Average Error: 11.6 → 0.2
Time: 25.2s
Precision: 64
\[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -338166549708266081878016:\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{elif}\;x \le 5.105764663642184685065153471050791722519 \cdot 10^{-5}:\\ \;\;\;\;\frac{e^{\left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right) \cdot x}}{\frac{x}{e^{\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right) \cdot x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-y}}{x}\\ \end{array}\]
\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -338166549708266081878016:\\
\;\;\;\;\frac{e^{-y}}{x}\\

\mathbf{elif}\;x \le 5.105764663642184685065153471050791722519 \cdot 10^{-5}:\\
\;\;\;\;\frac{e^{\left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right) \cdot x}}{\frac{x}{e^{\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right) \cdot x}}}\\

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

\end{array}
double f(double x, double y) {
        double r28608392 = x;
        double r28608393 = y;
        double r28608394 = r28608392 + r28608393;
        double r28608395 = r28608392 / r28608394;
        double r28608396 = log(r28608395);
        double r28608397 = r28608392 * r28608396;
        double r28608398 = exp(r28608397);
        double r28608399 = r28608398 / r28608392;
        return r28608399;
}
↓
double f(double x, double y) {
        double r28608400 = x;
        double r28608401 = -3.381665497082661e+23;
        bool r28608402 = r28608400 <= r28608401;
        double r28608403 = y;
        double r28608404 = -r28608403;
        double r28608405 = exp(r28608404);
        double r28608406 = r28608405 / r28608400;
        double r28608407 = 5.105764663642185e-05;
        bool r28608408 = r28608400 <= r28608407;
        double r28608409 = cbrt(r28608400);
        double r28608410 = r28608400 + r28608403;
        double r28608411 = cbrt(r28608410);
        double r28608412 = r28608409 / r28608411;
        double r28608413 = log(r28608412);
        double r28608414 = r28608413 + r28608413;
        double r28608415 = r28608414 * r28608400;
        double r28608416 = exp(r28608415);
        double r28608417 = r28608413 * r28608400;
        double r28608418 = exp(r28608417);
        double r28608419 = r28608400 / r28608418;
        double r28608420 = r28608416 / r28608419;
        double r28608421 = r28608408 ? r28608420 : r28608406;
        double r28608422 = r28608402 ? r28608406 : r28608421;
        return r28608422;
}

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
Target8.2
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;y \lt -3.73118442066479561492798134439269393419 \cdot 10^{94}:\\ \;\;\;\;\frac{e^{\frac{-1}{y}}}{x}\\ \mathbf{elif}\;y \lt 28179592427282878868860376020282245120:\\ \;\;\;\;\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}\\ \mathbf{elif}\;y \lt 2.347387415166997963747840232163110922613 \cdot 10^{178}:\\ \;\;\;\;\log \left(e^{\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{-1}{y}}}{x}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -3.381665497082661e+23 or 5.105764663642185e-05 < x

    1. Initial program 11.6

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

    2. Taylor expanded around inf 0.3

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

    3. Simplified0.3

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

    if -3.381665497082661e+23 < x < 5.105764663642185e-05

    1. Initial program 11.6

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

    3. Applied add-cube-cbrt12.4

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

    4. Applied add-cube-cbrt11.6

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

    5. Applied times-frac11.6

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

    6. Applied log-prod2.3

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

    7. Applied distribute-rgt-in2.3

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

    8. Applied exp-sum2.3

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

    9. Applied associate-/l*2.3

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

    11. Applied times-frac2.3

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

    12. Applied log-prod0.1

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -338166549708266081878016:\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{elif}\;x \le 5.105764663642184685065153471050791722519 \cdot 10^{-5}:\\ \;\;\;\;\frac{e^{\left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right) \cdot x}}{\frac{x}{e^{\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right) \cdot x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-y}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(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))