Average Error: 14.5 → 0.3
Time: 6.9s
Precision: binary64
\[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
↓
\[\begin{array}{l} t_0 := e^{-y}\\ \mathbf{if}\;x \leq -1.2194538249148998:\\ \;\;\;\;\frac{1}{\frac{x}{t_0}}\\ \mathbf{elif}\;x \leq 3.137771231155085 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, \frac{y}{\frac{x \cdot e^{y}}{y}}, t_0\right)}{x}\\ \end{array} \]
\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x}
↓
\begin{array}{l}
t_0 := e^{-y}\\
\mathbf{if}\;x \leq -1.2194538249148998:\\
\;\;\;\;\frac{1}{\frac{x}{t_0}}\\

\mathbf{elif}\;x \leq 3.137771231155085 \cdot 10^{-5}:\\
\;\;\;\;\frac{1}{x}\\

\mathbf{else}:\\
\;\;\;\;\frac{\mathsf{fma}\left(0.5, \frac{y}{\frac{x \cdot e^{y}}{y}}, t_0\right)}{x}\\


\end{array}

(FPCore (x y) :precision binary64 (/ (exp (* x (log (/ x (+ x y))))) x))
↓
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (exp (- y))))
   (if (<= x -1.2194538249148998)
     (/ 1.0 (/ x t_0))
     (if (<= x 3.137771231155085e-5)
       (/ 1.0 x)
       (/ (fma 0.5 (/ y (/ (* x (exp y)) y)) t_0) x)))))
double code(double x, double y) {
	return exp(x * log(x / (x + y))) / x;
}
↓
double code(double x, double y) {
	double t_0 = exp(-y);
	double tmp;
	if (x <= -1.2194538249148998) {
		tmp = 1.0 / (x / t_0);
	} else if (x <= 3.137771231155085e-5) {
		tmp = 1.0 / x;
	} else {
		tmp = fma(0.5, (y / ((x * exp(y)) / y)), t_0) / x;
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Target

Original14.5
Target14.2
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 3 regimes

  2. if x < -1.21945382491489984

    1. Initial program 15.9

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

    2. Simplified15.9

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

    3. Taylor expanded in x around inf 0.1

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

    4. Applied neg-sub0_binary640.1

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

    5. Applied exp-diff_binary640.1

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

    6. Applied associate-/l/_binary640.1

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

    7. Simplified0.1

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

    if -1.21945382491489984 < x < 3.13777123115508498e-5

    1. Initial program 11.5

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

    2. Simplified11.5

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

    3. Taylor expanded in x around 0 0.5

      \[\leadsto \frac{\color{blue}{1}}{x} \]

    if 3.13777123115508498e-5 < x

    1. Initial program 17.2

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

    2. Simplified17.2

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

    3. Taylor expanded in x around inf 9.4

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

    4. Simplified0.2

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(0.5, \frac{y}{\frac{x \cdot e^{y}}{y}}, e^{-y}\right)}}{x} \]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.2194538249148998:\\ \;\;\;\;\frac{1}{\frac{x}{e^{-y}}}\\ \mathbf{elif}\;x \leq 3.137771231155085 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, \frac{y}{\frac{x \cdot e^{y}}{y}}, e^{-y}\right)}{x}\\ \end{array} \]

Reproduce

herbie shell --seed 2022068 
(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))