Average Error: 17.1 → 13.1
Time: 21.0s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \frac{\frac{1}{F}}{\frac{1}{\sqrt[3]{1} \cdot \tan \left(\pi \cdot \ell\right)}}\]
\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 \frac{\frac{1}{F}}{\frac{1}{\sqrt[3]{1} \cdot \tan \left(\pi \cdot \ell\right)}}
double f(double F, double l) {
        double r27849 = atan2(1.0, 0.0);
        double r27850 = l;
        double r27851 = r27849 * r27850;
        double r27852 = 1.0;
        double r27853 = F;
        double r27854 = r27853 * r27853;
        double r27855 = r27852 / r27854;
        double r27856 = tan(r27851);
        double r27857 = r27855 * r27856;
        double r27858 = r27851 - r27857;
        return r27858;
}
↓
double f(double F, double l) {
        double r27859 = atan2(1.0, 0.0);
        double r27860 = l;
        double r27861 = r27859 * r27860;
        double r27862 = 1.0;
        double r27863 = cbrt(r27862);
        double r27864 = r27863 * r27863;
        double r27865 = F;
        double r27866 = r27864 / r27865;
        double r27867 = 1.0;
        double r27868 = r27867 / r27865;
        double r27869 = tan(r27861);
        double r27870 = r27863 * r27869;
        double r27871 = r27867 / r27870;
        double r27872 = r27868 / r27871;
        double r27873 = r27866 * r27872;
        double r27874 = r27861 - r27873;
        return r27874;
}

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 17.1

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

  3. Applied add-cube-cbrt17.1

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

  4. Applied times-frac17.2

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

  5. Applied associate-*l*13.1

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

  7. Applied *-un-lft-identity13.1

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

  8. Applied *-un-lft-identity13.1

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

  9. Applied cbrt-prod13.1

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

  10. Applied times-frac13.1

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

  11. Applied associate-*l*13.1

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

  12. Simplified13.0

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

  14. Applied clear-num13.0

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

  16. Applied div-inv13.0

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

  17. Applied associate-/r*13.1

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

  18. Final simplification13.1

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

Reproduce

herbie shell --seed 2019208 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))