Average Error: 11.6 → 5.1
Time: 6.1s
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 r514120 = x;
        double r514121 = y;
        double r514122 = r514120 + r514121;
        double r514123 = r514120 / r514122;
        double r514124 = log(r514123);
        double r514125 = r514120 * r514124;
        double r514126 = exp(r514125);
        double r514127 = r514126 / r514120;
        return r514127;
}
↓
double f(double x, double y) {
        double r514128 = y;
        double r514129 = 130.06933248317358;
        bool r514130 = r514128 <= r514129;
        double r514131 = 1.514187256420666e+102;
        bool r514132 = r514128 <= r514131;
        double r514133 = !r514132;
        bool r514134 = r514130 || r514133;
        double r514135 = x;
        double r514136 = 2.0;
        double r514137 = cbrt(r514135);
        double r514138 = r514135 + r514128;
        double r514139 = cbrt(r514138);
        double r514140 = r514137 / r514139;
        double r514141 = log(r514140);
        double r514142 = r514136 * r514141;
        double r514143 = r514142 + r514141;
        double r514144 = r514135 * r514143;
        double r514145 = exp(r514144);
        double r514146 = r514145 / r514135;
        double r514147 = r514139 * r514139;
        double r514148 = r514135 / r514147;
        double r514149 = r514148 / r514139;
        double r514150 = log(r514149);
        double r514151 = r514135 * r514150;
        double r514152 = exp(r514151);
        double r514153 = r514152 / r514135;
        double r514154 = r514134 ? r514146 : r514153;
        return r514154;
}

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))