Average Error: 16.6 → 12.4
Time: 29.1s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \left(\tan \left(\sqrt{\pi} \cdot \left(\ell \cdot \left(\sqrt{\sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}\right)\right)\right) \cdot \frac{\sqrt{1}}{F}\right) \cdot \frac{\sqrt{1}}{F}\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\pi \cdot \ell - \left(\tan \left(\sqrt{\pi} \cdot \left(\ell \cdot \left(\sqrt{\sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}\right)\right)\right) \cdot \frac{\sqrt{1}}{F}\right) \cdot \frac{\sqrt{1}}{F}
double f(double F, double l) {
        double r946358 = atan2(1.0, 0.0);
        double r946359 = l;
        double r946360 = r946358 * r946359;
        double r946361 = 1.0;
        double r946362 = F;
        double r946363 = r946362 * r946362;
        double r946364 = r946361 / r946363;
        double r946365 = tan(r946360);
        double r946366 = r946364 * r946365;
        double r946367 = r946360 - r946366;
        return r946367;
}
↓
double f(double F, double l) {
        double r946368 = atan2(1.0, 0.0);
        double r946369 = l;
        double r946370 = r946368 * r946369;
        double r946371 = sqrt(r946368);
        double r946372 = sqrt(r946371);
        double r946373 = r946372 * r946372;
        double r946374 = r946369 * r946373;
        double r946375 = r946371 * r946374;
        double r946376 = tan(r946375);
        double r946377 = 1.0;
        double r946378 = sqrt(r946377);
        double r946379 = F;
        double r946380 = r946378 / r946379;
        double r946381 = r946376 * r946380;
        double r946382 = r946381 * r946380;
        double r946383 = r946370 - r946382;
        return r946383;
}

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 16.6

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

  3. Applied add-sqr-sqrt16.6

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

  4. Applied times-frac16.6

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

  5. Applied associate-*l*12.4

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

  7. Applied add-sqr-sqrt12.5

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

  8. Applied associate-*l*12.5

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

  10. Applied add-sqr-sqrt12.5

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

  11. Applied sqrt-prod12.4

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

  12. Final simplification12.4

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

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))