Average Error: 10.8 → 0.2
Time: 23.8s
Precision: 64
\[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -2.122144077215159 \cdot 10^{+68}:\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{elif}\;x \le 12.791495951037414:\\ \;\;\;\;e^{x \cdot \left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y + x}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y + x}}\right)\right)} \cdot \frac{e^{\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y + x}}\right) \cdot x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-y}}{x}\\ \end{array}\]
\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -2.122144077215159 \cdot 10^{+68}:\\
\;\;\;\;\frac{e^{-y}}{x}\\

\mathbf{elif}\;x \le 12.791495951037414:\\
\;\;\;\;e^{x \cdot \left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y + x}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y + x}}\right)\right)} \cdot \frac{e^{\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y + x}}\right) \cdot x}}{x}\\

\mathbf{else}:\\
\;\;\;\;\frac{e^{-y}}{x}\\

\end{array}
double f(double x, double y) {
        double r15463869 = x;
        double r15463870 = y;
        double r15463871 = r15463869 + r15463870;
        double r15463872 = r15463869 / r15463871;
        double r15463873 = log(r15463872);
        double r15463874 = r15463869 * r15463873;
        double r15463875 = exp(r15463874);
        double r15463876 = r15463875 / r15463869;
        return r15463876;
}
↓
double f(double x, double y) {
        double r15463877 = x;
        double r15463878 = -2.122144077215159e+68;
        bool r15463879 = r15463877 <= r15463878;
        double r15463880 = y;
        double r15463881 = -r15463880;
        double r15463882 = exp(r15463881);
        double r15463883 = r15463882 / r15463877;
        double r15463884 = 12.791495951037414;
        bool r15463885 = r15463877 <= r15463884;
        double r15463886 = cbrt(r15463877);
        double r15463887 = r15463880 + r15463877;
        double r15463888 = cbrt(r15463887);
        double r15463889 = r15463886 / r15463888;
        double r15463890 = log(r15463889);
        double r15463891 = r15463890 + r15463890;
        double r15463892 = r15463877 * r15463891;
        double r15463893 = exp(r15463892);
        double r15463894 = r15463890 * r15463877;
        double r15463895 = exp(r15463894);
        double r15463896 = r15463895 / r15463877;
        double r15463897 = r15463893 * r15463896;
        double r15463898 = r15463885 ? r15463897 : r15463883;
        double r15463899 = r15463879 ? r15463883 : r15463898;
        return r15463899;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original10.8
Target7.7
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;y \lt -3.7311844206647956 \cdot 10^{+94}:\\ \;\;\;\;\frac{e^{\frac{-1}{y}}}{x}\\ \mathbf{elif}\;y \lt 2.817959242728288 \cdot 10^{+37}:\\ \;\;\;\;\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}\\ \mathbf{elif}\;y \lt 2.347387415166998 \cdot 10^{+178}:\\ \;\;\;\;\log \left(e^{\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{-1}{y}}}{x}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -2.122144077215159e+68 or 12.791495951037414 < x

    1. Initial program 11.3

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

    2. Taylor expanded around inf 0.0

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

    3. Simplified0.0

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

    if -2.122144077215159e+68 < x < 12.791495951037414

    1. Initial program 10.4

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

    3. Applied *-un-lft-identity10.4

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

    4. Applied add-cube-cbrt14.0

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

    5. Applied add-cube-cbrt10.4

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

    6. Applied times-frac10.4

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

    7. Applied log-prod2.5

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

    8. Applied distribute-lft-in2.5

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

    9. Applied exp-sum2.5

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

    10. Applied times-frac2.5

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

    12. Applied times-frac2.5

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

    13. Applied log-prod0.4

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.122144077215159 \cdot 10^{+68}:\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{elif}\;x \le 12.791495951037414:\\ \;\;\;\;e^{x \cdot \left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y + x}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y + x}}\right)\right)} \cdot \frac{e^{\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y + x}}\right) \cdot x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-y}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 +o rules:numerics
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, F"

  :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))