Average Error: 16.3 → 12.2
Time: 9.8s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \left(\sqrt[3]{1} \cdot \frac{1}{\frac{F}{\tan \left(\pi \cdot \ell\right)}}\right)\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \left(\sqrt[3]{1} \cdot \frac{1}{\frac{F}{\tan \left(\pi \cdot \ell\right)}}\right)
double f(double F, double l) {
        double r17476 = atan2(1.0, 0.0);
        double r17477 = l;
        double r17478 = r17476 * r17477;
        double r17479 = 1.0;
        double r17480 = F;
        double r17481 = r17480 * r17480;
        double r17482 = r17479 / r17481;
        double r17483 = tan(r17478);
        double r17484 = r17482 * r17483;
        double r17485 = r17478 - r17484;
        return r17485;
}
↓
double f(double F, double l) {
        double r17486 = atan2(1.0, 0.0);
        double r17487 = l;
        double r17488 = r17486 * r17487;
        double r17489 = 1.0;
        double r17490 = cbrt(r17489);
        double r17491 = r17490 * r17490;
        double r17492 = F;
        double r17493 = r17491 / r17492;
        double r17494 = 1.0;
        double r17495 = tan(r17488);
        double r17496 = r17492 / r17495;
        double r17497 = r17494 / r17496;
        double r17498 = r17490 * r17497;
        double r17499 = r17493 * r17498;
        double r17500 = r17488 - r17499;
        return r17500;
}

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 16.3

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

  3. Applied add-cube-cbrt16.3

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

  4. Applied times-frac16.3

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

  5. Applied associate-*l*12.3

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

  7. Applied div-inv12.3

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

  8. Applied associate-*l*12.3

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

  9. Simplified12.2

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

  11. Applied clear-num12.2

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

  12. Final simplification12.2

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

Reproduce

herbie shell --seed 2020036 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))