Average Error: 11.1 → 3.9
Time: 14.8s
Precision: 64
\[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le 1564109357.3757057:\\ \;\;\;\;\frac{e^{\left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right) \cdot x + x \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\left(2 \cdot \log \left(\frac{\left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}}{\sqrt[3]{x + y}}\right)\right) \cdot x + x \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}}{x}\\ \end{array}\]
\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x}
↓
\begin{array}{l}
\mathbf{if}\;y \le 1564109357.3757057:\\
\;\;\;\;\frac{e^{\left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right) \cdot x + x \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}}{x}\\

\mathbf{else}:\\
\;\;\;\;\frac{e^{\left(2 \cdot \log \left(\frac{\left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}}{\sqrt[3]{x + y}}\right)\right) \cdot x + x \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}}{x}\\

\end{array}
double f(double x, double y) {
        double r431411 = x;
        double r431412 = y;
        double r431413 = r431411 + r431412;
        double r431414 = r431411 / r431413;
        double r431415 = log(r431414);
        double r431416 = r431411 * r431415;
        double r431417 = exp(r431416);
        double r431418 = r431417 / r431411;
        return r431418;
}
↓
double f(double x, double y) {
        double r431419 = y;
        double r431420 = 1564109357.3757057;
        bool r431421 = r431419 <= r431420;
        double r431422 = 2.0;
        double r431423 = x;
        double r431424 = cbrt(r431423);
        double r431425 = r431423 + r431419;
        double r431426 = cbrt(r431425);
        double r431427 = r431424 / r431426;
        double r431428 = log(r431427);
        double r431429 = r431422 * r431428;
        double r431430 = r431429 * r431423;
        double r431431 = r431423 * r431428;
        double r431432 = r431430 + r431431;
        double r431433 = exp(r431432);
        double r431434 = r431433 / r431423;
        double r431435 = cbrt(r431424);
        double r431436 = r431435 * r431435;
        double r431437 = r431436 * r431435;
        double r431438 = r431437 / r431426;
        double r431439 = log(r431438);
        double r431440 = r431422 * r431439;
        double r431441 = r431440 * r431423;
        double r431442 = r431441 + r431431;
        double r431443 = exp(r431442);
        double r431444 = r431443 / r431423;
        double r431445 = r431421 ? r431434 : r431444;
        return r431445;
}

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
Target8.0
Herbie3.9
\[\begin{array}{l} \mathbf{if}\;y \lt -3.73118442066479561 \cdot 10^{94}:\\ \;\;\;\;\frac{e^{\frac{-1}{y}}}{x}\\ \mathbf{elif}\;y \lt 2.81795924272828789 \cdot 10^{37}:\\ \;\;\;\;\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}\\ \mathbf{elif}\;y \lt 2.347387415166998 \cdot 10^{178}:\\ \;\;\;\;\log \left(e^{\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{-1}{y}}}{x}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < 1564109357.3757057

    1. Initial program 4.6

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x}\]
    2. Using strategy rm

    3. Applied add-cube-cbrt28.7

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

    4. Applied add-cube-cbrt4.6

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

    5. Applied times-frac4.6

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

    6. Applied log-prod1.8

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

    7. Applied distribute-lft-in1.8

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

    8. Simplified1.2

      \[\leadsto \frac{e^{\color{blue}{\left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right) \cdot x} + x \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}}{x}\]

    if 1564109357.3757057 < y

    1. Initial program 33.1

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x}\]
    2. Using strategy rm

    3. Applied add-cube-cbrt25.5

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

    4. Applied add-cube-cbrt33.2

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

    5. Applied times-frac33.2

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

    6. Applied log-prod22.8

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

    7. Applied distribute-lft-in22.8

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

    8. Simplified20.6

      \[\leadsto \frac{e^{\color{blue}{\left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right) \cdot x} + x \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}}{x}\]
    9. Using strategy rm

    10. Applied add-cube-cbrt12.8

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

  4. Final simplification3.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le 1564109357.3757057:\\ \;\;\;\;\frac{e^{\left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right) \cdot x + x \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\left(2 \cdot \log \left(\frac{\left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}}{\sqrt[3]{x + y}}\right)\right) \cdot x + x \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020043 
(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))