VandenBroeck and Keller, Equation (6)

Percentage Accurate: 76.4% → 99.0%

Time: 21.5s

Alternatives: 12

Speedup: 1.0×

• could not determine a ground truth

  1. F = 4.484740164482843e-209
  2. l = -2920952525561893.5

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: 76.4% 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, 0.2× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := \sin \left(\pi \cdot l\_m\right)\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 2000000000000:\\ \;\;\;\;\pi \cdot l\_m + \frac{t\_0 \cdot \frac{-1}{F}}{F \cdot \cos \left(\pi \cdot l\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m + \frac{\frac{1}{F} \cdot t\_0}{F \cdot \mathsf{fma}\left(-0.5, {\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\pi \cdot l\_m\right)\right)\right)}^{2}, 1\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (let* ((t_0 (sin (* PI l_m))))
   (*
    l_s
    (if (<= (* PI l_m) 2000000000000.0)
      (+ (* PI l_m) (/ (* t_0 (/ -1.0 F)) (* F (cos (* PI l_m)))))
      (+
       (* PI l_m)
       (/
        (* (/ 1.0 F) t_0)
        (* F (fma -0.5 (pow (log1p (expm1 (* PI l_m))) 2.0) 1.0))))))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double t_0 = sin((((double) M_PI) * l_m));
	double tmp;
	if ((((double) M_PI) * l_m) <= 2000000000000.0) {
		tmp = (((double) M_PI) * l_m) + ((t_0 * (-1.0 / F)) / (F * cos((((double) M_PI) * l_m))));
	} else {
		tmp = (((double) M_PI) * l_m) + (((1.0 / F) * t_0) / (F * fma(-0.5, pow(log1p(expm1((((double) M_PI) * l_m))), 2.0), 1.0)));
	}
	return l_s * tmp;
}

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	t_0 = sin(Float64(pi * l_m))
	tmp = 0.0
	if (Float64(pi * l_m) <= 2000000000000.0)
		tmp = Float64(Float64(pi * l_m) + Float64(Float64(t_0 * Float64(-1.0 / F)) / Float64(F * cos(Float64(pi * l_m)))));
	else
		tmp = Float64(Float64(pi * l_m) + Float64(Float64(Float64(1.0 / F) * t_0) / Float64(F * fma(-0.5, (log1p(expm1(Float64(pi * l_m))) ^ 2.0), 1.0))));
	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_] := Block[{t$95$0 = N[Sin[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision]}, N[(l$95$s * If[LessEqual[N[(Pi * l$95$m), $MachinePrecision], 2000000000000.0], N[(N[(Pi * l$95$m), $MachinePrecision] + N[(N[(t$95$0 * N[(-1.0 / F), $MachinePrecision]), $MachinePrecision] / N[(F * N[Cos[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * l$95$m), $MachinePrecision] + N[(N[(N[(1.0 / F), $MachinePrecision] * t$95$0), $MachinePrecision] / N[(F * N[(-0.5 * N[Power[N[Log[1 + N[(Exp[N[(Pi * l$95$m), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 74.9%

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

      1. associate-/r* 74.9%

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

      2. metadata-eval 74.9%

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

      3. add-sqr-sqrt 41.3%

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

      4. sqrt-prod 66.8%

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

      5. sqrt-div 66.8%

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

      6. tan-quot 66.8%

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

      7. frac-times 66.8%

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

      8. sqrt-div 66.8%

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

      9. metadata-eval 66.8%

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

      10. sqrt-prod 46.3%

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

      11. add-sqr-sqrt 84.9%

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

   4. Applied egg-rr 84.9%

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

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

   1. Initial program 58.2%

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

      1. associate-/r* 58.2%

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

      2. metadata-eval 58.2%

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

      3. add-sqr-sqrt 27.5%

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

      4. sqrt-prod 57.9%

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

      5. sqrt-div 57.9%

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

      6. tan-quot 57.9%

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

      7. frac-times 57.9%

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

      8. sqrt-div 57.9%

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

      9. metadata-eval 57.9%

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

      10. sqrt-prod 27.5%

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

      11. add-sqr-sqrt 58.2%

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

   4. Applied egg-rr 58.2%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\frac{1}{F} \cdot \sin \left(\pi \cdot \ell\right)}{F \cdot \cos \left(\pi \cdot \ell\right)}} \]
   5. Step-by-step derivation

      1. add-sqr-sqrt 26.4%

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

      2. sqrt-unprod 58.2%

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

      3. sqr-neg 58.2%

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

      4. *-commutative 58.2%

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

      5. *-commutative 58.2%

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

      6. sqrt-unprod 31.9%

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

      7. add-sqr-sqrt 58.4%

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

      8. *-commutative 58.4%

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

   6. Applied egg-rr 58.4%

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

   7. Taylor expanded in l around 0 85.9%

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

      1. +-commutative 85.9%

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

      2. fma-define 85.9%

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

      3. *-commutative 85.9%

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

      4. unpow2 85.9%

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

      5. unpow2 85.9%

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

      6. swap-sqr 85.9%

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

      7. unpow2 85.9%

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

      8. *-commutative 85.9%

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

   9. Simplified 85.9%

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

       1. *-commutative 85.9%

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

       2. log1p-expm1-u 99.5%

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

   11. Applied egg-rr 99.5%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 2000000000000:\\ \;\;\;\;\pi \cdot \ell + \frac{\sin \left(\pi \cdot \ell\right) \cdot \frac{-1}{F}}{F \cdot \cos \left(\pi \cdot \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell + \frac{\frac{1}{F} \cdot \sin \left(\pi \cdot \ell\right)}{F \cdot \mathsf{fma}\left(-0.5, {\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\pi \cdot \ell\right)\right)\right)}^{2}, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := \sin \left(\pi \cdot l\_m\right)\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 10^{+18}:\\ \;\;\;\;\pi \cdot l\_m + \frac{t\_0 \cdot \frac{-1}{F}}{F \cdot \cos \left(\pi \cdot l\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m - \frac{\frac{t\_0}{F}}{F \cdot \left(\left({\pi}^{6} \cdot {l\_m}^{6}\right) \cdot -0.001388888888888889\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (let* ((t_0 (sin (* PI l_m))))
   (*
    l_s
    (if (<= (* PI l_m) 1e+18)
      (+ (* PI l_m) (/ (* t_0 (/ -1.0 F)) (* F (cos (* PI l_m)))))
      (-
       (* PI l_m)
       (/
        (/ t_0 F)
        (* F (* (* (pow PI 6.0) (pow l_m 6.0)) -0.001388888888888889))))))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double t_0 = sin((((double) M_PI) * l_m));
	double tmp;
	if ((((double) M_PI) * l_m) <= 1e+18) {
		tmp = (((double) M_PI) * l_m) + ((t_0 * (-1.0 / F)) / (F * cos((((double) M_PI) * l_m))));
	} else {
		tmp = (((double) M_PI) * l_m) - ((t_0 / F) / (F * ((pow(((double) M_PI), 6.0) * pow(l_m, 6.0)) * -0.001388888888888889)));
	}
	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 t_0 = Math.sin((Math.PI * l_m));
	double tmp;
	if ((Math.PI * l_m) <= 1e+18) {
		tmp = (Math.PI * l_m) + ((t_0 * (-1.0 / F)) / (F * Math.cos((Math.PI * l_m))));
	} else {
		tmp = (Math.PI * l_m) - ((t_0 / F) / (F * ((Math.pow(Math.PI, 6.0) * Math.pow(l_m, 6.0)) * -0.001388888888888889)));
	}
	return l_s * tmp;
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	t_0 = math.sin((math.pi * l_m))
	tmp = 0
	if (math.pi * l_m) <= 1e+18:
		tmp = (math.pi * l_m) + ((t_0 * (-1.0 / F)) / (F * math.cos((math.pi * l_m))))
	else:
		tmp = (math.pi * l_m) - ((t_0 / F) / (F * ((math.pow(math.pi, 6.0) * math.pow(l_m, 6.0)) * -0.001388888888888889)))
	return l_s * tmp

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	t_0 = sin(Float64(pi * l_m))
	tmp = 0.0
	if (Float64(pi * l_m) <= 1e+18)
		tmp = Float64(Float64(pi * l_m) + Float64(Float64(t_0 * Float64(-1.0 / F)) / Float64(F * cos(Float64(pi * l_m)))));
	else
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(t_0 / F) / Float64(F * Float64(Float64((pi ^ 6.0) * (l_m ^ 6.0)) * -0.001388888888888889))));
	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)
	t_0 = sin((pi * l_m));
	tmp = 0.0;
	if ((pi * l_m) <= 1e+18)
		tmp = (pi * l_m) + ((t_0 * (-1.0 / F)) / (F * cos((pi * l_m))));
	else
		tmp = (pi * l_m) - ((t_0 / F) / (F * (((pi ^ 6.0) * (l_m ^ 6.0)) * -0.001388888888888889)));
	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_] := Block[{t$95$0 = N[Sin[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision]}, N[(l$95$s * If[LessEqual[N[(Pi * l$95$m), $MachinePrecision], 1e+18], N[(N[(Pi * l$95$m), $MachinePrecision] + N[(N[(t$95$0 * N[(-1.0 / F), $MachinePrecision]), $MachinePrecision] / N[(F * N[Cos[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(t$95$0 / F), $MachinePrecision] / N[(F * N[(N[(N[Power[Pi, 6.0], $MachinePrecision] * N[Power[l$95$m, 6.0], $MachinePrecision]), $MachinePrecision] * -0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 74.9%

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

      1. associate-/r* 74.9%

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

      2. metadata-eval 74.9%

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

      3. add-sqr-sqrt 41.3%

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

      4. sqrt-prod 66.8%

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

      5. sqrt-div 66.8%

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

      6. tan-quot 66.8%

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

      7. frac-times 66.8%

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

      8. sqrt-div 66.8%

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

      9. metadata-eval 66.8%

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

      10. sqrt-prod 46.3%

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

      11. add-sqr-sqrt 84.9%

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

   4. Applied egg-rr 84.9%

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

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

   1. Initial program 58.2%

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

      1. associate-/r* 58.2%

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

      2. metadata-eval 58.2%

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

      3. add-sqr-sqrt 27.5%

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

      4. sqrt-prod 57.9%

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

      5. sqrt-div 57.9%

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

      6. tan-quot 57.9%

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

      7. frac-times 57.9%

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

      8. sqrt-div 57.9%

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

      9. metadata-eval 57.9%

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

      10. sqrt-prod 27.5%

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

      11. add-sqr-sqrt 58.2%

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

   4. Applied egg-rr 58.2%

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

   5. Taylor expanded in l around 0 96.8%

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

      1. associate-*l/ 96.8%

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

      2. *-un-lft-identity 96.8%

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

   7. Applied egg-rr 96.8%

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

   8. Taylor expanded in l around inf 96.8%

      \[\leadsto \pi \cdot \ell - \frac{\frac{\sin \left(\pi \cdot \ell\right)}{F}}{\color{blue}{-0.001388888888888889 \cdot \left(F \cdot \left({\ell}^{6} \cdot {\pi}^{6}\right)\right)}} \]
   9. Step-by-step derivation

      1. *-commutative 96.8%

         \[\leadsto \pi \cdot \ell - \frac{\frac{\sin \left(\pi \cdot \ell\right)}{F}}{\color{blue}{\left(F \cdot \left({\ell}^{6} \cdot {\pi}^{6}\right)\right) \cdot -0.001388888888888889}} \]

      2. associate-*l* 96.8%

         \[\leadsto \pi \cdot \ell - \frac{\frac{\sin \left(\pi \cdot \ell\right)}{F}}{\color{blue}{F \cdot \left(\left({\ell}^{6} \cdot {\pi}^{6}\right) \cdot -0.001388888888888889\right)}} \]

      3. *-commutative 96.8%

         \[\leadsto \pi \cdot \ell - \frac{\frac{\sin \left(\pi \cdot \ell\right)}{F}}{F \cdot \left(\color{blue}{\left({\pi}^{6} \cdot {\ell}^{6}\right)} \cdot -0.001388888888888889\right)} \]

   10. Simplified 96.8%

       \[\leadsto \pi \cdot \ell - \frac{\frac{\sin \left(\pi \cdot \ell\right)}{F}}{\color{blue}{F \cdot \left(\left({\pi}^{6} \cdot {\ell}^{6}\right) \cdot -0.001388888888888889\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 10^{+18}:\\ \;\;\;\;\pi \cdot \ell + \frac{\sin \left(\pi \cdot \ell\right) \cdot \frac{-1}{F}}{F \cdot \cos \left(\pi \cdot \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\sin \left(\pi \cdot \ell\right)}{F}}{F \cdot \left(\left({\pi}^{6} \cdot {\ell}^{6}\right) \cdot -0.001388888888888889\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.9% accurate, 0.4× speedup

Math

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

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) (* F (fma -0.5 (pow (* PI l_m) 2.0) 1.0))))))

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) / (F * fma(-0.5, pow((((double) M_PI) * l_m), 2.0), 1.0))));
}

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(F * fma(-0.5, (Float64(pi * l_m) ^ 2.0), 1.0)))))
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[(F * N[(-0.5 * N[Power[N[(Pi * l$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.3%

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

   1. associate-/r* 70.3%

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

   2. metadata-eval 70.3%

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

   3. add-sqr-sqrt 37.5%

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

   4. sqrt-prod 64.4%

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

   5. sqrt-div 64.4%

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

   6. tan-quot 64.4%

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

   7. frac-times 64.4%

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

   8. sqrt-div 64.4%

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

   9. metadata-eval 64.4%

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

   10. sqrt-prod 41.2%

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

   11. add-sqr-sqrt 77.6%

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

4. Applied egg-rr 77.6%

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

5. Taylor expanded in l around 0 97.0%

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

   1. associate-*l/ 97.0%

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

   2. *-un-lft-identity 97.0%

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

7. Applied egg-rr 97.0%

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

8. Taylor expanded in l around 0 91.1%

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

   1. +-commutative 91.1%

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

   2. *-commutative 91.1%

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

   3. unpow2 91.1%

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

   4. unpow2 91.1%

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

   5. swap-sqr 91.1%

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

   6. unpow2 91.1%

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

   7. associate-*r* 91.1%

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

   8. *-lft-identity 91.1%

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

   9. distribute-rgt-in 91.1%

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

   10. fma-undefine 91.1%

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

10. Simplified 91.1%

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

11. Final simplification 91.1%

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

Alternative 4: 92.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := \sin \left(\pi \cdot l\_m\right) \cdot \frac{-1}{F}\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 2000000000000:\\ \;\;\;\;\pi \cdot l\_m + \frac{t\_0}{F \cdot \cos \left(\pi \cdot l\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m + \frac{t\_0}{F \cdot \left({\left(\pi \cdot l\_m\right)}^{2} \cdot 0.5\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (let* ((t_0 (* (sin (* PI l_m)) (/ -1.0 F))))
   (*
    l_s
    (if (<= (* PI l_m) 2000000000000.0)
      (+ (* PI l_m) (/ t_0 (* F (cos (* PI l_m)))))
      (+ (* PI l_m) (/ t_0 (* F (* (pow (* PI l_m) 2.0) 0.5))))))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double t_0 = sin((((double) M_PI) * l_m)) * (-1.0 / F);
	double tmp;
	if ((((double) M_PI) * l_m) <= 2000000000000.0) {
		tmp = (((double) M_PI) * l_m) + (t_0 / (F * cos((((double) M_PI) * l_m))));
	} else {
		tmp = (((double) M_PI) * l_m) + (t_0 / (F * (pow((((double) M_PI) * l_m), 2.0) * 0.5)));
	}
	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 t_0 = Math.sin((Math.PI * l_m)) * (-1.0 / F);
	double tmp;
	if ((Math.PI * l_m) <= 2000000000000.0) {
		tmp = (Math.PI * l_m) + (t_0 / (F * Math.cos((Math.PI * l_m))));
	} else {
		tmp = (Math.PI * l_m) + (t_0 / (F * (Math.pow((Math.PI * l_m), 2.0) * 0.5)));
	}
	return l_s * tmp;
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	t_0 = math.sin((math.pi * l_m)) * (-1.0 / F)
	tmp = 0
	if (math.pi * l_m) <= 2000000000000.0:
		tmp = (math.pi * l_m) + (t_0 / (F * math.cos((math.pi * l_m))))
	else:
		tmp = (math.pi * l_m) + (t_0 / (F * (math.pow((math.pi * l_m), 2.0) * 0.5)))
	return l_s * tmp

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	t_0 = Float64(sin(Float64(pi * l_m)) * Float64(-1.0 / F))
	tmp = 0.0
	if (Float64(pi * l_m) <= 2000000000000.0)
		tmp = Float64(Float64(pi * l_m) + Float64(t_0 / Float64(F * cos(Float64(pi * l_m)))));
	else
		tmp = Float64(Float64(pi * l_m) + Float64(t_0 / Float64(F * Float64((Float64(pi * l_m) ^ 2.0) * 0.5))));
	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)
	t_0 = sin((pi * l_m)) * (-1.0 / F);
	tmp = 0.0;
	if ((pi * l_m) <= 2000000000000.0)
		tmp = (pi * l_m) + (t_0 / (F * cos((pi * l_m))));
	else
		tmp = (pi * l_m) + (t_0 / (F * (((pi * l_m) ^ 2.0) * 0.5)));
	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_] := Block[{t$95$0 = N[(N[Sin[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / F), $MachinePrecision]), $MachinePrecision]}, N[(l$95$s * If[LessEqual[N[(Pi * l$95$m), $MachinePrecision], 2000000000000.0], N[(N[(Pi * l$95$m), $MachinePrecision] + N[(t$95$0 / N[(F * N[Cos[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * l$95$m), $MachinePrecision] + N[(t$95$0 / N[(F * N[(N[Power[N[(Pi * l$95$m), $MachinePrecision], 2.0], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 74.9%

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

      1. associate-/r* 74.9%

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

      2. metadata-eval 74.9%

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

      3. add-sqr-sqrt 41.3%

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

      4. sqrt-prod 66.8%

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

      5. sqrt-div 66.8%

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

      6. tan-quot 66.8%

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

      7. frac-times 66.8%

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

      8. sqrt-div 66.8%

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

      9. metadata-eval 66.8%

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

      10. sqrt-prod 46.3%

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

      11. add-sqr-sqrt 84.9%

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

   4. Applied egg-rr 84.9%

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

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

   1. Initial program 58.2%

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

      1. associate-/r* 58.2%

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

      2. metadata-eval 58.2%

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

      3. add-sqr-sqrt 27.5%

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

      4. sqrt-prod 57.9%

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

      5. sqrt-div 57.9%

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

      6. tan-quot 57.9%

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

      7. frac-times 57.9%

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

      8. sqrt-div 57.9%

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

      9. metadata-eval 57.9%

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

      10. sqrt-prod 27.5%

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

      11. add-sqr-sqrt 58.2%

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

   4. Applied egg-rr 58.2%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\frac{1}{F} \cdot \sin \left(\pi \cdot \ell\right)}{F \cdot \cos \left(\pi \cdot \ell\right)}} \]
   5. Step-by-step derivation

      1. add-sqr-sqrt 26.4%

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

      2. sqrt-unprod 58.2%

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

      3. sqr-neg 58.2%

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

      4. *-commutative 58.2%

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

      5. *-commutative 58.2%

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

      6. sqrt-unprod 31.9%

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

      7. add-sqr-sqrt 58.4%

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

      8. *-commutative 58.4%

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

   6. Applied egg-rr 58.4%

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

   7. Taylor expanded in l around 0 85.9%

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

      1. +-commutative 85.9%

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

      2. fma-define 85.9%

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

      3. *-commutative 85.9%

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

      4. unpow2 85.9%

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

      5. unpow2 85.9%

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

      6. swap-sqr 85.9%

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

      7. unpow2 85.9%

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

      8. *-commutative 85.9%

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

   9. Simplified 85.9%

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

   10. Taylor expanded in l around inf 85.9%

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

       1. *-commutative 85.9%

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

       2. unpow2 85.9%

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

       3. unpow2 85.9%

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

       4. swap-sqr 85.9%

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

       5. unpow2 85.9%

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

       6. associate-*l* 85.9%

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

       7. unpow2 85.9%

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

       8. swap-sqr 85.9%

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

       9. unpow2 85.9%

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

       10. unpow2 85.9%

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

       11. *-commutative 85.9%

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

       12. unpow2 85.9%

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

       13. unpow2 85.9%

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

       14. swap-sqr 85.9%

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

       15. unpow2 85.9%

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

   12. Simplified 85.9%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 2000000000000:\\ \;\;\;\;\pi \cdot \ell + \frac{\sin \left(\pi \cdot \ell\right) \cdot \frac{-1}{F}}{F \cdot \cos \left(\pi \cdot \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell + \frac{\sin \left(\pi \cdot \ell\right) \cdot \frac{-1}{F}}{F \cdot \left({\left(\pi \cdot \ell\right)}^{2} \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.3% accurate, 0.5× speedup

Math

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

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)) (/ -1.0 F)) (* F (cos (* PI l_m)))))))

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)) * (-1.0 / F)) / (F * cos((((double) M_PI) * l_m)))));
}

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.sin((Math.PI * l_m)) * (-1.0 / F)) / (F * Math.cos((Math.PI * l_m)))));
}

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.sin((math.pi * l_m)) * (-1.0 / F)) / (F * math.cos((math.pi * l_m)))))

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)) * Float64(-1.0 / F)) / Float64(F * cos(Float64(pi * l_m))))))
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) + ((sin((pi * l_m)) * (-1.0 / F)) / (F * cos((pi * l_m)))));
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] * N[(-1.0 / F), $MachinePrecision]), $MachinePrecision] / N[(F * N[Cos[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.3%

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

   1. associate-/r* 70.3%

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

   2. metadata-eval 70.3%

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

   3. add-sqr-sqrt 37.5%

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

   4. sqrt-prod 64.4%

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

   5. sqrt-div 64.4%

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

   6. tan-quot 64.4%

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

   7. frac-times 64.4%

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

   8. sqrt-div 64.4%

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

   9. metadata-eval 64.4%

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

   10. sqrt-prod 41.2%

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

   11. add-sqr-sqrt 77.6%

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

4. Applied egg-rr 77.6%

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

5. Final simplification 77.6%

   \[\leadsto \pi \cdot \ell + \frac{\sin \left(\pi \cdot \ell\right) \cdot \frac{-1}{F}}{F \cdot \cos \left(\pi \cdot \ell\right)} \]
6. Add Preprocessing

Alternative 6: 82.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 2 \cdot 10^{-71}:\\ \;\;\;\;\pi \cdot l\_m - \frac{l\_m \cdot \frac{\pi}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m - \frac{\tan \left(\pi \cdot l\_m\right)}{F \cdot F}\\ \end{array} \end{array} \]

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-71)
    (- (* PI l_m) (/ (* l_m (/ PI F)) 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-71) {
		tmp = (((double) M_PI) * l_m) - ((l_m * (((double) M_PI) / F)) / 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-71) {
		tmp = (Math.PI * l_m) - ((l_m * (Math.PI / F)) / 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-71:
		tmp = (math.pi * l_m) - ((l_m * (math.pi / F)) / 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-71)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(l_m * Float64(pi / F)) / 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-71)
		tmp = (pi * l_m) - ((l_m * (pi / F)) / 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-71], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(l$95$m * N[(Pi / F), $MachinePrecision]), $MachinePrecision] / F), $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) < 1.9999999999999998e-71

   1. Initial program 73.2%

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

      1. *-commutative 73.2%

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

      2. sqr-neg 73.2%

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

      3. associate-*r/ 73.2%

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

      4. *-rgt-identity 73.2%

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

      5. sqr-neg 73.2%

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

   3. Simplified 73.2%

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

   5. Taylor expanded in l around 0 67.2%

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

      1. *-commutative 67.2%

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

      2. times-frac 77.8%

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

   7. Applied egg-rr 77.8%

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

      1. associate-*r/ 77.8%

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

   9. Applied egg-rr 77.8%

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

   if 1.9999999999999998e-71 < (*.f64 (PI.f64) l)

   1. Initial program 64.2%

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

      1. *-commutative 64.2%

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

      2. sqr-neg 64.2%

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

      3. associate-*r/ 64.2%

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

      4. *-rgt-identity 64.2%

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

      5. sqr-neg 64.2%

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

   3. Simplified 64.2%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 2 \cdot 10^{-71}:\\ \;\;\;\;\pi \cdot \ell - \frac{\ell \cdot \frac{\pi}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 2000000000000:\\ \;\;\;\;\pi \cdot l\_m - \frac{l\_m \cdot \frac{\pi}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m - \frac{\sin \left(\pi \cdot l\_m\right)}{F \cdot F}\\ \end{array} \end{array} \]

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) 2000000000000.0)
    (- (* PI l_m) (/ (* l_m (/ PI F)) F))
    (- (* PI l_m) (/ (sin (* 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) <= 2000000000000.0) {
		tmp = (((double) M_PI) * l_m) - ((l_m * (((double) M_PI) / F)) / F);
	} else {
		tmp = (((double) M_PI) * l_m) - (sin((((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) <= 2000000000000.0) {
		tmp = (Math.PI * l_m) - ((l_m * (Math.PI / F)) / F);
	} else {
		tmp = (Math.PI * l_m) - (Math.sin((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) <= 2000000000000.0:
		tmp = (math.pi * l_m) - ((l_m * (math.pi / F)) / F)
	else:
		tmp = (math.pi * l_m) - (math.sin((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) <= 2000000000000.0)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(l_m * Float64(pi / F)) / F));
	else
		tmp = Float64(Float64(pi * l_m) - Float64(sin(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) <= 2000000000000.0)
		tmp = (pi * l_m) - ((l_m * (pi / F)) / F);
	else
		tmp = (pi * l_m) - (sin((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], 2000000000000.0], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(l$95$m * N[(Pi / F), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[Sin[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) < 2e12

   1. Initial program 74.9%

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

      1. *-commutative 74.9%

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

      2. sqr-neg 74.9%

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

      3. associate-*r/ 74.9%

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

      4. *-rgt-identity 74.9%

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

      5. sqr-neg 74.9%

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

   3. Simplified 74.9%

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

   5. Taylor expanded in l around 0 68.9%

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

      1. *-commutative 68.9%

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

      2. times-frac 78.8%

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

   7. Applied egg-rr 78.8%

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

      1. associate-*r/ 78.8%

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

   9. Applied egg-rr 78.8%

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

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

   1. Initial program 58.2%

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

      1. *-commutative 58.2%

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

      2. sqr-neg 58.2%

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

      3. associate-*r/ 58.2%

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

      4. *-rgt-identity 58.2%

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

      5. sqr-neg 58.2%

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

   3. Simplified 58.2%

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

      1. tan-quot 58.2%

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

   6. Applied egg-rr 58.2%

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

   7. Taylor expanded in l around 0 58.2%

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

      1. /-rgt-identity 58.2%

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

   9. Applied egg-rr 58.2%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 2000000000000:\\ \;\;\;\;\pi \cdot \ell - \frac{\ell \cdot \frac{\pi}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sin \left(\pi \cdot \ell\right)}{F \cdot F}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 82.3% accurate, 1.0× speedup

Math

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

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)) (/ -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) {
	return l_s * ((((double) M_PI) * l_m) + ((tan((((double) M_PI) * l_m)) * (-1.0 / 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)) * (-1.0 / 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)) * (-1.0 / 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)) * Float64(-1.0 / 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)) * (-1.0 / 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] * N[(-1.0 / F), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.3%

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

   1. associate-/r* 70.3%

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

   2. metadata-eval 70.3%

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

   3. add-sqr-sqrt 37.5%

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

   4. sqrt-prod 64.4%

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

   5. sqrt-div 64.4%

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

   6. associate-*l/ 64.4%

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

   7. sqrt-div 64.4%

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

   8. metadata-eval 64.4%

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

   9. sqrt-prod 41.2%

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

   10. add-sqr-sqrt 77.6%

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

4. Applied egg-rr 77.6%

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

5. Final simplification 77.6%

   \[\leadsto \pi \cdot \ell + \frac{\tan \left(\pi \cdot \ell\right) \cdot \frac{-1}{F}}{F} \]
6. Add Preprocessing

Alternative 9: 82.3% accurate, 1.0× speedup

Math

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

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 70.3%

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

   1. associate-*l/ 70.4%

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

   2. *-un-lft-identity 70.4%

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

   3. associate-/r* 77.6%

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

4. Applied egg-rr 77.6%

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

Alternative 10: 81.5% accurate, 1.0× speedup

Math

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

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) (/ (/ (sin (* 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) - ((sin((((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.sin((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.sin((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(sin(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) - ((sin((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[Sin[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.3%

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

   1. associate-/r* 70.3%

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

   2. metadata-eval 70.3%

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

   3. add-sqr-sqrt 37.5%

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

   4. sqrt-prod 64.4%

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

   5. sqrt-div 64.4%

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

   6. tan-quot 64.4%

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

   7. frac-times 64.4%

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

   8. sqrt-div 64.4%

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

   9. metadata-eval 64.4%

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

   10. sqrt-prod 41.2%

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

   11. add-sqr-sqrt 77.6%

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

4. Applied egg-rr 77.6%

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

5. Taylor expanded in l around 0 97.0%

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

   1. associate-*l/ 97.0%

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

   2. *-un-lft-identity 97.0%

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

7. Applied egg-rr 97.0%

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

8. Taylor expanded in l around 0 77.0%

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

Alternative 11: 74.9% 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 70.3%

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

   1. *-commutative 70.3%

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

   2. sqr-neg 70.3%

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

   3. associate-*r/ 70.4%

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

   4. *-rgt-identity 70.4%

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

   5. sqr-neg 70.4%

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

3. Simplified 70.4%

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

5. Taylor expanded in l around 0 63.0%

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

   1. *-commutative 63.0%

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

   2. times-frac 70.3%

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

7. Applied egg-rr 70.3%

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

   1. associate-*r/ 70.3%

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

9. Applied egg-rr 70.3%

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

10. Final simplification 70.3%

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

Alternative 12: 74.9% accurate, 10.3× speedup

Math

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

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 F) (/ l_m 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) / F) * (l_m / 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 / F) * (l_m / 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 / F) * (l_m / 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 / F) * Float64(l_m / 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 / F) * (l_m / 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 / F), $MachinePrecision] * N[(l$95$m / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.3%

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

   1. *-commutative 70.3%

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

   2. sqr-neg 70.3%

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

   3. associate-*r/ 70.4%

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

   4. *-rgt-identity 70.4%

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

   5. sqr-neg 70.4%

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

3. Simplified 70.4%

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

5. Taylor expanded in l around 0 63.0%

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

   1. *-commutative 63.0%

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

   2. times-frac 70.3%

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

7. Applied egg-rr 70.3%

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

Reproduce

herbie shell --seed 2024131 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))