Average Error: 11.5 → 0.0
Time: 22.8s
Precision: 64
\[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -11.1311153737101254 \lor \neg \left(x \le 35.4023159063176607\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{1}{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}\right)}^{x} \cdot \left({\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{\sqrt[3]{x + y}} \cdot \sqrt[3]{\sqrt[3]{x + y}}}\right)}^{x} \cdot {\left(\frac{\sqrt[3]{x}}{\sqrt[3]{\sqrt[3]{x + y}}}\right)}^{x}\right)}{x}\\ \end{array}\]
\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -11.1311153737101254 \lor \neg \left(x \le 35.4023159063176607\right):\\
\;\;\;\;\frac{e^{-y}}{x}\\

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

\end{array}
double f(double x, double y) {
        double r334703 = x;
        double r334704 = y;
        double r334705 = r334703 + r334704;
        double r334706 = r334703 / r334705;
        double r334707 = log(r334706);
        double r334708 = r334703 * r334707;
        double r334709 = exp(r334708);
        double r334710 = r334709 / r334703;
        return r334710;
}
↓
double f(double x, double y) {
        double r334711 = x;
        double r334712 = -11.131115373710125;
        bool r334713 = r334711 <= r334712;
        double r334714 = 35.40231590631766;
        bool r334715 = r334711 <= r334714;
        double r334716 = !r334715;
        bool r334717 = r334713 || r334716;
        double r334718 = y;
        double r334719 = -r334718;
        double r334720 = exp(r334719);
        double r334721 = r334720 / r334711;
        double r334722 = 1.0;
        double r334723 = r334711 + r334718;
        double r334724 = cbrt(r334723);
        double r334725 = r334724 * r334724;
        double r334726 = r334722 / r334725;
        double r334727 = pow(r334726, r334711);
        double r334728 = cbrt(r334711);
        double r334729 = r334728 * r334728;
        double r334730 = cbrt(r334724);
        double r334731 = r334730 * r334730;
        double r334732 = r334729 / r334731;
        double r334733 = pow(r334732, r334711);
        double r334734 = r334728 / r334730;
        double r334735 = pow(r334734, r334711);
        double r334736 = r334733 * r334735;
        double r334737 = r334727 * r334736;
        double r334738 = r334737 / r334711;
        double r334739 = r334717 ? r334721 : r334738;
        return r334739;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original11.5
Target7.9
Herbie0.0
\[\begin{array}{l} \mathbf{if}\;y \lt -3.73118442066479561 \cdot 10^{94}:\\ \;\;\;\;\frac{e^{\frac{-1}{y}}}{x}\\ \mathbf{elif}\;y \lt 2.81795924272828789 \cdot 10^{37}:\\ \;\;\;\;\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}\\ \mathbf{elif}\;y \lt 2.347387415166998 \cdot 10^{178}:\\ \;\;\;\;\log \left(e^{\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{-1}{y}}}{x}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -11.131115373710125 or 35.40231590631766 < x

    1. Initial program 10.9

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

    2. Simplified10.9

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

    3. Taylor expanded around inf 0.0

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

    if -11.131115373710125 < x < 35.40231590631766

    1. Initial program 12.1

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

    2. Simplified12.1

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

    4. Applied add-cube-cbrt12.1

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

    5. Applied *-un-lft-identity12.1

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

    6. Applied times-frac12.1

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

    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

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

    10. Applied add-cube-cbrt3.0

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

    11. Applied times-frac3.0

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

    12. Applied unpow-prod-down0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -11.1311153737101254 \lor \neg \left(x \le 35.4023159063176607\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{1}{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}\right)}^{x} \cdot \left({\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{\sqrt[3]{x + y}} \cdot \sqrt[3]{\sqrt[3]{x + y}}}\right)}^{x} \cdot {\left(\frac{\sqrt[3]{x}}{\sqrt[3]{\sqrt[3]{x + y}}}\right)}^{x}\right)}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019199 
(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))