Average Error: 11.1 → 1.5
Time: 5.1s
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 r339429 = x;
        double r339430 = y;
        double r339431 = r339429 + r339430;
        double r339432 = r339429 / r339431;
        double r339433 = log(r339432);
        double r339434 = r339429 * r339433;
        double r339435 = exp(r339434);
        double r339436 = r339435 / r339429;
        return r339436;
}
↓
double f(double x, double y) {
        double r339437 = x;
        double r339438 = -5.89791977930625;
        bool r339439 = r339437 <= r339438;
        double r339440 = -1.0;
        double r339441 = y;
        double r339442 = r339440 * r339441;
        double r339443 = exp(r339442);
        double r339444 = r339443 / r339437;
        double r339445 = 0.0009297752561077733;
        bool r339446 = r339437 <= r339445;
        double r339447 = 1.0;
        double r339448 = r339437 + r339441;
        double r339449 = cbrt(r339448);
        double r339450 = r339449 * r339449;
        double r339451 = r339447 / r339450;
        double r339452 = pow(r339451, r339437);
        double r339453 = r339437 / r339449;
        double r339454 = pow(r339453, r339437);
        double r339455 = r339452 * r339454;
        double r339456 = r339455 / r339437;
        double r339457 = exp(r339441);
        double r339458 = r339437 * r339457;
        double r339459 = r339447 / r339458;
        double r339460 = r339446 ? r339456 : r339459;
        double r339461 = r339439 ? r339444 : r339460;
        return r339461;
}

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 
(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))