Average Error: 11.1 → 1.7
Time: 3.8s
Precision: binary64
\[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -5.660849715351774 \lor \neg \left(x \leq 9.896618786202479 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{1}{x \cdot e^{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{x} \cdot {\left(\frac{\sqrt[3]{x}}{x + y}\right)}^{x}}{x}\\ \end{array}\]
\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -5.660849715351774 \lor \neg \left(x \leq 9.896618786202479 \cdot 10^{-11}\right):\\
\;\;\;\;\frac{1}{x \cdot e^{y}}\\

\mathbf{else}:\\
\;\;\;\;\frac{{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{x} \cdot {\left(\frac{\sqrt[3]{x}}{x + y}\right)}^{x}}{x}\\

\end{array}
(FPCore (x y) :precision binary64 (/ (exp (* x (log (/ x (+ x y))))) x))
↓
(FPCore (x y)
 :precision binary64
 (if (or (<= x -5.660849715351774) (not (<= x 9.896618786202479e-11)))
   (/ 1.0 (* x (exp y)))
   (/ (* (pow (* (cbrt x) (cbrt x)) x) (pow (/ (cbrt x) (+ x y)) x)) 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 <= -5.660849715351774) || !(x <= 9.896618786202479e-11)) {
		tmp = 1.0 / (x * exp(y));
	} else {
		tmp = (pow((cbrt(x) * cbrt(x)), x) * pow((cbrt(x) / (x + y)), x)) / 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

Original11.1
Target7.9
Herbie1.7
\[\begin{array}{l} \mathbf{if}\;y < -3.7311844206647956 \cdot 10^{+94}:\\ \;\;\;\;\frac{e^{\frac{-1}{y}}}{x}\\ \mathbf{elif}\;y < 2.817959242728288 \cdot 10^{+37}:\\ \;\;\;\;\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}\\ \mathbf{elif}\;y < 2.347387415166998 \cdot 10^{+178}:\\ \;\;\;\;\log \left(e^{\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{-1}{y}}}{x}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -5.66084971535177406 or 9.8966187862024792e-11 < x

    1. Initial program 10.3

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

    2. Simplified10.3

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

    3. Taylor expanded around inf 0.4

      \[\leadsto \frac{\color{blue}{e^{-y}}}{x}\]
    4. Using strategy rm

    5. Applied clear-num_binary640.4

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

    6. Simplified0.4

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

    if -5.66084971535177406 < x < 9.8966187862024792e-11

    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. Using strategy rm

    4. Applied *-un-lft-identity_binary6412.0

      \[\leadsto \frac{{\left(\frac{x}{\color{blue}{1 \cdot \left(x + y\right)}}\right)}^{x}}{x}\]

    5. Applied add-cube-cbrt_binary6412.0

      \[\leadsto \frac{{\left(\frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}{1 \cdot \left(x + y\right)}\right)}^{x}}{x}\]

    6. Applied times-frac_binary6412.0

      \[\leadsto \frac{{\color{blue}{\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{1} \cdot \frac{\sqrt[3]{x}}{x + y}\right)}}^{x}}{x}\]

    7. Applied unpow-prod-down_binary643.1

      \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{1}\right)}^{x} \cdot {\left(\frac{\sqrt[3]{x}}{x + y}\right)}^{x}}}{x}\]

    8. Simplified3.1

      \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{x}} \cdot {\left(\frac{\sqrt[3]{x}}{x + y}\right)}^{x}}{x}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.660849715351774 \lor \neg \left(x \leq 9.896618786202479 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{1}{x \cdot e^{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{x} \cdot {\left(\frac{\sqrt[3]{x}}{x + y}\right)}^{x}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020224 
(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))