Average Error: 11.4 → 0.3
Time: 4.4s
Precision: 64
\[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.00731808408035590930024172351409106340725 \lor \neg \left(x \le 7.905402982476203987971530295908451080322\right):\\ \;\;\;\;\frac{e^{-1 \cdot y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{1}{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}\right)}^{x} \cdot \left({\left(\frac{1}{\sqrt[3]{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}}\right)}^{x} \cdot {\left(\frac{x}{\sqrt[3]{\sqrt[3]{x + y}}}\right)}^{x}\right)}{x}\\ \end{array}\]
\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.00731808408035590930024172351409106340725 \lor \neg \left(x \le 7.905402982476203987971530295908451080322\right):\\
\;\;\;\;\frac{e^{-1 \cdot y}}{x}\\

\mathbf{else}:\\
\;\;\;\;\frac{{\left(\frac{1}{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}\right)}^{x} \cdot \left({\left(\frac{1}{\sqrt[3]{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}}\right)}^{x} \cdot {\left(\frac{x}{\sqrt[3]{\sqrt[3]{x + y}}}\right)}^{x}\right)}{x}\\

\end{array}
double f(double x, double y) {
        double r470869 = x;
        double r470870 = y;
        double r470871 = r470869 + r470870;
        double r470872 = r470869 / r470871;
        double r470873 = log(r470872);
        double r470874 = r470869 * r470873;
        double r470875 = exp(r470874);
        double r470876 = r470875 / r470869;
        return r470876;
}
↓
double f(double x, double y) {
        double r470877 = x;
        double r470878 = -0.007318084080355909;
        bool r470879 = r470877 <= r470878;
        double r470880 = 7.905402982476204;
        bool r470881 = r470877 <= r470880;
        double r470882 = !r470881;
        bool r470883 = r470879 || r470882;
        double r470884 = -1.0;
        double r470885 = y;
        double r470886 = r470884 * r470885;
        double r470887 = exp(r470886);
        double r470888 = r470887 / r470877;
        double r470889 = 1.0;
        double r470890 = r470877 + r470885;
        double r470891 = cbrt(r470890);
        double r470892 = r470891 * r470891;
        double r470893 = r470889 / r470892;
        double r470894 = pow(r470893, r470877);
        double r470895 = cbrt(r470892);
        double r470896 = r470889 / r470895;
        double r470897 = pow(r470896, r470877);
        double r470898 = cbrt(r470891);
        double r470899 = r470877 / r470898;
        double r470900 = pow(r470899, r470877);
        double r470901 = r470897 * r470900;
        double r470902 = r470894 * r470901;
        double r470903 = r470902 / r470877;
        double r470904 = r470883 ? r470888 : r470903;
        return r470904;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original11.4
Target8.0
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;y \lt -3.73118442066479561492798134439269393419 \cdot 10^{94}:\\ \;\;\;\;\frac{e^{\frac{-1}{y}}}{x}\\ \mathbf{elif}\;y \lt 28179592427282878868860376020282245120:\\ \;\;\;\;\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}\\ \mathbf{elif}\;y \lt 2.347387415166997963747840232163110922613 \cdot 10^{178}:\\ \;\;\;\;\log \left(e^{\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{-1}{y}}}{x}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -0.007318084080355909 or 7.905402982476204 < x

    1. Initial program 10.8

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

    2. Simplified10.8

      \[\leadsto \color{blue}{\frac{{\left(\frac{x}{x + y}\right)}^{x}}{x}}\]

    3. Taylor expanded around inf 0.2

      \[\leadsto \frac{\color{blue}{e^{-y}}}{x}\]

    4. Simplified0.2

      \[\leadsto \frac{\color{blue}{e^{-1 \cdot y}}}{x}\]

    if -0.007318084080355909 < x < 7.905402982476204

    1. Initial program 12.0

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

    2. Simplified12.0

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

    4. Applied add-cube-cbrt12.0

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

    5. Applied *-un-lft-identity12.0

      \[\leadsto \frac{{\left(\frac{\color{blue}{1 \cdot x}}{\left(\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}\right) \cdot \sqrt[3]{x + y}}\right)}^{x}}{x}\]

    6. Applied times-frac12.0

      \[\leadsto \frac{{\color{blue}{\left(\frac{1}{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}} \cdot \frac{x}{\sqrt[3]{x + y}}\right)}}^{x}}{x}\]

    7. Applied unpow-prod-down3.0

      \[\leadsto \frac{\color{blue}{{\left(\frac{1}{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}\right)}^{x} \cdot {\left(\frac{x}{\sqrt[3]{x + y}}\right)}^{x}}}{x}\]
    8. Using strategy rm

    9. Applied add-cube-cbrt3.0

      \[\leadsto \frac{{\left(\frac{1}{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}\right)}^{x} \cdot {\left(\frac{x}{\sqrt[3]{\color{blue}{\left(\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}\right) \cdot \sqrt[3]{x + y}}}}\right)}^{x}}{x}\]

    10. Applied cbrt-prod3.0

      \[\leadsto \frac{{\left(\frac{1}{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}\right)}^{x} \cdot {\left(\frac{x}{\color{blue}{\sqrt[3]{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}} \cdot \sqrt[3]{\sqrt[3]{x + y}}}}\right)}^{x}}{x}\]

    11. Applied *-un-lft-identity3.0

      \[\leadsto \frac{{\left(\frac{1}{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}\right)}^{x} \cdot {\left(\frac{\color{blue}{1 \cdot x}}{\sqrt[3]{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}} \cdot \sqrt[3]{\sqrt[3]{x + y}}}\right)}^{x}}{x}\]

    12. Applied times-frac3.0

      \[\leadsto \frac{{\left(\frac{1}{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}\right)}^{x} \cdot {\color{blue}{\left(\frac{1}{\sqrt[3]{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}} \cdot \frac{x}{\sqrt[3]{\sqrt[3]{x + y}}}\right)}}^{x}}{x}\]

    13. Applied unpow-prod-down0.4

      \[\leadsto \frac{{\left(\frac{1}{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}\right)}^{x} \cdot \color{blue}{\left({\left(\frac{1}{\sqrt[3]{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}}\right)}^{x} \cdot {\left(\frac{x}{\sqrt[3]{\sqrt[3]{x + y}}}\right)}^{x}\right)}}{x}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.00731808408035590930024172351409106340725 \lor \neg \left(x \le 7.905402982476203987971530295908451080322\right):\\ \;\;\;\;\frac{e^{-1 \cdot y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{1}{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}\right)}^{x} \cdot \left({\left(\frac{1}{\sqrt[3]{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}}\right)}^{x} \cdot {\left(\frac{x}{\sqrt[3]{\sqrt[3]{x + y}}}\right)}^{x}\right)}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019354 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, F"
  :precision binary64

  :herbie-target
  (if (< y -3.7311844206647956e+94) (/ (exp (/ -1 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 y)) x))))

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