Average Error: 16.7 → 12.7
Time: 22.9s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \frac{\frac{\sqrt[3]{1}}{\sqrt[3]{F} \cdot \sqrt[3]{F}}}{\frac{\sqrt[3]{F}}{\tan \left(\pi \cdot \ell\right)}} \cdot \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F}\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\pi \cdot \ell - \frac{\frac{\sqrt[3]{1}}{\sqrt[3]{F} \cdot \sqrt[3]{F}}}{\frac{\sqrt[3]{F}}{\tan \left(\pi \cdot \ell\right)}} \cdot \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F}
double f(double F, double l) {
        double r29658 = atan2(1.0, 0.0);
        double r29659 = l;
        double r29660 = r29658 * r29659;
        double r29661 = 1.0;
        double r29662 = F;
        double r29663 = r29662 * r29662;
        double r29664 = r29661 / r29663;
        double r29665 = tan(r29660);
        double r29666 = r29664 * r29665;
        double r29667 = r29660 - r29666;
        return r29667;
}
↓
double f(double F, double l) {
        double r29668 = atan2(1.0, 0.0);
        double r29669 = l;
        double r29670 = r29668 * r29669;
        double r29671 = 1.0;
        double r29672 = cbrt(r29671);
        double r29673 = F;
        double r29674 = cbrt(r29673);
        double r29675 = r29674 * r29674;
        double r29676 = r29672 / r29675;
        double r29677 = tan(r29670);
        double r29678 = r29674 / r29677;
        double r29679 = r29676 / r29678;
        double r29680 = r29672 * r29672;
        double r29681 = r29680 / r29673;
        double r29682 = r29679 * r29681;
        double r29683 = r29670 - r29682;
        return r29683;
}

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 16.7

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

  3. Applied add-cube-cbrt16.7

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

  4. Applied times-frac16.7

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

  5. Applied associate-*l*12.6

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

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

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

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

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

  9. Applied cbrt-prod12.6

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

  10. Applied times-frac12.6

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

  11. Applied associate-*l*12.6

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

  12. Simplified12.5

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

  14. Applied associate-/l*12.5

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

  16. Applied *-un-lft-identity12.5

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

  17. Applied add-cube-cbrt12.7

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

  18. Applied times-frac12.7

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

  19. Applied associate-/r*12.7

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

  20. Simplified12.7

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

  21. Final simplification12.7

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

Reproduce

herbie shell --seed 2019305 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))