Average Error: 16.8 → 12.6
Time: 10.1s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \left(\left(\frac{\sqrt[3]{\sqrt[3]{1}}}{\sqrt[3]{F}} \cdot \sqrt[3]{\frac{\sqrt[3]{1}}{F}}\right) \cdot \left(\sqrt[3]{\frac{\sqrt[3]{1}}{F}} \cdot \tan \left(\pi \cdot \ell\right)\right)\right)\]
\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(\frac{\sqrt[3]{\sqrt[3]{1}}}{\sqrt[3]{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)
double f(double F, double l) {
        double r23817 = atan2(1.0, 0.0);
        double r23818 = l;
        double r23819 = r23817 * r23818;
        double r23820 = 1.0;
        double r23821 = F;
        double r23822 = r23821 * r23821;
        double r23823 = r23820 / r23822;
        double r23824 = tan(r23819);
        double r23825 = r23823 * r23824;
        double r23826 = r23819 - r23825;
        return r23826;
}
↓
double f(double F, double l) {
        double r23827 = atan2(1.0, 0.0);
        double r23828 = l;
        double r23829 = r23827 * r23828;
        double r23830 = 1.0;
        double r23831 = cbrt(r23830);
        double r23832 = r23831 * r23831;
        double r23833 = F;
        double r23834 = r23832 / r23833;
        double r23835 = cbrt(r23831);
        double r23836 = cbrt(r23833);
        double r23837 = r23835 / r23836;
        double r23838 = r23831 / r23833;
        double r23839 = cbrt(r23838);
        double r23840 = r23837 * r23839;
        double r23841 = tan(r23829);
        double r23842 = r23839 * r23841;
        double r23843 = r23840 * r23842;
        double r23844 = r23834 * r23843;
        double r23845 = r23829 - r23844;
        return r23845;
}

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 16.8

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

  3. Applied add-cube-cbrt16.8

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

  4. Applied times-frac16.8

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

  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 cbrt-div12.6

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

  11. Final simplification12.6

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

Reproduce

herbie shell --seed 2020024 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))