Average Error: 10.9 → 0.4
Time: 15.8s
Precision: 64
\[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -40676008707595812134875960694538240:\\ \;\;\;\;e^{-y} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \le 0.2043583175286355080313427379223867319524:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;e^{-y} \cdot \frac{1}{x}\\ \end{array}\]
\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -40676008707595812134875960694538240:\\
\;\;\;\;e^{-y} \cdot \frac{1}{x}\\

\mathbf{elif}\;x \le 0.2043583175286355080313427379223867319524:\\
\;\;\;\;\frac{1}{x}\\

\mathbf{else}:\\
\;\;\;\;e^{-y} \cdot \frac{1}{x}\\

\end{array}
double f(double x, double y) {
        double r15910381 = x;
        double r15910382 = y;
        double r15910383 = r15910381 + r15910382;
        double r15910384 = r15910381 / r15910383;
        double r15910385 = log(r15910384);
        double r15910386 = r15910381 * r15910385;
        double r15910387 = exp(r15910386);
        double r15910388 = r15910387 / r15910381;
        return r15910388;
}
↓
double f(double x, double y) {
        double r15910389 = x;
        double r15910390 = -4.067600870759581e+34;
        bool r15910391 = r15910389 <= r15910390;
        double r15910392 = y;
        double r15910393 = -r15910392;
        double r15910394 = exp(r15910393);
        double r15910395 = 1.0;
        double r15910396 = r15910395 / r15910389;
        double r15910397 = r15910394 * r15910396;
        double r15910398 = 0.2043583175286355;
        bool r15910399 = r15910389 <= r15910398;
        double r15910400 = r15910399 ? r15910396 : r15910397;
        double r15910401 = r15910391 ? r15910397 : r15910400;
        return r15910401;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original10.9
Target7.6
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;y \lt -3.73118442066479561492798134439269393419 \cdot 10^{94}:\\ \;\;\;\;\frac{e^{\frac{-1}{y}}}{x}\\ \mathbf{elif}\;y \lt 28179592427282878868860376020282245120:\\ \;\;\;\;\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}\\ \mathbf{elif}\;y \lt 2.347387415166997963747840232163110922613 \cdot 10^{178}:\\ \;\;\;\;\log \left(e^{\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{-1}{y}}}{x}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -4.067600870759581e+34 or 0.2043583175286355 < x

    1. Initial program 11.2

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

    2. Taylor expanded around inf 0.0

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

    3. Simplified0.0

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

    5. Applied div-inv0.0

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

    if -4.067600870759581e+34 < x < 0.2043583175286355

    1. Initial program 10.6

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

    2. Taylor expanded around inf 0.8

      \[\leadsto \frac{e^{\color{blue}{0}}}{x}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -40676008707595812134875960694538240:\\ \;\;\;\;e^{-y} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \le 0.2043583175286355080313427379223867319524:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;e^{-y} \cdot \frac{1}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 +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))