Average Error: 16.5 → 12.4
Time: 30.7s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \frac{\sqrt{1}}{F} \cdot \left(\left(\sqrt[3]{\frac{\sqrt{1}}{F} \cdot \tan \left(\pi \cdot \ell\right)} \cdot \sqrt[3]{\frac{\sqrt{1}}{F} \cdot \tan \left(\pi \cdot \ell\right)}\right) \cdot \left(\sqrt[3]{\frac{\sqrt{1}}{F}} \cdot \sqrt[3]{\tan \left(\pi \cdot \ell\right)}\right)\right)\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\pi \cdot \ell - \frac{\sqrt{1}}{F} \cdot \left(\left(\sqrt[3]{\frac{\sqrt{1}}{F} \cdot \tan \left(\pi \cdot \ell\right)} \cdot \sqrt[3]{\frac{\sqrt{1}}{F} \cdot \tan \left(\pi \cdot \ell\right)}\right) \cdot \left(\sqrt[3]{\frac{\sqrt{1}}{F}} \cdot \sqrt[3]{\tan \left(\pi \cdot \ell\right)}\right)\right)
double f(double F, double l) {
        double r31761 = atan2(1.0, 0.0);
        double r31762 = l;
        double r31763 = r31761 * r31762;
        double r31764 = 1.0;
        double r31765 = F;
        double r31766 = r31765 * r31765;
        double r31767 = r31764 / r31766;
        double r31768 = tan(r31763);
        double r31769 = r31767 * r31768;
        double r31770 = r31763 - r31769;
        return r31770;
}
↓
double f(double F, double l) {
        double r31771 = atan2(1.0, 0.0);
        double r31772 = l;
        double r31773 = r31771 * r31772;
        double r31774 = 1.0;
        double r31775 = sqrt(r31774);
        double r31776 = F;
        double r31777 = r31775 / r31776;
        double r31778 = tan(r31773);
        double r31779 = r31777 * r31778;
        double r31780 = cbrt(r31779);
        double r31781 = r31780 * r31780;
        double r31782 = cbrt(r31777);
        double r31783 = cbrt(r31778);
        double r31784 = r31782 * r31783;
        double r31785 = r31781 * r31784;
        double r31786 = r31777 * r31785;
        double r31787 = r31773 - r31786;
        return r31787;
}

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-sqr-sqrt16.5

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

  4. Applied times-frac16.5

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

  5. Applied associate-*l*12.3

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

  7. Applied add-cube-cbrt12.5

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

  9. Applied cbrt-prod12.4

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

  10. Final simplification12.4

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

Reproduce

herbie shell --seed 2019304 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))