Average Error: 11.1 → 1.3
Time: 17.7s
Precision: 64
\[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.782408069109436113491121240500612211051 \cdot 10^{86} \lor \neg \left(x \le 1719139.2202338078059256076812744140625\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{x} \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{y + x}}}{\sqrt[3]{y + x}}\right)}^{x} \cdot {\left(\frac{\sqrt[3]{x}}{\sqrt[3]{y + x}}\right)}^{x}}{x}\\ \end{array}\]
\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.782408069109436113491121240500612211051 \cdot 10^{86} \lor \neg \left(x \le 1719139.2202338078059256076812744140625\right):\\
\;\;\;\;\frac{e^{-y}}{x}\\

\mathbf{else}:\\
\;\;\;\;\frac{{\left(\frac{\sqrt[3]{x} \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{y + x}}}{\sqrt[3]{y + x}}\right)}^{x} \cdot {\left(\frac{\sqrt[3]{x}}{\sqrt[3]{y + x}}\right)}^{x}}{x}\\

\end{array}
double f(double x, double y) {
        double r280578 = x;
        double r280579 = y;
        double r280580 = r280578 + r280579;
        double r280581 = r280578 / r280580;
        double r280582 = log(r280581);
        double r280583 = r280578 * r280582;
        double r280584 = exp(r280583);
        double r280585 = r280584 / r280578;
        return r280585;
}
↓
double f(double x, double y) {
        double r280586 = x;
        double r280587 = -1.782408069109436e+86;
        bool r280588 = r280586 <= r280587;
        double r280589 = 1719139.2202338078;
        bool r280590 = r280586 <= r280589;
        double r280591 = !r280590;
        bool r280592 = r280588 || r280591;
        double r280593 = y;
        double r280594 = -r280593;
        double r280595 = exp(r280594);
        double r280596 = r280595 / r280586;
        double r280597 = cbrt(r280586);
        double r280598 = r280593 + r280586;
        double r280599 = cbrt(r280598);
        double r280600 = r280597 / r280599;
        double r280601 = r280597 * r280600;
        double r280602 = r280601 / r280599;
        double r280603 = pow(r280602, r280586);
        double r280604 = pow(r280600, r280586);
        double r280605 = r280603 * r280604;
        double r280606 = r280605 / r280586;
        double r280607 = r280592 ? r280596 : r280606;
        return r280607;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original11.1
Target7.9
Herbie1.3
\[\begin{array}{l} \mathbf{if}\;y \lt -3.73118442066479561492798134439269393419 \cdot 10^{94}:\\ \;\;\;\;\frac{e^{\frac{-1}{y}}}{x}\\ \mathbf{elif}\;y \lt 28179592427282878868860376020282245120:\\ \;\;\;\;\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}\\ \mathbf{elif}\;y \lt 2.347387415166997963747840232163110922613 \cdot 10^{178}:\\ \;\;\;\;\log \left(e^{\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{-1}{y}}}{x}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -1.782408069109436e+86 or 1719139.2202338078 < x

    1. Initial program 11.4

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

    2. Simplified11.4

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

    3. Taylor expanded around inf 0.0

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

    if -1.782408069109436e+86 < x < 1719139.2202338078

    1. Initial program 10.8

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

    2. Simplified10.8

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

    4. Applied add-cube-cbrt15.3

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

    5. Applied add-cube-cbrt10.8

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

    6. Applied times-frac10.8

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

    7. Applied unpow-prod-down2.4

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

    8. Simplified2.4

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

    9. Simplified2.4

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

  4. Final simplification1.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.782408069109436113491121240500612211051 \cdot 10^{86} \lor \neg \left(x \le 1719139.2202338078059256076812744140625\right):\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{x} \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{y + x}}}{\sqrt[3]{y + x}}\right)}^{x} \cdot {\left(\frac{\sqrt[3]{x}}{\sqrt[3]{y + x}}\right)}^{x}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 +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))