Average Error: 11.1 → 1.5
Time: 5.2s
Precision: 64
\[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -5.89791977930625:\\ \;\;\;\;\frac{e^{-1 \cdot y}}{x}\\ \mathbf{elif}\;x \le 9.2977525610777332 \cdot 10^{-4}:\\ \;\;\;\;\frac{{\left(\frac{1}{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}\right)}^{x} \cdot {\left(\frac{x}{\sqrt[3]{x + y}}\right)}^{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot e^{y}}\\ \end{array}\]
\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -5.89791977930625:\\
\;\;\;\;\frac{e^{-1 \cdot y}}{x}\\

\mathbf{elif}\;x \le 9.2977525610777332 \cdot 10^{-4}:\\
\;\;\;\;\frac{{\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}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{x \cdot e^{y}}\\

\end{array}
double f(double x, double y) {
        double r317981 = x;
        double r317982 = y;
        double r317983 = r317981 + r317982;
        double r317984 = r317981 / r317983;
        double r317985 = log(r317984);
        double r317986 = r317981 * r317985;
        double r317987 = exp(r317986);
        double r317988 = r317987 / r317981;
        return r317988;
}
↓
double f(double x, double y) {
        double r317989 = x;
        double r317990 = -5.89791977930625;
        bool r317991 = r317989 <= r317990;
        double r317992 = -1.0;
        double r317993 = y;
        double r317994 = r317992 * r317993;
        double r317995 = exp(r317994);
        double r317996 = r317995 / r317989;
        double r317997 = 0.0009297752561077733;
        bool r317998 = r317989 <= r317997;
        double r317999 = 1.0;
        double r318000 = r317989 + r317993;
        double r318001 = cbrt(r318000);
        double r318002 = r318001 * r318001;
        double r318003 = r317999 / r318002;
        double r318004 = pow(r318003, r317989);
        double r318005 = r317989 / r318001;
        double r318006 = pow(r318005, r317989);
        double r318007 = r318004 * r318006;
        double r318008 = r318007 / r317989;
        double r318009 = exp(r317993);
        double r318010 = r317989 * r318009;
        double r318011 = r317999 / r318010;
        double r318012 = r317998 ? r318008 : r318011;
        double r318013 = r317991 ? r317996 : r318012;
        return r318013;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original11.1
Target8.1
Herbie1.5
\[\begin{array}{l} \mathbf{if}\;y \lt -3.73118442066479561 \cdot 10^{94}:\\ \;\;\;\;\frac{e^{\frac{-1}{y}}}{x}\\ \mathbf{elif}\;y \lt 2.81795924272828789 \cdot 10^{37}:\\ \;\;\;\;\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}\\ \mathbf{elif}\;y \lt 2.347387415166998 \cdot 10^{178}:\\ \;\;\;\;\log \left(e^{\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{-1}{y}}}{x}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -5.89791977930625

    1. Initial program 11.6

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

    2. Simplified11.6

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

    3. Taylor expanded around inf 0.1

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

    4. Simplified0.1

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

    if -5.89791977930625 < x < 0.0009297752561077733

    1. Initial program 11.6

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

    2. Simplified11.6

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

    4. Applied add-cube-cbrt11.6

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

    5. Applied *-un-lft-identity11.6

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

    6. Applied times-frac11.6

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

    7. Applied unpow-prod-down2.9

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

    if 0.0009297752561077733 < x

    1. Initial program 9.9

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

    2. Simplified9.9

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

    3. Taylor expanded around inf 0.3

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

    4. Simplified0.3

      \[\leadsto \frac{\color{blue}{e^{-1 \cdot y}}}{x}\]
    5. Using strategy rm

    6. Applied clear-num0.3

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

    7. Simplified0.3

      \[\leadsto \frac{1}{\color{blue}{x \cdot e^{y}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification1.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -5.89791977930625:\\ \;\;\;\;\frac{e^{-1 \cdot y}}{x}\\ \mathbf{elif}\;x \le 9.2977525610777332 \cdot 10^{-4}:\\ \;\;\;\;\frac{{\left(\frac{1}{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}\right)}^{x} \cdot {\left(\frac{x}{\sqrt[3]{x + y}}\right)}^{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot e^{y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020027 +o rules:numerics
(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))