Average Error: 15.8 → 8.1
Time: 36.5s
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 r985243 = atan2(1.0, 0.0);
        double r985244 = l;
        double r985245 = r985243 * r985244;
        double r985246 = 1.0;
        double r985247 = F;
        double r985248 = r985247 * r985247;
        double r985249 = r985246 / r985248;
        double r985250 = tan(r985245);
        double r985251 = r985249 * r985250;
        double r985252 = r985245 - r985251;
        return r985252;
}
↓
double f(double F, double l) {
        double r985253 = atan2(1.0, 0.0);
        double r985254 = l;
        double r985255 = r985253 * r985254;
        double r985256 = -169813308136631.62;
        bool r985257 = r985255 <= r985256;
        double r985258 = tan(r985255);
        double r985259 = F;
        double r985260 = r985258 / r985259;
        double r985261 = /* ERROR: no posit support in C */;
        double r985262 = /* ERROR: no posit support in C */;
        double r985263 = 1.0;
        double r985264 = r985263 / r985259;
        double r985265 = r985262 * r985264;
        double r985266 = r985255 - r985265;
        double r985267 = 3.493536384384748e+16;
        bool r985268 = r985255 <= r985267;
        double r985269 = log1p(r985255);
        double r985270 = expm1(r985269);
        double r985271 = tan(r985270);
        double r985272 = r985271 / r985259;
        double r985273 = r985272 * r985264;
        double r985274 = r985255 - r985273;
        double r985275 = 2.4253442475351736e+118;
        bool r985276 = r985255 <= r985275;
        double r985277 = r985259 * r985259;
        double r985278 = r985258 / r985277;
        double r985279 = /* ERROR: no posit support in C */;
        double r985280 = /* ERROR: no posit support in C */;
        double r985281 = r985255 - r985280;
        double r985282 = r985276 ? r985281 : r985266;
        double r985283 = r985268 ? r985274 : r985282;
        double r985284 = r985257 ? r985266 : r985283;
        return r985284;
}

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)))))