Average Error: 11.4 → 0.5
Time: 28.1s
Precision: 64
\[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.825603746180263343792126173792163470345 \cdot 10^{117}:\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{elif}\;x \le 5.955847209664487884595018840627744793892:\\ \;\;\;\;\frac{{\left(e^{x}\right)}^{\left(\log \left(\frac{x}{x + y}\right)\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-y}}{x}\\ \end{array}\]
\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.825603746180263343792126173792163470345 \cdot 10^{117}:\\
\;\;\;\;\frac{e^{-y}}{x}\\

\mathbf{elif}\;x \le 5.955847209664487884595018840627744793892:\\
\;\;\;\;\frac{{\left(e^{x}\right)}^{\left(\log \left(\frac{x}{x + y}\right)\right)}}{x}\\

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

\end{array}
double f(double x, double y) {
        double r92198090 = x;
        double r92198091 = y;
        double r92198092 = r92198090 + r92198091;
        double r92198093 = r92198090 / r92198092;
        double r92198094 = log(r92198093);
        double r92198095 = r92198090 * r92198094;
        double r92198096 = exp(r92198095);
        double r92198097 = r92198096 / r92198090;
        return r92198097;
}
↓
double f(double x, double y) {
        double r92198098 = x;
        double r92198099 = -1.8256037461802633e+117;
        bool r92198100 = r92198098 <= r92198099;
        double r92198101 = y;
        double r92198102 = -r92198101;
        double r92198103 = exp(r92198102);
        double r92198104 = r92198103 / r92198098;
        double r92198105 = 5.955847209664488;
        bool r92198106 = r92198098 <= r92198105;
        double r92198107 = exp(r92198098);
        double r92198108 = r92198098 + r92198101;
        double r92198109 = r92198098 / r92198108;
        double r92198110 = log(r92198109);
        double r92198111 = pow(r92198107, r92198110);
        double r92198112 = r92198111 / r92198098;
        double r92198113 = r92198106 ? r92198112 : r92198104;
        double r92198114 = r92198100 ? r92198104 : r92198113;
        return r92198114;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original11.4
Target7.8
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;y \lt -3.73118442066479561492798134439269393419 \cdot 10^{94}:\\ \;\;\;\;\frac{e^{\frac{-1}{y}}}{x}\\ \mathbf{elif}\;y \lt 28179592427282878868860376020282245120:\\ \;\;\;\;\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}\\ \mathbf{elif}\;y \lt 2.347387415166997963747840232163110922613 \cdot 10^{178}:\\ \;\;\;\;\log \left(e^{\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{-1}{y}}}{x}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -1.8256037461802633e+117 or 5.955847209664488 < x

    1. Initial program 11.7

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

    2. Taylor expanded around inf 0.0

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

    3. Simplified0.0

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

    if -1.8256037461802633e+117 < x < 5.955847209664488

    1. Initial program 11.1

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

    3. Applied add-log-exp19.2

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

    4. Applied exp-to-pow0.9

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.825603746180263343792126173792163470345 \cdot 10^{117}:\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{elif}\;x \le 5.955847209664487884595018840627744793892:\\ \;\;\;\;\frac{{\left(e^{x}\right)}^{\left(\log \left(\frac{x}{x + y}\right)\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-y}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019173 +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))