Average Error: 16.1 → 8.6
Time: 25.4s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \le -2.7973049399812435 \cdot 10^{+30}:\\ \;\;\;\;\pi \cdot \ell - \left(\left(\frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}\right)\right)\\ \mathbf{elif}\;\pi \cdot \ell \le 2.4432018004276666 \cdot 10^{+27}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{F} \cdot \frac{\tan \left(\sqrt{\pi} \cdot \left(\sqrt{\pi} \cdot \ell\right)\right)}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\left(\left(\frac{\tan \left(\pi \cdot \ell\right)}{F}\right)\right)}{F}\\ \end{array}\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\begin{array}{l}
\mathbf{if}\;\pi \cdot \ell \le -2.7973049399812435 \cdot 10^{+30}:\\
\;\;\;\;\pi \cdot \ell - \left(\left(\frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}\right)\right)\\

\mathbf{elif}\;\pi \cdot \ell \le 2.4432018004276666 \cdot 10^{+27}:\\
\;\;\;\;\pi \cdot \ell - \frac{1}{F} \cdot \frac{\tan \left(\sqrt{\pi} \cdot \left(\sqrt{\pi} \cdot \ell\right)\right)}{F}\\

\mathbf{else}:\\
\;\;\;\;\pi \cdot \ell - \frac{\left(\left(\frac{\tan \left(\pi \cdot \ell\right)}{F}\right)\right)}{F}\\

\end{array}
double f(double F, double l) {
        double r457787 = atan2(1.0, 0.0);
        double r457788 = l;
        double r457789 = r457787 * r457788;
        double r457790 = 1.0;
        double r457791 = F;
        double r457792 = r457791 * r457791;
        double r457793 = r457790 / r457792;
        double r457794 = tan(r457789);
        double r457795 = r457793 * r457794;
        double r457796 = r457789 - r457795;
        return r457796;
}
↓
double f(double F, double l) {
        double r457797 = atan2(1.0, 0.0);
        double r457798 = l;
        double r457799 = r457797 * r457798;
        double r457800 = -2.7973049399812435e+30;
        bool r457801 = r457799 <= r457800;
        double r457802 = tan(r457799);
        double r457803 = F;
        double r457804 = r457803 * r457803;
        double r457805 = r457802 / r457804;
        double r457806 = /* ERROR: no posit support in C */;
        double r457807 = /* ERROR: no posit support in C */;
        double r457808 = r457799 - r457807;
        double r457809 = 2.4432018004276666e+27;
        bool r457810 = r457799 <= r457809;
        double r457811 = 1.0;
        double r457812 = r457811 / r457803;
        double r457813 = sqrt(r457797);
        double r457814 = r457813 * r457798;
        double r457815 = r457813 * r457814;
        double r457816 = tan(r457815);
        double r457817 = r457816 / r457803;
        double r457818 = r457812 * r457817;
        double r457819 = r457799 - r457818;
        double r457820 = r457802 / r457803;
        double r457821 = /* ERROR: no posit support in C */;
        double r457822 = /* ERROR: no posit support in C */;
        double r457823 = r457822 / r457803;
        double r457824 = r457799 - r457823;
        double r457825 = r457810 ? r457819 : r457824;
        double r457826 = r457801 ? r457808 : r457825;
        return r457826;
}

Error

Bits error versus F

Bits error versus l

Derivation

  1. Split input into 3 regimes

  2. if (* PI l) < -2.7973049399812435e+30

    1. Initial program 23.5

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

    2. Simplified23.5

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

    4. Applied insert-posit1615.8

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

    if -2.7973049399812435e+30 < (* PI l) < 2.4432018004276666e+27

    1. Initial program 9.8

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

    2. Simplified9.4

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

    4. Applied *-un-lft-identity9.4

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

    5. Applied times-frac2.1

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

    7. Applied add-sqr-sqrt2.3

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

    8. Applied associate-*l*2.3

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

    if 2.4432018004276666e+27 < (* PI l)

    1. Initial program 22.5

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

    2. Simplified22.5

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

    4. Applied associate-/r*22.5

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

    6. Applied insert-posit1615.2

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

  4. Final simplification8.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \le -2.7973049399812435 \cdot 10^{+30}:\\ \;\;\;\;\pi \cdot \ell - \left(\left(\frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}\right)\right)\\ \mathbf{elif}\;\pi \cdot \ell \le 2.4432018004276666 \cdot 10^{+27}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{F} \cdot \frac{\tan \left(\sqrt{\pi} \cdot \left(\sqrt{\pi} \cdot \ell\right)\right)}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\left(\left(\frac{\tan \left(\pi \cdot \ell\right)}{F}\right)\right)}{F}\\ \end{array}\]

Reproduce

herbie shell --seed 2019154 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))