Average Error: 17.1 → 9.0
Time: 12.8s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \le -3.07123983171505107 \cdot 10^{154}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\sqrt{\pi} \cdot \left(\sqrt{\pi} \cdot \ell\right)\right)\\ \mathbf{elif}\;\pi \cdot \ell \le 3.0447161306430852 \cdot 10^{131}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \frac{\sqrt[3]{1} \cdot \frac{\sin \left(\pi \cdot \ell\right)}{F}}{\left(\frac{1}{24} \cdot \left({\pi}^{4} \cdot {\ell}^{4}\right) + 1\right) - \frac{1}{2} \cdot \left({\pi}^{2} \cdot {\ell}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \left(\frac{\sqrt[3]{1}}{F} \cdot \tan \left(\left(\pi \cdot \left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right)\right) \cdot \sqrt[3]{\ell}\right)\right)\\ \end{array}\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\begin{array}{l}
\mathbf{if}\;\pi \cdot \ell \le -3.07123983171505107 \cdot 10^{154}:\\
\;\;\;\;\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\sqrt{\pi} \cdot \left(\sqrt{\pi} \cdot \ell\right)\right)\\

\mathbf{elif}\;\pi \cdot \ell \le 3.0447161306430852 \cdot 10^{131}:\\
\;\;\;\;\pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \frac{\sqrt[3]{1} \cdot \frac{\sin \left(\pi \cdot \ell\right)}{F}}{\left(\frac{1}{24} \cdot \left({\pi}^{4} \cdot {\ell}^{4}\right) + 1\right) - \frac{1}{2} \cdot \left({\pi}^{2} \cdot {\ell}^{2}\right)}\\

\mathbf{else}:\\
\;\;\;\;\pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \left(\frac{\sqrt[3]{1}}{F} \cdot \tan \left(\left(\pi \cdot \left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right)\right) \cdot \sqrt[3]{\ell}\right)\right)\\

\end{array}
double f(double F, double l) {
        double r13582 = atan2(1.0, 0.0);
        double r13583 = l;
        double r13584 = r13582 * r13583;
        double r13585 = 1.0;
        double r13586 = F;
        double r13587 = r13586 * r13586;
        double r13588 = r13585 / r13587;
        double r13589 = tan(r13584);
        double r13590 = r13588 * r13589;
        double r13591 = r13584 - r13590;
        return r13591;
}
↓
double f(double F, double l) {
        double r13592 = atan2(1.0, 0.0);
        double r13593 = l;
        double r13594 = r13592 * r13593;
        double r13595 = -3.071239831715051e+154;
        bool r13596 = r13594 <= r13595;
        double r13597 = 1.0;
        double r13598 = F;
        double r13599 = r13598 * r13598;
        double r13600 = r13597 / r13599;
        double r13601 = sqrt(r13592);
        double r13602 = r13601 * r13593;
        double r13603 = r13601 * r13602;
        double r13604 = tan(r13603);
        double r13605 = r13600 * r13604;
        double r13606 = r13594 - r13605;
        double r13607 = 3.0447161306430852e+131;
        bool r13608 = r13594 <= r13607;
        double r13609 = cbrt(r13597);
        double r13610 = r13609 * r13609;
        double r13611 = r13610 / r13598;
        double r13612 = sin(r13594);
        double r13613 = r13612 / r13598;
        double r13614 = r13609 * r13613;
        double r13615 = 0.041666666666666664;
        double r13616 = 4.0;
        double r13617 = pow(r13592, r13616);
        double r13618 = pow(r13593, r13616);
        double r13619 = r13617 * r13618;
        double r13620 = r13615 * r13619;
        double r13621 = 1.0;
        double r13622 = r13620 + r13621;
        double r13623 = 0.5;
        double r13624 = 2.0;
        double r13625 = pow(r13592, r13624);
        double r13626 = pow(r13593, r13624);
        double r13627 = r13625 * r13626;
        double r13628 = r13623 * r13627;
        double r13629 = r13622 - r13628;
        double r13630 = r13614 / r13629;
        double r13631 = r13611 * r13630;
        double r13632 = r13594 - r13631;
        double r13633 = r13609 / r13598;
        double r13634 = cbrt(r13593);
        double r13635 = r13634 * r13634;
        double r13636 = r13592 * r13635;
        double r13637 = r13636 * r13634;
        double r13638 = tan(r13637);
        double r13639 = r13633 * r13638;
        double r13640 = r13611 * r13639;
        double r13641 = r13594 - r13640;
        double r13642 = r13608 ? r13632 : r13641;
        double r13643 = r13596 ? r13606 : r13642;
        return r13643;
}

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. Split input into 3 regimes

  2. if (* PI l) < -3.071239831715051e+154

    1. Initial program 20.9

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

    3. Applied add-sqr-sqrt20.9

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

    4. Applied associate-*l*20.9

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

    if -3.071239831715051e+154 < (* PI l) < 3.0447161306430852e+131

    1. Initial program 15.5

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

    3. Applied add-cube-cbrt15.5

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

    4. Applied times-frac15.5

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

    5. Applied associate-*l*9.5

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

    7. Applied tan-quot9.5

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

    8. Applied associate-*r/9.5

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

    9. Simplified9.4

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

    10. Taylor expanded around 0 4.2

      \[\leadsto \pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \frac{\sqrt[3]{1} \cdot \frac{\sin \left(\pi \cdot \ell\right)}{F}}{\color{blue}{\left(\frac{1}{24} \cdot \left({\pi}^{4} \cdot {\ell}^{4}\right) + 1\right) - \frac{1}{2} \cdot \left({\pi}^{2} \cdot {\ell}^{2}\right)}}\]

    if 3.0447161306430852e+131 < (* PI l)

    1. Initial program 21.1

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

    3. Applied add-cube-cbrt21.1

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

    4. Applied times-frac21.1

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

    5. Applied associate-*l*21.1

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

    7. Applied add-cube-cbrt21.2

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

    8. Applied associate-*r*21.2

      \[\leadsto \pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \left(\frac{\sqrt[3]{1}}{F} \cdot \tan \color{blue}{\left(\left(\pi \cdot \left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right)\right) \cdot \sqrt[3]{\ell}\right)}\right)\]
  3. Recombined 3 regimes into one program.

  4. Final simplification9.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \le -3.07123983171505107 \cdot 10^{154}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\sqrt{\pi} \cdot \left(\sqrt{\pi} \cdot \ell\right)\right)\\ \mathbf{elif}\;\pi \cdot \ell \le 3.0447161306430852 \cdot 10^{131}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \frac{\sqrt[3]{1} \cdot \frac{\sin \left(\pi \cdot \ell\right)}{F}}{\left(\frac{1}{24} \cdot \left({\pi}^{4} \cdot {\ell}^{4}\right) + 1\right) - \frac{1}{2} \cdot \left({\pi}^{2} \cdot {\ell}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \left(\frac{\sqrt[3]{1}}{F} \cdot \tan \left(\left(\pi \cdot \left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right)\right) \cdot \sqrt[3]{\ell}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))