Average Error: 11.6 → 5.1
Time: 6.3s
Precision: 64
\[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le 130.06933248317358 \lor \neg \left(y \le 1.514187256420666 \cdot 10^{102}\right):\\ \;\;\;\;\frac{e^{x \cdot \left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \log \left(\frac{\frac{x}{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}}{\sqrt[3]{x + y}}\right)}}{x}\\ \end{array}\]
\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x}
↓
\begin{array}{l}
\mathbf{if}\;y \le 130.06933248317358 \lor \neg \left(y \le 1.514187256420666 \cdot 10^{102}\right):\\
\;\;\;\;\frac{e^{x \cdot \left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right)}}{x}\\

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

\end{array}
double f(double x, double y) {
        double r423331 = x;
        double r423332 = y;
        double r423333 = r423331 + r423332;
        double r423334 = r423331 / r423333;
        double r423335 = log(r423334);
        double r423336 = r423331 * r423335;
        double r423337 = exp(r423336);
        double r423338 = r423337 / r423331;
        return r423338;
}
↓
double f(double x, double y) {
        double r423339 = y;
        double r423340 = 130.06933248317358;
        bool r423341 = r423339 <= r423340;
        double r423342 = 1.514187256420666e+102;
        bool r423343 = r423339 <= r423342;
        double r423344 = !r423343;
        bool r423345 = r423341 || r423344;
        double r423346 = x;
        double r423347 = 2.0;
        double r423348 = cbrt(r423346);
        double r423349 = r423346 + r423339;
        double r423350 = cbrt(r423349);
        double r423351 = r423348 / r423350;
        double r423352 = log(r423351);
        double r423353 = r423347 * r423352;
        double r423354 = r423353 + r423352;
        double r423355 = r423346 * r423354;
        double r423356 = exp(r423355);
        double r423357 = r423356 / r423346;
        double r423358 = r423350 * r423350;
        double r423359 = r423346 / r423358;
        double r423360 = r423359 / r423350;
        double r423361 = log(r423360);
        double r423362 = r423346 * r423361;
        double r423363 = exp(r423362);
        double r423364 = r423363 / r423346;
        double r423365 = r423345 ? r423357 : r423364;
        return r423365;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original11.6
Target8.1
Herbie5.1
\[\begin{array}{l} \mathbf{if}\;y \lt -3.73118442066479561 \cdot 10^{94}:\\ \;\;\;\;\frac{e^{\frac{-1}{y}}}{x}\\ \mathbf{elif}\;y \lt 2.81795924272828789 \cdot 10^{37}:\\ \;\;\;\;\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}\\ \mathbf{elif}\;y \lt 2.347387415166998 \cdot 10^{178}:\\ \;\;\;\;\log \left(e^{\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{-1}{y}}}{x}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < 130.06933248317358 or 1.514187256420666e+102 < y

    1. Initial program 9.0

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

    3. Applied add-cube-cbrt28.3

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

    4. Applied add-cube-cbrt9.0

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

    5. Applied times-frac9.0

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

    6. Applied log-prod4.5

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

    7. Simplified3.3

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

    if 130.06933248317358 < y < 1.514187256420666e+102

    1. Initial program 36.4

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

    3. Applied add-cube-cbrt22.5

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

    4. Applied associate-/r*22.9

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

  4. Final simplification5.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le 130.06933248317358 \lor \neg \left(y \le 1.514187256420666 \cdot 10^{102}\right):\\ \;\;\;\;\frac{e^{x \cdot \left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \log \left(\frac{\frac{x}{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}}{\sqrt[3]{x + y}}\right)}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 
(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))