Average Error: 17.3 → 12.8
Time: 22.5s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \frac{\sqrt{1}}{F} \cdot \frac{1}{\frac{F}{\sqrt{1} \cdot \tan \left(\sqrt{\pi} \cdot \left(\sqrt{\pi} \cdot \ell\right)\right)}}\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\pi \cdot \ell - \frac{\sqrt{1}}{F} \cdot \frac{1}{\frac{F}{\sqrt{1} \cdot \tan \left(\sqrt{\pi} \cdot \left(\sqrt{\pi} \cdot \ell\right)\right)}}
double f(double F, double l) {
        double r26555 = atan2(1.0, 0.0);
        double r26556 = l;
        double r26557 = r26555 * r26556;
        double r26558 = 1.0;
        double r26559 = F;
        double r26560 = r26559 * r26559;
        double r26561 = r26558 / r26560;
        double r26562 = tan(r26557);
        double r26563 = r26561 * r26562;
        double r26564 = r26557 - r26563;
        return r26564;
}
↓
double f(double F, double l) {
        double r26565 = atan2(1.0, 0.0);
        double r26566 = l;
        double r26567 = r26565 * r26566;
        double r26568 = 1.0;
        double r26569 = sqrt(r26568);
        double r26570 = F;
        double r26571 = r26569 / r26570;
        double r26572 = 1.0;
        double r26573 = sqrt(r26565);
        double r26574 = r26573 * r26566;
        double r26575 = r26573 * r26574;
        double r26576 = tan(r26575);
        double r26577 = r26569 * r26576;
        double r26578 = r26570 / r26577;
        double r26579 = r26572 / r26578;
        double r26580 = r26571 * r26579;
        double r26581 = r26567 - r26580;
        return r26581;
}

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 17.3

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

  3. Applied add-sqr-sqrt17.3

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

  4. Applied times-frac17.3

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

  5. Applied associate-*l*12.7

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

  7. Applied associate-*l/12.6

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

  9. Applied clear-num12.6

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

  11. Applied add-sqr-sqrt12.8

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

  12. Applied associate-*l*12.8

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

  13. Final simplification12.8

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

Reproduce

herbie shell --seed 2019198 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))