Average Error: 11.2 → 0.0
Time: 19.7s
Precision: 64
\[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -10.11942408653544234198307094629853963852 \lor \neg \left(x \le 0.1204014281115151213663594376157561782748\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{1}{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}\right)}^{x} \cdot \left({\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{\sqrt[3]{x + y}} \cdot \sqrt[3]{\sqrt[3]{x + y}}}\right)}^{x} \cdot {\left(\frac{\sqrt[3]{x}}{\sqrt[3]{\sqrt[3]{x + y}}}\right)}^{x}\right)}{x}\\ \end{array}\]
\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -10.11942408653544234198307094629853963852 \lor \neg \left(x \le 0.1204014281115151213663594376157561782748\right):\\
\;\;\;\;\frac{e^{-y}}{x}\\

\mathbf{else}:\\
\;\;\;\;\frac{{\left(\frac{1}{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}\right)}^{x} \cdot \left({\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{\sqrt[3]{x + y}} \cdot \sqrt[3]{\sqrt[3]{x + y}}}\right)}^{x} \cdot {\left(\frac{\sqrt[3]{x}}{\sqrt[3]{\sqrt[3]{x + y}}}\right)}^{x}\right)}{x}\\

\end{array}
double f(double x, double y) {
        double r292140 = x;
        double r292141 = y;
        double r292142 = r292140 + r292141;
        double r292143 = r292140 / r292142;
        double r292144 = log(r292143);
        double r292145 = r292140 * r292144;
        double r292146 = exp(r292145);
        double r292147 = r292146 / r292140;
        return r292147;
}
↓
double f(double x, double y) {
        double r292148 = x;
        double r292149 = -10.119424086535442;
        bool r292150 = r292148 <= r292149;
        double r292151 = 0.12040142811151512;
        bool r292152 = r292148 <= r292151;
        double r292153 = !r292152;
        bool r292154 = r292150 || r292153;
        double r292155 = y;
        double r292156 = -r292155;
        double r292157 = exp(r292156);
        double r292158 = r292157 / r292148;
        double r292159 = 1.0;
        double r292160 = r292148 + r292155;
        double r292161 = cbrt(r292160);
        double r292162 = r292161 * r292161;
        double r292163 = r292159 / r292162;
        double r292164 = pow(r292163, r292148);
        double r292165 = cbrt(r292148);
        double r292166 = r292165 * r292165;
        double r292167 = cbrt(r292161);
        double r292168 = r292167 * r292167;
        double r292169 = r292166 / r292168;
        double r292170 = pow(r292169, r292148);
        double r292171 = r292165 / r292167;
        double r292172 = pow(r292171, r292148);
        double r292173 = r292170 * r292172;
        double r292174 = r292164 * r292173;
        double r292175 = r292174 / r292148;
        double r292176 = r292154 ? r292158 : r292175;
        return r292176;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original11.2
Target8.2
Herbie0.0
\[\begin{array}{l} \mathbf{if}\;y \lt -3.73118442066479561492798134439269393419 \cdot 10^{94}:\\ \;\;\;\;\frac{e^{\frac{-1}{y}}}{x}\\ \mathbf{elif}\;y \lt 28179592427282878868860376020282245120:\\ \;\;\;\;\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}\\ \mathbf{elif}\;y \lt 2.347387415166997963747840232163110922613 \cdot 10^{178}:\\ \;\;\;\;\log \left(e^{\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{-1}{y}}}{x}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -10.119424086535442 or 0.12040142811151512 < x

    1. Initial program 10.9

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

    2. Simplified10.9

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

    3. Taylor expanded around inf 0.1

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

    if -10.119424086535442 < x < 0.12040142811151512

    1. Initial program 11.6

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

    2. Simplified11.6

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

    4. Applied add-cube-cbrt11.6

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

    5. Applied *-un-lft-identity11.6

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

    6. Applied times-frac11.6

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

    7. Applied unpow-prod-down3.3

      \[\leadsto \frac{\color{blue}{{\left(\frac{1}{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}\right)}^{x} \cdot {\left(\frac{x}{\sqrt[3]{x + y}}\right)}^{x}}}{x}\]
    8. Using strategy rm

    9. Applied add-cube-cbrt3.3

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

    10. Applied add-cube-cbrt3.3

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

    11. Applied times-frac3.3

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

    12. Applied unpow-prod-down0.0

      \[\leadsto \frac{{\left(\frac{1}{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}\right)}^{x} \cdot \color{blue}{\left({\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{\sqrt[3]{x + y}} \cdot \sqrt[3]{\sqrt[3]{x + y}}}\right)}^{x} \cdot {\left(\frac{\sqrt[3]{x}}{\sqrt[3]{\sqrt[3]{x + y}}}\right)}^{x}\right)}}{x}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -10.11942408653544234198307094629853963852 \lor \neg \left(x \le 0.1204014281115151213663594376157561782748\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{1}{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}\right)}^{x} \cdot \left({\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{\sqrt[3]{x + y}} \cdot \sqrt[3]{\sqrt[3]{x + y}}}\right)}^{x} \cdot {\left(\frac{\sqrt[3]{x}}{\sqrt[3]{\sqrt[3]{x + y}}}\right)}^{x}\right)}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(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))