Average Error: 10.9 → 1.9
Time: 5.1s
Precision: 64
\[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -7.3893711893619755 \cdot 10^{115} \lor \neg \left(x \le 8.4574501416202074 \cdot 10^{-18}\right):\\ \;\;\;\;\frac{e^{-1 \cdot y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}\right)}^{x} \cdot {\left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}^{x}}{x}\\ \end{array}\]
\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -7.3893711893619755 \cdot 10^{115} \lor \neg \left(x \le 8.4574501416202074 \cdot 10^{-18}\right):\\
\;\;\;\;\frac{e^{-1 \cdot y}}{x}\\

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

\end{array}
double f(double x, double y) {
        double r377016 = x;
        double r377017 = y;
        double r377018 = r377016 + r377017;
        double r377019 = r377016 / r377018;
        double r377020 = log(r377019);
        double r377021 = r377016 * r377020;
        double r377022 = exp(r377021);
        double r377023 = r377022 / r377016;
        return r377023;
}
↓
double f(double x, double y) {
        double r377024 = x;
        double r377025 = -7.3893711893619755e+115;
        bool r377026 = r377024 <= r377025;
        double r377027 = 8.457450141620207e-18;
        bool r377028 = r377024 <= r377027;
        double r377029 = !r377028;
        bool r377030 = r377026 || r377029;
        double r377031 = -1.0;
        double r377032 = y;
        double r377033 = r377031 * r377032;
        double r377034 = exp(r377033);
        double r377035 = r377034 / r377024;
        double r377036 = cbrt(r377024);
        double r377037 = r377036 * r377036;
        double r377038 = r377024 + r377032;
        double r377039 = cbrt(r377038);
        double r377040 = r377039 * r377039;
        double r377041 = r377037 / r377040;
        double r377042 = pow(r377041, r377024);
        double r377043 = r377036 / r377039;
        double r377044 = pow(r377043, r377024);
        double r377045 = r377042 * r377044;
        double r377046 = r377045 / r377024;
        double r377047 = r377030 ? r377035 : r377046;
        return r377047;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original10.9
Target7.9
Herbie1.9
\[\begin{array}{l} \mathbf{if}\;y \lt -3.73118442066479561 \cdot 10^{94}:\\ \;\;\;\;\frac{e^{\frac{-1}{y}}}{x}\\ \mathbf{elif}\;y \lt 2.81795924272828789 \cdot 10^{37}:\\ \;\;\;\;\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}\\ \mathbf{elif}\;y \lt 2.347387415166998 \cdot 10^{178}:\\ \;\;\;\;\log \left(e^{\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{-1}{y}}}{x}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -7.3893711893619755e+115 or 8.457450141620207e-18 < 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 0.9

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

    4. Simplified0.9

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

    if -7.3893711893619755e+115 < x < 8.457450141620207e-18

    1. Initial program 11.0

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

    2. Simplified11.0

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

    4. Applied add-cube-cbrt17.2

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

    5. Applied add-cube-cbrt11.0

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

    6. Applied times-frac11.0

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

    7. Applied unpow-prod-down2.8

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

  4. Final simplification1.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -7.3893711893619755 \cdot 10^{115} \lor \neg \left(x \le 8.4574501416202074 \cdot 10^{-18}\right):\\ \;\;\;\;\frac{e^{-1 \cdot y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}\right)}^{x} \cdot {\left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}^{x}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020081 
(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))