Average Error: 15.8 → 8.1
Time: 33.8s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \le -169813308136631.62:\\ \;\;\;\;\pi \cdot \ell - \left(\left(\frac{\tan \left(\pi \cdot \ell\right)}{F}\right)\right) \cdot \frac{1}{F}\\ \mathbf{elif}\;\pi \cdot \ell \le 3.493536384384748 \cdot 10^{+16}:\\ \;\;\;\;\pi \cdot \ell - \frac{\tan \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \ell\right)\right)\right)}{F} \cdot \frac{1}{F}\\ \mathbf{elif}\;\pi \cdot \ell \le 2.4253442475351736 \cdot 10^{+118}:\\ \;\;\;\;\pi \cdot \ell - \left(\left(\frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \left(\left(\frac{\tan \left(\pi \cdot \ell\right)}{F}\right)\right) \cdot \frac{1}{F}\\ \end{array}\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\begin{array}{l}
\mathbf{if}\;\pi \cdot \ell \le -169813308136631.62:\\
\;\;\;\;\pi \cdot \ell - \left(\left(\frac{\tan \left(\pi \cdot \ell\right)}{F}\right)\right) \cdot \frac{1}{F}\\

\mathbf{elif}\;\pi \cdot \ell \le 3.493536384384748 \cdot 10^{+16}:\\
\;\;\;\;\pi \cdot \ell - \frac{\tan \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \ell\right)\right)\right)}{F} \cdot \frac{1}{F}\\

\mathbf{elif}\;\pi \cdot \ell \le 2.4253442475351736 \cdot 10^{+118}:\\
\;\;\;\;\pi \cdot \ell - \left(\left(\frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}\right)\right)\\

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

\end{array}
double f(double F, double l) {
        double r1002653 = atan2(1.0, 0.0);
        double r1002654 = l;
        double r1002655 = r1002653 * r1002654;
        double r1002656 = 1.0;
        double r1002657 = F;
        double r1002658 = r1002657 * r1002657;
        double r1002659 = r1002656 / r1002658;
        double r1002660 = tan(r1002655);
        double r1002661 = r1002659 * r1002660;
        double r1002662 = r1002655 - r1002661;
        return r1002662;
}
↓
double f(double F, double l) {
        double r1002663 = atan2(1.0, 0.0);
        double r1002664 = l;
        double r1002665 = r1002663 * r1002664;
        double r1002666 = -169813308136631.62;
        bool r1002667 = r1002665 <= r1002666;
        double r1002668 = tan(r1002665);
        double r1002669 = F;
        double r1002670 = r1002668 / r1002669;
        double r1002671 = /* ERROR: no posit support in C */;
        double r1002672 = /* ERROR: no posit support in C */;
        double r1002673 = 1.0;
        double r1002674 = r1002673 / r1002669;
        double r1002675 = r1002672 * r1002674;
        double r1002676 = r1002665 - r1002675;
        double r1002677 = 3.493536384384748e+16;
        bool r1002678 = r1002665 <= r1002677;
        double r1002679 = log1p(r1002665);
        double r1002680 = expm1(r1002679);
        double r1002681 = tan(r1002680);
        double r1002682 = r1002681 / r1002669;
        double r1002683 = r1002682 * r1002674;
        double r1002684 = r1002665 - r1002683;
        double r1002685 = 2.4253442475351736e+118;
        bool r1002686 = r1002665 <= r1002685;
        double r1002687 = r1002669 * r1002669;
        double r1002688 = r1002668 / r1002687;
        double r1002689 = /* ERROR: no posit support in C */;
        double r1002690 = /* ERROR: no posit support in C */;
        double r1002691 = r1002665 - r1002690;
        double r1002692 = r1002686 ? r1002691 : r1002676;
        double r1002693 = r1002678 ? r1002684 : r1002692;
        double r1002694 = r1002667 ? r1002676 : r1002693;
        return r1002694;
}

Error

Bits error versus F

Bits error versus l

Derivation

  1. Split input into 3 regimes

  2. if (* PI l) < -169813308136631.62 or 2.4253442475351736e+118 < (* PI l)

    1. Initial program 21.2

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

    2. Simplified21.2

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

    4. Applied *-un-lft-identity21.2

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

    5. Applied times-frac21.2

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

    7. Applied insert-posit1613.9

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

    if -169813308136631.62 < (* PI l) < 3.493536384384748e+16

    1. Initial program 9.1

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

    2. Simplified8.7

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

    4. Applied *-un-lft-identity8.7

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

    5. Applied times-frac0.7

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

    7. Applied expm1-log1p-u1.8

      \[\leadsto \pi \cdot \ell - \frac{1}{F} \cdot \frac{\tan \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \ell\right)\right)\right)}}{F}\]

    if 3.493536384384748e+16 < (* PI l) < 2.4253442475351736e+118

    1. Initial program 28.4

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

    2. Simplified28.4

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

    4. Applied insert-posit1616.3

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

  4. Final simplification8.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \le -169813308136631.62:\\ \;\;\;\;\pi \cdot \ell - \left(\left(\frac{\tan \left(\pi \cdot \ell\right)}{F}\right)\right) \cdot \frac{1}{F}\\ \mathbf{elif}\;\pi \cdot \ell \le 3.493536384384748 \cdot 10^{+16}:\\ \;\;\;\;\pi \cdot \ell - \frac{\tan \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \ell\right)\right)\right)}{F} \cdot \frac{1}{F}\\ \mathbf{elif}\;\pi \cdot \ell \le 2.4253442475351736 \cdot 10^{+118}:\\ \;\;\;\;\pi \cdot \ell - \left(\left(\frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \left(\left(\frac{\tan \left(\pi \cdot \ell\right)}{F}\right)\right) \cdot \frac{1}{F}\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))