Average Error: 16.5 → 12.4
Time: 30.7s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \left(\left(\sqrt[3]{\frac{\sqrt[3]{1}}{F} \cdot \tan \left(\pi \cdot \ell\right)} \cdot \sqrt[3]{\frac{\sqrt[3]{1}}{F} \cdot \tan \left(\pi \cdot \ell\right)}\right) \cdot \left(\sqrt[3]{\frac{\sqrt[3]{1}}{F}} \cdot \sqrt[3]{\tan \left(\pi \cdot \ell\right)}\right)\right)\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \left(\left(\sqrt[3]{\frac{\sqrt[3]{1}}{F} \cdot \tan \left(\pi \cdot \ell\right)} \cdot \sqrt[3]{\frac{\sqrt[3]{1}}{F} \cdot \tan \left(\pi \cdot \ell\right)}\right) \cdot \left(\sqrt[3]{\frac{\sqrt[3]{1}}{F}} \cdot \sqrt[3]{\tan \left(\pi \cdot \ell\right)}\right)\right)
double f(double F, double l) {
        double r32900 = atan2(1.0, 0.0);
        double r32901 = l;
        double r32902 = r32900 * r32901;
        double r32903 = 1.0;
        double r32904 = F;
        double r32905 = r32904 * r32904;
        double r32906 = r32903 / r32905;
        double r32907 = tan(r32902);
        double r32908 = r32906 * r32907;
        double r32909 = r32902 - r32908;
        return r32909;
}
↓
double f(double F, double l) {
        double r32910 = atan2(1.0, 0.0);
        double r32911 = l;
        double r32912 = r32910 * r32911;
        double r32913 = 1.0;
        double r32914 = cbrt(r32913);
        double r32915 = r32914 * r32914;
        double r32916 = F;
        double r32917 = r32915 / r32916;
        double r32918 = r32914 / r32916;
        double r32919 = tan(r32912);
        double r32920 = r32918 * r32919;
        double r32921 = cbrt(r32920);
        double r32922 = r32921 * r32921;
        double r32923 = cbrt(r32918);
        double r32924 = cbrt(r32919);
        double r32925 = r32923 * r32924;
        double r32926 = r32922 * r32925;
        double r32927 = r32917 * r32926;
        double r32928 = r32912 - r32927;
        return r32928;
}

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-frac16.5

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

  5. Applied associate-*l*12.3

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

  7. Applied add-cube-cbrt12.5

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

  9. Applied cbrt-prod12.4

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

  10. Final simplification12.4

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

Reproduce

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