Average Error: 16.6 → 12.7
Time: 24.6s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \frac{\frac{\sqrt[3]{\tan \left(\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \left(\sqrt[3]{\ell} \cdot \pi\right)\right)} \cdot \left(\sqrt[3]{\tan \left(\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \left(\sqrt[3]{\ell} \cdot \pi\right)\right)} \cdot \sqrt[3]{\tan \left(\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \left(\sqrt[3]{\ell} \cdot \pi\right)\right)}\right)}{F}}{F} \cdot 1\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\pi \cdot \ell - \frac{\frac{\sqrt[3]{\tan \left(\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \left(\sqrt[3]{\ell} \cdot \pi\right)\right)} \cdot \left(\sqrt[3]{\tan \left(\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \left(\sqrt[3]{\ell} \cdot \pi\right)\right)} \cdot \sqrt[3]{\tan \left(\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \left(\sqrt[3]{\ell} \cdot \pi\right)\right)}\right)}{F}}{F} \cdot 1
double f(double F, double l) {
        double r789332 = atan2(1.0, 0.0);
        double r789333 = l;
        double r789334 = r789332 * r789333;
        double r789335 = 1.0;
        double r789336 = F;
        double r789337 = r789336 * r789336;
        double r789338 = r789335 / r789337;
        double r789339 = tan(r789334);
        double r789340 = r789338 * r789339;
        double r789341 = r789334 - r789340;
        return r789341;
}
↓
double f(double F, double l) {
        double r789342 = atan2(1.0, 0.0);
        double r789343 = l;
        double r789344 = r789342 * r789343;
        double r789345 = cbrt(r789343);
        double r789346 = r789345 * r789345;
        double r789347 = r789345 * r789342;
        double r789348 = r789346 * r789347;
        double r789349 = tan(r789348);
        double r789350 = cbrt(r789349);
        double r789351 = r789350 * r789350;
        double r789352 = r789350 * r789351;
        double r789353 = F;
        double r789354 = r789352 / r789353;
        double r789355 = r789354 / r789353;
        double r789356 = 1.0;
        double r789357 = r789355 * r789356;
        double r789358 = r789344 - r789357;
        return r789358;
}

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 16.6

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

  3. Applied div-inv16.6

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

  4. Applied associate-*l*16.6

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

  5. Simplified12.4

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

  7. Applied add-cube-cbrt12.6

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

  8. Applied associate-*l*12.6

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

  10. Applied add-cube-cbrt12.7

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

  11. Final simplification12.7

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

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))