Average Error: 11.3 → 7.0
Time: 19.7s
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 r23335697 = x;
        double r23335698 = y;
        double r23335699 = r23335697 + r23335698;
        double r23335700 = r23335697 / r23335699;
        double r23335701 = log(r23335700);
        double r23335702 = r23335697 * r23335701;
        double r23335703 = exp(r23335702);
        double r23335704 = r23335703 / r23335697;
        return r23335704;
}
↓
double f(double x, double y) {
        double r23335705 = y;
        double r23335706 = 5.639442099966964;
        bool r23335707 = r23335705 <= r23335706;
        double r23335708 = 1.0;
        double r23335709 = x;
        double r23335710 = r23335708 / r23335709;
        double r23335711 = cbrt(r23335709);
        double r23335712 = r23335711 * r23335711;
        double r23335713 = cbrt(r23335712);
        double r23335714 = cbrt(r23335711);
        double r23335715 = r23335713 * r23335714;
        double r23335716 = r23335715 * r23335711;
        double r23335717 = r23335709 + r23335705;
        double r23335718 = r23335711 / r23335717;
        double r23335719 = r23335716 * r23335718;
        double r23335720 = log(r23335719);
        double r23335721 = r23335720 * r23335709;
        double r23335722 = exp(r23335721);
        double r23335723 = r23335722 / r23335709;
        double r23335724 = r23335707 ? r23335710 : r23335723;
        return r23335724;
}

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))