Average Error: 11.5 → 0.4
Time: 7.1s
Precision: binary64
\[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -1126.4152090125688 \lor \neg \left(x \leq 0.06506087734989285\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, \frac{y}{\frac{x \cdot e^{y}}{y}}, e^{-y}\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]
\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -1126.4152090125688 \lor \neg \left(x \leq 0.06506087734989285\right):\\
\;\;\;\;\frac{\mathsf{fma}\left(0.5, \frac{y}{\frac{x \cdot e^{y}}{y}}, e^{-y}\right)}{x}\\

\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 -1126.4152090125688) (not (<= x 0.06506087734989285)))
   (/ (fma 0.5 (/ y (/ (* x (exp y)) y)) (exp (- y))) x)
   (/ 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 <= -1126.4152090125688) || !(x <= 0.06506087734989285)) {
		tmp = fma(0.5, (y / ((x * exp(y)) / y)), exp(-y)) / x;
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Target

Original11.5
Target8.1
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;y < -3.7311844206647956 \cdot 10^{+94}:\\ \;\;\;\;\frac{e^{\frac{-1}{y}}}{x}\\ \mathbf{elif}\;y < 2.817959242728288 \cdot 10^{+37}:\\ \;\;\;\;\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}\\ \mathbf{elif}\;y < 2.347387415166998 \cdot 10^{+178}:\\ \;\;\;\;\log \left(e^{\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{-1}{y}}}{x}\\ \end{array} \]

Derivation

  1. Split input into 2 regimes

  2. if x < -1126.41520901256877 or 0.065060877349892854 < x

    1. Initial program 11.2

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

    2. Simplified11.2

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

    3. Taylor expanded in x around inf 7.5

      \[\leadsto \frac{\color{blue}{0.5 \cdot \frac{{y}^{2} \cdot e^{-y}}{x} + e^{-y}}}{x} \]

    4. Simplified0.1

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(0.5, \frac{y}{\frac{x \cdot e^{y}}{y}}, e^{-y}\right)}}{x} \]

    if -1126.41520901256877 < x < 0.065060877349892854

    1. Initial program 12.0

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

    2. Simplified12.0

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

    3. Taylor expanded in x around 0 0.7

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1126.4152090125688 \lor \neg \left(x \leq 0.06506087734989285\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, \frac{y}{\frac{x \cdot e^{y}}{y}}, e^{-y}\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

Reproduce

herbie shell --seed 2021340 
(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))