Average Error: 15.7 → 12.0
Time: 31.4s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \frac{\frac{1}{F} \cdot \tan \left(\left(\sqrt{\sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}\right) \cdot \left(\sqrt{\pi} \cdot \ell\right)\right)}{F}\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\pi \cdot \ell - \frac{\frac{1}{F} \cdot \tan \left(\left(\sqrt{\sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}\right) \cdot \left(\sqrt{\pi} \cdot \ell\right)\right)}{F}
double f(double F, double l) {
        double r565578 = atan2(1.0, 0.0);
        double r565579 = l;
        double r565580 = r565578 * r565579;
        double r565581 = 1.0;
        double r565582 = F;
        double r565583 = r565582 * r565582;
        double r565584 = r565581 / r565583;
        double r565585 = tan(r565580);
        double r565586 = r565584 * r565585;
        double r565587 = r565580 - r565586;
        return r565587;
}
↓
double f(double F, double l) {
        double r565588 = atan2(1.0, 0.0);
        double r565589 = l;
        double r565590 = r565588 * r565589;
        double r565591 = 1.0;
        double r565592 = F;
        double r565593 = r565591 / r565592;
        double r565594 = sqrt(r565588);
        double r565595 = sqrt(r565594);
        double r565596 = r565595 * r565595;
        double r565597 = r565594 * r565589;
        double r565598 = r565596 * r565597;
        double r565599 = tan(r565598);
        double r565600 = r565593 * r565599;
        double r565601 = r565600 / r565592;
        double r565602 = r565590 - r565601;
        return r565602;
}

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 15.7

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

  2. Simplified11.9

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

  4. Applied div-inv12.0

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

  6. Applied add-sqr-sqrt12.1

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

  7. Applied associate-*l*12.1

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

  9. Applied add-sqr-sqrt12.1

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

  10. Applied sqrt-prod12.0

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

  11. Final simplification12.0

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

Reproduce

herbie shell --seed 2019143 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))