Average Error: 16.5 → 12.5
Time: 9.1s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \frac{1}{F} \cdot \left(\frac{\sqrt{1}}{\sqrt[3]{F} \cdot \sqrt[3]{F}} \cdot \left(\frac{\sqrt{1}}{\sqrt[3]{F}} \cdot \tan \left(\pi \cdot \ell\right)\right)\right)\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\pi \cdot \ell - \frac{1}{F} \cdot \left(\frac{\sqrt{1}}{\sqrt[3]{F} \cdot \sqrt[3]{F}} \cdot \left(\frac{\sqrt{1}}{\sqrt[3]{F}} \cdot \tan \left(\pi \cdot \ell\right)\right)\right)
double f(double F, double l) {
        double r17986 = atan2(1.0, 0.0);
        double r17987 = l;
        double r17988 = r17986 * r17987;
        double r17989 = 1.0;
        double r17990 = F;
        double r17991 = r17990 * r17990;
        double r17992 = r17989 / r17991;
        double r17993 = tan(r17988);
        double r17994 = r17992 * r17993;
        double r17995 = r17988 - r17994;
        return r17995;
}
↓
double f(double F, double l) {
        double r17996 = atan2(1.0, 0.0);
        double r17997 = l;
        double r17998 = r17996 * r17997;
        double r17999 = 1.0;
        double r18000 = F;
        double r18001 = r17999 / r18000;
        double r18002 = 1.0;
        double r18003 = sqrt(r18002);
        double r18004 = cbrt(r18000);
        double r18005 = r18004 * r18004;
        double r18006 = r18003 / r18005;
        double r18007 = r18003 / r18004;
        double r18008 = tan(r17998);
        double r18009 = r18007 * r18008;
        double r18010 = r18006 * r18009;
        double r18011 = r18001 * r18010;
        double r18012 = r17998 - r18011;
        return r18012;
}

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 *-un-lft-identity16.5

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

  4. Applied times-frac16.5

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

  5. Applied associate-*l*12.3

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

  7. Applied add-cube-cbrt12.5

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

  8. Applied add-sqr-sqrt12.5

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

  9. Applied times-frac12.5

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

  10. Applied associate-*l*12.5

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

  11. Final simplification12.5

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

Reproduce

herbie shell --seed 2020057 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))