Average Error: 15.8 → 8.2
Time: 28.1s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \le -3.061897402634554 \cdot 10^{+20}:\\ \;\;\;\;\pi \cdot \ell - \left(\left(\frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}\right)\right)\\ \mathbf{elif}\;\pi \cdot \ell \le 89374916.04315051:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{\sqrt[3]{F}} \cdot \frac{1}{\sqrt[3]{F} \cdot \sqrt[3]{F}}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \left(\left(\frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}\right)\right)\\ \end{array}\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\begin{array}{l}
\mathbf{if}\;\pi \cdot \ell \le -3.061897402634554 \cdot 10^{+20}:\\
\;\;\;\;\pi \cdot \ell - \left(\left(\frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}\right)\right)\\

\mathbf{elif}\;\pi \cdot \ell \le 89374916.04315051:\\
\;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{\sqrt[3]{F}} \cdot \frac{1}{\sqrt[3]{F} \cdot \sqrt[3]{F}}}{F}\\

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

\end{array}
double f(double F, double l) {
        double r378241 = atan2(1.0, 0.0);
        double r378242 = l;
        double r378243 = r378241 * r378242;
        double r378244 = 1.0;
        double r378245 = F;
        double r378246 = r378245 * r378245;
        double r378247 = r378244 / r378246;
        double r378248 = tan(r378243);
        double r378249 = r378247 * r378248;
        double r378250 = r378243 - r378249;
        return r378250;
}
↓
double f(double F, double l) {
        double r378251 = atan2(1.0, 0.0);
        double r378252 = l;
        double r378253 = r378251 * r378252;
        double r378254 = -3.061897402634554e+20;
        bool r378255 = r378253 <= r378254;
        double r378256 = tan(r378253);
        double r378257 = F;
        double r378258 = r378257 * r378257;
        double r378259 = r378256 / r378258;
        double r378260 = /* ERROR: no posit support in C */;
        double r378261 = /* ERROR: no posit support in C */;
        double r378262 = r378253 - r378261;
        double r378263 = 89374916.04315051;
        bool r378264 = r378253 <= r378263;
        double r378265 = cbrt(r378257);
        double r378266 = r378256 / r378265;
        double r378267 = 1.0;
        double r378268 = r378265 * r378265;
        double r378269 = r378267 / r378268;
        double r378270 = r378266 * r378269;
        double r378271 = r378270 / r378257;
        double r378272 = r378253 - r378271;
        double r378273 = r378264 ? r378272 : r378262;
        double r378274 = r378255 ? r378262 : r378273;
        return r378274;
}

Error

Bits error versus F

Bits error versus l

Derivation

  1. Split input into 2 regimes

  2. if (* PI l) < -3.061897402634554e+20 or 89374916.04315051 < (* 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 insert-posit1615.2

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

    if -3.061897402634554e+20 < (* PI l) < 89374916.04315051

    1. Initial program 8.9

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

    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 associate-/r*0.8

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

    6. Applied add-cube-cbrt1.2

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

    7. Applied *-un-lft-identity1.2

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

    8. Applied times-frac1.2

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

  4. Final simplification8.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \le -3.061897402634554 \cdot 10^{+20}:\\ \;\;\;\;\pi \cdot \ell - \left(\left(\frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}\right)\right)\\ \mathbf{elif}\;\pi \cdot \ell \le 89374916.04315051:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{\sqrt[3]{F}} \cdot \frac{1}{\sqrt[3]{F} \cdot \sqrt[3]{F}}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \left(\left(\frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019151 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))