Average Error: 11.0 → 4.9
Time: 5.9s
Precision: binary64
\[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le 11.394391641767648 \lor \neg \left(y \le 6.542989163601907 \cdot 10^{137}\right):\\ \;\;\;\;\frac{{\left(e^{x}\right)}^{\left(\log \left(\frac{x}{x + y}\right)\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \log \left(\frac{1}{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}} \cdot \frac{x}{\sqrt[3]{x + y}}\right)}}{x}\\ \end{array}\]
\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x}
↓
\begin{array}{l}
\mathbf{if}\;y \le 11.394391641767648 \lor \neg \left(y \le 6.542989163601907 \cdot 10^{137}\right):\\
\;\;\;\;\frac{{\left(e^{x}\right)}^{\left(\log \left(\frac{x}{x + y}\right)\right)}}{x}\\

\mathbf{else}:\\
\;\;\;\;\frac{e^{x \cdot \log \left(\frac{1}{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}} \cdot \frac{x}{\sqrt[3]{x + y}}\right)}}{x}\\

\end{array}
double code(double x, double y) {
	return ((double) (((double) exp(((double) (x * ((double) log(((double) (x / ((double) (x + y)))))))))) / x));
}
↓
double code(double x, double y) {
	double VAR;
	if (((y <= 11.394391641767648) || !(y <= 6.542989163601907e+137))) {
		VAR = ((double) (((double) pow(((double) exp(x)), ((double) log(((double) (x / ((double) (x + y)))))))) / x));
	} else {
		VAR = ((double) (((double) exp(((double) (x * ((double) log(((double) (((double) (1.0 / ((double) (((double) cbrt(((double) (x + y)))) * ((double) cbrt(((double) (x + y)))))))) * ((double) (x / ((double) cbrt(((double) (x + y)))))))))))))) / x));
	}
	return VAR;
}

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
Target7.8
Herbie4.9
\[\begin{array}{l} \mathbf{if}\;y \lt -3.73118442066479561 \cdot 10^{94}:\\ \;\;\;\;\frac{e^{\frac{-1}{y}}}{x}\\ \mathbf{elif}\;y \lt 2.81795924272828789 \cdot 10^{37}:\\ \;\;\;\;\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}\\ \mathbf{elif}\;y \lt 2.347387415166998 \cdot 10^{178}:\\ \;\;\;\;\log \left(e^{\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{-1}{y}}}{x}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < 11.394391641767648 or 6.542989163601907e137 < y

    1. Initial program 7.9

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

    3. Applied add-log-exp35.1

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

    4. Applied exp-to-pow2.6

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

    if 11.394391641767648 < y < 6.542989163601907e137

    1. Initial program 36.1

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

    3. Applied add-cube-cbrt22.5

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

    4. Applied *-un-lft-identity22.5

      \[\leadsto \frac{e^{x \cdot \log \left(\frac{\color{blue}{1 \cdot x}}{\left(\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}\right) \cdot \sqrt[3]{x + y}}\right)}}{x}\]

    5. Applied times-frac23.2

      \[\leadsto \frac{e^{x \cdot \log \color{blue}{\left(\frac{1}{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}} \cdot \frac{x}{\sqrt[3]{x + y}}\right)}}}{x}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification4.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le 11.394391641767648 \lor \neg \left(y \le 6.542989163601907 \cdot 10^{137}\right):\\ \;\;\;\;\frac{{\left(e^{x}\right)}^{\left(\log \left(\frac{x}{x + y}\right)\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \log \left(\frac{1}{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}} \cdot \frac{x}{\sqrt[3]{x + y}}\right)}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020173 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, F"
  :precision binary64

  :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))