Average Error: 16.7 → 12.4
Time: 26.9s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \frac{\sqrt{1}}{F} \cdot \left(\left(\sqrt[3]{\frac{\sqrt{1} \cdot \tan \left(\pi \cdot \ell\right)}{F}} \cdot \sqrt[3]{\frac{\sqrt{1} \cdot \tan \left(\pi \cdot \ell\right)}{F}}\right) \cdot \frac{\sqrt[3]{\sqrt{1} \cdot \tan \left(\pi \cdot \ell\right)}}{\sqrt[3]{F}}\right)\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\pi \cdot \ell - \frac{\sqrt{1}}{F} \cdot \left(\left(\sqrt[3]{\frac{\sqrt{1} \cdot \tan \left(\pi \cdot \ell\right)}{F}} \cdot \sqrt[3]{\frac{\sqrt{1} \cdot \tan \left(\pi \cdot \ell\right)}{F}}\right) \cdot \frac{\sqrt[3]{\sqrt{1} \cdot \tan \left(\pi \cdot \ell\right)}}{\sqrt[3]{F}}\right)
double f(double F, double l) {
        double r28360 = atan2(1.0, 0.0);
        double r28361 = l;
        double r28362 = r28360 * r28361;
        double r28363 = 1.0;
        double r28364 = F;
        double r28365 = r28364 * r28364;
        double r28366 = r28363 / r28365;
        double r28367 = tan(r28362);
        double r28368 = r28366 * r28367;
        double r28369 = r28362 - r28368;
        return r28369;
}
↓
double f(double F, double l) {
        double r28370 = atan2(1.0, 0.0);
        double r28371 = l;
        double r28372 = r28370 * r28371;
        double r28373 = 1.0;
        double r28374 = sqrt(r28373);
        double r28375 = F;
        double r28376 = r28374 / r28375;
        double r28377 = tan(r28372);
        double r28378 = r28374 * r28377;
        double r28379 = r28378 / r28375;
        double r28380 = cbrt(r28379);
        double r28381 = r28380 * r28380;
        double r28382 = cbrt(r28378);
        double r28383 = cbrt(r28375);
        double r28384 = r28382 / r28383;
        double r28385 = r28381 * r28384;
        double r28386 = r28376 * r28385;
        double r28387 = r28372 - r28386;
        return r28387;
}

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 16.7

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

  3. Applied add-sqr-sqrt16.7

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

  4. Applied times-frac16.7

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

  5. Applied associate-*l*12.2

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

  7. Applied associate-*l/12.2

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

  9. Applied add-cube-cbrt12.4

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

  11. Applied cbrt-div12.4

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

  12. Final simplification12.4

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

Reproduce

herbie shell --seed 2019326 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))