Average Error: 16.8 → 12.7
Time: 13.9s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \left(\left(\sqrt[3]{\frac{\sqrt[3]{1}}{F}} \cdot \frac{\sqrt[3]{\sqrt[3]{1}}}{\sqrt[3]{F}}\right) \cdot \left(\sqrt[3]{\frac{\sqrt[3]{1}}{F}} \cdot \tan \left(\pi \cdot \ell\right)\right)\right)\]
\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(\left(\sqrt[3]{\frac{\sqrt[3]{1}}{F}} \cdot \frac{\sqrt[3]{\sqrt[3]{1}}}{\sqrt[3]{F}}\right) \cdot \left(\sqrt[3]{\frac{\sqrt[3]{1}}{F}} \cdot \tan \left(\pi \cdot \ell\right)\right)\right)
double f(double F, double l) {
        double r37351 = atan2(1.0, 0.0);
        double r37352 = l;
        double r37353 = r37351 * r37352;
        double r37354 = 1.0;
        double r37355 = F;
        double r37356 = r37355 * r37355;
        double r37357 = r37354 / r37356;
        double r37358 = tan(r37353);
        double r37359 = r37357 * r37358;
        double r37360 = r37353 - r37359;
        return r37360;
}
↓
double f(double F, double l) {
        double r37361 = atan2(1.0, 0.0);
        double r37362 = l;
        double r37363 = r37361 * r37362;
        double r37364 = 1.0;
        double r37365 = cbrt(r37364);
        double r37366 = r37365 * r37365;
        double r37367 = F;
        double r37368 = r37366 / r37367;
        double r37369 = r37365 / r37367;
        double r37370 = cbrt(r37369);
        double r37371 = cbrt(r37365);
        double r37372 = cbrt(r37367);
        double r37373 = r37371 / r37372;
        double r37374 = r37370 * r37373;
        double r37375 = tan(r37363);
        double r37376 = r37370 * r37375;
        double r37377 = r37374 * r37376;
        double r37378 = r37368 * r37377;
        double r37379 = r37363 - r37378;
        return r37379;
}

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 16.8

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

  3. Applied add-cube-cbrt16.8

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

  4. Applied times-frac16.8

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

  5. Applied associate-*l*12.6

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

  7. Applied add-cube-cbrt12.7

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

  8. Applied associate-*l*12.7

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

  10. Applied cbrt-div12.7

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

  11. Final simplification12.7

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

Reproduce

herbie shell --seed 2019322 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))