Average Error: 11.2 → 0.3
Time: 43.1s
Precision: 64
\[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -4.909420533601188192839755443280075130357 \cdot 10^{97}:\\ \;\;\;\;\frac{1}{e^{y} \cdot x}\\ \mathbf{elif}\;x \le 11.69456786955556992779747815802693367004:\\ \;\;\;\;\frac{{\left(e^{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right)}^{\left(\sqrt[3]{x} \cdot \log \left(\frac{x}{x + y}\right)\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-y}}{x}\\ \end{array}\]
\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -4.909420533601188192839755443280075130357 \cdot 10^{97}:\\
\;\;\;\;\frac{1}{e^{y} \cdot x}\\

\mathbf{elif}\;x \le 11.69456786955556992779747815802693367004:\\
\;\;\;\;\frac{{\left(e^{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right)}^{\left(\sqrt[3]{x} \cdot \log \left(\frac{x}{x + y}\right)\right)}}{x}\\

\mathbf{else}:\\
\;\;\;\;\frac{e^{-y}}{x}\\

\end{array}
double f(double x, double y) {
        double r19506954 = x;
        double r19506955 = y;
        double r19506956 = r19506954 + r19506955;
        double r19506957 = r19506954 / r19506956;
        double r19506958 = log(r19506957);
        double r19506959 = r19506954 * r19506958;
        double r19506960 = exp(r19506959);
        double r19506961 = r19506960 / r19506954;
        return r19506961;
}
↓
double f(double x, double y) {
        double r19506962 = x;
        double r19506963 = -4.909420533601188e+97;
        bool r19506964 = r19506962 <= r19506963;
        double r19506965 = 1.0;
        double r19506966 = y;
        double r19506967 = exp(r19506966);
        double r19506968 = r19506967 * r19506962;
        double r19506969 = r19506965 / r19506968;
        double r19506970 = 11.69456786955557;
        bool r19506971 = r19506962 <= r19506970;
        double r19506972 = cbrt(r19506962);
        double r19506973 = r19506972 * r19506972;
        double r19506974 = exp(r19506973);
        double r19506975 = r19506962 + r19506966;
        double r19506976 = r19506962 / r19506975;
        double r19506977 = log(r19506976);
        double r19506978 = r19506972 * r19506977;
        double r19506979 = pow(r19506974, r19506978);
        double r19506980 = r19506979 / r19506962;
        double r19506981 = -r19506966;
        double r19506982 = exp(r19506981);
        double r19506983 = r19506982 / r19506962;
        double r19506984 = r19506971 ? r19506980 : r19506983;
        double r19506985 = r19506964 ? r19506969 : r19506984;
        return r19506985;
}

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
Target8.0
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;y \lt -3.73118442066479561492798134439269393419 \cdot 10^{94}:\\ \;\;\;\;\frac{e^{\frac{-1}{y}}}{x}\\ \mathbf{elif}\;y \lt 28179592427282878868860376020282245120:\\ \;\;\;\;\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}\\ \mathbf{elif}\;y \lt 2.347387415166997963747840232163110922613 \cdot 10^{178}:\\ \;\;\;\;\log \left(e^{\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{-1}{y}}}{x}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -4.909420533601188e+97

    1. Initial program 15.2

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

    2. Taylor expanded around inf 0.0

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

    3. Simplified0.0

      \[\leadsto \frac{\color{blue}{e^{-y}}}{x}\]
    4. Using strategy rm

    5. Applied clear-num0.0

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

    6. Simplified0.0

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

    if -4.909420533601188e+97 < x < 11.69456786955557

    1. Initial program 11.0

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

    3. Applied add-log-exp18.1

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

    4. Applied exp-to-pow0.6

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

    6. Applied add-cube-cbrt0.6

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

    7. Applied exp-prod0.6

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

    8. Applied pow-pow0.5

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

    if 11.69456786955557 < x

    1. Initial program 9.3

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

    2. Taylor expanded around inf 0.0

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

    3. Simplified0.0

      \[\leadsto \frac{\color{blue}{e^{-y}}}{x}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4.909420533601188192839755443280075130357 \cdot 10^{97}:\\ \;\;\;\;\frac{1}{e^{y} \cdot x}\\ \mathbf{elif}\;x \le 11.69456786955556992779747815802693367004:\\ \;\;\;\;\frac{{\left(e^{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right)}^{\left(\sqrt[3]{x} \cdot \log \left(\frac{x}{x + y}\right)\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-y}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(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))