Average Error: 16.5 → 12.5
Time: 9.8s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \frac{\sqrt{1}}{F} \cdot \left(\frac{\sqrt{1}}{F} \cdot \tan \left(\left({\left(\sqrt[3]{\sqrt[3]{\pi}}\right)}^{5} \cdot \sqrt[3]{\sqrt[3]{\pi}}\right) \cdot \left(\sqrt[3]{\pi} \cdot \ell\right)\right)\right)\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\pi \cdot \ell - \frac{\sqrt{1}}{F} \cdot \left(\frac{\sqrt{1}}{F} \cdot \tan \left(\left({\left(\sqrt[3]{\sqrt[3]{\pi}}\right)}^{5} \cdot \sqrt[3]{\sqrt[3]{\pi}}\right) \cdot \left(\sqrt[3]{\pi} \cdot \ell\right)\right)\right)
double f(double F, double l) {
        double r18631 = atan2(1.0, 0.0);
        double r18632 = l;
        double r18633 = r18631 * r18632;
        double r18634 = 1.0;
        double r18635 = F;
        double r18636 = r18635 * r18635;
        double r18637 = r18634 / r18636;
        double r18638 = tan(r18633);
        double r18639 = r18637 * r18638;
        double r18640 = r18633 - r18639;
        return r18640;
}
↓
double f(double F, double l) {
        double r18641 = atan2(1.0, 0.0);
        double r18642 = l;
        double r18643 = r18641 * r18642;
        double r18644 = 1.0;
        double r18645 = sqrt(r18644);
        double r18646 = F;
        double r18647 = r18645 / r18646;
        double r18648 = cbrt(r18641);
        double r18649 = cbrt(r18648);
        double r18650 = 5.0;
        double r18651 = pow(r18649, r18650);
        double r18652 = r18651 * r18649;
        double r18653 = r18648 * r18642;
        double r18654 = r18652 * r18653;
        double r18655 = tan(r18654);
        double r18656 = r18647 * r18655;
        double r18657 = r18647 * r18656;
        double r18658 = r18643 - r18657;
        return r18658;
}

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 add-sqr-sqrt16.5

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

  4. Applied times-frac16.5

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

  5. Applied associate-*l*12.5

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

  7. Applied add-cube-cbrt12.7

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

  8. Applied associate-*l*12.7

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

  10. Applied add-cube-cbrt12.5

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

  11. Applied associate-*r*12.5

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

  12. Simplified12.5

    \[\leadsto \pi \cdot \ell - \frac{\sqrt{1}}{F} \cdot \left(\frac{\sqrt{1}}{F} \cdot \tan \left(\left(\color{blue}{{\left(\sqrt[3]{\sqrt[3]{\pi}}\right)}^{5}} \cdot \sqrt[3]{\sqrt[3]{\pi}}\right) \cdot \left(\sqrt[3]{\pi} \cdot \ell\right)\right)\right)\]

  13. Final simplification12.5

    \[\leadsto \pi \cdot \ell - \frac{\sqrt{1}}{F} \cdot \left(\frac{\sqrt{1}}{F} \cdot \tan \left(\left({\left(\sqrt[3]{\sqrt[3]{\pi}}\right)}^{5} \cdot \sqrt[3]{\sqrt[3]{\pi}}\right) \cdot \left(\sqrt[3]{\pi} \cdot \ell\right)\right)\right)\]

Reproduce

herbie shell --seed 2020020 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))