Average Error: 11.2 → 7.0
Time: 17.8s
Precision: 64
\[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le 7.289916210907466442847635335056111216545:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log \left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\frac{x + y}{\sqrt[3]{x}}}\right) \cdot x}}{x}\\ \end{array}\]
\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x}
↓
\begin{array}{l}
\mathbf{if}\;y \le 7.289916210907466442847635335056111216545:\\
\;\;\;\;\frac{1}{x}\\

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

\end{array}
double f(double x, double y) {
        double r18200209 = x;
        double r18200210 = y;
        double r18200211 = r18200209 + r18200210;
        double r18200212 = r18200209 / r18200211;
        double r18200213 = log(r18200212);
        double r18200214 = r18200209 * r18200213;
        double r18200215 = exp(r18200214);
        double r18200216 = r18200215 / r18200209;
        return r18200216;
}
↓
double f(double x, double y) {
        double r18200217 = y;
        double r18200218 = 7.289916210907466;
        bool r18200219 = r18200217 <= r18200218;
        double r18200220 = 1.0;
        double r18200221 = x;
        double r18200222 = r18200220 / r18200221;
        double r18200223 = cbrt(r18200221);
        double r18200224 = r18200223 * r18200223;
        double r18200225 = r18200221 + r18200217;
        double r18200226 = r18200225 / r18200223;
        double r18200227 = r18200224 / r18200226;
        double r18200228 = log(r18200227);
        double r18200229 = r18200228 * r18200221;
        double r18200230 = exp(r18200229);
        double r18200231 = r18200230 / r18200221;
        double r18200232 = r18200219 ? r18200222 : r18200231;
        return r18200232;
}

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
Herbie7.0
\[\begin{array}{l} \mathbf{if}\;y \lt -3.73118442066479561492798134439269393419 \cdot 10^{94}:\\ \;\;\;\;\frac{e^{\frac{-1}{y}}}{x}\\ \mathbf{elif}\;y \lt 28179592427282878868860376020282245120:\\ \;\;\;\;\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}\\ \mathbf{elif}\;y \lt 2.347387415166997963747840232163110922613 \cdot 10^{178}:\\ \;\;\;\;\log \left(e^{\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{-1}{y}}}{x}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < 7.289916210907466

    1. Initial program 4.6

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

    2. Taylor expanded around inf 1.1

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

    if 7.289916210907466 < y

    1. Initial program 31.8

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

    3. Applied add-cube-cbrt25.4

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

    4. Applied associate-/l*25.5

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

  4. Final simplification7.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le 7.289916210907466442847635335056111216545:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log \left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\frac{x + y}{\sqrt[3]{x}}}\right) \cdot x}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019192 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, F"

  :herbie-target
  (if (< y -3.7311844206647956e+94) (/ (exp (/ -1.0 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 (/ -1.0 y)) x))))

  (/ (exp (* x (log (/ x (+ x y))))) x))