Average Error: 11.3 → 7.0
Time: 20.1s
Precision: 64
\[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le 5.639442099966964:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log \left(\left(\left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{x}\right) \cdot \frac{\sqrt[3]{x}}{x + y}\right) \cdot x}}{x}\\ \end{array}\]
\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x}
↓
\begin{array}{l}
\mathbf{if}\;y \le 5.639442099966964:\\
\;\;\;\;\frac{1}{x}\\

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

\end{array}
double f(double x, double y) {
        double r16713303 = x;
        double r16713304 = y;
        double r16713305 = r16713303 + r16713304;
        double r16713306 = r16713303 / r16713305;
        double r16713307 = log(r16713306);
        double r16713308 = r16713303 * r16713307;
        double r16713309 = exp(r16713308);
        double r16713310 = r16713309 / r16713303;
        return r16713310;
}
↓
double f(double x, double y) {
        double r16713311 = y;
        double r16713312 = 5.639442099966964;
        bool r16713313 = r16713311 <= r16713312;
        double r16713314 = 1.0;
        double r16713315 = x;
        double r16713316 = r16713314 / r16713315;
        double r16713317 = cbrt(r16713315);
        double r16713318 = r16713317 * r16713317;
        double r16713319 = cbrt(r16713318);
        double r16713320 = cbrt(r16713317);
        double r16713321 = r16713319 * r16713320;
        double r16713322 = r16713321 * r16713317;
        double r16713323 = r16713315 + r16713311;
        double r16713324 = r16713317 / r16713323;
        double r16713325 = r16713322 * r16713324;
        double r16713326 = log(r16713325);
        double r16713327 = r16713326 * r16713315;
        double r16713328 = exp(r16713327);
        double r16713329 = r16713328 / r16713315;
        double r16713330 = r16713313 ? r16713316 : r16713329;
        return r16713330;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original11.3
Target7.9
Herbie7.0
\[\begin{array}{l} \mathbf{if}\;y \lt -3.7311844206647956 \cdot 10^{+94}:\\ \;\;\;\;\frac{e^{\frac{-1}{y}}}{x}\\ \mathbf{elif}\;y \lt 2.817959242728288 \cdot 10^{+37}:\\ \;\;\;\;\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}\\ \mathbf{elif}\;y \lt 2.347387415166998 \cdot 10^{+178}:\\ \;\;\;\;\log \left(e^{\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{-1}{y}}}{x}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < 5.639442099966964

    1. Initial program 4.5

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

    2. Taylor expanded around inf 1.2

      \[\leadsto \frac{e^{\color{blue}{0}}}{x}\]

    if 5.639442099966964 < y

    1. Initial program 33.3

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

    3. Applied *-un-lft-identity33.3

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

    4. Applied add-cube-cbrt25.6

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

    5. Applied times-frac25.5

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

    6. Simplified25.5

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

    8. Applied add-cube-cbrt25.7

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

    9. Applied cbrt-prod25.6

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

  4. Final simplification7.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le 5.639442099966964:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log \left(\left(\left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{x}\right) \cdot \frac{\sqrt[3]{x}}{x + y}\right) \cdot x}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 +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 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))