Average Error: 8.5 → 0.7
Time: 35.0s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \frac{\frac{\sin \left(\pi \cdot \ell\right)}{F}}{F \cdot \cos \left(\left(\sqrt[3]{\pi \cdot \ell} \cdot \sqrt[3]{\pi \cdot \ell}\right) \cdot \left(\sqrt[3]{\ell} \cdot \sqrt[3]{\pi}\right)\right)}\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\pi \cdot \ell - \frac{\frac{\sin \left(\pi \cdot \ell\right)}{F}}{F \cdot \cos \left(\left(\sqrt[3]{\pi \cdot \ell} \cdot \sqrt[3]{\pi \cdot \ell}\right) \cdot \left(\sqrt[3]{\ell} \cdot \sqrt[3]{\pi}\right)\right)}
double f(double F, double l) {
        double r658335 = atan2(1.0, 0.0);
        double r658336 = l;
        double r658337 = r658335 * r658336;
        double r658338 = 1.0;
        double r658339 = F;
        double r658340 = r658339 * r658339;
        double r658341 = r658338 / r658340;
        double r658342 = tan(r658337);
        double r658343 = r658341 * r658342;
        double r658344 = r658337 - r658343;
        return r658344;
}
↓
double f(double F, double l) {
        double r658345 = atan2(1.0, 0.0);
        double r658346 = l;
        double r658347 = r658345 * r658346;
        double r658348 = sin(r658347);
        double r658349 = F;
        double r658350 = r658348 / r658349;
        double r658351 = cbrt(r658347);
        double r658352 = r658351 * r658351;
        double r658353 = cbrt(r658346);
        double r658354 = cbrt(r658345);
        double r658355 = r658353 * r658354;
        double r658356 = r658352 * r658355;
        double r658357 = cos(r658356);
        double r658358 = r658349 * r658357;
        double r658359 = r658350 / r658358;
        double r658360 = r658347 - r658359;
        return r658360;
}

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. Taylor expanded around -inf 8.0

    \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\sin \left(\pi \cdot \ell\right)}{{F}^{2} \cdot \cos \left(\pi \cdot \ell\right)}}\]

  4. Simplified0.7

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

  6. Applied add-cube-cbrt0.7

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

  8. Applied cbrt-prod0.7

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

  9. Final simplification0.7

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

Reproduce

herbie shell --seed 2019133 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))