Average Error: 11.5 → 0.0
Time: 29.0s
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 r303493 = x;
        double r303494 = y;
        double r303495 = r303493 + r303494;
        double r303496 = r303493 / r303495;
        double r303497 = log(r303496);
        double r303498 = r303493 * r303497;
        double r303499 = exp(r303498);
        double r303500 = r303499 / r303493;
        return r303500;
}
↓
double f(double x, double y) {
        double r303501 = x;
        double r303502 = -11.131115373710125;
        bool r303503 = r303501 <= r303502;
        double r303504 = 35.40231590631766;
        bool r303505 = r303501 <= r303504;
        double r303506 = !r303505;
        bool r303507 = r303503 || r303506;
        double r303508 = y;
        double r303509 = -r303508;
        double r303510 = exp(r303509);
        double r303511 = r303510 / r303501;
        double r303512 = 1.0;
        double r303513 = r303501 + r303508;
        double r303514 = cbrt(r303513);
        double r303515 = r303514 * r303514;
        double r303516 = r303512 / r303515;
        double r303517 = pow(r303516, r303501);
        double r303518 = cbrt(r303501);
        double r303519 = r303518 * r303518;
        double r303520 = cbrt(r303514);
        double r303521 = r303520 * r303520;
        double r303522 = r303519 / r303521;
        double r303523 = pow(r303522, r303501);
        double r303524 = r303518 / r303520;
        double r303525 = pow(r303524, r303501);
        double r303526 = r303523 * r303525;
        double r303527 = r303517 * r303526;
        double r303528 = r303527 / r303501;
        double r303529 = r303507 ? r303511 : r303528;
        return r303529;
}

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 +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 (/ -1.0 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))