VandenBroeck and Keller, Equation (6)

Percentage Accurate: 76.5% → 99.3%

Time: 38.8s

Alternatives: 12

Speedup: 1.0×

• could not determine a ground truth

  1. F = 1.8069888913226996e-31
  2. l = -2449258903056635.5

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: 76.5% 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: 99.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ l_s = \mathsf{copysign}\left(1, \ell\right) \\ l_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l_m \leq 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(\pi, l_m, \frac{\frac{\tan \left(\pi \cdot l_m\right)}{F}}{-F}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l_m + \frac{\frac{-1}{F}}{\frac{F}{\log \left(e^{\frac{\sin \left(\pi \cdot l_m\right)}{\mathsf{fma}\left(-0.5, {\left(\pi \cdot l_m\right)}^{2}, 1\right)}}\right)}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
l_s = (copysign.f64 1 l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= (* PI l_m) 1e+16)
    (fma PI l_m (/ (/ (tan (* PI l_m)) F) (- F)))
    (+
     (* PI l_m)
     (/
      (/ -1.0 F)
      (/
       F
       (log
        (exp (/ (sin (* PI l_m)) (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) {
	double tmp;
	if ((((double) M_PI) * l_m) <= 1e+16) {
		tmp = fma(((double) M_PI), l_m, ((tan((((double) M_PI) * l_m)) / F) / -F));
	} else {
		tmp = (((double) M_PI) * l_m) + ((-1.0 / F) / (F / log(exp((sin((((double) M_PI) * l_m)) / fma(-0.5, pow((((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)
	tmp = 0.0
	if (Float64(pi * l_m) <= 1e+16)
		tmp = fma(pi, l_m, Float64(Float64(tan(Float64(pi * l_m)) / F) / Float64(-F)));
	else
		tmp = Float64(Float64(pi * l_m) + Float64(Float64(-1.0 / F) / Float64(F / log(exp(Float64(sin(Float64(pi * l_m)) / fma(-0.5, (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_] := N[(l$95$s * If[LessEqual[N[(Pi * l$95$m), $MachinePrecision], 1e+16], N[(Pi * l$95$m + 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[(-1.0 / F), $MachinePrecision] / N[(F / N[Log[N[Exp[N[(N[Sin[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision] / N[(-0.5 * N[Power[N[(Pi * l$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 1e16

   1. Initial program 85.4%

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

      1. fma-neg 85.4%

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

      2. distribute-lft-neg-in 85.4%

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

      3. sqr-neg 85.4%

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

      4. distribute-neg-frac 85.4%

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

      5. metadata-eval 85.4%

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

      6. distribute-lft-neg-out 85.4%

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

      7. neg-mul-1 85.4%

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

      8. associate-/r* 85.4%

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

      9. metadata-eval 85.4%

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

      10. associate-*l/ 85.9%

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

      11. *-lft-identity 85.9%

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

      12. associate-/r* 91.5%

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

   3. Simplified 91.5%

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

   if 1e16 < (*.f64 (PI.f64) l)

   1. Initial program 62.5%

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

      1. associate-/r/ 62.5%

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

      2. associate-/l* 62.5%

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

      3. clear-num 62.5%

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

      4. add-sqr-sqrt 23.7%

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

      5. sqrt-prod 62.8%

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

      6. sqr-neg 62.8%

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

      7. sqrt-unprod 39.0%

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

      8. add-sqr-sqrt 62.6%

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

      9. div-inv 62.6%

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

      10. clear-num 62.6%

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

      11. associate-*l/ 62.6%

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

      12. *-un-lft-identity 62.6%

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

      13. add-sqr-sqrt 39.0%

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

      14. sqrt-unprod 62.8%

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

      15. sqr-neg 62.8%

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

      16. sqrt-prod 23.7%

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

      17. add-sqr-sqrt 62.5%

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

   4. Applied egg-rr 62.5%

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

      1. tan-quot 62.5%

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

      2. clear-num 62.5%

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

   6. Applied egg-rr 62.5%

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

   7. Taylor expanded in l around 0 86.6%

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

      1. +-commutative 86.6%

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

      2. fma-def 86.6%

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

      3. *-commutative 86.6%

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

      4. unpow2 86.6%

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

      5. unpow2 86.6%

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

      6. swap-sqr 86.6%

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

      7. unpow2 86.6%

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

      8. *-commutative 86.6%

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

   9. Simplified 86.6%

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

       1. clear-num 86.6%

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

       2. add-log-exp 99.6%

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

       3. *-commutative 99.6%

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

   11. Applied egg-rr 99.6%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(\pi, \ell, \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{-F}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell + \frac{\frac{-1}{F}}{\frac{F}{\log \left(e^{\frac{\sin \left(\pi \cdot \ell\right)}{\mathsf{fma}\left(-0.5, {\left(\pi \cdot \ell\right)}^{2}, 1\right)}}\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ l_s = \mathsf{copysign}\left(1, \ell\right) \\ l_s \cdot \left(\pi \cdot l_m + \frac{\frac{-1}{F}}{\frac{F}{\frac{1}{\frac{\mathsf{fma}\left(-0.5, \sqrt{{\left(\pi \cdot l_m\right)}^{4}}, 1\right)}{\sin \left(\pi \cdot l_m\right)}}}}\right) \end{array} \]

FPCore

l_m = (fabs.f64 l)
l_s = (copysign.f64 1 l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (+
   (* PI l_m)
   (/
    (/ -1.0 F)
    (/
     F
     (/
      1.0
      (/ (fma -0.5 (sqrt (pow (* PI l_m) 4.0)) 1.0) (sin (* 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) + ((-1.0 / F) / (F / (1.0 / (fma(-0.5, sqrt(pow((((double) M_PI) * l_m), 4.0)), 1.0) / sin((((double) M_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(-1.0 / F) / Float64(F / Float64(1.0 / Float64(fma(-0.5, sqrt((Float64(pi * l_m) ^ 4.0)), 1.0) / sin(Float64(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[(-1.0 / F), $MachinePrecision] / N[(F / N[(1.0 / N[(N[(-0.5 * N[Sqrt[N[Power[N[(Pi * l$95$m), $MachinePrecision], 4.0], $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision] / N[Sin[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.0%

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

   1. associate-/r/ 80.4%

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

   2. associate-/l* 84.7%

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

   3. clear-num 84.7%

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

   4. add-sqr-sqrt 40.7%

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

   5. sqrt-prod 65.7%

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

   6. sqr-neg 65.7%

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

   7. sqrt-unprod 27.1%

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

   8. add-sqr-sqrt 54.3%

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

   9. div-inv 54.3%

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

   10. clear-num 54.3%

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

   11. associate-*l/ 54.3%

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

   12. *-un-lft-identity 54.3%

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

   13. add-sqr-sqrt 27.1%

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

   14. sqrt-unprod 65.7%

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

   15. sqr-neg 65.7%

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

   16. sqrt-prod 40.7%

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

   17. add-sqr-sqrt 84.7%

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

4. Applied egg-rr 84.7%

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

   1. tan-quot 84.7%

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

   2. clear-num 84.7%

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

6. Applied egg-rr 84.7%

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

7. Taylor expanded in l around 0 93.5%

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

   1. +-commutative 93.5%

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

   2. fma-def 93.5%

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

   3. *-commutative 93.5%

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

   4. unpow2 93.5%

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

   5. unpow2 93.5%

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

   6. swap-sqr 93.5%

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

   7. unpow2 93.5%

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

   8. *-commutative 93.5%

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

9. Simplified 93.5%

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

    1. add-sqr-sqrt 93.5%

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

    2. sqrt-unprod 95.0%

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

    3. pow-prod-up 95.0%

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

    4. metadata-eval 95.0%

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

11. Applied egg-rr 95.0%

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

12. Final simplification 95.0%

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

Alternative 3: 92.0% accurate, 0.4× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
l_s = (copysign.f64 1 l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (+
   (* PI l_m)
   (/
    (/ -1.0 F)
    (/ F (/ 1.0 (/ (fma -0.5 (pow (* PI l_m) 2.0) 1.0) (sin (* 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) + ((-1.0 / F) / (F / (1.0 / (fma(-0.5, pow((((double) M_PI) * l_m), 2.0), 1.0) / sin((((double) M_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(-1.0 / F) / Float64(F / Float64(1.0 / Float64(fma(-0.5, (Float64(pi * l_m) ^ 2.0), 1.0) / sin(Float64(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[(-1.0 / F), $MachinePrecision] / N[(F / N[(1.0 / N[(N[(-0.5 * N[Power[N[(Pi * l$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision] / N[Sin[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.0%

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

   1. associate-/r/ 80.4%

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

   2. associate-/l* 84.7%

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

   3. clear-num 84.7%

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

   4. add-sqr-sqrt 40.7%

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

   5. sqrt-prod 65.7%

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

   6. sqr-neg 65.7%

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

   7. sqrt-unprod 27.1%

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

   8. add-sqr-sqrt 54.3%

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

   9. div-inv 54.3%

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

   10. clear-num 54.3%

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

   11. associate-*l/ 54.3%

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

   12. *-un-lft-identity 54.3%

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

   13. add-sqr-sqrt 27.1%

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

   14. sqrt-unprod 65.7%

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

   15. sqr-neg 65.7%

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

   16. sqrt-prod 40.7%

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

   17. add-sqr-sqrt 84.7%

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

4. Applied egg-rr 84.7%

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

   1. tan-quot 84.7%

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

   2. clear-num 84.7%

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

6. Applied egg-rr 84.7%

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

7. Taylor expanded in l around 0 93.5%

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

   1. +-commutative 93.5%

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

   2. fma-def 93.5%

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

   3. *-commutative 93.5%

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

   4. unpow2 93.5%

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

   5. unpow2 93.5%

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

   6. swap-sqr 93.5%

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

   7. unpow2 93.5%

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

   8. *-commutative 93.5%

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

9. Simplified 93.5%

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

10. Final simplification 93.5%

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

Alternative 4: 92.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ l_s = \mathsf{copysign}\left(1, \ell\right) \\ l_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l_m \leq 2 \cdot 10^{+56}:\\ \;\;\;\;\mathsf{fma}\left(\pi, l_m, \frac{\frac{\tan \left(\pi \cdot l_m\right)}{F}}{-F}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l_m + \frac{\frac{-1}{F}}{\frac{F}{\sin \left(\pi \cdot l_m\right) \cdot \frac{-2}{{\left(\pi \cdot l_m\right)}^{2}}}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
l_s = (copysign.f64 1 l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= (* PI l_m) 2e+56)
    (fma PI l_m (/ (/ (tan (* PI l_m)) F) (- F)))
    (+
     (* PI l_m)
     (/
      (/ -1.0 F)
      (/ F (* (sin (* PI l_m)) (/ -2.0 (pow (* PI l_m) 2.0)))))))))

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) <= 2e+56) {
		tmp = fma(((double) M_PI), l_m, ((tan((((double) M_PI) * l_m)) / F) / -F));
	} else {
		tmp = (((double) M_PI) * l_m) + ((-1.0 / F) / (F / (sin((((double) M_PI) * l_m)) * (-2.0 / pow((((double) M_PI) * l_m), 2.0)))));
	}
	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) <= 2e+56)
		tmp = fma(pi, l_m, Float64(Float64(tan(Float64(pi * l_m)) / F) / Float64(-F)));
	else
		tmp = Float64(Float64(pi * l_m) + Float64(Float64(-1.0 / F) / Float64(F / Float64(sin(Float64(pi * l_m)) * Float64(-2.0 / (Float64(pi * l_m) ^ 2.0))))));
	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_] := N[(l$95$s * If[LessEqual[N[(Pi * l$95$m), $MachinePrecision], 2e+56], N[(Pi * l$95$m + 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[(-1.0 / F), $MachinePrecision] / N[(F / N[(N[Sin[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision] * N[(-2.0 / N[Power[N[(Pi * l$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 2.00000000000000018e56

   1. Initial program 83.0%

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

      1. fma-neg 83.0%

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

      2. distribute-lft-neg-in 83.0%

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

      3. sqr-neg 83.0%

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

      4. distribute-neg-frac 83.0%

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

      5. metadata-eval 83.0%

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

      6. distribute-lft-neg-out 83.0%

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

      7. neg-mul-1 83.0%

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

      8. associate-/r* 83.0%

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

      9. metadata-eval 83.0%

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

      10. associate-*l/ 83.5%

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

      11. *-lft-identity 83.5%

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

      12. associate-/r* 88.9%

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

   3. Simplified 88.9%

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

   if 2.00000000000000018e56 < (*.f64 (PI.f64) l)

   1. Initial program 68.9%

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

      1. associate-/r/ 68.9%

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

      2. associate-/l* 68.9%

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

      3. clear-num 68.9%

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

      4. add-sqr-sqrt 26.1%

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

      5. sqrt-prod 69.4%

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

      6. sqr-neg 69.4%

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

      7. sqrt-unprod 43.3%

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

      8. add-sqr-sqrt 69.4%

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

      9. div-inv 69.4%

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

      10. clear-num 69.4%

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

      11. associate-*l/ 69.4%

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

      12. *-un-lft-identity 69.4%

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

      13. add-sqr-sqrt 43.3%

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

      14. sqrt-unprod 69.4%

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

      15. sqr-neg 69.4%

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

      16. sqrt-prod 26.1%

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

      17. add-sqr-sqrt 68.9%

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

   4. Applied egg-rr 68.9%

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

      1. tan-quot 68.9%

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

      2. clear-num 68.9%

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

   6. Applied egg-rr 68.9%

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

   7. Taylor expanded in l around 0 95.9%

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

      1. +-commutative 95.9%

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

      2. fma-def 95.9%

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

      3. *-commutative 95.9%

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

      4. unpow2 95.9%

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

      5. unpow2 95.9%

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

      6. swap-sqr 95.9%

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

      7. unpow2 95.9%

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

      8. *-commutative 95.9%

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

   9. Simplified 95.9%

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

   10. Taylor expanded in l around inf 95.9%

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

       1. unpow2 95.9%

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

       2. unpow2 95.9%

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

       3. swap-sqr 95.9%

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

       4. unpow2 95.9%

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

       5. associate-*r/ 95.9%

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

       6. *-rgt-identity 95.9%

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

       7. times-frac 95.9%

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

       8. rem-square-sqrt 50.0%

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

       9. associate-*r/ 50.0%

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

       10. /-rgt-identity 50.0%

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

       11. rem-square-sqrt 95.9%

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

   12. Simplified 95.9%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 2 \cdot 10^{+56}:\\ \;\;\;\;\mathsf{fma}\left(\pi, \ell, \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{-F}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell + \frac{\frac{-1}{F}}{\frac{F}{\sin \left(\pi \cdot \ell\right) \cdot \frac{-2}{{\left(\pi \cdot \ell\right)}^{2}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.3% accurate, 0.5× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
l_s = (copysign.f64 1 l)
(FPCore (l_s F l_m)
 :precision binary64
 (* l_s (fma 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 * fma(((double) M_PI), l_m, ((tan((((double) M_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 * fma(pi, l_m, Float64(Float64(tan(Float64(pi * l_m)) / F) / Float64(-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[(Pi * l$95$m + N[(N[(N[Tan[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision] / (-F)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.0%

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

   1. fma-neg 80.0%

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

   2. distribute-lft-neg-in 80.0%

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

   3. sqr-neg 80.0%

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

   4. distribute-neg-frac 80.0%

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

   5. metadata-eval 80.0%

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

   6. distribute-lft-neg-out 80.0%

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

   7. neg-mul-1 80.0%

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

   8. associate-/r* 80.0%

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

   9. metadata-eval 80.0%

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

   10. associate-*l/ 80.4%

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

   11. *-lft-identity 80.4%

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

   12. associate-/r* 84.7%

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

3. Simplified 84.7%

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

5. Final simplification 84.7%

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

Alternative 6: 82.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ l_s = \mathsf{copysign}\left(1, \ell\right) \\ l_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l_m \leq 1.5 \cdot 10^{-27}:\\ \;\;\;\;\pi \cdot l_m + \frac{l_m}{\frac{F}{\pi}} \cdot \frac{-1}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l_m - \tan \left(\pi \cdot l_m\right) \cdot \frac{1}{F \cdot F}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
l_s = (copysign.f64 1 l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= (* PI l_m) 1.5e-27)
    (+ (* PI l_m) (* (/ l_m (/ F PI)) (/ -1.0 F)))
    (- (* 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) {
	double tmp;
	if ((((double) M_PI) * l_m) <= 1.5e-27) {
		tmp = (((double) M_PI) * l_m) + ((l_m / (F / ((double) M_PI))) * (-1.0 / F));
	} else {
		tmp = (((double) M_PI) * l_m) - (tan((((double) M_PI) * l_m)) * (1.0 / (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) <= 1.5e-27) {
		tmp = (Math.PI * l_m) + ((l_m / (F / Math.PI)) * (-1.0 / F));
	} else {
		tmp = (Math.PI * l_m) - (Math.tan((Math.PI * l_m)) * (1.0 / (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) <= 1.5e-27:
		tmp = (math.pi * l_m) + ((l_m / (F / math.pi)) * (-1.0 / F))
	else:
		tmp = (math.pi * l_m) - (math.tan((math.pi * l_m)) * (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) <= 1.5e-27)
		tmp = Float64(Float64(pi * l_m) + Float64(Float64(l_m / Float64(F / pi)) * Float64(-1.0 / F)));
	else
		tmp = Float64(Float64(pi * l_m) - Float64(tan(Float64(pi * l_m)) * Float64(1.0 / 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) <= 1.5e-27)
		tmp = (pi * l_m) + ((l_m / (F / pi)) * (-1.0 / F));
	else
		tmp = (pi * l_m) - (tan((pi * l_m)) * (1.0 / (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], 1.5e-27], N[(N[(Pi * l$95$m), $MachinePrecision] + N[(N[(l$95$m / N[(F / Pi), $MachinePrecision]), $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[(1.0 / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 1.5000000000000001e-27

   1. Initial program 85.0%

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

      1. sqr-neg 85.0%

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

      2. associate-*l/ 85.5%

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

      3. *-lft-identity 85.5%

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

      4. sqr-neg 85.5%

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

   3. Simplified 85.5%

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

   5. Taylor expanded in l around 0 82.4%

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

      1. associate-/r* 88.3%

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

      2. div-inv 88.3%

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

      3. associate-/l* 88.4%

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

   7. Applied egg-rr 88.4%

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

   if 1.5000000000000001e-27 < (*.f64 (PI.f64) l)

   1. Initial program 66.6%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 1.5 \cdot 10^{-27}:\\ \;\;\;\;\pi \cdot \ell + \frac{\ell}{\frac{F}{\pi}} \cdot \frac{-1}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \tan \left(\pi \cdot \ell\right) \cdot \frac{1}{F \cdot F}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ l_s = \mathsf{copysign}\left(1, \ell\right) \\ l_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l_m \leq 2 \cdot 10^{-28}:\\ \;\;\;\;\pi \cdot l_m + \frac{l_m}{\frac{F}{\pi}} \cdot \frac{-1}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l_m - \frac{\tan \left(\pi \cdot l_m\right)}{F \cdot F}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
l_s = (copysign.f64 1 l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= (* PI l_m) 2e-28)
    (+ (* PI l_m) (* (/ l_m (/ F PI)) (/ -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-28) {
		tmp = (((double) M_PI) * l_m) + ((l_m / (F / ((double) M_PI))) * (-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-28) {
		tmp = (Math.PI * l_m) + ((l_m / (F / Math.PI)) * (-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-28:
		tmp = (math.pi * l_m) + ((l_m / (F / math.pi)) * (-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-28)
		tmp = Float64(Float64(pi * l_m) + Float64(Float64(l_m / Float64(F / pi)) * 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-28)
		tmp = (pi * l_m) + ((l_m / (F / pi)) * (-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-28], N[(N[(Pi * l$95$m), $MachinePrecision] + N[(N[(l$95$m / N[(F / Pi), $MachinePrecision]), $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) < 1.99999999999999994e-28

   1. Initial program 84.9%

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

      1. sqr-neg 84.9%

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

      2. associate-*l/ 85.4%

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

      3. *-lft-identity 85.4%

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

      4. sqr-neg 85.4%

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

   3. Simplified 85.4%

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

   5. Taylor expanded in l around 0 82.3%

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

      1. associate-/r* 88.3%

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

      2. div-inv 88.3%

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

      3. associate-/l* 88.3%

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

   7. Applied egg-rr 88.3%

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

   if 1.99999999999999994e-28 < (*.f64 (PI.f64) l)

   1. Initial program 67.0%

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

      1. sqr-neg 67.0%

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

      2. associate-*l/ 67.0%

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

      3. *-lft-identity 67.0%

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

      4. sqr-neg 67.0%

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

   3. Simplified 67.0%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 2 \cdot 10^{-28}:\\ \;\;\;\;\pi \cdot \ell + \frac{\ell}{\frac{F}{\pi}} \cdot \frac{-1}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 82.2% accurate, 1.0× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
l_s = (copysign.f64 1 l)
(FPCore (l_s F l_m)
 :precision binary64
 (* l_s (+ (* PI l_m) (/ (/ -1.0 F) (/ F (tan (* 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) + ((-1.0 / F) / (F / tan((((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) + ((-1.0 / F) / (F / Math.tan((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) + ((-1.0 / F) / (F / math.tan((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(-1.0 / F) / Float64(F / tan(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) + ((-1.0 / F) / (F / tan((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[(-1.0 / F), $MachinePrecision] / N[(F / N[Tan[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.0%

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

   1. associate-/r/ 80.4%

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

   2. associate-/l* 84.7%

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

   3. clear-num 84.7%

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

   4. add-sqr-sqrt 40.7%

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

   5. sqrt-prod 65.7%

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

   6. sqr-neg 65.7%

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

   7. sqrt-unprod 27.1%

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

   8. add-sqr-sqrt 54.3%

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

   9. div-inv 54.3%

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

   10. clear-num 54.3%

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

   11. associate-*l/ 54.3%

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

   12. *-un-lft-identity 54.3%

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

   13. add-sqr-sqrt 27.1%

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

   14. sqrt-unprod 65.7%

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

   15. sqr-neg 65.7%

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

   16. sqrt-prod 40.7%

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

   17. add-sqr-sqrt 84.7%

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

4. Applied egg-rr 84.7%

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

5. Final simplification 84.7%

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

Alternative 9: 75.0% accurate, 8.7× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
l_s = (copysign.f64 1 l)
(FPCore (l_s F l_m)
 :precision binary64
 (* l_s (+ (* PI l_m) (* (/ l_m (/ F PI)) (/ -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) + ((l_m / (F / ((double) M_PI))) * (-1.0 / 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)) * (-1.0 / 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)) * (-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(l_m / Float64(F / pi)) * Float64(-1.0 / 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)) * (-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[(l$95$m / N[(F / Pi), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.0%

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

   1. sqr-neg 80.0%

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

   2. associate-*l/ 80.4%

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

   3. *-lft-identity 80.4%

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

   4. sqr-neg 80.4%

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

3. Simplified 80.4%

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

5. Taylor expanded in l around 0 74.4%

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

   1. associate-/r* 78.7%

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

   2. div-inv 78.7%

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

   3. associate-/l* 78.7%

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

7. Applied egg-rr 78.7%

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

8. Final simplification 78.7%

   \[\leadsto \pi \cdot \ell + \frac{\ell}{\frac{F}{\pi}} \cdot \frac{-1}{F} \]
9. Add Preprocessing

Alternative 10: 75.0% 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 1 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 80.0%

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

   1. sqr-neg 80.0%

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

   2. associate-*l/ 80.4%

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

   3. *-lft-identity 80.4%

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

   4. sqr-neg 80.4%

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

3. Simplified 80.4%

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

5. Taylor expanded in l around 0 74.4%

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

   1. *-commutative 74.4%

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

   2. times-frac 78.6%

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

7. Applied egg-rr 78.6%

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

8. Final simplification 78.6%

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

Alternative 11: 75.0% accurate, 10.3× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
l_s = (copysign.f64 1 l)
(FPCore (l_s F l_m)
 :precision binary64
 (* l_s (- (* PI l_m) (/ PI (* F (/ F 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) - (((double) M_PI) / (F * (F / 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.PI / (F * (F / 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.pi / (F * (F / 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(pi / Float64(F * Float64(F / 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) - (pi / (F * (F / 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[(Pi / N[(F * N[(F / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.0%

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

   1. sqr-neg 80.0%

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

   2. associate-*l/ 80.4%

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

   3. *-lft-identity 80.4%

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

   4. sqr-neg 80.4%

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

3. Simplified 80.4%

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

5. Taylor expanded in l around 0 74.4%

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

   1. *-commutative 74.4%

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

   2. times-frac 78.6%

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

7. Applied egg-rr 78.6%

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

   1. *-commutative 78.6%

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

   2. clear-num 78.6%

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

   3. frac-times 78.7%

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

   4. *-un-lft-identity 78.7%

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

9. Applied egg-rr 78.7%

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

10. Final simplification 78.7%

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

Alternative 12: 75.0% accurate, 10.3× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
l_s = (copysign.f64 1 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(pi / Float64(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[(Pi / N[(F / N[(l$95$m / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.0%

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

   1. sqr-neg 80.0%

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

   2. associate-*l/ 80.4%

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

   3. *-lft-identity 80.4%

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

   4. sqr-neg 80.4%

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

3. Simplified 80.4%

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

5. Taylor expanded in l around 0 74.4%

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

   1. *-commutative 74.4%

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

   2. times-frac 78.6%

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

7. Applied egg-rr 78.6%

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

   1. associate-*l/ 78.7%

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

   2. associate-/l* 78.7%

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

9. Applied egg-rr 78.7%

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

10. Final simplification 78.7%

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

Reproduce

herbie shell --seed 2024019 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))