Average Error: 16.6 → 12.3
Time: 9.8s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \frac{\sqrt{1}}{F} \cdot \frac{1}{\frac{F}{\sqrt{1} \cdot \tan \left(\pi \cdot \ell\right)}}\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\pi \cdot \ell - \frac{\sqrt{1}}{F} \cdot \frac{1}{\frac{F}{\sqrt{1} \cdot \tan \left(\pi \cdot \ell\right)}}
double f(double F, double l) {
        double r17691 = atan2(1.0, 0.0);
        double r17692 = l;
        double r17693 = r17691 * r17692;
        double r17694 = 1.0;
        double r17695 = F;
        double r17696 = r17695 * r17695;
        double r17697 = r17694 / r17696;
        double r17698 = tan(r17693);
        double r17699 = r17697 * r17698;
        double r17700 = r17693 - r17699;
        return r17700;
}
↓
double f(double F, double l) {
        double r17701 = atan2(1.0, 0.0);
        double r17702 = l;
        double r17703 = r17701 * r17702;
        double r17704 = 1.0;
        double r17705 = sqrt(r17704);
        double r17706 = F;
        double r17707 = r17705 / r17706;
        double r17708 = 1.0;
        double r17709 = tan(r17703);
        double r17710 = r17705 * r17709;
        double r17711 = r17706 / r17710;
        double r17712 = r17708 / r17711;
        double r17713 = r17707 * r17712;
        double r17714 = r17703 - r17713;
        return r17714;
}

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-frac16.6

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

  5. Applied associate-*l*12.3

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

  7. Applied associate-*l/12.3

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

  9. Applied clear-num12.3

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

  10. Final simplification12.3

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

Reproduce

herbie shell --seed 2020035 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))