Average Error: 16.7 → 8.7
Time: 9.4s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \le -4.6380634062953026 \cdot 10^{154}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \frac{\sin \left(\pi \cdot \ell\right) \cdot \sqrt[3]{1}}{\cos \left(\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right) \cdot \left(\left(\sqrt[3]{\sqrt{\pi}} \cdot \sqrt[3]{\sqrt{\pi}}\right) \cdot \ell\right)\right) \cdot F}\\ \mathbf{elif}\;\pi \cdot \ell \le 1.8434257583154768 \cdot 10^{144}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \frac{\sin \left(\pi \cdot \ell\right) \cdot \sqrt[3]{1}}{\left(\left(\frac{1}{24} \cdot \left({\pi}^{4} \cdot {\ell}^{4}\right) + 1\right) - \frac{1}{2} \cdot \left({\pi}^{2} \cdot {\ell}^{2}\right)\right) \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\left(\sqrt[3]{\pi \cdot \ell} \cdot \sqrt[3]{\pi \cdot \ell}\right) \cdot \sqrt[3]{\pi \cdot \ell}\right)\\ \end{array}\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\begin{array}{l}
\mathbf{if}\;\pi \cdot \ell \le -4.6380634062953026 \cdot 10^{154}:\\
\;\;\;\;\pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \frac{\sin \left(\pi \cdot \ell\right) \cdot \sqrt[3]{1}}{\cos \left(\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right) \cdot \left(\left(\sqrt[3]{\sqrt{\pi}} \cdot \sqrt[3]{\sqrt{\pi}}\right) \cdot \ell\right)\right) \cdot F}\\

\mathbf{elif}\;\pi \cdot \ell \le 1.8434257583154768 \cdot 10^{144}:\\
\;\;\;\;\pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \frac{\sin \left(\pi \cdot \ell\right) \cdot \sqrt[3]{1}}{\left(\left(\frac{1}{24} \cdot \left({\pi}^{4} \cdot {\ell}^{4}\right) + 1\right) - \frac{1}{2} \cdot \left({\pi}^{2} \cdot {\ell}^{2}\right)\right) \cdot F}\\

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

\end{array}
double f(double F, double l) {
        double r15874 = atan2(1.0, 0.0);
        double r15875 = l;
        double r15876 = r15874 * r15875;
        double r15877 = 1.0;
        double r15878 = F;
        double r15879 = r15878 * r15878;
        double r15880 = r15877 / r15879;
        double r15881 = tan(r15876);
        double r15882 = r15880 * r15881;
        double r15883 = r15876 - r15882;
        return r15883;
}
↓
double f(double F, double l) {
        double r15884 = atan2(1.0, 0.0);
        double r15885 = l;
        double r15886 = r15884 * r15885;
        double r15887 = -4.6380634062953026e+154;
        bool r15888 = r15886 <= r15887;
        double r15889 = 1.0;
        double r15890 = cbrt(r15889);
        double r15891 = r15890 * r15890;
        double r15892 = F;
        double r15893 = r15891 / r15892;
        double r15894 = sin(r15886);
        double r15895 = r15894 * r15890;
        double r15896 = cbrt(r15884);
        double r15897 = r15896 * r15896;
        double r15898 = sqrt(r15884);
        double r15899 = cbrt(r15898);
        double r15900 = r15899 * r15899;
        double r15901 = r15900 * r15885;
        double r15902 = r15897 * r15901;
        double r15903 = cos(r15902);
        double r15904 = r15903 * r15892;
        double r15905 = r15895 / r15904;
        double r15906 = r15893 * r15905;
        double r15907 = r15886 - r15906;
        double r15908 = 1.8434257583154768e+144;
        bool r15909 = r15886 <= r15908;
        double r15910 = 0.041666666666666664;
        double r15911 = 4.0;
        double r15912 = pow(r15884, r15911);
        double r15913 = pow(r15885, r15911);
        double r15914 = r15912 * r15913;
        double r15915 = r15910 * r15914;
        double r15916 = 1.0;
        double r15917 = r15915 + r15916;
        double r15918 = 0.5;
        double r15919 = 2.0;
        double r15920 = pow(r15884, r15919);
        double r15921 = pow(r15885, r15919);
        double r15922 = r15920 * r15921;
        double r15923 = r15918 * r15922;
        double r15924 = r15917 - r15923;
        double r15925 = r15924 * r15892;
        double r15926 = r15895 / r15925;
        double r15927 = r15893 * r15926;
        double r15928 = r15886 - r15927;
        double r15929 = r15892 * r15892;
        double r15930 = r15889 / r15929;
        double r15931 = cbrt(r15886);
        double r15932 = r15931 * r15931;
        double r15933 = r15932 * r15931;
        double r15934 = tan(r15933);
        double r15935 = r15930 * r15934;
        double r15936 = r15886 - r15935;
        double r15937 = r15909 ? r15928 : r15936;
        double r15938 = r15888 ? r15907 : r15937;
        return r15938;
}

Error

Bits error versus F

Bits error versus l

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if (* PI l) < -4.6380634062953026e+154

    1. Initial program 21.4

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

    3. Applied add-cube-cbrt21.4

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

    4. Applied times-frac21.4

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

    5. Applied associate-*l*21.4

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

    6. Taylor expanded around inf 21.4

      \[\leadsto \pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \color{blue}{\frac{\sin \left(\pi \cdot \ell\right) \cdot \sqrt[3]{1}}{\cos \left(\pi \cdot \ell\right) \cdot F}}\]
    7. Using strategy rm

    8. Applied add-cube-cbrt21.4

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

    9. Applied associate-*l*21.4

      \[\leadsto \pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \frac{\sin \left(\pi \cdot \ell\right) \cdot \sqrt[3]{1}}{\cos \color{blue}{\left(\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right) \cdot \left(\sqrt[3]{\pi} \cdot \ell\right)\right)} \cdot F}\]
    10. Using strategy rm

    11. Applied add-sqr-sqrt21.4

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

    12. Applied cbrt-prod21.4

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

    if -4.6380634062953026e+154 < (* PI l) < 1.8434257583154768e+144

    1. Initial program 14.8

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

    3. Applied add-cube-cbrt14.8

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

    4. Applied times-frac14.8

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

    5. Applied associate-*l*9.0

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

    6. Taylor expanded around inf 9.0

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

    7. Taylor expanded around 0 3.7

      \[\leadsto \pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \frac{\sin \left(\pi \cdot \ell\right) \cdot \sqrt[3]{1}}{\color{blue}{\left(\left(\frac{1}{24} \cdot \left({\pi}^{4} \cdot {\ell}^{4}\right) + 1\right) - \frac{1}{2} \cdot \left({\pi}^{2} \cdot {\ell}^{2}\right)\right)} \cdot F}\]

    if 1.8434257583154768e+144 < (* PI l)

    1. Initial program 21.6

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

    3. Applied add-cube-cbrt21.6

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

  4. Final simplification8.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \le -4.6380634062953026 \cdot 10^{154}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \frac{\sin \left(\pi \cdot \ell\right) \cdot \sqrt[3]{1}}{\cos \left(\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right) \cdot \left(\left(\sqrt[3]{\sqrt{\pi}} \cdot \sqrt[3]{\sqrt{\pi}}\right) \cdot \ell\right)\right) \cdot F}\\ \mathbf{elif}\;\pi \cdot \ell \le 1.8434257583154768 \cdot 10^{144}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{F} \cdot \frac{\sin \left(\pi \cdot \ell\right) \cdot \sqrt[3]{1}}{\left(\left(\frac{1}{24} \cdot \left({\pi}^{4} \cdot {\ell}^{4}\right) + 1\right) - \frac{1}{2} \cdot \left({\pi}^{2} \cdot {\ell}^{2}\right)\right) \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\left(\sqrt[3]{\pi \cdot \ell} \cdot \sqrt[3]{\pi \cdot \ell}\right) \cdot \sqrt[3]{\pi \cdot \ell}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))