Average Error: 16.7 → 8.9
Time: 8.2s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \le -2.30427372910525123 \cdot 10^{156}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \frac{\frac{\sin \left(\pi \cdot \ell\right) \cdot \sqrt[3]{1}}{F}}{\cos \left(\sqrt{\pi} \cdot \left(\left(\left(\sqrt[3]{\sqrt{\pi}} \cdot \sqrt[3]{\sqrt{\pi}}\right) \cdot \sqrt[3]{\sqrt{\pi}}\right) \cdot \ell\right)\right)}\\ \mathbf{elif}\;\pi \cdot \ell \le 6.8293540204626205 \cdot 10^{135}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \frac{\frac{\sin \left(\pi \cdot \ell\right) \cdot \sqrt[3]{1}}{F}}{\left(\frac{1}{24} \cdot \left({\pi}^{4} \cdot {\ell}^{4}\right) + 1\right) - \frac{1}{2} \cdot \left({\pi}^{2} \cdot {\ell}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\left(\sqrt[3]{\pi \cdot \ell} \cdot \sqrt[3]{\pi \cdot \ell}\right) \cdot \sqrt[3]{\pi \cdot \ell}\right)\\ \end{array}\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\begin{array}{l}
\mathbf{if}\;\pi \cdot \ell \le -2.30427372910525123 \cdot 10^{156}:\\
\;\;\;\;\pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \frac{\frac{\sin \left(\pi \cdot \ell\right) \cdot \sqrt[3]{1}}{F}}{\cos \left(\sqrt{\pi} \cdot \left(\left(\left(\sqrt[3]{\sqrt{\pi}} \cdot \sqrt[3]{\sqrt{\pi}}\right) \cdot \sqrt[3]{\sqrt{\pi}}\right) \cdot \ell\right)\right)}\\

\mathbf{elif}\;\pi \cdot \ell \le 6.8293540204626205 \cdot 10^{135}:\\
\;\;\;\;\pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \frac{\frac{\sin \left(\pi \cdot \ell\right) \cdot \sqrt[3]{1}}{F}}{\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)}\\

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

\end{array}
double f(double F, double l) {
        double r15421 = atan2(1.0, 0.0);
        double r15422 = l;
        double r15423 = r15421 * r15422;
        double r15424 = 1.0;
        double r15425 = F;
        double r15426 = r15425 * r15425;
        double r15427 = r15424 / r15426;
        double r15428 = tan(r15423);
        double r15429 = r15427 * r15428;
        double r15430 = r15423 - r15429;
        return r15430;
}
↓
double f(double F, double l) {
        double r15431 = atan2(1.0, 0.0);
        double r15432 = l;
        double r15433 = r15431 * r15432;
        double r15434 = -2.3042737291052512e+156;
        bool r15435 = r15433 <= r15434;
        double r15436 = 1.0;
        double r15437 = cbrt(r15436);
        double r15438 = r15437 * r15437;
        double r15439 = F;
        double r15440 = r15438 / r15439;
        double r15441 = sin(r15433);
        double r15442 = r15441 * r15437;
        double r15443 = r15442 / r15439;
        double r15444 = sqrt(r15431);
        double r15445 = cbrt(r15444);
        double r15446 = r15445 * r15445;
        double r15447 = r15446 * r15445;
        double r15448 = r15447 * r15432;
        double r15449 = r15444 * r15448;
        double r15450 = cos(r15449);
        double r15451 = r15443 / r15450;
        double r15452 = r15440 * r15451;
        double r15453 = r15433 - r15452;
        double r15454 = 6.82935402046262e+135;
        bool r15455 = r15433 <= r15454;
        double r15456 = 0.041666666666666664;
        double r15457 = 4.0;
        double r15458 = pow(r15431, r15457);
        double r15459 = pow(r15432, r15457);
        double r15460 = r15458 * r15459;
        double r15461 = r15456 * r15460;
        double r15462 = 1.0;
        double r15463 = r15461 + r15462;
        double r15464 = 0.5;
        double r15465 = 2.0;
        double r15466 = pow(r15431, r15465);
        double r15467 = pow(r15432, r15465);
        double r15468 = r15466 * r15467;
        double r15469 = r15464 * r15468;
        double r15470 = r15463 - r15469;
        double r15471 = r15443 / r15470;
        double r15472 = r15440 * r15471;
        double r15473 = r15433 - r15472;
        double r15474 = r15439 * r15439;
        double r15475 = r15436 / r15474;
        double r15476 = cbrt(r15433);
        double r15477 = r15476 * r15476;
        double r15478 = r15477 * r15476;
        double r15479 = tan(r15478);
        double r15480 = r15475 * r15479;
        double r15481 = r15433 - r15480;
        double r15482 = r15455 ? r15473 : r15481;
        double r15483 = r15435 ? r15453 : r15482;
        return r15483;
}

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 (* PI l) < -2.3042737291052512e+156

    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 add-cube-cbrt20.8

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

    4. Applied times-frac20.8

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

    5. Applied associate-*l*20.7

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

    7. Applied add-sqr-sqrt20.8

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

    8. Applied associate-*l*20.8

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

    10. Applied tan-quot20.8

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

    11. Applied associate-*r/20.8

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

    12. Simplified20.8

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

    14. Applied add-cube-cbrt20.8

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

    if -2.3042737291052512e+156 < (* PI l) < 6.82935402046262e+135

    1. Initial program 14.9

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

    3. Applied add-cube-cbrt14.9

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

    4. Applied times-frac14.9

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

    5. Applied associate-*l*9.0

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

    7. Applied add-sqr-sqrt9.1

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

    8. Applied associate-*l*9.1

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

    10. Applied tan-quot9.1

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

    11. Applied associate-*r/9.1

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

    12. Simplified8.9

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

    13. Taylor expanded around 0 4.1

      \[\leadsto \pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \frac{\frac{\sin \left(\pi \cdot \ell\right) \cdot \sqrt[3]{1}}{F}}{\color{blue}{\left(\frac{1}{24} \cdot \left({\pi}^{4} \cdot {\ell}^{4}\right) + 1\right) - \frac{1}{2} \cdot \left({\pi}^{2} \cdot {\ell}^{2}\right)}}\]

    if 6.82935402046262e+135 < (* PI l)

    1. Initial program 21.4

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

    3. Applied add-cube-cbrt21.4

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

  4. Final simplification8.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \le -2.30427372910525123 \cdot 10^{156}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \frac{\frac{\sin \left(\pi \cdot \ell\right) \cdot \sqrt[3]{1}}{F}}{\cos \left(\sqrt{\pi} \cdot \left(\left(\left(\sqrt[3]{\sqrt{\pi}} \cdot \sqrt[3]{\sqrt{\pi}}\right) \cdot \sqrt[3]{\sqrt{\pi}}\right) \cdot \ell\right)\right)}\\ \mathbf{elif}\;\pi \cdot \ell \le 6.8293540204626205 \cdot 10^{135}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \frac{\frac{\sin \left(\pi \cdot \ell\right) \cdot \sqrt[3]{1}}{F}}{\left(\frac{1}{24} \cdot \left({\pi}^{4} \cdot {\ell}^{4}\right) + 1\right) - \frac{1}{2} \cdot \left({\pi}^{2} \cdot {\ell}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\left(\sqrt[3]{\pi \cdot \ell} \cdot \sqrt[3]{\pi \cdot \ell}\right) \cdot \sqrt[3]{\pi \cdot \ell}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))