Average Error: 16.7 → 12.9
Time: 10.0s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \frac{\sqrt[3]{1} \cdot \tan \left(\left(\pi \cdot \left(\sqrt[3]{\ell} \cdot \left(\sqrt[3]{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \sqrt[3]{\sqrt[3]{\ell}}\right)\right)\right) \cdot \sqrt[3]{\ell}\right)}{F}\]
\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 \frac{\sqrt[3]{1} \cdot \tan \left(\left(\pi \cdot \left(\sqrt[3]{\ell} \cdot \left(\sqrt[3]{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \sqrt[3]{\sqrt[3]{\ell}}\right)\right)\right) \cdot \sqrt[3]{\ell}\right)}{F}
double f(double F, double l) {
        double r20967 = atan2(1.0, 0.0);
        double r20968 = l;
        double r20969 = r20967 * r20968;
        double r20970 = 1.0;
        double r20971 = F;
        double r20972 = r20971 * r20971;
        double r20973 = r20970 / r20972;
        double r20974 = tan(r20969);
        double r20975 = r20973 * r20974;
        double r20976 = r20969 - r20975;
        return r20976;
}
↓
double f(double F, double l) {
        double r20977 = atan2(1.0, 0.0);
        double r20978 = l;
        double r20979 = r20977 * r20978;
        double r20980 = 1.0;
        double r20981 = cbrt(r20980);
        double r20982 = r20981 * r20981;
        double r20983 = F;
        double r20984 = r20982 / r20983;
        double r20985 = cbrt(r20978);
        double r20986 = r20985 * r20985;
        double r20987 = cbrt(r20986);
        double r20988 = cbrt(r20985);
        double r20989 = r20987 * r20988;
        double r20990 = r20985 * r20989;
        double r20991 = r20977 * r20990;
        double r20992 = r20991 * r20985;
        double r20993 = tan(r20992);
        double r20994 = r20981 * r20993;
        double r20995 = r20994 / r20983;
        double r20996 = r20984 * r20995;
        double r20997 = r20979 - r20996;
        return r20997;
}

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 associate-*l/12.6

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

  9. Applied add-cube-cbrt12.8

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

  10. Applied associate-*r*12.8

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

  12. Applied add-cube-cbrt12.8

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

  13. Applied cbrt-prod12.9

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

  14. Final simplification12.9

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

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))