Average Error: 16.3 → 9.1
Time: 27.8s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \le -1.1470128979354866 \cdot 10^{+18}:\\ \;\;\;\;\pi \cdot \ell - \left(\left(\frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}\right)\right)\\ \mathbf{elif}\;\pi \cdot \ell \le 6.176150626079544 \cdot 10^{-13}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{F} \cdot \frac{\tan \left(\pi \cdot \ell\right)}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\left(\left(\frac{\tan \left(\pi \cdot \ell\right)}{F}\right)\right)}{F}\\ \end{array}\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\begin{array}{l}
\mathbf{if}\;\pi \cdot \ell \le -1.1470128979354866 \cdot 10^{+18}:\\
\;\;\;\;\pi \cdot \ell - \left(\left(\frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}\right)\right)\\

\mathbf{elif}\;\pi \cdot \ell \le 6.176150626079544 \cdot 10^{-13}:\\
\;\;\;\;\pi \cdot \ell - \frac{1}{F} \cdot \frac{\tan \left(\pi \cdot \ell\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 r386670 = atan2(1.0, 0.0);
        double r386671 = l;
        double r386672 = r386670 * r386671;
        double r386673 = 1.0;
        double r386674 = F;
        double r386675 = r386674 * r386674;
        double r386676 = r386673 / r386675;
        double r386677 = tan(r386672);
        double r386678 = r386676 * r386677;
        double r386679 = r386672 - r386678;
        return r386679;
}
↓
double f(double F, double l) {
        double r386680 = atan2(1.0, 0.0);
        double r386681 = l;
        double r386682 = r386680 * r386681;
        double r386683 = -1.1470128979354866e+18;
        bool r386684 = r386682 <= r386683;
        double r386685 = tan(r386682);
        double r386686 = F;
        double r386687 = r386686 * r386686;
        double r386688 = r386685 / r386687;
        double r386689 = /* ERROR: no posit support in C */;
        double r386690 = /* ERROR: no posit support in C */;
        double r386691 = r386682 - r386690;
        double r386692 = 6.176150626079544e-13;
        bool r386693 = r386682 <= r386692;
        double r386694 = 1.0;
        double r386695 = r386694 / r386686;
        double r386696 = r386685 / r386686;
        double r386697 = r386695 * r386696;
        double r386698 = r386682 - r386697;
        double r386699 = /* ERROR: no posit support in C */;
        double r386700 = /* ERROR: no posit support in C */;
        double r386701 = r386700 / r386686;
        double r386702 = r386682 - r386701;
        double r386703 = r386693 ? r386698 : r386702;
        double r386704 = r386684 ? r386691 : r386703;
        return r386704;
}

Error

Bits error versus F

Bits error versus l

Derivation

  1. Split input into 3 regimes

  2. if (* PI l) < -1.1470128979354866e+18

    1. Initial program 23.3

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

    2. Simplified23.3

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

    4. Applied insert-posit1615.5

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

    if -1.1470128979354866e+18 < (* PI l) < 6.176150626079544e-13

    1. Initial program 9.1

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

    2. Simplified8.6

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

    4. Applied *-un-lft-identity8.6

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

    5. Applied times-frac0.8

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

    if 6.176150626079544e-13 < (* PI l)

    1. Initial program 22.0

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

    2. Simplified22.0

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

    4. Applied associate-/r*22.0

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

    6. Applied insert-posit1617.2

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

  4. Final simplification9.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \le -1.1470128979354866 \cdot 10^{+18}:\\ \;\;\;\;\pi \cdot \ell - \left(\left(\frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}\right)\right)\\ \mathbf{elif}\;\pi \cdot \ell \le 6.176150626079544 \cdot 10^{-13}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{F} \cdot \frac{\tan \left(\pi \cdot \ell\right)}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\left(\left(\frac{\tan \left(\pi \cdot \ell\right)}{F}\right)\right)}{F}\\ \end{array}\]

Reproduce

herbie shell --seed 2019155 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))