expm1(pow(((double) M_PI), 6.0)))))))))))) / (F * F));
	}
	return l_s * tmp;
}

Java

l\_m = Math.abs(l);
l\_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	double tmp;
	if ((Math.PI * l_m) <= 4e+33) {
		tmp = (Math.PI * l_m) - ((Math.tan((Math.PI * l_m)) / F) / F);
	} else {
		tmp = (Math.PI * l_m) + ((Math.sin((Math.PI * l_m)) * (-1.0 / (1.0 + (Math.pow(l_m, 2.0) * ((-0.5 * Math.pow(Math.PI, 2.0)) + (Math.pow(l_m, 2.0) * ((0.041666666666666664 * Math.pow(Math.PI, 4.0)) + (-0.001388888888888889 * (Math.pow(l_m, 2.0) * Math.log1p(Math.expm1(Math.pow(Math.PI, 6.0)))))))))))) / (F * F));
	}
	return l_s * tmp;
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	tmp = 0
	if (math.pi * l_m) <= 4e+33:
		tmp = (math.pi * l_m) - ((math.tan((math.pi * l_m)) / F) / F)
	else:
		tmp = (math.pi * l_m) + ((math.sin((math.pi * l_m)) * (-1.0 / (1.0 + (math.pow(l_m, 2.0) * ((-0.5 * math.pow(math.pi, 2.0)) + (math.pow(l_m, 2.0) * ((0.041666666666666664 * math.pow(math.pi, 4.0)) + (-0.001388888888888889 * (math.pow(l_m, 2.0) * math.log1p(math.expm1(math.pow(math.pi, 6.0)))))))))))) / (F * F))
	return l_s * tmp

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	tmp = 0.0
	if (Float64(pi * l_m) <= 4e+33)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(tan(Float64(pi * l_m)) / F) / F));
	else
		tmp = Float64(Float64(pi * l_m) + Float64(Float64(sin(Float64(pi * l_m)) * Float64(-1.0 / Float64(1.0 + Float64((l_m ^ 2.0) * Float64(Float64(-0.5 * (pi ^ 2.0)) + Float64((l_m ^ 2.0) * Float64(Float64(0.041666666666666664 * (pi ^ 4.0)) + Float64(-0.001388888888888889 * Float64((l_m ^ 2.0) * log1p(expm1((pi ^ 6.0)))))))))))) / Float64(F * F)));
	end
	return Float64(l_s * tmp)
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * If[LessEqual[N[(Pi * l$95$m), $MachinePrecision], 4e+33], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(N[Tan[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * l$95$m), $MachinePrecision] + N[(N[(N[Sin[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / N[(1.0 + N[(N[Power[l$95$m, 2.0], $MachinePrecision] * N[(N[(-0.5 * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[Power[l$95$m, 2.0], $MachinePrecision] * N[(N[(0.041666666666666664 * N[Power[Pi, 4.0], $MachinePrecision]), $MachinePrecision] + N[(-0.001388888888888889 * N[(N[Power[l$95$m, 2.0], $MachinePrecision] * N[Log[1 + N[(Exp[N[Power[Pi, 6.0], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 3.9999999999999998e33

   1. Initial program 77.4%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l/ 77.4%

         \[\leadsto \pi \cdot \ell - \color{blue}{\frac{1 \cdot \tan \left(\pi \cdot \ell\right)}{F \cdot F}} \]

      2. *-un-lft-identity 77.4%

         \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\tan \left(\pi \cdot \ell\right)}}{F \cdot F} \]

      3. associate-/r* 83.4%

         \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}} \]

   4. Applied egg-rr 83.4%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}} \]

   if 3.9999999999999998e33 < (*.f64 (PI.f64) l)

   1. Initial program 52.9%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   2. Step-by-step derivation

      1. *-commutative 52.9%

         \[\leadsto \pi \cdot \ell - \color{blue}{\tan \left(\pi \cdot \ell\right) \cdot \frac{1}{F \cdot F}} \]

      2. sqr-neg 52.9%

         \[\leadsto \pi \cdot \ell - \tan \left(\pi \cdot \ell\right) \cdot \frac{1}{\color{blue}{\left(-F\right) \cdot \left(-F\right)}} \]

      3. associate-*r/ 52.9%

         \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\tan \left(\pi \cdot \ell\right) \cdot 1}{\left(-F\right) \cdot \left(-F\right)}} \]

      4. sqr-neg 52.9%

         \[\leadsto \pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right) \cdot 1}{\color{blue}{F \cdot F}} \]

      5. *-rgt-identity 52.9%

         \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\tan \left(\pi \cdot \ell\right)}}{F \cdot F} \]

   3. Simplified 52.9%

      \[\leadsto \color{blue}{\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. tan-quot 52.9%

         \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\frac{\sin \left(\pi \cdot \ell\right)}{\cos \left(\pi \cdot \ell\right)}}}{F \cdot F} \]

      2. div-inv 52.9%

         \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\sin \left(\pi \cdot \ell\right) \cdot \frac{1}{\cos \left(\pi \cdot \ell\right)}}}{F \cdot F} \]

   6. Applied egg-rr 52.9%

      \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\sin \left(\pi \cdot \ell\right) \cdot \frac{1}{\cos \left(\pi \cdot \ell\right)}}}{F \cdot F} \]

   7. Taylor expanded in l around 0 63.7%

      \[\leadsto \pi \cdot \ell - \frac{\sin \left(\pi \cdot \ell\right) \cdot \frac{1}{\color{blue}{1 + {\ell}^{2} \cdot \left(-0.5 \cdot {\pi}^{2} + {\ell}^{2} \cdot \left(-0.001388888888888889 \cdot \left({\ell}^{2} \cdot {\pi}^{6}\right) + 0.041666666666666664 \cdot {\pi}^{4}\right)\right)}}}{F \cdot F} \]
   8. Step-by-step derivation

      1. log1p-expm1-u 99.5%

         \[\leadsto \pi \cdot \ell - \frac{\sin \left(\pi \cdot \ell\right)}{F} \cdot \frac{\frac{1}{\mathsf{fma}\left({\ell}^{2}, \mathsf{fma}\left({\ell}^{2}, \mathsf{fma}\left(-0.001388888888888889, {\ell}^{2} \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left({\pi}^{6}\right)\right)}, 0.041666666666666664 \cdot {\pi}^{4}\right), -0.5 \cdot {\pi}^{2}\right), 1\right)}}{F} \]

   9. Applied egg-rr 68.9%

      \[\leadsto \pi \cdot \ell - \frac{\sin \left(\pi \cdot \ell\right) \cdot \frac{1}{1 + {\ell}^{2} \cdot \left(-0.5 \cdot {\pi}^{2} + {\ell}^{2} \cdot \left(-0.001388888888888889 \cdot \left({\ell}^{2} \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left({\pi}^{6}\right)\right)}\right) + 0.041666666666666664 \cdot {\pi}^{4}\right)\right)}}{F \cdot F} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 4 \cdot 10^{+33}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell + \frac{\sin \left(\pi \cdot \ell\right) \cdot \frac{-1}{1 + {\ell}^{2} \cdot \left(-0.5 \cdot {\pi}^{2} + {\ell}^{2} \cdot \left(0.041666666666666664 \cdot {\pi}^{4} + -0.001388888888888889 \cdot \left({\ell}^{2} \cdot \mathsf{log1p}\left(\mathsf{expm1}\left({\pi}^{6}\right)\right)\right)\right)\right)}}{F \cdot F}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 10^{+18}:\\ \;\;\;\;\pi \cdot l\_m - \frac{\frac{\tan \left(\pi \cdot l\_m\right)}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m - \frac{\sin \left(\pi \cdot l\_m\right) \cdot -720}{{F}^{2} \cdot \left({\pi}^{6} \cdot {l\_m}^{6}\right)}\\ \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= (* PI l_m) 1e+18)
    (- (* PI l_m) (/ (/ (tan (* PI l_m)) F) F))
    (-
     (* PI l_m)
     (/
      (* (sin (* PI l_m)) -720.0)
      (* (pow F 2.0) (* (pow PI 6.0) (pow l_m 6.0))))))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if ((((double) M_PI) * l_m) <= 1e+18) {
		tmp = (((double) M_PI) * l_m) - ((tan((((double) M_PI) * l_m)) / F) / F);
	} else {
		tmp = (((double) M_PI) * l_m) - ((sin((((double) M_PI) * l_m)) * -720.0) / (pow(F, 2.0) * (pow(((double) M_PI), 6.0) * pow(l_m, 6.0))));
	}
	return l_s * tmp;
}

Java

l\_m = Math.abs(l);
l\_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	double tmp;
	if ((Math.PI * l_m) <= 1e+18) {
		tmp = (Math.PI * l_m) - ((Math.tan((Math.PI * l_m)) / F) / F);
	} else {
		tmp = (Math.PI * l_m) - ((Math.sin((Math.PI * l_m)) * -720.0) / (Math.pow(F, 2.0) * (Math.pow(Math.PI, 6.0) * Math.pow(l_m, 6.0))));
	}
	return l_s * tmp;
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	tmp = 0
	if (math.pi * l_m) <= 1e+18:
		tmp = (math.pi * l_m) - ((math.tan((math.pi * l_m)) / F) / F)
	else:
		tmp = (math.pi * l_m) - ((math.sin((math.pi * l_m)) * -720.0) / (math.pow(F, 2.0) * (math.pow(math.pi, 6.0) * math.pow(l_m, 6.0))))
	return l_s * tmp

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	tmp = 0.0
	if (Float64(pi * l_m) <= 1e+18)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(tan(Float64(pi * l_m)) / F) / F));
	else
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(sin(Float64(pi * l_m)) * -720.0) / Float64((F ^ 2.0) * Float64((pi ^ 6.0) * (l_m ^ 6.0)))));
	end
	return Float64(l_s * tmp)
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp_2 = code(l_s, F, l_m)
	tmp = 0.0;
	if ((pi * l_m) <= 1e+18)
		tmp = (pi * l_m) - ((tan((pi * l_m)) / F) / F);
	else
		tmp = (pi * l_m) - ((sin((pi * l_m)) * -720.0) / ((F ^ 2.0) * ((pi ^ 6.0) * (l_m ^ 6.0))));
	end
	tmp_2 = l_s * tmp;
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * If[LessEqual[N[(Pi * l$95$m), $MachinePrecision], 1e+18], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(N[Tan[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(N[Sin[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision] * -720.0), $MachinePrecision] / N[(N[Power[F, 2.0], $MachinePrecision] * N[(N[Power[Pi, 6.0], $MachinePrecision] * N[Power[l$95$m, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 1e18

   1. Initial program 78.7%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l/ 78.8%

         \[\leadsto \pi \cdot \ell - \color{blue}{\frac{1 \cdot \tan \left(\pi \cdot \ell\right)}{F \cdot F}} \]

      2. *-un-lft-identity 78.8%

         \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\tan \left(\pi \cdot \ell\right)}}{F \cdot F} \]

      3. associate-/r* 84.9%

         \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}} \]

   4. Applied egg-rr 84.9%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}} \]

   if 1e18 < (*.f64 (PI.f64) l)

   1. Initial program 50.9%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   2. Step-by-step derivation

      1. *-commutative 50.9%

         \[\leadsto \pi \cdot \ell - \color{blue}{\tan \left(\pi \cdot \ell\right) \cdot \frac{1}{F \cdot F}} \]

      2. sqr-neg 50.9%

         \[\leadsto \pi \cdot \ell - \tan \left(\pi \cdot \ell\right) \cdot \frac{1}{\color{blue}{\left(-F\right) \cdot \left(-F\right)}} \]

      3. associate-*r/ 50.9%

         \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\tan \left(\pi \cdot \ell\right) \cdot 1}{\left(-F\right) \cdot \left(-F\right)}} \]

      4. sqr-neg 50.9%

         \[\leadsto \pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right) \cdot 1}{\color{blue}{F \cdot F}} \]

      5. *-rgt-identity 50.9%

         \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\tan \left(\pi \cdot \ell\right)}}{F \cdot F} \]

   3. Simplified 50.9%

      \[\leadsto \color{blue}{\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. tan-quot 50.9%

         \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\frac{\sin \left(\pi \cdot \ell\right)}{\cos \left(\pi \cdot \ell\right)}}}{F \cdot F} \]

      2. div-inv 50.9%

         \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\sin \left(\pi \cdot \ell\right) \cdot \frac{1}{\cos \left(\pi \cdot \ell\right)}}}{F \cdot F} \]

   6. Applied egg-rr 50.9%

      \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\sin \left(\pi \cdot \ell\right) \cdot \frac{1}{\cos \left(\pi \cdot \ell\right)}}}{F \cdot F} \]

   7. Taylor expanded in l around 0 60.7%

      \[\leadsto \pi \cdot \ell - \frac{\sin \left(\pi \cdot \ell\right) \cdot \frac{1}{\color{blue}{1 + {\ell}^{2} \cdot \left(-0.5 \cdot {\pi}^{2} + {\ell}^{2} \cdot \left(-0.001388888888888889 \cdot \left({\ell}^{2} \cdot {\pi}^{6}\right) + 0.041666666666666664 \cdot {\pi}^{4}\right)\right)}}}{F \cdot F} \]

   8. Taylor expanded in l around inf 60.7%

      \[\leadsto \pi \cdot \ell - \color{blue}{-720 \cdot \frac{\sin \left(\ell \cdot \pi\right)}{{F}^{2} \cdot \left({\ell}^{6} \cdot {\pi}^{6}\right)}} \]
   9. Step-by-step derivation

      1. associate-*r/ 60.7%

         \[\leadsto \pi \cdot \ell - \color{blue}{\frac{-720 \cdot \sin \left(\ell \cdot \pi\right)}{{F}^{2} \cdot \left({\ell}^{6} \cdot {\pi}^{6}\right)}} \]

   10. Simplified 60.7%

       \[\leadsto \pi \cdot \ell - \color{blue}{\frac{-720 \cdot \sin \left(\ell \cdot \pi\right)}{{F}^{2} \cdot \left({\ell}^{6} \cdot {\pi}^{6}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 10^{+18}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sin \left(\pi \cdot \ell\right) \cdot -720}{{F}^{2} \cdot \left({\pi}^{6} \cdot {\ell}^{6}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 10^{+18}:\\ \;\;\;\;\pi \cdot l\_m - \frac{\frac{\tan \left(\pi \cdot l\_m\right)}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m + \frac{\sin \left(\pi \cdot l\_m\right) \cdot \frac{-1}{\mathsf{fma}\left(-0.5, {\left(\pi \cdot l\_m\right)}^{2}, 1\right)}}{F \cdot F}\\ \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= (* PI l_m) 1e+18)
    (- (* PI l_m) (/ (/ (tan (* PI l_m)) F) F))
    (+
     (* PI l_m)
     (/
      (* (sin (* PI l_m)) (/ -1.0 (fma -0.5 (pow (* PI l_m) 2.0) 1.0)))
      (* F F))))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if ((((double) M_PI) * l_m) <= 1e+18) {
		tmp = (((double) M_PI) * l_m) - ((tan((((double) M_PI) * l_m)) / F) / F);
	} else {
		tmp = (((double) M_PI) * l_m) + ((sin((((double) M_PI) * l_m)) * (-1.0 / fma(-0.5, pow((((double) M_PI) * l_m), 2.0), 1.0))) / (F * F));
	}
	return l_s * tmp;
}

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	tmp = 0.0
	if (Float64(pi * l_m) <= 1e+18)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(tan(Float64(pi * l_m)) / F) / F));
	else
		tmp = Float64(Float64(pi * l_m) + Float64(Float64(sin(Float64(pi * l_m)) * Float64(-1.0 / fma(-0.5, (Float64(pi * l_m) ^ 2.0), 1.0))) / Float64(F * F)));
	end
	return Float64(l_s * tmp)
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * If[LessEqual[N[(Pi * l$95$m), $MachinePrecision], 1e+18], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(N[Tan[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * l$95$m), $MachinePrecision] + N[(N[(N[Sin[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / N[(-0.5 * N[Power[N[(Pi * l$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 1e18

   1. Initial program 78.7%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l/ 78.8%

         \[\leadsto \pi \cdot \ell - \color{blue}{\frac{1 \cdot \tan \left(\pi \cdot \ell\right)}{F \cdot F}} \]

      2. *-un-lft-identity 78.8%

         \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\tan \left(\pi \cdot \ell\right)}}{F \cdot F} \]

      3. associate-/r* 84.9%

         \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}} \]

   4. Applied egg-rr 84.9%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}} \]

   if 1e18 < (*.f64 (PI.f64) l)

   1. Initial program 50.9%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   2. Step-by-step derivation

      1. *-commutative 50.9%

         \[\leadsto \pi \cdot \ell - \color{blue}{\tan \left(\pi \cdot \ell\right) \cdot \frac{1}{F \cdot F}} \]

      2. sqr-neg 50.9%

         \[\leadsto \pi \cdot \ell - \tan \left(\pi \cdot \ell\right) \cdot \frac{1}{\color{blue}{\left(-F\right) \cdot \left(-F\right)}} \]

      3. associate-*r/ 50.9%

         \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\tan \left(\pi \cdot \ell\right) \cdot 1}{\left(-F\right) \cdot \left(-F\right)}} \]

      4. sqr-neg 50.9%

         \[\leadsto \pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right) \cdot 1}{\color{blue}{F \cdot F}} \]

      5. *-rgt-identity 50.9%

         \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\tan \left(\pi \cdot \ell\right)}}{F \cdot F} \]

   3. Simplified 50.9%

      \[\leadsto \color{blue}{\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. tan-quot 50.9%

         \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\frac{\sin \left(\pi \cdot \ell\right)}{\cos \left(\pi \cdot \ell\right)}}}{F \cdot F} \]

      2. div-inv 50.9%

         \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\sin \left(\pi \cdot \ell\right) \cdot \frac{1}{\cos \left(\pi \cdot \ell\right)}}}{F \cdot F} \]

   6. Applied egg-rr 50.9%

      \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\sin \left(\pi \cdot \ell\right) \cdot \frac{1}{\cos \left(\pi \cdot \ell\right)}}}{F \cdot F} \]

   7. Taylor expanded in l around 0 58.5%

      \[\leadsto \pi \cdot \ell - \frac{\sin \left(\pi \cdot \ell\right) \cdot \frac{1}{\color{blue}{1 + -0.5 \cdot \left({\ell}^{2} \cdot {\pi}^{2}\right)}}}{F \cdot F} \]
   8. Step-by-step derivation

      1. +-commutative 58.5%

         \[\leadsto \pi \cdot \ell - \frac{\sin \left(\pi \cdot \ell\right) \cdot \frac{1}{\color{blue}{-0.5 \cdot \left({\ell}^{2} \cdot {\pi}^{2}\right) + 1}}}{F \cdot F} \]

      2. fma-define 58.5%

         \[\leadsto \pi \cdot \ell - \frac{\sin \left(\pi \cdot \ell\right) \cdot \frac{1}{\color{blue}{\mathsf{fma}\left(-0.5, {\ell}^{2} \cdot {\pi}^{2}, 1\right)}}}{F \cdot F} \]

      3. *-commutative 58.5%

         \[\leadsto \pi \cdot \ell - \frac{\sin \left(\pi \cdot \ell\right) \cdot \frac{1}{\mathsf{fma}\left(-0.5, \color{blue}{{\pi}^{2} \cdot {\ell}^{2}}, 1\right)}}{F \cdot F} \]

      4. unpow2 58.5%

         \[\leadsto \pi \cdot \ell - \frac{\sin \left(\pi \cdot \ell\right) \cdot \frac{1}{\mathsf{fma}\left(-0.5, \color{blue}{\left(\pi \cdot \pi\right)} \cdot {\ell}^{2}, 1\right)}}{F \cdot F} \]

      5. unpow2 58.5%

         \[\leadsto \pi \cdot \ell - \frac{\sin \left(\pi \cdot \ell\right) \cdot \frac{1}{\mathsf{fma}\left(-0.5, \left(\pi \cdot \pi\right) \cdot \color{blue}{\left(\ell \cdot \ell\right)}, 1\right)}}{F \cdot F} \]

      6. swap-sqr 58.5%

         \[\leadsto \pi \cdot \ell - \frac{\sin \left(\pi \cdot \ell\right) \cdot \frac{1}{\mathsf{fma}\left(-0.5, \color{blue}{\left(\pi \cdot \ell\right) \cdot \left(\pi \cdot \ell\right)}, 1\right)}}{F \cdot F} \]

      7. unpow2 58.5%

         \[\leadsto \pi \cdot \ell - \frac{\sin \left(\pi \cdot \ell\right) \cdot \frac{1}{\mathsf{fma}\left(-0.5, \color{blue}{{\left(\pi \cdot \ell\right)}^{2}}, 1\right)}}{F \cdot F} \]

      8. *-commutative 58.5%

         \[\leadsto \pi \cdot \ell - \frac{\sin \left(\pi \cdot \ell\right) \cdot \frac{1}{\mathsf{fma}\left(-0.5, {\color{blue}{\left(\ell \cdot \pi\right)}}^{2}, 1\right)}}{F \cdot F} \]

   9. Simplified 58.5%

      \[\leadsto \pi \cdot \ell - \frac{\sin \left(\pi \cdot \ell\right) \cdot \frac{1}{\color{blue}{\mathsf{fma}\left(-0.5, {\left(\ell \cdot \pi\right)}^{2}, 1\right)}}}{F \cdot F} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 10^{+18}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell + \frac{\sin \left(\pi \cdot \ell\right) \cdot \frac{-1}{\mathsf{fma}\left(-0.5, {\left(\pi \cdot \ell\right)}^{2}, 1\right)}}{F \cdot F}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 2 \cdot 10^{-157}:\\ \;\;\;\;\pi \cdot l\_m + \frac{\pi \cdot l\_m}{F} \cdot \frac{-1}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m - \frac{\tan \left(\pi \cdot l\_m\right)}{F \cdot F}\\ \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= (* PI l_m) 2e-157)
    (+ (* PI l_m) (* (/ (* PI l_m) F) (/ -1.0 F)))
    (- (* PI l_m) (/ (tan (* PI l_m)) (* F F))))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if ((((double) M_PI) * l_m) <= 2e-157) {
		tmp = (((double) M_PI) * l_m) + (((((double) M_PI) * l_m) / F) * (-1.0 / F));
	} else {
		tmp = (((double) M_PI) * l_m) - (tan((((double) M_PI) * l_m)) / (F * F));
	}
	return l_s * tmp;
}

Java

l\_m = Math.abs(l);
l\_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	double tmp;
	if ((Math.PI * l_m) <= 2e-157) {
		tmp = (Math.PI * l_m) + (((Math.PI * l_m) / F) * (-1.0 / F));
	} else {
		tmp = (Math.PI * l_m) - (Math.tan((Math.PI * l_m)) / (F * F));
	}
	return l_s * tmp;
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	tmp = 0
	if (math.pi * l_m) <= 2e-157:
		tmp = (math.pi * l_m) + (((math.pi * l_m) / F) * (-1.0 / F))
	else:
		tmp = (math.pi * l_m) - (math.tan((math.pi * l_m)) / (F * F))
	return l_s * tmp

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	tmp = 0.0
	if (Float64(pi * l_m) <= 2e-157)
		tmp = Float64(Float64(pi * l_m) + Float64(Float64(Float64(pi * l_m) / F) * Float64(-1.0 / F)));
	else
		tmp = Float64(Float64(pi * l_m) - Float64(tan(Float64(pi * l_m)) / Float64(F * F)));
	end
	return Float64(l_s * tmp)
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp_2 = code(l_s, F, l_m)
	tmp = 0.0;
	if ((pi * l_m) <= 2e-157)
		tmp = (pi * l_m) + (((pi * l_m) / F) * (-1.0 / F));
	else
		tmp = (pi * l_m) - (tan((pi * l_m)) / (F * F));
	end
	tmp_2 = l_s * tmp;
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * If[LessEqual[N[(Pi * l$95$m), $MachinePrecision], 2e-157], N[(N[(Pi * l$95$m), $MachinePrecision] + N[(N[(N[(Pi * l$95$m), $MachinePrecision] / F), $MachinePrecision] * N[(-1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[Tan[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 1.99999999999999989e-157

   1. Initial program 74.6%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   2. Step-by-step derivation

      1. *-commutative 74.6%

         \[\leadsto \pi \cdot \ell - \color{blue}{\tan \left(\pi \cdot \ell\right) \cdot \frac{1}{F \cdot F}} \]

      2. sqr-neg 74.6%

         \[\leadsto \pi \cdot \ell - \tan \left(\pi \cdot \ell\right) \cdot \frac{1}{\color{blue}{\left(-F\right) \cdot \left(-F\right)}} \]

      3. associate-*r/ 74.6%

         \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\tan \left(\pi \cdot \ell\right) \cdot 1}{\left(-F\right) \cdot \left(-F\right)}} \]

      4. sqr-neg 74.6%

         \[\leadsto \pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right) \cdot 1}{\color{blue}{F \cdot F}} \]

      5. *-rgt-identity 74.6%

         \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\tan \left(\pi \cdot \ell\right)}}{F \cdot F} \]

   3. Simplified 74.6%

      \[\leadsto \color{blue}{\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}} \]
   4. Add Preprocessing

   5. Taylor expanded in l around 0 65.9%

      \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\ell \cdot \pi}}{F \cdot F} \]
   6. Step-by-step derivation

      1. associate-/r* 73.3%

         \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\frac{\ell \cdot \pi}{F}}{F}} \]

      2. div-inv 73.3%

         \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\ell \cdot \pi}{F} \cdot \frac{1}{F}} \]

      3. *-commutative 73.3%

         \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\pi \cdot \ell}}{F} \cdot \frac{1}{F} \]

   7. Applied egg-rr 73.3%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\pi \cdot \ell}{F} \cdot \frac{1}{F}} \]

   if 1.99999999999999989e-157 < (*.f64 (PI.f64) l)

   1. Initial program 68.6%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   2. Step-by-step derivation

      1. *-commutative 68.6%

         \[\leadsto \pi \cdot \ell - \color{blue}{\tan \left(\pi \cdot \ell\right) \cdot \frac{1}{F \cdot F}} \]

      2. sqr-neg 68.6%

         \[\leadsto \pi \cdot \ell - \tan \left(\pi \cdot \ell\right) \cdot \frac{1}{\color{blue}{\left(-F\right) \cdot \left(-F\right)}} \]

      3. associate-*r/ 68.6%

         \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\tan \left(\pi \cdot \ell\right) \cdot 1}{\left(-F\right) \cdot \left(-F\right)}} \]

      4. sqr-neg 68.6%

         \[\leadsto \pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right) \cdot 1}{\color{blue}{F \cdot F}} \]

      5. *-rgt-identity 68.6%

         \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\tan \left(\pi \cdot \ell\right)}}{F \cdot F} \]

   3. Simplified 68.6%

      \[\leadsto \color{blue}{\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}} \]
   4. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 2 \cdot 10^{-157}:\\ \;\;\;\;\pi \cdot \ell + \frac{\pi \cdot \ell}{F} \cdot \frac{-1}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \left(\pi \cdot l\_m - \frac{\frac{\tan \left(\pi \cdot l\_m\right)}{F}}{F}\right) \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (* l_s (- (* PI l_m) (/ (/ (tan (* PI l_m)) F) F))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	return l_s * ((((double) M_PI) * l_m) - ((tan((((double) M_PI) * l_m)) / F) / F));
}

Java

l\_m = Math.abs(l);
l\_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	return l_s * ((Math.PI * l_m) - ((Math.tan((Math.PI * l_m)) / F) / F));
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	return l_s * ((math.pi * l_m) - ((math.tan((math.pi * l_m)) / F) / F))

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	return Float64(l_s * Float64(Float64(pi * l_m) - Float64(Float64(tan(Float64(pi * l_m)) / F) / F)))
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp = code(l_s, F, l_m)
	tmp = l_s * ((pi * l_m) - ((tan((pi * l_m)) / F) / F));
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(N[Tan[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.4%

   \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. associate-*l/ 72.5%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{1 \cdot \tan \left(\pi \cdot \ell\right)}{F \cdot F}} \]

   2. *-un-lft-identity 72.5%

      \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\tan \left(\pi \cdot \ell\right)}}{F \cdot F} \]

   3. associate-/r* 77.2%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}} \]

4. Applied egg-rr 77.2%

   \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}} \]
5. Add Preprocessing

Alternative 8: 75.1% accurate, 10.3× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \left(\pi \cdot l\_m - \frac{\frac{\pi \cdot l\_m}{F}}{F}\right) \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (* l_s (- (* PI l_m) (/ (/ (* PI l_m) F) F))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	return l_s * ((((double) M_PI) * l_m) - (((((double) M_PI) * l_m) / F) / F));
}

Java

l\_m = Math.abs(l);
l\_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	return l_s * ((Math.PI * l_m) - (((Math.PI * l_m) / F) / F));
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	return l_s * ((math.pi * l_m) - (((math.pi * l_m) / F) / F))

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	return Float64(l_s * Float64(Float64(pi * l_m) - Float64(Float64(Float64(pi * l_m) / F) / F)))
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp = code(l_s, F, l_m)
	tmp = l_s * ((pi * l_m) - (((pi * l_m) / F) / F));
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(N[(Pi * l$95$m), $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.4%

   \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. associate-*l/ 72.5%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{1 \cdot \tan \left(\pi \cdot \ell\right)}{F \cdot F}} \]

   2. *-un-lft-identity 72.5%

      \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\tan \left(\pi \cdot \ell\right)}}{F \cdot F} \]

   3. associate-/r* 77.2%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}} \]

4. Applied egg-rr 77.2%

   \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}} \]

5. Taylor expanded in l around 0 69.1%

   \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\frac{\ell \cdot \pi}{F}}}{F} \]

6. Final simplification 69.1%

   \[\leadsto \pi \cdot \ell - \frac{\frac{\pi \cdot \ell}{F}}{F} \]
7. Add Preprocessing

Alternative 9: 75.2% accurate, 10.3× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \left(\pi \cdot l\_m - \frac{\pi \cdot \frac{l\_m}{F}}{F}\right) \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (* l_s (- (* PI l_m) (/ (* PI (/ l_m F)) F))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	return l_s * ((((double) M_PI) * l_m) - ((((double) M_PI) * (l_m / F)) / F));
}

Java

l\_m = Math.abs(l);
l\_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	return l_s * ((Math.PI * l_m) - ((Math.PI * (l_m / F)) / F));
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	return l_s * ((math.pi * l_m) - ((math.pi * (l_m / F)) / F))

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	return Float64(l_s * Float64(Float64(pi * l_m) - Float64(Float64(pi * Float64(l_m / F)) / F)))
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp = code(l_s, F, l_m)
	tmp = l_s * ((pi * l_m) - ((pi * (l_m / F)) / F));
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(Pi * N[(l$95$m / F), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.4%

   \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
2. Step-by-step derivation

   1. *-commutative 72.4%

      \[\leadsto \pi \cdot \ell - \color{blue}{\tan \left(\pi \cdot \ell\right) \cdot \frac{1}{F \cdot F}} \]

   2. sqr-neg 72.4%

      \[\leadsto \pi \cdot \ell - \tan \left(\pi \cdot \ell\right) \cdot \frac{1}{\color{blue}{\left(-F\right) \cdot \left(-F\right)}} \]

   3. associate-*r/ 72.5%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\tan \left(\pi \cdot \ell\right) \cdot 1}{\left(-F\right) \cdot \left(-F\right)}} \]

   4. sqr-neg 72.5%

      \[\leadsto \pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right) \cdot 1}{\color{blue}{F \cdot F}} \]

   5. *-rgt-identity 72.5%

      \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\tan \left(\pi \cdot \ell\right)}}{F \cdot F} \]

3. Simplified 72.5%

   \[\leadsto \color{blue}{\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}} \]
4. Add Preprocessing

5. Taylor expanded in l around 0 64.4%

   \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\ell \cdot \pi}}{F \cdot F} \]
6. Step-by-step derivation

   1. *-commutative 64.4%

      \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\pi \cdot \ell}}{F \cdot F} \]

   2. times-frac 69.0%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\pi}{F} \cdot \frac{\ell}{F}} \]

7. Applied egg-rr 69.0%

   \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\pi}{F} \cdot \frac{\ell}{F}} \]
8. Step-by-step derivation

   1. associate-*l/ 69.1%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\pi \cdot \frac{\ell}{F}}{F}} \]

9. Applied egg-rr 69.1%

   \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\pi \cdot \frac{\ell}{F}}{F}} \]
10. Add Preprocessing

Alternative 10: 75.2% accurate, 10.3× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \left(\pi \cdot l\_m - \frac{l\_m \cdot \frac{\pi}{F}}{F}\right) \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (* l_s (- (* PI l_m) (/ (* l_m (/ PI F)) F))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	return l_s * ((((double) M_PI) * l_m) - ((l_m * (((double) M_PI) / F)) / F));
}

Java

l\_m = Math.abs(l);
l\_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	return l_s * ((Math.PI * l_m) - ((l_m * (Math.PI / F)) / F));
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	return l_s * ((math.pi * l_m) - ((l_m * (math.pi / F)) / F))

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	return Float64(l_s * Float64(Float64(pi * l_m) - Float64(Float64(l_m * Float64(pi / F)) / F)))
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp = code(l_s, F, l_m)
	tmp = l_s * ((pi * l_m) - ((l_m * (pi / F)) / F));
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(l$95$m * N[(Pi / F), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.4%

   \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. associate-*l/ 72.5%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{1 \cdot \tan \left(\pi \cdot \ell\right)}{F \cdot F}} \]

   2. *-un-lft-identity 72.5%

      \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\tan \left(\pi \cdot \ell\right)}}{F \cdot F} \]

   3. associate-/r* 77.2%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}} \]

4. Applied egg-rr 77.2%

   \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}} \]

5. Taylor expanded in l around 0 69.1%

   \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\frac{\ell \cdot \pi}{F}}}{F} \]
6. Step-by-step derivation

   1. associate-/l* 69.1%

      \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\ell \cdot \frac{\pi}{F}}}{F} \]

7. Simplified 69.1%

   \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\ell \cdot \frac{\pi}{F}}}{F} \]
8. Add Preprocessing

Alternative 11: 75.2% accurate, 10.3× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \left(\pi \cdot l\_m - \frac{l\_m}{F} \cdot \frac{\pi}{F}\right) \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (* l_s (- (* PI l_m) (* (/ l_m F) (/ PI F)))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	return l_s * ((((double) M_PI) * l_m) - ((l_m / F) * (((double) M_PI) / F)));
}

Java

l\_m = Math.abs(l);
l\_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	return l_s * ((Math.PI * l_m) - ((l_m / F) * (Math.PI / F)));
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	return l_s * ((math.pi * l_m) - ((l_m / F) * (math.pi / F)))

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	return Float64(l_s * Float64(Float64(pi * l_m) - Float64(Float64(l_m / F) * Float64(pi / F))))
end

MATLAB

l\_m = abs(l);
l\_s = sign(l) * abs(1.0);
function tmp = code(l_s, F, l_m)
	tmp = l_s * ((pi * l_m) - ((l_m / F) * (pi / F)));
end

Wolfram

l\_m = N[Abs[l], $MachinePrecision]
l\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(l$95$m / F), $MachinePrecision] * N[(Pi / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.4%

   \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
2. Step-by-step derivation

   1. *-commutative 72.4%

      \[\leadsto \pi \cdot \ell - \color{blue}{\tan \left(\pi \cdot \ell\right) \cdot \frac{1}{F \cdot F}} \]

   2. sqr-neg 72.4%

      \[\leadsto \pi \cdot \ell - \tan \left(\pi \cdot \ell\right) \cdot \frac{1}{\color{blue}{\left(-F\right) \cdot \left(-F\right)}} \]

   3. associate-*r/ 72.5%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\tan \left(\pi \cdot \ell\right) \cdot 1}{\left(-F\right) \cdot \left(-F\right)}} \]

   4. sqr-neg 72.5%

      \[\leadsto \pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right) \cdot 1}{\color{blue}{F \cdot F}} \]

   5. *-rgt-identity 72.5%

      \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\tan \left(\pi \cdot \ell\right)}}{F \cdot F} \]

3. Simplified 72.5%

   \[\leadsto \color{blue}{\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}} \]
4. Add Preprocessing

5. Taylor expanded in l around 0 64.4%

   \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\ell \cdot \pi}}{F \cdot F} \]
6. Step-by-step derivation

   1. *-commutative 64.4%

      \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\pi \cdot \ell}}{F \cdot F} \]

   2. times-frac 69.0%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\pi}{F} \cdot \frac{\ell}{F}} \]

7. Applied egg-rr 69.0%

   \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\pi}{F} \cdot \frac{\ell}{F}} \]

8. Final simplification 69.0%

   \[\leadsto \pi \cdot \ell - \frac{\ell}{F} \cdot \frac{\pi}{F} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024100 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))