Average Error: 16.5 → 8.5
Time: 23.3s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \frac{\frac{\sqrt[3]{1}}{\frac{F}{\pi \cdot \ell} - \frac{1}{3} \cdot \left(F \cdot \left(\pi \cdot \ell\right)\right)}}{F}\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\pi \cdot \ell - \left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \frac{\frac{\sqrt[3]{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 r28464 = atan2(1.0, 0.0);
        double r28465 = l;
        double r28466 = r28464 * r28465;
        double r28467 = 1.0;
        double r28468 = F;
        double r28469 = r28468 * r28468;
        double r28470 = r28467 / r28469;
        double r28471 = tan(r28466);
        double r28472 = r28470 * r28471;
        double r28473 = r28466 - r28472;
        return r28473;
}
↓
double f(double F, double l) {
        double r28474 = atan2(1.0, 0.0);
        double r28475 = l;
        double r28476 = r28474 * r28475;
        double r28477 = 1.0;
        double r28478 = cbrt(r28477);
        double r28479 = r28478 * r28478;
        double r28480 = F;
        double r28481 = r28480 / r28476;
        double r28482 = 0.3333333333333333;
        double r28483 = r28480 * r28476;
        double r28484 = r28482 * r28483;
        double r28485 = r28481 - r28484;
        double r28486 = r28478 / r28485;
        double r28487 = r28486 / r28480;
        double r28488 = r28479 * r28487;
        double r28489 = r28476 - r28488;
        return r28489;
}

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 16.5

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

  3. Applied add-cube-cbrt16.5

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

  4. Applied times-frac16.6

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

  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 div-inv12.5

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

  10. Applied associate-*l*12.5

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

  11. Simplified12.5

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

  12. Taylor expanded around 0 8.5

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

  13. Final simplification8.5

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

Reproduce

herbie shell --seed 2019347 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))