Average Error: 16.5 → 12.6
Time: 10.3s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \left(\left(\sqrt[3]{\frac{\sqrt[3]{1}}{F}} \cdot \left(\sqrt[3]{\frac{1}{\sqrt[3]{F} \cdot \sqrt[3]{F}}} \cdot \sqrt[3]{\frac{\sqrt[3]{1}}{\sqrt[3]{\sqrt[3]{F} \cdot \sqrt[3]{F}} \cdot \sqrt[3]{\sqrt[3]{F}}}}\right)\right) \cdot \left(\sqrt[3]{\frac{\sqrt[3]{1}}{F}} \cdot \tan \left(\pi \cdot \ell\right)\right)\right)\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \left(\left(\sqrt[3]{\frac{\sqrt[3]{1}}{F}} \cdot \left(\sqrt[3]{\frac{1}{\sqrt[3]{F} \cdot \sqrt[3]{F}}} \cdot \sqrt[3]{\frac{\sqrt[3]{1}}{\sqrt[3]{\sqrt[3]{F} \cdot \sqrt[3]{F}} \cdot \sqrt[3]{\sqrt[3]{F}}}}\right)\right) \cdot \left(\sqrt[3]{\frac{\sqrt[3]{1}}{F}} \cdot \tan \left(\pi \cdot \ell\right)\right)\right)
double f(double F, double l) {
        double r25424 = atan2(1.0, 0.0);
        double r25425 = l;
        double r25426 = r25424 * r25425;
        double r25427 = 1.0;
        double r25428 = F;
        double r25429 = r25428 * r25428;
        double r25430 = r25427 / r25429;
        double r25431 = tan(r25426);
        double r25432 = r25430 * r25431;
        double r25433 = r25426 - r25432;
        return r25433;
}
↓
double f(double F, double l) {
        double r25434 = atan2(1.0, 0.0);
        double r25435 = l;
        double r25436 = r25434 * r25435;
        double r25437 = 1.0;
        double r25438 = cbrt(r25437);
        double r25439 = r25438 * r25438;
        double r25440 = F;
        double r25441 = r25439 / r25440;
        double r25442 = r25438 / r25440;
        double r25443 = cbrt(r25442);
        double r25444 = 1.0;
        double r25445 = cbrt(r25440);
        double r25446 = r25445 * r25445;
        double r25447 = r25444 / r25446;
        double r25448 = cbrt(r25447);
        double r25449 = cbrt(r25446);
        double r25450 = cbrt(r25445);
        double r25451 = r25449 * r25450;
        double r25452 = r25438 / r25451;
        double r25453 = cbrt(r25452);
        double r25454 = r25448 * r25453;
        double r25455 = r25443 * r25454;
        double r25456 = tan(r25436);
        double r25457 = r25443 * r25456;
        double r25458 = r25455 * r25457;
        double r25459 = r25441 * r25458;
        double r25460 = r25436 - r25459;
        return r25460;
}

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. Initial program 16.5

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

  3. Applied add-cube-cbrt16.5

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

  4. Applied times-frac16.6

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

  5. Applied associate-*l*12.5

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

  7. Applied add-cube-cbrt12.6

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

  8. Applied associate-*l*12.6

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

  10. Applied add-cube-cbrt12.6

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

  11. Applied *-un-lft-identity12.6

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

  12. Applied cbrt-prod12.6

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

  13. Applied times-frac12.6

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

  14. Applied cbrt-prod12.6

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

  15. Simplified12.6

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

  17. Applied add-cube-cbrt12.6

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

  18. Applied cbrt-prod12.6

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

  19. Final simplification12.6

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

Reproduce

herbie shell --seed 2020021 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))