Average Error: 11.1 → 1.5
Time: 6.1s
Precision: binary64
\[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]
↓
\[\begin{array}{l} t_0 := \frac{e^{-y}}{x}\\ \mathbf{if}\;x \leq -18.571610858935497:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 4.426045253680128:\\ \;\;\;\;\frac{{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{x}}{\frac{x}{{\left(\frac{\sqrt[3]{x}}{x + y}\right)}^{x}}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x}
↓
\begin{array}{l}
t_0 := \frac{e^{-y}}{x}\\
\mathbf{if}\;x \leq -18.571610858935497:\\
\;\;\;\;t_0\\

\mathbf{elif}\;x \leq 4.426045253680128:\\
\;\;\;\;\frac{{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{x}}{\frac{x}{{\left(\frac{\sqrt[3]{x}}{x + y}\right)}^{x}}}\\

\mathbf{else}:\\
\;\;\;\;t_0\\


\end{array}

(FPCore (x y) :precision binary64 (/ (exp (* x (log (/ x (+ x y))))) x))
↓
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (exp (- y)) x)))
   (if (<= x -18.571610858935497)
     t_0
     (if (<= x 4.426045253680128)
       (/ (pow (* (cbrt x) (cbrt x)) x) (/ x (pow (/ (cbrt x) (+ x y)) x)))
       t_0))))
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) / x;
	double tmp;
	if (x <= -18.571610858935497) {
		tmp = t_0;
	} else if (x <= 4.426045253680128) {
		tmp = pow((cbrt(x) * cbrt(x)), x) / (x / pow((cbrt(x) / (x + y)), x));
	} else {
		tmp = t_0;
	}
	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
Target8.0
Herbie1.5
\[\begin{array}{l} \mathbf{if}\;y < -3.7311844206647956 \cdot 10^{+94}:\\ \;\;\;\;\frac{e^{\frac{-1}{y}}}{x}\\ \mathbf{elif}\;y < 2.817959242728288 \cdot 10^{+37}:\\ \;\;\;\;\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}\\ \mathbf{elif}\;y < 2.347387415166998 \cdot 10^{+178}:\\ \;\;\;\;\log \left(e^{\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{-1}{y}}}{x}\\ \end{array} \]

Derivation

  1. Split input into 2 regimes

  2. if x < -18.5716108589354967 or 4.4260452536801278 < x

    1. Initial program 10.8

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

    2. Simplified10.8

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

    3. Taylor expanded in x around inf 0.0

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

    if -18.5716108589354967 < x < 4.4260452536801278

    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. Applied *-un-lft-identity_binary6411.5

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

    4. Applied add-cube-cbrt_binary6411.5

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

    5. Applied times-frac_binary6411.5

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

    6. Applied unpow-prod-down_binary643.2

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

    7. Applied associate-/l*_binary643.2

      \[\leadsto \color{blue}{\frac{{\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{1}\right)}^{x}}{\frac{x}{{\left(\frac{\sqrt[3]{x}}{x + y}\right)}^{x}}}} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -18.571610858935497:\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{elif}\;x \leq 4.426045253680128:\\ \;\;\;\;\frac{{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{x}}{\frac{x}{{\left(\frac{\sqrt[3]{x}}{x + y}\right)}^{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-y}}{x}\\ \end{array} \]

Reproduce

herbie shell --seed 2022117 
(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))