Average Error: 16.6 → 12.4
Time: 27.7s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \left(\tan \left(\left(\sqrt{\sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}\right) \cdot \left(\ell \cdot \sqrt{\pi}\right)\right) \cdot \frac{\sqrt{1}}{F}\right) \cdot \frac{\sqrt{1}}{F}\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\pi \cdot \ell - \left(\tan \left(\left(\sqrt{\sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}\right) \cdot \left(\ell \cdot \sqrt{\pi}\right)\right) \cdot \frac{\sqrt{1}}{F}\right) \cdot \frac{\sqrt{1}}{F}
double f(double F, double l) {
        double r999858 = atan2(1.0, 0.0);
        double r999859 = l;
        double r999860 = r999858 * r999859;
        double r999861 = 1.0;
        double r999862 = F;
        double r999863 = r999862 * r999862;
        double r999864 = r999861 / r999863;
        double r999865 = tan(r999860);
        double r999866 = r999864 * r999865;
        double r999867 = r999860 - r999866;
        return r999867;
}
↓
double f(double F, double l) {
        double r999868 = atan2(1.0, 0.0);
        double r999869 = l;
        double r999870 = r999868 * r999869;
        double r999871 = sqrt(r999868);
        double r999872 = sqrt(r999871);
        double r999873 = r999872 * r999872;
        double r999874 = r999869 * r999871;
        double r999875 = r999873 * r999874;
        double r999876 = tan(r999875);
        double r999877 = 1.0;
        double r999878 = sqrt(r999877);
        double r999879 = F;
        double r999880 = r999878 / r999879;
        double r999881 = r999876 * r999880;
        double r999882 = r999881 * r999880;
        double r999883 = r999870 - r999882;
        return r999883;
}

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 16.6

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

  3. Applied add-sqr-sqrt16.6

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

  4. Applied times-frac16.7

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

  5. Applied associate-*l*12.4

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

  7. Applied add-sqr-sqrt12.5

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

  8. Applied associate-*l*12.5

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

  10. Applied add-sqr-sqrt12.5

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

  11. Applied sqrt-prod12.4

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

  12. Final simplification12.4

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

Reproduce

herbie shell --seed 2019170 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))