Average Error: 16.5 → 12.5
Time: 9.2s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \frac{1}{\frac{F}{\tan \left(\pi \cdot \ell\right)}} \cdot \frac{1}{F}\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\pi \cdot \ell - \frac{1}{\frac{F}{\tan \left(\pi \cdot \ell\right)}} \cdot \frac{1}{F}
double f(double F, double l) {
        double r17398 = atan2(1.0, 0.0);
        double r17399 = l;
        double r17400 = r17398 * r17399;
        double r17401 = 1.0;
        double r17402 = F;
        double r17403 = r17402 * r17402;
        double r17404 = r17401 / r17403;
        double r17405 = tan(r17400);
        double r17406 = r17404 * r17405;
        double r17407 = r17400 - r17406;
        return r17407;
}
↓
double f(double F, double l) {
        double r17408 = atan2(1.0, 0.0);
        double r17409 = l;
        double r17410 = r17408 * r17409;
        double r17411 = 1.0;
        double r17412 = F;
        double r17413 = tan(r17410);
        double r17414 = r17412 / r17413;
        double r17415 = r17411 / r17414;
        double r17416 = 1.0;
        double r17417 = r17416 / r17412;
        double r17418 = r17415 * r17417;
        double r17419 = r17410 - r17418;
        return r17419;
}

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

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

  10. Applied associate-*r*12.5

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

  11. Simplified12.5

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

  12. Final simplification12.5

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

Reproduce

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