Average Error: 11.6 → 5.2
Time: 6.3s
Precision: 64
\[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le 1077.4618968781419 \lor \neg \left(y \le 4.93738979423122685 \cdot 10^{97}\right):\\ \;\;\;\;\frac{e^{x \cdot \left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right) + x \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(2 \cdot \log \left(\frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt[3]{x}\right)\right)}{\sqrt[3]{x + y}}\right)\right) + x \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}}{x}\\ \end{array}\]
\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x}
↓
\begin{array}{l}
\mathbf{if}\;y \le 1077.4618968781419 \lor \neg \left(y \le 4.93738979423122685 \cdot 10^{97}\right):\\
\;\;\;\;\frac{e^{x \cdot \left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right) + x \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}}{x}\\

\mathbf{else}:\\
\;\;\;\;\frac{e^{x \cdot \left(2 \cdot \log \left(\frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt[3]{x}\right)\right)}{\sqrt[3]{x + y}}\right)\right) + x \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}}{x}\\

\end{array}
double f(double x, double y) {
        double r428929 = x;
        double r428930 = y;
        double r428931 = r428929 + r428930;
        double r428932 = r428929 / r428931;
        double r428933 = log(r428932);
        double r428934 = r428929 * r428933;
        double r428935 = exp(r428934);
        double r428936 = r428935 / r428929;
        return r428936;
}
↓
double f(double x, double y) {
        double r428937 = y;
        double r428938 = 1077.461896878142;
        bool r428939 = r428937 <= r428938;
        double r428940 = 4.937389794231227e+97;
        bool r428941 = r428937 <= r428940;
        double r428942 = !r428941;
        bool r428943 = r428939 || r428942;
        double r428944 = x;
        double r428945 = 2.0;
        double r428946 = cbrt(r428944);
        double r428947 = r428944 + r428937;
        double r428948 = cbrt(r428947);
        double r428949 = r428946 / r428948;
        double r428950 = log(r428949);
        double r428951 = r428945 * r428950;
        double r428952 = r428944 * r428951;
        double r428953 = r428944 * r428950;
        double r428954 = r428952 + r428953;
        double r428955 = exp(r428954);
        double r428956 = r428955 / r428944;
        double r428957 = expm1(r428946);
        double r428958 = log1p(r428957);
        double r428959 = r428958 / r428948;
        double r428960 = log(r428959);
        double r428961 = r428945 * r428960;
        double r428962 = r428944 * r428961;
        double r428963 = r428962 + r428953;
        double r428964 = exp(r428963);
        double r428965 = r428964 / r428944;
        double r428966 = r428943 ? r428956 : r428965;
        return r428966;
}

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
Herbie5.2
\[\begin{array}{l} \mathbf{if}\;y \lt -3.73118442066479561 \cdot 10^{94}:\\ \;\;\;\;\frac{e^{\frac{-1}{y}}}{x}\\ \mathbf{elif}\;y \lt 2.81795924272828789 \cdot 10^{37}:\\ \;\;\;\;\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}\\ \mathbf{elif}\;y \lt 2.347387415166998 \cdot 10^{178}:\\ \;\;\;\;\log \left(e^{\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{-1}{y}}}{x}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < 1077.461896878142 or 4.937389794231227e+97 < y

    1. Initial program 9.2

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x}\]
    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-cbrt9.2

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

    5. Applied times-frac9.2

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

    6. Applied log-prod4.7

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

    7. Applied distribute-lft-in4.7

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

    8. Simplified3.5

      \[\leadsto \frac{e^{\color{blue}{x \cdot \left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right)} + x \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}}{x}\]

    if 1077.461896878142 < y < 4.937389794231227e+97

    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-cbrt22.3

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

    4. Applied add-cube-cbrt36.4

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

    5. Applied times-frac36.4

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

    6. Applied log-prod33.8

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

    7. Applied distribute-lft-in33.8

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

    8. Simplified33.8

      \[\leadsto \frac{e^{\color{blue}{x \cdot \left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right)} + x \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}}{x}\]
    9. Using strategy rm

    10. Applied log1p-expm1-u23.0

      \[\leadsto \frac{e^{x \cdot \left(2 \cdot \log \left(\frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt[3]{x}\right)\right)}}{\sqrt[3]{x + y}}\right)\right) + x \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}}{x}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification5.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le 1077.4618968781419 \lor \neg \left(y \le 4.93738979423122685 \cdot 10^{97}\right):\\ \;\;\;\;\frac{e^{x \cdot \left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right) + x \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(2 \cdot \log \left(\frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt[3]{x}\right)\right)}{\sqrt[3]{x + y}}\right)\right) + x \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 +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))