Average Error: 17.1 → 8.6
Time: 7.9s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \le -4.22162737413892601 \cdot 10^{157} \lor \neg \left(\pi \cdot \ell \le 2.0152910594483865 \cdot 10^{142}\right):\\ \;\;\;\;\pi \cdot \ell - \frac{1}{F} \cdot \frac{1 \cdot \left(\left(\sqrt[3]{\sin \left(\pi \cdot \ell\right)} \cdot \sqrt[3]{\sin \left(\pi \cdot \ell\right)}\right) \cdot \sqrt[3]{\sin \left(\pi \cdot \ell\right)}\right)}{F \cdot \cos \left(\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right) \cdot \left(\sqrt[3]{\pi} \cdot \ell\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{F} \cdot \frac{1 \cdot \sin \left(\pi \cdot \ell\right)}{F \cdot \left(\left(\frac{1}{24} \cdot \left({\pi}^{4} \cdot {\ell}^{4}\right) + 1\right) - \frac{1}{2} \cdot \left({\pi}^{2} \cdot {\ell}^{2}\right)\right)}\\ \end{array}\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\begin{array}{l}
\mathbf{if}\;\pi \cdot \ell \le -4.22162737413892601 \cdot 10^{157} \lor \neg \left(\pi \cdot \ell \le 2.0152910594483865 \cdot 10^{142}\right):\\
\;\;\;\;\pi \cdot \ell - \frac{1}{F} \cdot \frac{1 \cdot \left(\left(\sqrt[3]{\sin \left(\pi \cdot \ell\right)} \cdot \sqrt[3]{\sin \left(\pi \cdot \ell\right)}\right) \cdot \sqrt[3]{\sin \left(\pi \cdot \ell\right)}\right)}{F \cdot \cos \left(\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right) \cdot \left(\sqrt[3]{\pi} \cdot \ell\right)\right)}\\

\mathbf{else}:\\
\;\;\;\;\pi \cdot \ell - \frac{1}{F} \cdot \frac{1 \cdot \sin \left(\pi \cdot \ell\right)}{F \cdot \left(\left(\frac{1}{24} \cdot \left({\pi}^{4} \cdot {\ell}^{4}\right) + 1\right) - \frac{1}{2} \cdot \left({\pi}^{2} \cdot {\ell}^{2}\right)\right)}\\

\end{array}
double code(double F, double l) {
	return ((double) (((double) (((double) M_PI) * l)) - ((double) (((double) (1.0 / ((double) (F * F)))) * ((double) tan(((double) (((double) M_PI) * l))))))));
}
↓
double code(double F, double l) {
	double VAR;
	if (((((double) (((double) M_PI) * l)) <= -4.221627374138926e+157) || !(((double) (((double) M_PI) * l)) <= 2.0152910594483865e+142))) {
		VAR = ((double) (((double) (((double) M_PI) * l)) - ((double) (((double) (1.0 / F)) * ((double) (((double) (1.0 * ((double) (((double) (((double) cbrt(((double) sin(((double) (((double) M_PI) * l)))))) * ((double) cbrt(((double) sin(((double) (((double) M_PI) * l)))))))) * ((double) cbrt(((double) sin(((double) (((double) M_PI) * l)))))))))) / ((double) (F * ((double) cos(((double) (((double) (((double) cbrt(((double) M_PI))) * ((double) cbrt(((double) M_PI))))) * ((double) (((double) cbrt(((double) M_PI))) * l))))))))))))));
	} else {
		VAR = ((double) (((double) (((double) M_PI) * l)) - ((double) (((double) (1.0 / F)) * ((double) (((double) (1.0 * ((double) sin(((double) (((double) M_PI) * l)))))) / ((double) (F * ((double) (((double) (((double) (0.041666666666666664 * ((double) (((double) pow(((double) M_PI), 4.0)) * ((double) pow(l, 4.0)))))) + 1.0)) - ((double) (0.5 * ((double) (((double) pow(((double) M_PI), 2.0)) * ((double) pow(l, 2.0))))))))))))))));
	}
	return VAR;
}

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 2 regimes

  2. if (* PI l) < -4.221627374138926e+157 or 2.0152910594483865e+142 < (* PI l)

    1. Initial program 20.8

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

    3. Applied *-un-lft-identity20.8

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

    4. Applied times-frac20.8

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

    5. Applied associate-*l*20.8

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

    7. Applied tan-quot20.8

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

    8. Applied frac-times20.8

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

    10. Applied add-cube-cbrt20.8

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

    11. Applied associate-*l*20.8

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

    13. Applied add-cube-cbrt20.8

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

    if -4.221627374138926e+157 < (* PI l) < 2.0152910594483865e+142

    1. Initial program 15.7

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

    3. Applied *-un-lft-identity15.7

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

    4. Applied times-frac15.8

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

    5. Applied associate-*l*9.9

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

    7. Applied tan-quot9.9

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

    8. Applied frac-times9.9

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

    9. Taylor expanded around 0 4.1

      \[\leadsto \pi \cdot \ell - \frac{1}{F} \cdot \frac{1 \cdot \sin \left(\pi \cdot \ell\right)}{F \cdot \color{blue}{\left(\left(\frac{1}{24} \cdot \left({\pi}^{4} \cdot {\ell}^{4}\right) + 1\right) - \frac{1}{2} \cdot \left({\pi}^{2} \cdot {\ell}^{2}\right)\right)}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification8.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \le -4.22162737413892601 \cdot 10^{157} \lor \neg \left(\pi \cdot \ell \le 2.0152910594483865 \cdot 10^{142}\right):\\ \;\;\;\;\pi \cdot \ell - \frac{1}{F} \cdot \frac{1 \cdot \left(\left(\sqrt[3]{\sin \left(\pi \cdot \ell\right)} \cdot \sqrt[3]{\sin \left(\pi \cdot \ell\right)}\right) \cdot \sqrt[3]{\sin \left(\pi \cdot \ell\right)}\right)}{F \cdot \cos \left(\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right) \cdot \left(\sqrt[3]{\pi} \cdot \ell\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{F} \cdot \frac{1 \cdot \sin \left(\pi \cdot \ell\right)}{F \cdot \left(\left(\frac{1}{24} \cdot \left({\pi}^{4} \cdot {\ell}^{4}\right) + 1\right) - \frac{1}{2} \cdot \left({\pi}^{2} \cdot {\ell}^{2}\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020130 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))