Average Error: 16.5 → 12.4
Time: 17.1s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \frac{1}{F} \cdot \left(\left(\sqrt[3]{\frac{1}{F} \cdot \tan \left(\pi \cdot \ell\right)} \cdot \sqrt[3]{\frac{1}{F} \cdot \tan \left(\pi \cdot \ell\right)}\right) \cdot \left(\sqrt[3]{\frac{1}{F}} \cdot \sqrt[3]{\tan \left(\pi \cdot \ell\right)}\right)\right)\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\pi \cdot \ell - \frac{1}{F} \cdot \left(\left(\sqrt[3]{\frac{1}{F} \cdot \tan \left(\pi \cdot \ell\right)} \cdot \sqrt[3]{\frac{1}{F} \cdot \tan \left(\pi \cdot \ell\right)}\right) \cdot \left(\sqrt[3]{\frac{1}{F}} \cdot \sqrt[3]{\tan \left(\pi \cdot \ell\right)}\right)\right)
double f(double F, double l) {
        double r28363 = atan2(1.0, 0.0);
        double r28364 = l;
        double r28365 = r28363 * r28364;
        double r28366 = 1.0;
        double r28367 = F;
        double r28368 = r28367 * r28367;
        double r28369 = r28366 / r28368;
        double r28370 = tan(r28365);
        double r28371 = r28369 * r28370;
        double r28372 = r28365 - r28371;
        return r28372;
}
↓
double f(double F, double l) {
        double r28373 = atan2(1.0, 0.0);
        double r28374 = l;
        double r28375 = r28373 * r28374;
        double r28376 = 1.0;
        double r28377 = F;
        double r28378 = r28376 / r28377;
        double r28379 = 1.0;
        double r28380 = r28379 / r28377;
        double r28381 = tan(r28375);
        double r28382 = r28380 * r28381;
        double r28383 = cbrt(r28382);
        double r28384 = r28383 * r28383;
        double r28385 = cbrt(r28380);
        double r28386 = cbrt(r28381);
        double r28387 = r28385 * r28386;
        double r28388 = r28384 * r28387;
        double r28389 = r28378 * r28388;
        double r28390 = r28375 - r28389;
        return r28390;
}

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 *-un-lft-identity16.5

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

  4. Applied times-frac16.5

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

  5. Applied associate-*l*12.3

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

  7. Applied add-cube-cbrt12.5

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

  9. Applied cbrt-prod12.4

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

  10. Final simplification12.4

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

Reproduce

herbie shell --seed 2019304 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))