Average Error: 16.9 → 12.9
Time: 26.6s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \left(\sqrt[3]{\frac{1}{F}} \cdot \sqrt[3]{\frac{1}{F}}\right) \cdot \left(\sqrt[3]{\frac{1}{F}} \cdot \frac{1 \cdot \tan \left(\pi \cdot \ell\right)}{F}\right)\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\pi \cdot \ell - \left(\sqrt[3]{\frac{1}{F}} \cdot \sqrt[3]{\frac{1}{F}}\right) \cdot \left(\sqrt[3]{\frac{1}{F}} \cdot \frac{1 \cdot \tan \left(\pi \cdot \ell\right)}{F}\right)
double f(double F, double l) {
        double r25851 = atan2(1.0, 0.0);
        double r25852 = l;
        double r25853 = r25851 * r25852;
        double r25854 = 1.0;
        double r25855 = F;
        double r25856 = r25855 * r25855;
        double r25857 = r25854 / r25856;
        double r25858 = tan(r25853);
        double r25859 = r25857 * r25858;
        double r25860 = r25853 - r25859;
        return r25860;
}
↓
double f(double F, double l) {
        double r25861 = atan2(1.0, 0.0);
        double r25862 = l;
        double r25863 = r25861 * r25862;
        double r25864 = 1.0;
        double r25865 = F;
        double r25866 = r25864 / r25865;
        double r25867 = cbrt(r25866);
        double r25868 = r25867 * r25867;
        double r25869 = 1.0;
        double r25870 = tan(r25863);
        double r25871 = r25869 * r25870;
        double r25872 = r25871 / r25865;
        double r25873 = r25867 * r25872;
        double r25874 = r25868 * r25873;
        double r25875 = r25863 - r25874;
        return r25875;
}

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 16.9

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

  3. Applied *-un-lft-identity16.9

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

  4. Applied times-frac16.9

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

  5. Applied associate-*l*12.7

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

  7. Applied associate-*l/12.7

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

  9. Applied add-cube-cbrt12.9

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

  10. Applied associate-*l*12.9

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

  11. Final simplification12.9

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

Reproduce

herbie shell --seed 2019325 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))