Average Error: 16.9 → 12.5
Time: 29.5s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \frac{1}{F} \cdot \left(\frac{1}{F} \cdot \tan \left(\sqrt{\pi} \cdot \left(\left(\sqrt{\sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}\right) \cdot \ell\right)\right)\right)\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\pi \cdot \ell - \frac{1}{F} \cdot \left(\frac{1}{F} \cdot \tan \left(\sqrt{\pi} \cdot \left(\left(\sqrt{\sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}\right) \cdot \ell\right)\right)\right)
double f(double F, double l) {
        double r23511 = atan2(1.0, 0.0);
        double r23512 = l;
        double r23513 = r23511 * r23512;
        double r23514 = 1.0;
        double r23515 = F;
        double r23516 = r23515 * r23515;
        double r23517 = r23514 / r23516;
        double r23518 = tan(r23513);
        double r23519 = r23517 * r23518;
        double r23520 = r23513 - r23519;
        return r23520;
}
↓
double f(double F, double l) {
        double r23521 = atan2(1.0, 0.0);
        double r23522 = l;
        double r23523 = r23521 * r23522;
        double r23524 = 1.0;
        double r23525 = F;
        double r23526 = r23524 / r23525;
        double r23527 = 1.0;
        double r23528 = r23527 / r23525;
        double r23529 = sqrt(r23521);
        double r23530 = sqrt(r23529);
        double r23531 = r23530 * r23530;
        double r23532 = r23531 * r23522;
        double r23533 = r23529 * r23532;
        double r23534 = tan(r23533);
        double r23535 = r23528 * r23534;
        double r23536 = r23526 * r23535;
        double r23537 = r23523 - r23536;
        return r23537;
}

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 16.9

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

  3. Applied *-un-lft-identity16.9

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

  4. Applied times-frac16.9

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

  5. Applied associate-*l*12.5

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

  7. Applied add-sqr-sqrt12.6

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

  8. Applied associate-*l*12.6

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

  10. Applied add-sqr-sqrt12.6

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

  11. Applied sqrt-prod12.5

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

  12. Final simplification12.5

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

Reproduce

herbie shell --seed 2019303 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))