Average Error: 11.0 → 0.3
Time: 20.9s
Precision: 64
\[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -4.909420533601188 \cdot 10^{+97}:\\ \;\;\;\;\frac{1}{e^{y} \cdot x}\\ \mathbf{elif}\;x \le 11.69456786955557:\\ \;\;\;\;\frac{{\left(e^{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right)}^{\left(\sqrt[3]{x} \cdot \log \left(\frac{x}{x + y}\right)\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-y}}{x}\\ \end{array}\]
\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -4.909420533601188 \cdot 10^{+97}:\\
\;\;\;\;\frac{1}{e^{y} \cdot x}\\

\mathbf{elif}\;x \le 11.69456786955557:\\
\;\;\;\;\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 r21317981 = x;
        double r21317982 = y;
        double r21317983 = r21317981 + r21317982;
        double r21317984 = r21317981 / r21317983;
        double r21317985 = log(r21317984);
        double r21317986 = r21317981 * r21317985;
        double r21317987 = exp(r21317986);
        double r21317988 = r21317987 / r21317981;
        return r21317988;
}
↓
double f(double x, double y) {
        double r21317989 = x;
        double r21317990 = -4.909420533601188e+97;
        bool r21317991 = r21317989 <= r21317990;
        double r21317992 = 1.0;
        double r21317993 = y;
        double r21317994 = exp(r21317993);
        double r21317995 = r21317994 * r21317989;
        double r21317996 = r21317992 / r21317995;
        double r21317997 = 11.69456786955557;
        bool r21317998 = r21317989 <= r21317997;
        double r21317999 = cbrt(r21317989);
        double r21318000 = r21317999 * r21317999;
        double r21318001 = exp(r21318000);
        double r21318002 = r21317989 + r21317993;
        double r21318003 = r21317989 / r21318002;
        double r21318004 = log(r21318003);
        double r21318005 = r21317999 * r21318004;
        double r21318006 = pow(r21318001, r21318005);
        double r21318007 = r21318006 / r21317989;
        double r21318008 = -r21317993;
        double r21318009 = exp(r21318008);
        double r21318010 = r21318009 / r21317989;
        double r21318011 = r21317998 ? r21318007 : r21318010;
        double r21318012 = r21317991 ? r21317996 : r21318011;
        return r21318012;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original11.0
Target7.9
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;y \lt -3.7311844206647956 \cdot 10^{+94}:\\ \;\;\;\;\frac{e^{\frac{-1}{y}}}{x}\\ \mathbf{elif}\;y \lt 2.817959242728288 \cdot 10^{+37}:\\ \;\;\;\;\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}\\ \mathbf{elif}\;y \lt 2.347387415166998 \cdot 10^{+178}:\\ \;\;\;\;\log \left(e^{\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{-1}{y}}}{x}\\ \end{array}\]

Derivation

  1. Split input into 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 exp-neg0.0

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

    6. Applied associate-/l/0.0

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

    if -4.909420533601188e+97 < x < 11.69456786955557

    1. Initial program 10.6

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

    3. Applied add-log-exp17.7

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

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4.909420533601188 \cdot 10^{+97}:\\ \;\;\;\;\frac{1}{e^{y} \cdot x}\\ \mathbf{elif}\;x \le 11.69456786955557:\\ \;\;\;\;\frac{{\left(e^{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right)}^{\left(\sqrt[3]{x} \cdot \log \left(\frac{x}{x + y}\right)\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-y}}{x}\\ \end{array}\]

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 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))