Average Error: 16.8 → 8.4
Time: 8.1s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \le -1.3614300485963869 \cdot 10^{155} \lor \neg \left(\pi \cdot \ell \le 3.5842679016593094 \cdot 10^{146}\right):\\ \;\;\;\;\pi \cdot \ell - \left(\sqrt[3]{\frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)} \cdot \sqrt[3]{\frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)}\right) \cdot \sqrt[3]{\frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \frac{\sqrt[3]{1} \cdot \sin \left(\pi \cdot \ell\right)}{F \cdot \left(\left(\frac{1}{24} \cdot \left({\pi}^{4} \cdot {\ell}^{4}\right) + 1\right) - \frac{1}{2} \cdot \left({\pi}^{2} \cdot {\ell}^{2}\right)\right)}\\ \end{array}\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\begin{array}{l}
\mathbf{if}\;\pi \cdot \ell \le -1.3614300485963869 \cdot 10^{155} \lor \neg \left(\pi \cdot \ell \le 3.5842679016593094 \cdot 10^{146}\right):\\
\;\;\;\;\pi \cdot \ell - \left(\sqrt[3]{\frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)} \cdot \sqrt[3]{\frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)}\right) \cdot \sqrt[3]{\frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)}\\

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

\end{array}
double f(double F, double l) {
        double r13636 = atan2(1.0, 0.0);
        double r13637 = l;
        double r13638 = r13636 * r13637;
        double r13639 = 1.0;
        double r13640 = F;
        double r13641 = r13640 * r13640;
        double r13642 = r13639 / r13641;
        double r13643 = tan(r13638);
        double r13644 = r13642 * r13643;
        double r13645 = r13638 - r13644;
        return r13645;
}
↓
double f(double F, double l) {
        double r13646 = atan2(1.0, 0.0);
        double r13647 = l;
        double r13648 = r13646 * r13647;
        double r13649 = -1.361430048596387e+155;
        bool r13650 = r13648 <= r13649;
        double r13651 = 3.5842679016593094e+146;
        bool r13652 = r13648 <= r13651;
        double r13653 = !r13652;
        bool r13654 = r13650 || r13653;
        double r13655 = 1.0;
        double r13656 = F;
        double r13657 = r13656 * r13656;
        double r13658 = r13655 / r13657;
        double r13659 = tan(r13648);
        double r13660 = r13658 * r13659;
        double r13661 = cbrt(r13660);
        double r13662 = r13661 * r13661;
        double r13663 = r13662 * r13661;
        double r13664 = r13648 - r13663;
        double r13665 = cbrt(r13655);
        double r13666 = r13665 * r13665;
        double r13667 = r13666 / r13656;
        double r13668 = sin(r13648);
        double r13669 = r13665 * r13668;
        double r13670 = 0.041666666666666664;
        double r13671 = 4.0;
        double r13672 = pow(r13646, r13671);
        double r13673 = pow(r13647, r13671);
        double r13674 = r13672 * r13673;
        double r13675 = r13670 * r13674;
        double r13676 = 1.0;
        double r13677 = r13675 + r13676;
        double r13678 = 0.5;
        double r13679 = 2.0;
        double r13680 = pow(r13646, r13679);
        double r13681 = pow(r13647, r13679);
        double r13682 = r13680 * r13681;
        double r13683 = r13678 * r13682;
        double r13684 = r13677 - r13683;
        double r13685 = r13656 * r13684;
        double r13686 = r13669 / r13685;
        double r13687 = r13667 * r13686;
        double r13688 = r13648 - r13687;
        double r13689 = r13654 ? r13664 : r13688;
        return r13689;
}

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 2 regimes

  2. if (* PI l) < -1.361430048596387e+155 or 3.5842679016593094e+146 < (* PI l)

    1. Initial program 20.1

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

    3. Applied add-cube-cbrt20.1

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

    if -1.361430048596387e+155 < (* PI l) < 3.5842679016593094e+146

    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-frac15.6

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

    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 frac-times9.5

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

    9. Taylor expanded around 0 4.1

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

  4. Final simplification8.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \le -1.3614300485963869 \cdot 10^{155} \lor \neg \left(\pi \cdot \ell \le 3.5842679016593094 \cdot 10^{146}\right):\\ \;\;\;\;\pi \cdot \ell - \left(\sqrt[3]{\frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)} \cdot \sqrt[3]{\frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)}\right) \cdot \sqrt[3]{\frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \frac{\sqrt[3]{1} \cdot \sin \left(\pi \cdot \ell\right)}{F \cdot \left(\left(\frac{1}{24} \cdot \left({\pi}^{4} \cdot {\ell}^{4}\right) + 1\right) - \frac{1}{2} \cdot \left({\pi}^{2} \cdot {\ell}^{2}\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020064 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))