Average Error: 11.5 → 5.8
Time: 5.2s
Precision: binary64
\[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \leq -2.7506651735342347 \cdot 10^{+132}:\\ \;\;\;\;0.5 \cdot \left(\frac{{\left(-x\right)}^{x} \cdot {\left(\frac{-1}{y}\right)}^{x}}{y \cdot y} \cdot \left(x \cdot x + {x}^{3}\right)\right) + \left(\frac{{\left(-x\right)}^{x} \cdot {\left(\frac{-1}{y}\right)}^{x}}{x} - x \cdot \frac{{\left(-x\right)}^{x} \cdot {\left(\frac{-1}{y}\right)}^{x}}{y}\right)\\ \mathbf{elif}\;y \leq 7.745671602444141 \cdot 10^{+18}:\\ \;\;\;\;\frac{1}{\frac{x}{{\left(\frac{x}{y + x}\right)}^{x}}}\\ \mathbf{elif}\;y \leq 1.2585632689113307 \cdot 10^{+119}:\\ \;\;\;\;\log \left(e^{\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\sqrt{x}}{\sqrt{y + x}}\right)}^{x} \cdot {\left(\frac{\sqrt{x}}{\sqrt{y + x}}\right)}^{x}}{x}\\ \end{array}\]
\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x}
↓
\begin{array}{l}
\mathbf{if}\;y \leq -2.7506651735342347 \cdot 10^{+132}:\\
\;\;\;\;0.5 \cdot \left(\frac{{\left(-x\right)}^{x} \cdot {\left(\frac{-1}{y}\right)}^{x}}{y \cdot y} \cdot \left(x \cdot x + {x}^{3}\right)\right) + \left(\frac{{\left(-x\right)}^{x} \cdot {\left(\frac{-1}{y}\right)}^{x}}{x} - x \cdot \frac{{\left(-x\right)}^{x} \cdot {\left(\frac{-1}{y}\right)}^{x}}{y}\right)\\

\mathbf{elif}\;y \leq 7.745671602444141 \cdot 10^{+18}:\\
\;\;\;\;\frac{1}{\frac{x}{{\left(\frac{x}{y + x}\right)}^{x}}}\\

\mathbf{elif}\;y \leq 1.2585632689113307 \cdot 10^{+119}:\\
\;\;\;\;\log \left(e^{\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}}\right)\\

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

\end{array}
double code(double x, double y) {
	return (((double) exp(((double) (x * ((double) log((x / ((double) (x + y))))))))) / x);
}
↓
double code(double x, double y) {
	double VAR;
	if ((y <= -2.7506651735342347e+132)) {
		VAR = ((double) (((double) (0.5 * ((double) ((((double) (((double) pow(((double) -(x)), x)) * ((double) pow((-1.0 / y), x)))) / ((double) (y * y))) * ((double) (((double) (x * x)) + ((double) pow(x, 3.0)))))))) + ((double) ((((double) (((double) pow(((double) -(x)), x)) * ((double) pow((-1.0 / y), x)))) / x) - ((double) (x * (((double) (((double) pow(((double) -(x)), x)) * ((double) pow((-1.0 / y), x)))) / y)))))));
	} else {
		double VAR_1;
		if ((y <= 7.745671602444141e+18)) {
			VAR_1 = (1.0 / (x / ((double) pow((x / ((double) (y + x))), x))));
		} else {
			double VAR_2;
			if ((y <= 1.2585632689113307e+119)) {
				VAR_2 = ((double) log(((double) exp((((double) pow((x / ((double) (y + x))), x)) / x)))));
			} else {
				VAR_2 = (((double) (((double) pow((((double) sqrt(x)) / ((double) sqrt(((double) (y + x))))), x)) * ((double) pow((((double) sqrt(x)) / ((double) sqrt(((double) (y + x))))), x)))) / x);
			}
			VAR_1 = VAR_2;
		}
		VAR = VAR_1;
	}
	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

Original11.5
Target7.9
Herbie5.8
\[\begin{array}{l} \mathbf{if}\;y < -3.7311844206647956 \cdot 10^{+94}:\\ \;\;\;\;\frac{e^{\frac{-1}{y}}}{x}\\ \mathbf{elif}\;y < 2.817959242728288 \cdot 10^{+37}:\\ \;\;\;\;\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}\\ \mathbf{elif}\;y < 2.347387415166998 \cdot 10^{+178}:\\ \;\;\;\;\log \left(e^{\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{-1}{y}}}{x}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if y < -2.7506651735342347e132

    1. Initial program Error: 44.3 bits

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

    2. SimplifiedError: 44.3 bits

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

    3. Taylor expanded around -inf Error: 0.1 bits

      \[\leadsto \color{blue}{\left(0.5 \cdot \frac{e^{x \cdot \left(\log \left(-1 \cdot x\right) + \log \left(\frac{-1}{y}\right)\right)} \cdot {x}^{2}}{{y}^{2}} + \left(0.5 \cdot \frac{e^{x \cdot \left(\log \left(-1 \cdot x\right) + \log \left(\frac{-1}{y}\right)\right)} \cdot {x}^{3}}{{y}^{2}} + \frac{e^{x \cdot \left(\log \left(-1 \cdot x\right) + \log \left(\frac{-1}{y}\right)\right)}}{x}\right)\right) - \frac{e^{x \cdot \left(\log \left(-1 \cdot x\right) + \log \left(\frac{-1}{y}\right)\right)} \cdot x}{y}}\]

    4. SimplifiedError: 0.0 bits

      \[\leadsto \color{blue}{0.5 \cdot \left(\frac{{\left(-x\right)}^{x} \cdot {\left(\frac{-1}{y}\right)}^{x}}{y \cdot y} \cdot \left(x \cdot x + {x}^{3}\right)\right) + \left(\frac{{\left(-x\right)}^{x} \cdot {\left(\frac{-1}{y}\right)}^{x}}{x} - \frac{{\left(-x\right)}^{x} \cdot {\left(\frac{-1}{y}\right)}^{x}}{y} \cdot x\right)}\]

    if -2.7506651735342347e132 < y < 7745671602444141000

    1. Initial program Error: 2.4 bits

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

    2. SimplifiedError: 2.4 bits

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

    4. Applied clear-numError: 2.4 bits

      \[\leadsto \color{blue}{\frac{1}{\frac{x}{{\left(\frac{x}{x + y}\right)}^{x}}}}\]

    if 7745671602444141000 < y < 1.25856326891133072e119

    1. Initial program Error: 35.9 bits

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

    2. SimplifiedError: 35.9 bits

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

    4. Applied add-log-expError: 20.9 bits

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

    if 1.25856326891133072e119 < y

    1. Initial program Error: 31.3 bits

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

    2. SimplifiedError: 31.3 bits

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

    4. Applied add-sqr-sqrtError: 31.4 bits

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

    5. Applied add-sqr-sqrtError: 33.1 bits

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

    6. Applied times-fracError: 33.1 bits

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

    7. Applied unpow-prod-downError: 16.0 bits

      \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt{x}}{\sqrt{x + y}}\right)}^{x} \cdot {\left(\frac{\sqrt{x}}{\sqrt{x + y}}\right)}^{x}}}{x}\]
  3. Recombined 4 regimes into one program.

  4. Final simplificationError: 5.8 bits

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.7506651735342347 \cdot 10^{+132}:\\ \;\;\;\;0.5 \cdot \left(\frac{{\left(-x\right)}^{x} \cdot {\left(\frac{-1}{y}\right)}^{x}}{y \cdot y} \cdot \left(x \cdot x + {x}^{3}\right)\right) + \left(\frac{{\left(-x\right)}^{x} \cdot {\left(\frac{-1}{y}\right)}^{x}}{x} - x \cdot \frac{{\left(-x\right)}^{x} \cdot {\left(\frac{-1}{y}\right)}^{x}}{y}\right)\\ \mathbf{elif}\;y \leq 7.745671602444141 \cdot 10^{+18}:\\ \;\;\;\;\frac{1}{\frac{x}{{\left(\frac{x}{y + x}\right)}^{x}}}\\ \mathbf{elif}\;y \leq 1.2585632689113307 \cdot 10^{+119}:\\ \;\;\;\;\log \left(e^{\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\sqrt{x}}{\sqrt{y + x}}\right)}^{x} \cdot {\left(\frac{\sqrt{x}}{\sqrt{y + x}}\right)}^{x}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020200 
(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))