Average Error: 11.5 → 0.8
Time: 5.8s
Precision: binary64
\[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -9229002597363212:\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{elif}\;x \leq 3.700480968675798 \cdot 10^{-25}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{e^{y}}}{-x}\\ \end{array} \]
\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -9229002597363212:\\
\;\;\;\;\frac{e^{-y}}{x}\\

\mathbf{elif}\;x \leq 3.700480968675798 \cdot 10^{-25}:\\
\;\;\;\;\frac{1}{x}\\

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


\end{array}

(FPCore (x y) :precision binary64 (/ (exp (* x (log (/ x (+ x y))))) x))
↓
(FPCore (x y)
 :precision binary64
 (if (<= x -9229002597363212.0)
   (/ (exp (- y)) x)
   (if (<= x 3.700480968675798e-25) (/ 1.0 x) (/ (/ -1.0 (exp y)) (- 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 <= -9229002597363212.0) {
		tmp = exp(-y) / x;
	} else if (x <= 3.700480968675798e-25) {
		tmp = 1.0 / x;
	} else {
		tmp = (-1.0 / exp(y)) / -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

Original11.5
Target7.5
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;y < -3.7311844206647956 \cdot 10^{+94}:\\ \;\;\;\;\frac{e^{\frac{-1}{y}}}{x}\\ \mathbf{elif}\;y < 2.817959242728288 \cdot 10^{+37}:\\ \;\;\;\;\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}\\ \mathbf{elif}\;y < 2.347387415166998 \cdot 10^{+178}:\\ \;\;\;\;\log \left(e^{\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{-1}{y}}}{x}\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

  2. if x < -9229002597363212

    1. Initial program 12.2

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

    2. Simplified12.2

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

    3. Taylor expanded in x around inf 0.1

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

    if -9229002597363212 < x < 3.70048096867579791e-25

    1. Initial program 12.6

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

    2. Simplified12.6

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

    3. Taylor expanded in x around 0 0.4

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

    if 3.70048096867579791e-25 < x

    1. Initial program 9.3

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

    2. Simplified9.3

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

    3. Taylor expanded in x around inf 1.7

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

    4. Applied frac-2neg_binary641.7

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

    5. Simplified1.7

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9229002597363212:\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{elif}\;x \leq 3.700480968675798 \cdot 10^{-25}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{e^{y}}}{-x}\\ \end{array} \]

Reproduce

herbie shell --seed 2022081 
(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))