Average Error: 16.1 → 7.9
Time: 30.4s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \le -1.3881414212069512 \cdot 10^{+177}:\\ \;\;\;\;\pi \cdot \ell - \frac{\left(\left(\frac{\tan \left(\pi \cdot \ell\right)}{F}\right)\right)}{F}\\ \mathbf{elif}\;\pi \cdot \ell \le -8.794843946240566 \cdot 10^{+22}:\\ \;\;\;\;\pi \cdot \ell - \left(\left(\frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}\right)\right)\\ \mathbf{elif}\;\pi \cdot \ell \le 229490.3220043475:\\ \;\;\;\;\pi \cdot \ell - \frac{\sqrt[3]{\frac{\tan \left(\pi \cdot \ell\right)}{F}} \cdot \left(\sqrt[3]{\frac{\tan \left(\pi \cdot \ell\right)}{F}} \cdot \sqrt[3]{\frac{\tan \left(\pi \cdot \ell\right)}{F}}\right)}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \left(\left(\frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}\right)\right)\\ \end{array}\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\begin{array}{l}
\mathbf{if}\;\pi \cdot \ell \le -1.3881414212069512 \cdot 10^{+177}:\\
\;\;\;\;\pi \cdot \ell - \frac{\left(\left(\frac{\tan \left(\pi \cdot \ell\right)}{F}\right)\right)}{F}\\

\mathbf{elif}\;\pi \cdot \ell \le -8.794843946240566 \cdot 10^{+22}:\\
\;\;\;\;\pi \cdot \ell - \left(\left(\frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}\right)\right)\\

\mathbf{elif}\;\pi \cdot \ell \le 229490.3220043475:\\
\;\;\;\;\pi \cdot \ell - \frac{\sqrt[3]{\frac{\tan \left(\pi \cdot \ell\right)}{F}} \cdot \left(\sqrt[3]{\frac{\tan \left(\pi \cdot \ell\right)}{F}} \cdot \sqrt[3]{\frac{\tan \left(\pi \cdot \ell\right)}{F}}\right)}{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 r986808 = atan2(1.0, 0.0);
        double r986809 = l;
        double r986810 = r986808 * r986809;
        double r986811 = 1.0;
        double r986812 = F;
        double r986813 = r986812 * r986812;
        double r986814 = r986811 / r986813;
        double r986815 = tan(r986810);
        double r986816 = r986814 * r986815;
        double r986817 = r986810 - r986816;
        return r986817;
}
↓
double f(double F, double l) {
        double r986818 = atan2(1.0, 0.0);
        double r986819 = l;
        double r986820 = r986818 * r986819;
        double r986821 = -1.3881414212069512e+177;
        bool r986822 = r986820 <= r986821;
        double r986823 = tan(r986820);
        double r986824 = F;
        double r986825 = r986823 / r986824;
        double r986826 = /* ERROR: no posit support in C */;
        double r986827 = /* ERROR: no posit support in C */;
        double r986828 = r986827 / r986824;
        double r986829 = r986820 - r986828;
        double r986830 = -8.794843946240566e+22;
        bool r986831 = r986820 <= r986830;
        double r986832 = r986824 * r986824;
        double r986833 = r986823 / r986832;
        double r986834 = /* ERROR: no posit support in C */;
        double r986835 = /* ERROR: no posit support in C */;
        double r986836 = r986820 - r986835;
        double r986837 = 229490.3220043475;
        bool r986838 = r986820 <= r986837;
        double r986839 = cbrt(r986825);
        double r986840 = r986839 * r986839;
        double r986841 = r986839 * r986840;
        double r986842 = r986841 / r986824;
        double r986843 = r986820 - r986842;
        double r986844 = r986838 ? r986843 : r986836;
        double r986845 = r986831 ? r986836 : r986844;
        double r986846 = r986822 ? r986829 : r986845;
        return r986846;
}

Error

Bits error versus F

Bits error versus l

Derivation

  1. Split input into 3 regimes

  2. if (* PI l) < -1.3881414212069512e+177

    1. Initial program 20.4

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

    2. Simplified20.4

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

    4. Applied associate-/r*20.4

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

    6. Applied insert-posit168.7

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

    if -1.3881414212069512e+177 < (* PI l) < -8.794843946240566e+22 or 229490.3220043475 < (* PI l)

    1. Initial program 24.6

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

    2. Simplified24.6

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

    4. Applied insert-posit1616.5

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

    if -8.794843946240566e+22 < (* PI l) < 229490.3220043475

    1. Initial program 8.7

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

    2. Simplified8.2

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

    4. Applied associate-/r*0.7

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

    6. Applied add-cube-cbrt1.1

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

    8. Applied *-un-lft-identity1.1

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

    9. Applied *-un-lft-identity1.1

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

    10. Applied times-frac1.1

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

    11. Applied cbrt-prod1.1

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

    12. Simplified1.1

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

  4. Final simplification7.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \le -1.3881414212069512 \cdot 10^{+177}:\\ \;\;\;\;\pi \cdot \ell - \frac{\left(\left(\frac{\tan \left(\pi \cdot \ell\right)}{F}\right)\right)}{F}\\ \mathbf{elif}\;\pi \cdot \ell \le -8.794843946240566 \cdot 10^{+22}:\\ \;\;\;\;\pi \cdot \ell - \left(\left(\frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}\right)\right)\\ \mathbf{elif}\;\pi \cdot \ell \le 229490.3220043475:\\ \;\;\;\;\pi \cdot \ell - \frac{\sqrt[3]{\frac{\tan \left(\pi \cdot \ell\right)}{F}} \cdot \left(\sqrt[3]{\frac{\tan \left(\pi \cdot \ell\right)}{F}} \cdot \sqrt[3]{\frac{\tan \left(\pi \cdot \ell\right)}{F}}\right)}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \left(\left(\frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019164 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))