Average Error: 16.5 → 12.3
Time: 44.2s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\ell \cdot \pi - 1 \cdot \frac{\frac{1}{F} \cdot \tan \left(\left(\left(\left(\sqrt{\sqrt[3]{\pi}} \cdot \sqrt{\sqrt[3]{\pi}}\right) \cdot \left(\sqrt{\sqrt[3]{\pi}} \cdot \sqrt{\sqrt[3]{\pi}}\right)\right) \cdot \ell\right) \cdot \sqrt[3]{\pi}\right)}{F}\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\ell \cdot \pi - 1 \cdot \frac{\frac{1}{F} \cdot \tan \left(\left(\left(\left(\sqrt{\sqrt[3]{\pi}} \cdot \sqrt{\sqrt[3]{\pi}}\right) \cdot \left(\sqrt{\sqrt[3]{\pi}} \cdot \sqrt{\sqrt[3]{\pi}}\right)\right) \cdot \ell\right) \cdot \sqrt[3]{\pi}\right)}{F}
double f(double F, double l) {
        double r746744 = atan2(1.0, 0.0);
        double r746745 = l;
        double r746746 = r746744 * r746745;
        double r746747 = 1.0;
        double r746748 = F;
        double r746749 = r746748 * r746748;
        double r746750 = r746747 / r746749;
        double r746751 = tan(r746746);
        double r746752 = r746750 * r746751;
        double r746753 = r746746 - r746752;
        return r746753;
}
↓
double f(double F, double l) {
        double r746754 = l;
        double r746755 = atan2(1.0, 0.0);
        double r746756 = r746754 * r746755;
        double r746757 = 1.0;
        double r746758 = 1.0;
        double r746759 = F;
        double r746760 = r746758 / r746759;
        double r746761 = cbrt(r746755);
        double r746762 = sqrt(r746761);
        double r746763 = r746762 * r746762;
        double r746764 = r746763 * r746763;
        double r746765 = r746764 * r746754;
        double r746766 = r746765 * r746761;
        double r746767 = tan(r746766);
        double r746768 = r746760 * r746767;
        double r746769 = r746768 / r746759;
        double r746770 = r746757 * r746769;
        double r746771 = r746756 - r746770;
        return r746771;
}

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 div-inv16.5

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

  4. Applied associate-*l*16.5

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

  5. Simplified12.3

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

  7. Applied div-inv12.3

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

  9. Applied add-cube-cbrt12.6

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

  10. Applied associate-*r*12.6

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

  12. Applied add-sqr-sqrt12.4

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

  13. Applied add-sqr-sqrt12.3

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

  14. Applied swap-sqr12.3

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

  15. Final simplification12.3

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

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))