Average Error: 8.4 → 0.7
Time: 41.3s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \frac{\frac{1}{F}}{\cos \left(\pi \cdot \ell\right)} \cdot \left(\sin \left(\pi \cdot \ell\right) \cdot \frac{1}{F}\right)\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\pi \cdot \ell - \frac{\frac{1}{F}}{\cos \left(\pi \cdot \ell\right)} \cdot \left(\sin \left(\pi \cdot \ell\right) \cdot \frac{1}{F}\right)
double f(double F, double l) {
        double r585859 = atan2(1.0, 0.0);
        double r585860 = l;
        double r585861 = r585859 * r585860;
        double r585862 = 1.0;
        double r585863 = F;
        double r585864 = r585863 * r585863;
        double r585865 = r585862 / r585864;
        double r585866 = tan(r585861);
        double r585867 = r585865 * r585866;
        double r585868 = r585861 - r585867;
        return r585868;
}
↓
double f(double F, double l) {
        double r585869 = atan2(1.0, 0.0);
        double r585870 = l;
        double r585871 = r585869 * r585870;
        double r585872 = 1.0;
        double r585873 = F;
        double r585874 = r585872 / r585873;
        double r585875 = cos(r585871);
        double r585876 = r585874 / r585875;
        double r585877 = sin(r585871);
        double r585878 = r585877 * r585874;
        double r585879 = r585876 * r585878;
        double r585880 = r585871 - r585879;
        return r585880;
}

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 8.4

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

  2. Simplified0.7

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

  4. Applied clear-num0.7

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

  6. Applied *-un-lft-identity0.7

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

  7. Applied associate-/l*0.7

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

  8. Simplified0.7

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

  10. Applied associate-/r*0.7

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

  12. Applied tan-quot0.7

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

  13. Applied associate-/r/0.7

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

  14. Applied *-un-lft-identity0.7

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

  15. Applied add-cube-cbrt0.7

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

  16. Applied times-frac0.7

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

  17. Applied times-frac0.7

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

  18. Simplified0.7

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

  19. Simplified0.7

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

  20. Final simplification0.7

    \[\leadsto \pi \cdot \ell - \frac{\frac{1}{F}}{\cos \left(\pi \cdot \ell\right)} \cdot \left(\sin \left(\pi \cdot \ell\right) \cdot \frac{1}{F}\right)\]

Reproduce

herbie shell --seed 2019138 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))