Average Error: 16.8 → 8.5
Time: 21.1s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \frac{\frac{1}{\frac{F}{\pi \cdot \ell} - \frac{1}{3} \cdot \left(F \cdot \left(\pi \cdot \ell\right)\right)}}{F}\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\pi \cdot \ell - \frac{\frac{1}{\frac{F}{\pi \cdot \ell} - \frac{1}{3} \cdot \left(F \cdot \left(\pi \cdot \ell\right)\right)}}{F}
double f(double F, double l) {
        double r26613 = atan2(1.0, 0.0);
        double r26614 = l;
        double r26615 = r26613 * r26614;
        double r26616 = 1.0;
        double r26617 = F;
        double r26618 = r26617 * r26617;
        double r26619 = r26616 / r26618;
        double r26620 = tan(r26615);
        double r26621 = r26619 * r26620;
        double r26622 = r26615 - r26621;
        return r26622;
}
↓
double f(double F, double l) {
        double r26623 = atan2(1.0, 0.0);
        double r26624 = l;
        double r26625 = r26623 * r26624;
        double r26626 = 1.0;
        double r26627 = F;
        double r26628 = r26627 / r26625;
        double r26629 = 0.3333333333333333;
        double r26630 = r26627 * r26625;
        double r26631 = r26629 * r26630;
        double r26632 = r26628 - r26631;
        double r26633 = r26626 / r26632;
        double r26634 = r26633 / r26627;
        double r26635 = r26625 - r26634;
        return r26635;
}

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 16.8

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

  3. Applied add-cube-cbrt16.8

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

  4. Applied times-frac16.8

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

  5. Applied associate-*l*12.5

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

  7. Applied associate-*l/12.5

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

  9. Applied associate-*r/12.5

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

  10. Simplified12.5

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

  11. Taylor expanded around 0 8.5

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

  12. Final simplification8.5

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

Reproduce

herbie shell --seed 2019298 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))