Average Error: 11.3 → 0.7
Time: 4.7s
Precision: binary64
\[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
↓
\[\begin{array}{l} t_0 := \sqrt{e^{-y}}\\ \mathbf{if}\;x \leq -4.4195987562608264 \cdot 10^{+73}:\\ \;\;\;\;t_0 \cdot \frac{t_0}{x}\\ \mathbf{elif}\;x \leq 0.6658307847861302:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot e^{y}}\\ \end{array} \]
\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x}
↓
\begin{array}{l}
t_0 := \sqrt{e^{-y}}\\
\mathbf{if}\;x \leq -4.4195987562608264 \cdot 10^{+73}:\\
\;\;\;\;t_0 \cdot \frac{t_0}{x}\\

\mathbf{elif}\;x \leq 0.6658307847861302:\\
\;\;\;\;\frac{1}{x}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{x \cdot e^{y}}\\


\end{array}

(FPCore (x y) :precision binary64 (/ (exp (* x (log (/ x (+ x y))))) x))
↓
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (sqrt (exp (- y)))))
   (if (<= x -4.4195987562608264e+73)
     (* t_0 (/ t_0 x))
     (if (<= x 0.6658307847861302) (/ 1.0 x) (/ 1.0 (* x (exp y)))))))
double code(double x, double y) {
	return exp((x * log((x / (x + y))))) / x;
}
↓
double code(double x, double y) {
	double t_0 = sqrt(exp(-y));
	double tmp;
	if (x <= -4.4195987562608264e+73) {
		tmp = t_0 * (t_0 / x);
	} else if (x <= 0.6658307847861302) {
		tmp = 1.0 / x;
	} else {
		tmp = 1.0 / (x * exp(y));
	}
	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

Original11.3
Target8.0
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;y < -3.7311844206647956 \cdot 10^{+94}:\\ \;\;\;\;\frac{e^{\frac{-1}{y}}}{x}\\ \mathbf{elif}\;y < 2.817959242728288 \cdot 10^{+37}:\\ \;\;\;\;\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}\\ \mathbf{elif}\;y < 2.347387415166998 \cdot 10^{+178}:\\ \;\;\;\;\log \left(e^{\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{-1}{y}}}{x}\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

  2. if x < -4.41959875626082636e73

    1. Initial program 13.7

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

    2. Simplified13.7

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

    3. Taylor expanded in x around inf 0.1

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

    4. Applied *-un-lft-identity_binary640.1

      \[\leadsto \frac{e^{-y}}{\color{blue}{1 \cdot x}} \]

    5. Applied add-sqr-sqrt_binary640.1

      \[\leadsto \frac{\color{blue}{\sqrt{e^{-y}} \cdot \sqrt{e^{-y}}}}{1 \cdot x} \]

    6. Applied times-frac_binary640.1

      \[\leadsto \color{blue}{\frac{\sqrt{e^{-y}}}{1} \cdot \frac{\sqrt{e^{-y}}}{x}} \]

    if -4.41959875626082636e73 < x < 0.66583078478613023

    1. Initial program 11.1

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

    2. Simplified11.1

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

    3. Taylor expanded in x around 0 1.2

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

    if 0.66583078478613023 < x

    1. Initial program 9.9

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

    2. Simplified9.9

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

    3. Taylor expanded in x around inf 0.1

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

    4. Applied clear-num_binary640.1

      \[\leadsto \color{blue}{\frac{1}{\frac{x}{e^{-y}}}} \]

    5. Simplified0.1

      \[\leadsto \frac{1}{\color{blue}{x \cdot e^{y}}} \]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.4195987562608264 \cdot 10^{+73}:\\ \;\;\;\;\sqrt{e^{-y}} \cdot \frac{\sqrt{e^{-y}}}{x}\\ \mathbf{elif}\;x \leq 0.6658307847861302:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot e^{y}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022125 
(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))