Average Error: 14.6 → 0.3
Time: 10.3s
Precision: binary64
\[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -0.18501999673040256 \lor \neg \left(x \leq 1.1474851571757078 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{1}{x \cdot e^{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]
\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -0.18501999673040256 \lor \neg \left(x \leq 1.1474851571757078 \cdot 10^{-7}\right):\\
\;\;\;\;\frac{1}{x \cdot e^{y}}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{x}\\


\end{array}

(FPCore (x y) :precision binary64 (/ (exp (* x (log (/ x (+ x y))))) x))
↓
(FPCore (x y)
 :precision binary64
 (if (or (<= x -0.18501999673040256) (not (<= x 1.1474851571757078e-7)))
   (/ 1.0 (* x (exp y)))
   (/ 1.0 x)))
double code(double x, double y) {
	return exp(x * log(x / (x + y))) / x;
}
↓
double code(double x, double y) {
	double tmp;
	if ((x <= -0.18501999673040256) || !(x <= 1.1474851571757078e-7)) {
		tmp = 1.0 / (x * exp(y));
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original14.6
Target14.0
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;y < -3.7311844206647956 \cdot 10^{+94}:\\ \;\;\;\;\frac{e^{\frac{-1}{y}}}{x}\\ \mathbf{elif}\;y < 2.817959242728288 \cdot 10^{+37}:\\ \;\;\;\;\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}\\ \mathbf{elif}\;y < 2.347387415166998 \cdot 10^{+178}:\\ \;\;\;\;\log \left(e^{\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{-1}{y}}}{x}\\ \end{array} \]

Derivation

  1. Split input into 2 regimes

  2. if x < -0.185019996730402564 or 1.14748515717570782e-7 < x

    1. Initial program 16.8

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

    2. Simplified16.8

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

    3. Taylor expanded in x around inf 0.2

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

    4. Applied neg-sub0_binary640.2

      \[\leadsto \frac{e^{\color{blue}{0 - y}}}{x} \]

    5. Applied exp-diff_binary640.2

      \[\leadsto \frac{\color{blue}{\frac{e^{0}}{e^{y}}}}{x} \]

    6. Applied associate-/l/_binary640.2

      \[\leadsto \color{blue}{\frac{e^{0}}{x \cdot e^{y}}} \]

    if -0.185019996730402564 < x < 1.14748515717570782e-7

    1. Initial program 11.4

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

    2. Simplified11.4

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

    3. Taylor expanded in x around 0 0.5

      \[\leadsto \frac{\color{blue}{1}}{x} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.18501999673040256 \lor \neg \left(x \leq 1.1474851571757078 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{1}{x \cdot e^{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

Reproduce

herbie shell --seed 2022088 
(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))