Average Error: 11.3 → 7.0
Time: 18.8s
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 r16153135 = x;
        double r16153136 = y;
        double r16153137 = r16153135 + r16153136;
        double r16153138 = r16153135 / r16153137;
        double r16153139 = log(r16153138);
        double r16153140 = r16153135 * r16153139;
        double r16153141 = exp(r16153140);
        double r16153142 = r16153141 / r16153135;
        return r16153142;
}
↓
double f(double x, double y) {
        double r16153143 = y;
        double r16153144 = 5.639442099966964;
        bool r16153145 = r16153143 <= r16153144;
        double r16153146 = 1.0;
        double r16153147 = x;
        double r16153148 = r16153146 / r16153147;
        double r16153149 = cbrt(r16153147);
        double r16153150 = r16153149 * r16153149;
        double r16153151 = cbrt(r16153150);
        double r16153152 = cbrt(r16153149);
        double r16153153 = r16153151 * r16153152;
        double r16153154 = r16153153 * r16153149;
        double r16153155 = r16153147 + r16153143;
        double r16153156 = r16153149 / r16153155;
        double r16153157 = r16153154 * r16153156;
        double r16153158 = log(r16153157);
        double r16153159 = r16153158 * r16153147;
        double r16153160 = exp(r16153159);
        double r16153161 = r16153160 / r16153147;
        double r16153162 = r16153145 ? r16153148 : r16153161;
        return r16153162;
}

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

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

    9. Applied cbrt-prod25.6

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