Average Error: 8.5 → 1.0
Time: 36.0s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \frac{\left(\sqrt[3]{\frac{\tan \left(\pi \cdot \ell\right)}{F}} \cdot \sqrt[3]{\frac{\tan \left(\pi \cdot \ell\right)}{F}}\right) \cdot \sqrt[3]{\frac{\tan \left(\pi \cdot \ell\right)}{F}}}{F}\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\pi \cdot \ell - \frac{\left(\sqrt[3]{\frac{\tan \left(\pi \cdot \ell\right)}{F}} \cdot \sqrt[3]{\frac{\tan \left(\pi \cdot \ell\right)}{F}}\right) \cdot \sqrt[3]{\frac{\tan \left(\pi \cdot \ell\right)}{F}}}{F}
double f(double F, double l) {
        double r574577 = atan2(1.0, 0.0);
        double r574578 = l;
        double r574579 = r574577 * r574578;
        double r574580 = 1.0;
        double r574581 = F;
        double r574582 = r574581 * r574581;
        double r574583 = r574580 / r574582;
        double r574584 = tan(r574579);
        double r574585 = r574583 * r574584;
        double r574586 = r574579 - r574585;
        return r574586;
}
↓
double f(double F, double l) {
        double r574587 = atan2(1.0, 0.0);
        double r574588 = l;
        double r574589 = r574587 * r574588;
        double r574590 = tan(r574589);
        double r574591 = F;
        double r574592 = r574590 / r574591;
        double r574593 = cbrt(r574592);
        double r574594 = r574593 * r574593;
        double r574595 = r574594 * r574593;
        double r574596 = r574595 / r574591;
        double r574597 = r574589 - r574596;
        return r574597;
}

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 8.5

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

  2. Simplified8.0

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

  4. Applied associate-/r*0.6

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

  6. Applied add-cube-cbrt1.0

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

  7. Final simplification1.0

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

Reproduce

herbie shell --seed 2019132 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))