Average Error: 16.3 → 8.0
Time: 9.6s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \le -6.18571587100333497 \cdot 10^{154} \lor \neg \left(\pi \cdot \ell \le 1.0127806780932582 \cdot 10^{146}\right):\\ \;\;\;\;\pi \cdot \ell - \frac{\sqrt{1}}{F} \cdot \frac{\sqrt{1} \cdot \sin \left(\pi \cdot \ell\right)}{F \cdot \left(\left(\sqrt[3]{\cos \left(\sqrt{\pi} \cdot \left(\sqrt{\pi} \cdot \ell\right)\right)} \cdot \sqrt[3]{\cos \left(\sqrt{\pi} \cdot \left(\sqrt{\pi} \cdot \ell\right)\right)}\right) \cdot \sqrt[3]{\cos \left(\sqrt{\pi} \cdot \left(\sqrt{\pi} \cdot \ell\right)\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sqrt{1}}{F} \cdot \frac{\sqrt{1} \cdot \sin \left(\pi \cdot \ell\right)}{F \cdot \left(\left(\frac{1}{24} \cdot \left({\pi}^{4} \cdot {\ell}^{4}\right) + 1\right) - \frac{1}{2} \cdot \left({\pi}^{2} \cdot {\ell}^{2}\right)\right)}\\ \end{array}\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\begin{array}{l}
\mathbf{if}\;\pi \cdot \ell \le -6.18571587100333497 \cdot 10^{154} \lor \neg \left(\pi \cdot \ell \le 1.0127806780932582 \cdot 10^{146}\right):\\
\;\;\;\;\pi \cdot \ell - \frac{\sqrt{1}}{F} \cdot \frac{\sqrt{1} \cdot \sin \left(\pi \cdot \ell\right)}{F \cdot \left(\left(\sqrt[3]{\cos \left(\sqrt{\pi} \cdot \left(\sqrt{\pi} \cdot \ell\right)\right)} \cdot \sqrt[3]{\cos \left(\sqrt{\pi} \cdot \left(\sqrt{\pi} \cdot \ell\right)\right)}\right) \cdot \sqrt[3]{\cos \left(\sqrt{\pi} \cdot \left(\sqrt{\pi} \cdot \ell\right)\right)}\right)}\\

\mathbf{else}:\\
\;\;\;\;\pi \cdot \ell - \frac{\sqrt{1}}{F} \cdot \frac{\sqrt{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 f(double F, double l) {
        double r13419 = atan2(1.0, 0.0);
        double r13420 = l;
        double r13421 = r13419 * r13420;
        double r13422 = 1.0;
        double r13423 = F;
        double r13424 = r13423 * r13423;
        double r13425 = r13422 / r13424;
        double r13426 = tan(r13421);
        double r13427 = r13425 * r13426;
        double r13428 = r13421 - r13427;
        return r13428;
}
↓
double f(double F, double l) {
        double r13429 = atan2(1.0, 0.0);
        double r13430 = l;
        double r13431 = r13429 * r13430;
        double r13432 = -6.185715871003335e+154;
        bool r13433 = r13431 <= r13432;
        double r13434 = 1.0127806780932582e+146;
        bool r13435 = r13431 <= r13434;
        double r13436 = !r13435;
        bool r13437 = r13433 || r13436;
        double r13438 = 1.0;
        double r13439 = sqrt(r13438);
        double r13440 = F;
        double r13441 = r13439 / r13440;
        double r13442 = sin(r13431);
        double r13443 = r13439 * r13442;
        double r13444 = sqrt(r13429);
        double r13445 = r13444 * r13430;
        double r13446 = r13444 * r13445;
        double r13447 = cos(r13446);
        double r13448 = cbrt(r13447);
        double r13449 = r13448 * r13448;
        double r13450 = r13449 * r13448;
        double r13451 = r13440 * r13450;
        double r13452 = r13443 / r13451;
        double r13453 = r13441 * r13452;
        double r13454 = r13431 - r13453;
        double r13455 = 0.041666666666666664;
        double r13456 = 4.0;
        double r13457 = pow(r13429, r13456);
        double r13458 = pow(r13430, r13456);
        double r13459 = r13457 * r13458;
        double r13460 = r13455 * r13459;
        double r13461 = 1.0;
        double r13462 = r13460 + r13461;
        double r13463 = 0.5;
        double r13464 = 2.0;
        double r13465 = pow(r13429, r13464);
        double r13466 = pow(r13430, r13464);
        double r13467 = r13465 * r13466;
        double r13468 = r13463 * r13467;
        double r13469 = r13462 - r13468;
        double r13470 = r13440 * r13469;
        double r13471 = r13443 / r13470;
        double r13472 = r13441 * r13471;
        double r13473 = r13431 - r13472;
        double r13474 = r13437 ? r13454 : r13473;
        return r13474;
}

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) < -6.185715871003335e+154 or 1.0127806780932582e+146 < (* PI l)

    1. Initial program 19.7

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

    3. Applied add-sqr-sqrt19.7

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

    4. Applied times-frac19.7

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

    5. Applied associate-*l*19.7

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

    7. Applied tan-quot19.7

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

    8. Applied frac-times19.7

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

    10. Applied add-sqr-sqrt19.7

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

    11. Applied associate-*l*19.7

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

    13. Applied add-cube-cbrt19.7

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

    if -6.185715871003335e+154 < (* PI l) < 1.0127806780932582e+146

    1. Initial program 15.1

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

    3. Applied add-sqr-sqrt15.1

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

    4. Applied times-frac15.1

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

    5. Applied associate-*l*9.5

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

    7. Applied tan-quot9.5

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

    8. Applied frac-times9.4

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

    9. Taylor expanded around 0 3.6

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

  4. Final simplification8.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \le -6.18571587100333497 \cdot 10^{154} \lor \neg \left(\pi \cdot \ell \le 1.0127806780932582 \cdot 10^{146}\right):\\ \;\;\;\;\pi \cdot \ell - \frac{\sqrt{1}}{F} \cdot \frac{\sqrt{1} \cdot \sin \left(\pi \cdot \ell\right)}{F \cdot \left(\left(\sqrt[3]{\cos \left(\sqrt{\pi} \cdot \left(\sqrt{\pi} \cdot \ell\right)\right)} \cdot \sqrt[3]{\cos \left(\sqrt{\pi} \cdot \left(\sqrt{\pi} \cdot \ell\right)\right)}\right) \cdot \sqrt[3]{\cos \left(\sqrt{\pi} \cdot \left(\sqrt{\pi} \cdot \ell\right)\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sqrt{1}}{F} \cdot \frac{\sqrt{1} \cdot \sin \left(\pi \cdot \ell\right)}{F \cdot \left(\left(\frac{1}{24} \cdot \left({\pi}^{4} \cdot {\ell}^{4}\right) + 1\right) - \frac{1}{2} \cdot \left({\pi}^{2} \cdot {\ell}^{2}\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020036 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))