Average Error: 11.3 → 0.2
Time: 23.5s
Precision: 64
\[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -10594210996052847989843034112 \lor \neg \left(x \le 0.003027614112253133524244042007467214716598\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right)}}{x}\\ \end{array}\]
\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -10594210996052847989843034112 \lor \neg \left(x \le 0.003027614112253133524244042007467214716598\right):\\
\;\;\;\;\frac{e^{-y}}{x}\\

\mathbf{else}:\\
\;\;\;\;\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}\\

\end{array}
double f(double x, double y) {
        double r243299 = x;
        double r243300 = y;
        double r243301 = r243299 + r243300;
        double r243302 = r243299 / r243301;
        double r243303 = log(r243302);
        double r243304 = r243299 * r243303;
        double r243305 = exp(r243304);
        double r243306 = r243305 / r243299;
        return r243306;
}
↓
double f(double x, double y) {
        double r243307 = x;
        double r243308 = -1.0594210996052848e+28;
        bool r243309 = r243307 <= r243308;
        double r243310 = 0.0030276141122531335;
        bool r243311 = r243307 <= r243310;
        double r243312 = !r243311;
        bool r243313 = r243309 || r243312;
        double r243314 = y;
        double r243315 = -r243314;
        double r243316 = exp(r243315);
        double r243317 = r243316 / r243307;
        double r243318 = 2.0;
        double r243319 = cbrt(r243307);
        double r243320 = r243307 + r243314;
        double r243321 = cbrt(r243320);
        double r243322 = r243319 / r243321;
        double r243323 = log(r243322);
        double r243324 = r243318 * r243323;
        double r243325 = r243324 + r243323;
        double r243326 = r243307 * r243325;
        double r243327 = exp(r243326);
        double r243328 = r243327 / r243307;
        double r243329 = r243313 ? r243317 : r243328;
        return r243329;
}

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

Derivation

  1. Split input into 2 regimes

  2. if x < -1.0594210996052848e+28 or 0.0030276141122531335 < x

    1. Initial program 11.9

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

    2. Taylor expanded around inf 0.2

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot y}}{x}}\]

    3. Simplified0.2

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

    if -1.0594210996052848e+28 < x < 0.0030276141122531335

    1. Initial program 10.7

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

    3. Applied add-cube-cbrt11.9

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

    4. Applied add-cube-cbrt10.7

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

    5. Applied times-frac10.7

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

    6. Applied log-prod2.3

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

    7. Simplified0.1

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -10594210996052847989843034112 \lor \neg \left(x \le 0.003027614112253133524244042007467214716598\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right)}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019306 +o rules:numerics
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, F"
  :precision binary64

  :herbie-target
  (if (< y -3.73118442066479561e94) (/ (exp (/ -1 y)) x) (if (< y 2.81795924272828789e37) (/ (pow (/ x (+ y x)) x) x) (if (< y 2.347387415166998e178) (log (exp (/ (pow (/ x (+ y x)) x) x))) (/ (exp (/ -1 y)) x))))

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