Average Error: 16.5 → 8.9
Time: 7.4s
Precision: binary64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5.498102941521268 \cdot 10^{+158}:\\ \;\;\;\;\pi \cdot \ell - \frac{\tan \left(\sqrt[3]{\ell} \cdot \left(\pi \cdot \left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right)\right)\right)}{F \cdot F}\\ \mathbf{elif}\;\pi \cdot \ell \leq 1.0547115320298808 \cdot 10^{+132}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{F} \cdot \frac{\frac{\sin \left(\pi \cdot \ell\right)}{\left(1 + 0.041666666666666664 \cdot {\left(\pi \cdot \ell\right)}^{4}\right) - {\pi}^{2} \cdot \left(\left(\ell \cdot \ell\right) \cdot 0.5\right)}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{F} \cdot \frac{\frac{\sin \left(\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right) \cdot \left(\ell \cdot \sqrt[3]{\pi}\right)\right)}{\cos \left(\pi \cdot \ell\right)}}{F}\\ \end{array}\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\begin{array}{l}
\mathbf{if}\;\pi \cdot \ell \leq -5.498102941521268 \cdot 10^{+158}:\\
\;\;\;\;\pi \cdot \ell - \frac{\tan \left(\sqrt[3]{\ell} \cdot \left(\pi \cdot \left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right)\right)\right)}{F \cdot F}\\

\mathbf{elif}\;\pi \cdot \ell \leq 1.0547115320298808 \cdot 10^{+132}:\\
\;\;\;\;\pi \cdot \ell - \frac{1}{F} \cdot \frac{\frac{\sin \left(\pi \cdot \ell\right)}{\left(1 + 0.041666666666666664 \cdot {\left(\pi \cdot \ell\right)}^{4}\right) - {\pi}^{2} \cdot \left(\left(\ell \cdot \ell\right) \cdot 0.5\right)}}{F}\\

\mathbf{else}:\\
\;\;\;\;\pi \cdot \ell - \frac{1}{F} \cdot \frac{\frac{\sin \left(\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right) \cdot \left(\ell \cdot \sqrt[3]{\pi}\right)\right)}{\cos \left(\pi \cdot \ell\right)}}{F}\\

\end{array}
(FPCore (F l)
 :precision binary64
 (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))
↓
(FPCore (F l)
 :precision binary64
 (if (<= (* PI l) -5.498102941521268e+158)
   (- (* PI l) (/ (tan (* (cbrt l) (* PI (* (cbrt l) (cbrt l))))) (* F F)))
   (if (<= (* PI l) 1.0547115320298808e+132)
     (-
      (* PI l)
      (*
       (/ 1.0 F)
       (/
        (/
         (sin (* PI l))
         (-
          (+ 1.0 (* 0.041666666666666664 (pow (* PI l) 4.0)))
          (* (pow PI 2.0) (* (* l l) 0.5))))
        F)))
     (-
      (* PI l)
      (*
       (/ 1.0 F)
       (/
        (/ (sin (* (* (cbrt PI) (cbrt PI)) (* l (cbrt PI)))) (cos (* PI l)))
        F))))))
double code(double F, double l) {
	return (((double) M_PI) * l) - ((1.0 / (F * F)) * tan(((double) M_PI) * l));
}
↓
double code(double F, double l) {
	double tmp;
	if ((((double) M_PI) * l) <= -5.498102941521268e+158) {
		tmp = (((double) M_PI) * l) - (tan(cbrt(l) * (((double) M_PI) * (cbrt(l) * cbrt(l)))) / (F * F));
	} else if ((((double) M_PI) * l) <= 1.0547115320298808e+132) {
		tmp = (((double) M_PI) * l) - ((1.0 / F) * ((sin(((double) M_PI) * l) / ((1.0 + (0.041666666666666664 * pow((((double) M_PI) * l), 4.0))) - (pow(((double) M_PI), 2.0) * ((l * l) * 0.5)))) / F));
	} else {
		tmp = (((double) M_PI) * l) - ((1.0 / F) * ((sin((cbrt((double) M_PI) * cbrt((double) M_PI)) * (l * cbrt((double) M_PI))) / cos(((double) M_PI) * l)) / F));
	}
	return tmp;
}

Error

Bits error versus F

Bits error versus l

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if (*.f64 PI.f64 l) < -5.49810294152126846e158

    1. Initial program 21.7

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

    2. Simplified21.7

      \[\leadsto \color{blue}{\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}}\]
    3. Using strategy rm

    4. Applied add-cube-cbrt_binary6421.7

      \[\leadsto \pi \cdot \ell - \frac{\tan \left(\pi \cdot \color{blue}{\left(\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}\right)}\right)}{F \cdot F}\]

    5. Applied associate-*r*_binary6421.7

      \[\leadsto \pi \cdot \ell - \frac{\tan \color{blue}{\left(\left(\pi \cdot \left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right)\right) \cdot \sqrt[3]{\ell}\right)}}{F \cdot F}\]

    if -5.49810294152126846e158 < (*.f64 PI.f64 l) < 1.05471153202988076e132

    1. Initial program 14.9

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

    2. Simplified14.5

      \[\leadsto \color{blue}{\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}}\]
    3. Using strategy rm

    4. Applied *-un-lft-identity_binary6414.5

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

    5. Applied times-frac_binary648.9

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

    7. Applied tan-quot_binary648.9

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

    8. Taylor expanded around 0 4.3

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

    9. Simplified4.3

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

    if 1.05471153202988076e132 < (*.f64 PI.f64 l)

    1. Initial program 19.7

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

    2. Simplified19.7

      \[\leadsto \color{blue}{\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}}\]
    3. Using strategy rm

    4. Applied *-un-lft-identity_binary6419.7

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

    5. Applied times-frac_binary6419.7

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

    7. Applied tan-quot_binary6419.7

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

    9. Applied add-cube-cbrt_binary6419.7

      \[\leadsto \pi \cdot \ell - \frac{1}{F} \cdot \frac{\frac{\sin \left(\color{blue}{\left(\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right) \cdot \sqrt[3]{\pi}\right)} \cdot \ell\right)}{\cos \left(\pi \cdot \ell\right)}}{F}\]

    10. Applied associate-*l*_binary6419.7

      \[\leadsto \pi \cdot \ell - \frac{1}{F} \cdot \frac{\frac{\sin \color{blue}{\left(\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right) \cdot \left(\sqrt[3]{\pi} \cdot \ell\right)\right)}}{\cos \left(\pi \cdot \ell\right)}}{F}\]

    11. Simplified19.7

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

  4. Final simplification8.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5.498102941521268 \cdot 10^{+158}:\\ \;\;\;\;\pi \cdot \ell - \frac{\tan \left(\sqrt[3]{\ell} \cdot \left(\pi \cdot \left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right)\right)\right)}{F \cdot F}\\ \mathbf{elif}\;\pi \cdot \ell \leq 1.0547115320298808 \cdot 10^{+132}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{F} \cdot \frac{\frac{\sin \left(\pi \cdot \ell\right)}{\left(1 + 0.041666666666666664 \cdot {\left(\pi \cdot \ell\right)}^{4}\right) - {\pi}^{2} \cdot \left(\left(\ell \cdot \ell\right) \cdot 0.5\right)}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{F} \cdot \frac{\frac{\sin \left(\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right) \cdot \left(\ell \cdot \sqrt[3]{\pi}\right)\right)}{\cos \left(\pi \cdot \ell\right)}}{F}\\ \end{array}\]

Reproduce

herbie shell --seed 2020346 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))