Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, F

Average Error: 11.3 → 3.6

Time: 28.6s

Precision: 64

Math

\[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x}\]

↓

\[\begin{array}{l} \mathbf{if}\;y \le 607910.67095027677714824676513671875:\\ \;\;\;\;\frac{e^{\left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right) + \left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right)\right) \cdot x}}{x}\\ \mathbf{elif}\;y \le 1.244799935864629687771983408648295509411 \cdot 10^{53}:\\ \;\;\;\;\frac{e^{x \cdot \left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right) + \left(\left(\left(\log \left(\frac{\sqrt[3]{\sqrt[3]{\sqrt[3]{x}}}}{\sqrt[3]{\sqrt[3]{\sqrt[3]{x + y}}}}\right) + \log \left(\frac{\sqrt[3]{\sqrt[3]{\sqrt[3]{x}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{x}}}}{\sqrt[3]{\sqrt[3]{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}}}\right)\right) + \log \left(\frac{\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}}{\sqrt[3]{\sqrt[3]{x + y}} \cdot \sqrt[3]{\sqrt[3]{x + y}}}\right)\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right)\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right) + \left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right) + \left(\left(\log \left(\frac{\sqrt[3]{\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}}}{\sqrt[3]{\sqrt[3]{\sqrt[3]{x + y}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{x + y}}}}\right) + \log \left(\frac{\sqrt[3]{\sqrt[3]{\sqrt[3]{x}}}}{\sqrt[3]{\sqrt[3]{\sqrt[3]{x + y}}}}\right)\right) + \log \left(\frac{\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}}{\sqrt[3]{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}}\right)\right)\right)\right) \cdot x}}{x}\\ \end{array}\]

C

double f(double x, double y) {
        double r21670799 = x;
        double r21670800 = y;
        double r21670801 = r21670799 + r21670800;
        double r21670802 = r21670799 / r21670801;
        double r21670803 = log(r21670802);
        double r21670804 = r21670799 * r21670803;
        double r21670805 = exp(r21670804);
        double r21670806 = r21670805 / r21670799;
        return r21670806;
}

↓

double f(double x, double y) {
        double r21670807 = y;
        double r21670808 = 607910.6709502768;
        bool r21670809 = r21670807 <= r21670808;
        double r21670810 = x;
        double r21670811 = cbrt(r21670810);
        double r21670812 = r21670810 + r21670807;
        double r21670813 = cbrt(r21670812);
        double r21670814 = r21670811 / r21670813;
        double r21670815 = log(r21670814);
        double r21670816 = r21670815 + r21670815;
        double r21670817 = r21670815 + r21670816;
        double r21670818 = r21670817 * r21670810;
        double r21670819 = exp(r21670818);
        double r21670820 = r21670819 / r21670810;
        double r21670821 = 1.2447999358646297e+53;
        bool r21670822 = r21670807 <= r21670821;
        double r21670823 = cbrt(r21670811);
        double r21670824 = cbrt(r21670823);
        double r21670825 = cbrt(r21670813);
        double r21670826 = cbrt(r21670825);
        double r21670827 = r21670824 / r21670826;
        double r21670828 = log(r21670827);
        double r21670829 = r21670824 * r21670824;
        double r21670830 = r21670813 * r21670813;
        double r21670831 = cbrt(r21670830);
        double r21670832 = cbrt(r21670831);
        double r21670833 = r21670829 / r21670832;
        double r21670834 = log(r21670833);
        double r21670835 = r21670828 + r21670834;
        double r21670836 = r21670811 * r21670811;
        double r21670837 = cbrt(r21670836);
        double r21670838 = r21670825 * r21670825;
        double r21670839 = r21670837 / r21670838;
        double r21670840 = log(r21670839);
        double r21670841 = r21670835 + r21670840;
        double r21670842 = r21670841 + r21670815;
        double r21670843 = r21670815 + r21670842;
        double r21670844 = r21670810 * r21670843;
        double r21670845 = exp(r21670844);
        double r21670846 = r21670845 / r21670810;
        double r21670847 = cbrt(r21670837);
        double r21670848 = r21670826 * r21670826;
        double r21670849 = r21670847 / r21670848;
        double r21670850 = log(r21670849);
        double r21670851 = r21670850 + r21670828;
        double r21670852 = r21670823 * r21670823;
        double r21670853 = r21670852 / r21670831;
        double r21670854 = log(r21670853);
        double r21670855 = r21670851 + r21670854;
        double r21670856 = r21670815 + r21670855;
        double r21670857 = r21670815 + r21670856;
        double r21670858 = r21670857 * r21670810;
        double r21670859 = exp(r21670858);
        double r21670860 = r21670859 / r21670810;
        double r21670861 = r21670822 ? r21670846 : r21670860;
        double r21670862 = r21670809 ? r21670820 : r21670861;
        return r21670862;
}

Error

Bits error versus x

Bits error versus y

Target

Original	11.3
Target	7.8
Herbie	3.6

\[\begin{array}{l} \mathbf{if}\;y \lt -3.73118442066479561492798134439269393419 \cdot 10^{94}:\\ \;\;\;\;\frac{e^{\frac{-1}{y}}}{x}\\ \mathbf{elif}\;y \lt 28179592427282878868860376020282245120:\\ \;\;\;\;\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}\\ \mathbf{elif}\;y \lt 2.347387415166997963747840232163110922613 \cdot 10^{178}:\\ \;\;\;\;\log \left(e^{\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{-1}{y}}}{x}\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if y < 607910.6709502768

   1. Initial program 4.8

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x}\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 29.1

      \[\leadsto \frac{e^{x \cdot \log \left(\frac{x}{\color{blue}{\left(\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}\right) \cdot \sqrt[3]{x + y}}}\right)}}{x}\]

   4. Applied add-cube-cbrt 4.8

      \[\leadsto \frac{e^{x \cdot \log \left(\frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}{\left(\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}\right) \cdot \sqrt[3]{x + y}}\right)}}{x}\]

   5. Applied times-frac 4.8

      \[\leadsto \frac{e^{x \cdot \log \color{blue}{\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}} \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}}}{x}\]

   6. Applied log-prod 1.8

      \[\leadsto \frac{e^{x \cdot \color{blue}{\left(\log \left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right)}}}{x}\]

   7. Simplified 1.1

      \[\leadsto \frac{e^{x \cdot \left(\color{blue}{\left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right)} + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right)}}{x}\]

   if 607910.6709502768 < y < 1.2447999358646297e+53

   1. Initial program 33.6

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x}\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 21.2

      \[\leadsto \frac{e^{x \cdot \log \left(\frac{x}{\color{blue}{\left(\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}\right) \cdot \sqrt[3]{x + y}}}\right)}}{x}\]

   4. Applied add-cube-cbrt 33.6

      \[\leadsto \frac{e^{x \cdot \log \left(\frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}{\left(\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}\right) \cdot \sqrt[3]{x + y}}\right)}}{x}\]

   5. Applied times-frac 33.6

      \[\leadsto \frac{e^{x \cdot \log \color{blue}{\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}} \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}}}{x}\]

   6. Applied log-prod 32.4

      \[\leadsto \frac{e^{x \cdot \color{blue}{\left(\log \left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right)}}}{x}\]

   7. Simplified 32.4

      \[\leadsto \frac{e^{x \cdot \left(\color{blue}{\left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right)} + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right)}}{x}\]
   8. Using strategy rm

   9. Applied add-cube-cbrt 18.3

      \[\leadsto \frac{e^{x \cdot \left(\left(\log \left(\frac{\sqrt[3]{x}}{\color{blue}{\left(\sqrt[3]{\sqrt[3]{x + y}} \cdot \sqrt[3]{\sqrt[3]{x + y}}\right) \cdot \sqrt[3]{\sqrt[3]{x + y}}}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right)}}{x}\]

   10. Applied add-cube-cbrt 19.2

       \[\leadsto \frac{e^{x \cdot \left(\left(\log \left(\frac{\sqrt[3]{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}}{\left(\sqrt[3]{\sqrt[3]{x + y}} \cdot \sqrt[3]{\sqrt[3]{x + y}}\right) \cdot \sqrt[3]{\sqrt[3]{x + y}}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right)}}{x}\]

   11. Applied cbrt-prod 20.8

       \[\leadsto \frac{e^{x \cdot \left(\left(\log \left(\frac{\color{blue}{\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}}}{\left(\sqrt[3]{\sqrt[3]{x + y}} \cdot \sqrt[3]{\sqrt[3]{x + y}}\right) \cdot \sqrt[3]{\sqrt[3]{x + y}}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right)}}{x}\]

   12. Applied times-frac 20.4

       \[\leadsto \frac{e^{x \cdot \left(\left(\log \color{blue}{\left(\frac{\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}}{\sqrt[3]{\sqrt[3]{x + y}} \cdot \sqrt[3]{\sqrt[3]{x + y}}} \cdot \frac{\sqrt[3]{\sqrt[3]{x}}}{\sqrt[3]{\sqrt[3]{x + y}}}\right)} + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right)}}{x}\]

   13. Applied log-prod 20.4

       \[\leadsto \frac{e^{x \cdot \left(\left(\color{blue}{\left(\log \left(\frac{\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}}{\sqrt[3]{\sqrt[3]{x + y}} \cdot \sqrt[3]{\sqrt[3]{x + y}}}\right) + \log \left(\frac{\sqrt[3]{\sqrt[3]{x}}}{\sqrt[3]{\sqrt[3]{x + y}}}\right)\right)} + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right)}}{x}\]
   14. Using strategy rm

   15. Applied add-cube-cbrt 20.1

       \[\leadsto \frac{e^{x \cdot \left(\left(\left(\log \left(\frac{\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}}{\sqrt[3]{\sqrt[3]{x + y}} \cdot \sqrt[3]{\sqrt[3]{x + y}}}\right) + \log \left(\frac{\sqrt[3]{\sqrt[3]{x}}}{\sqrt[3]{\sqrt[3]{\color{blue}{\left(\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}\right) \cdot \sqrt[3]{x + y}}}}}\right)\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right)}}{x}\]

   16. Applied cbrt-prod 19.6

       \[\leadsto \frac{e^{x \cdot \left(\left(\left(\log \left(\frac{\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}}{\sqrt[3]{\sqrt[3]{x + y}} \cdot \sqrt[3]{\sqrt[3]{x + y}}}\right) + \log \left(\frac{\sqrt[3]{\sqrt[3]{x}}}{\sqrt[3]{\color{blue}{\sqrt[3]{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}} \cdot \sqrt[3]{\sqrt[3]{x + y}}}}}\right)\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right)}}{x}\]

   17. Applied cbrt-prod 20.1

       \[\leadsto \frac{e^{x \cdot \left(\left(\left(\log \left(\frac{\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}}{\sqrt[3]{\sqrt[3]{x + y}} \cdot \sqrt[3]{\sqrt[3]{x + y}}}\right) + \log \left(\frac{\sqrt[3]{\sqrt[3]{x}}}{\color{blue}{\sqrt[3]{\sqrt[3]{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{x + y}}}}}\right)\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right)}}{x}\]

   18. Applied add-cube-cbrt 15.7

       \[\leadsto \frac{e^{x \cdot \left(\left(\left(\log \left(\frac{\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}}{\sqrt[3]{\sqrt[3]{x + y}} \cdot \sqrt[3]{\sqrt[3]{x + y}}}\right) + \log \left(\frac{\color{blue}{\left(\sqrt[3]{\sqrt[3]{\sqrt[3]{x}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{x}}}\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{x}}}}}{\sqrt[3]{\sqrt[3]{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{x + y}}}}\right)\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right)}}{x}\]

   19. Applied times-frac 15.3

       \[\leadsto \frac{e^{x \cdot \left(\left(\left(\log \left(\frac{\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}}{\sqrt[3]{\sqrt[3]{x + y}} \cdot \sqrt[3]{\sqrt[3]{x + y}}}\right) + \log \color{blue}{\left(\frac{\sqrt[3]{\sqrt[3]{\sqrt[3]{x}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{x}}}}{\sqrt[3]{\sqrt[3]{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}}} \cdot \frac{\sqrt[3]{\sqrt[3]{\sqrt[3]{x}}}}{\sqrt[3]{\sqrt[3]{\sqrt[3]{x + y}}}}\right)}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right)}}{x}\]

   20. Applied log-prod 15.3

       \[\leadsto \frac{e^{x \cdot \left(\left(\left(\log \left(\frac{\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}}{\sqrt[3]{\sqrt[3]{x + y}} \cdot \sqrt[3]{\sqrt[3]{x + y}}}\right) + \color{blue}{\left(\log \left(\frac{\sqrt[3]{\sqrt[3]{\sqrt[3]{x}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{x}}}}{\sqrt[3]{\sqrt[3]{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}}}\right) + \log \left(\frac{\sqrt[3]{\sqrt[3]{\sqrt[3]{x}}}}{\sqrt[3]{\sqrt[3]{\sqrt[3]{x + y}}}}\right)\right)}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right)}}{x}\]

   if 1.2447999358646297e+53 < y

   1. Initial program 32.6

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x}\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 26.0

      \[\leadsto \frac{e^{x \cdot \log \left(\frac{x}{\color{blue}{\left(\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}\right) \cdot \sqrt[3]{x + y}}}\right)}}{x}\]

   4. Applied add-cube-cbrt 32.7

      \[\leadsto \frac{e^{x \cdot \log \left(\frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}{\left(\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}\right) \cdot \sqrt[3]{x + y}}\right)}}{x}\]

   5. Applied times-frac 32.7

      \[\leadsto \frac{e^{x \cdot \log \color{blue}{\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}} \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}}}{x}\]

   6. Applied log-prod 21.4

      \[\leadsto \frac{e^{x \cdot \color{blue}{\left(\log \left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right)}}}{x}\]

   7. Simplified 18.6

      \[\leadsto \frac{e^{x \cdot \left(\color{blue}{\left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right)} + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right)}}{x}\]
   8. Using strategy rm

   9. Applied add-cube-cbrt 16.0

      \[\leadsto \frac{e^{x \cdot \left(\left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{\color{blue}{\left(\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}\right) \cdot \sqrt[3]{x + y}}}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right)}}{x}\]

   10. Applied cbrt-prod 12.7

       \[\leadsto \frac{e^{x \cdot \left(\left(\log \left(\frac{\sqrt[3]{x}}{\color{blue}{\sqrt[3]{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}} \cdot \sqrt[3]{\sqrt[3]{x + y}}}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right)}}{x}\]

   11. Applied add-cube-cbrt 13.4

       \[\leadsto \frac{e^{x \cdot \left(\left(\log \left(\frac{\color{blue}{\left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}}}{\sqrt[3]{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}} \cdot \sqrt[3]{\sqrt[3]{x + y}}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right)}}{x}\]

   12. Applied times-frac 12.9

       \[\leadsto \frac{e^{x \cdot \left(\left(\log \color{blue}{\left(\frac{\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}}{\sqrt[3]{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}} \cdot \frac{\sqrt[3]{\sqrt[3]{x}}}{\sqrt[3]{\sqrt[3]{x + y}}}\right)} + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right)}}{x}\]

   13. Applied log-prod 12.9

       \[\leadsto \frac{e^{x \cdot \left(\left(\color{blue}{\left(\log \left(\frac{\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}}{\sqrt[3]{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}}\right) + \log \left(\frac{\sqrt[3]{\sqrt[3]{x}}}{\sqrt[3]{\sqrt[3]{x + y}}}\right)\right)} + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right)}}{x}\]
   14. Using strategy rm

   15. Applied add-cube-cbrt 10.2

       \[\leadsto \frac{e^{x \cdot \left(\left(\left(\log \left(\frac{\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}}{\sqrt[3]{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}}\right) + \log \left(\frac{\sqrt[3]{\sqrt[3]{x}}}{\color{blue}{\left(\sqrt[3]{\sqrt[3]{\sqrt[3]{x + y}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{x + y}}}\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{x + y}}}}}\right)\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right)}}{x}\]

   16. Applied add-cube-cbrt 10.2

       \[\leadsto \frac{e^{x \cdot \left(\left(\left(\log \left(\frac{\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}}{\sqrt[3]{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}}\right) + \log \left(\frac{\sqrt[3]{\sqrt[3]{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}}}{\left(\sqrt[3]{\sqrt[3]{\sqrt[3]{x + y}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{x + y}}}\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{x + y}}}}\right)\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right)}}{x}\]

   17. Applied cbrt-prod 10.1

       \[\leadsto \frac{e^{x \cdot \left(\left(\left(\log \left(\frac{\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}}{\sqrt[3]{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}}\right) + \log \left(\frac{\sqrt[3]{\color{blue}{\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}}}}{\left(\sqrt[3]{\sqrt[3]{\sqrt[3]{x + y}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{x + y}}}\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{x + y}}}}\right)\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right)}}{x}\]

   18. Applied cbrt-prod 11.0

       \[\leadsto \frac{e^{x \cdot \left(\left(\left(\log \left(\frac{\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}}{\sqrt[3]{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}}\right) + \log \left(\frac{\color{blue}{\sqrt[3]{\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{x}}}}}{\left(\sqrt[3]{\sqrt[3]{\sqrt[3]{x + y}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{x + y}}}\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{x + y}}}}\right)\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right)}}{x}\]

   19. Applied times-frac 10.9

       \[\leadsto \frac{e^{x \cdot \left(\left(\left(\log \left(\frac{\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}}{\sqrt[3]{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}}\right) + \log \color{blue}{\left(\frac{\sqrt[3]{\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}}}{\sqrt[3]{\sqrt[3]{\sqrt[3]{x + y}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{x + y}}}} \cdot \frac{\sqrt[3]{\sqrt[3]{\sqrt[3]{x}}}}{\sqrt[3]{\sqrt[3]{\sqrt[3]{x + y}}}}\right)}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right)}}{x}\]

   20. Applied log-prod 10.9

       \[\leadsto \frac{e^{x \cdot \left(\left(\left(\log \left(\frac{\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}}{\sqrt[3]{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}}\right) + \color{blue}{\left(\log \left(\frac{\sqrt[3]{\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}}}{\sqrt[3]{\sqrt[3]{\sqrt[3]{x + y}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{x + y}}}}\right) + \log \left(\frac{\sqrt[3]{\sqrt[3]{\sqrt[3]{x}}}}{\sqrt[3]{\sqrt[3]{\sqrt[3]{x + y}}}}\right)\right)}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right)}}{x}\]
3. Recombined 3 regimes into one program.

4. Final simplification 3.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le 607910.67095027677714824676513671875:\\ \;\;\;\;\frac{e^{\left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right) + \left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right)\right) \cdot x}}{x}\\ \mathbf{elif}\;y \le 1.244799935864629687771983408648295509411 \cdot 10^{53}:\\ \;\;\;\;\frac{e^{x \cdot \left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right) + \left(\left(\left(\log \left(\frac{\sqrt[3]{\sqrt[3]{\sqrt[3]{x}}}}{\sqrt[3]{\sqrt[3]{\sqrt[3]{x + y}}}}\right) + \log \left(\frac{\sqrt[3]{\sqrt[3]{\sqrt[3]{x}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{x}}}}{\sqrt[3]{\sqrt[3]{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}}}\right)\right) + \log \left(\frac{\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}}{\sqrt[3]{\sqrt[3]{x + y}} \cdot \sqrt[3]{\sqrt[3]{x + y}}}\right)\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right)\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right) + \left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right) + \left(\left(\log \left(\frac{\sqrt[3]{\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}}}{\sqrt[3]{\sqrt[3]{\sqrt[3]{x + y}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{x + y}}}}\right) + \log \left(\frac{\sqrt[3]{\sqrt[3]{\sqrt[3]{x}}}}{\sqrt[3]{\sqrt[3]{\sqrt[3]{x + y}}}}\right)\right) + \log \left(\frac{\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}}{\sqrt[3]{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}}\right)\right)\right)\right) \cdot x}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, F"

  :herbie-target
  (if (< y -3.7311844206647956e+94) (/ (exp (/ -1.0 y)) x) (if (< y 2.817959242728288e+37) (/ (pow (/ x (+ y x)) x) x) (if (< y 2.347387415166998e+178) (log (exp (/ (pow (/ x (+ y x)) x) x))) (/ (exp (/ -1.0 y)) x))))

  (/ (exp (* x (log (/ x (+ x y))))) x))