Average Error: 11.7 → 0.1
Time: 5.7s
Precision: binary64
\[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -12.735787451030035:\\ \;\;\;\;e^{-y} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq 0.28438769866631414:\\ \;\;\;\;\frac{e^{x \cdot \left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right) + 2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;e^{-y} \cdot \frac{1}{x}\\ \end{array}\]
\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -12.735787451030035:\\
\;\;\;\;e^{-y} \cdot \frac{1}{x}\\

\mathbf{elif}\;x \leq 0.28438769866631414:\\
\;\;\;\;\frac{e^{x \cdot \left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right) + 2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right)}}{x}\\

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

\end{array}
(FPCore (x y) :precision binary64 (/ (exp (* x (log (/ x (+ x y))))) x))
↓
(FPCore (x y)
 :precision binary64
 (if (<= x -12.735787451030035)
   (* (exp (- y)) (/ 1.0 x))
   (if (<= x 0.28438769866631414)
     (/
      (exp
       (*
        x
        (+
         (log (/ (cbrt x) (cbrt (+ x y))))
         (* 2.0 (log (/ (cbrt x) (cbrt (+ x y))))))))
      x)
     (* (exp (- y)) (/ 1.0 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 <= -12.735787451030035) {
		tmp = exp(-y) * (1.0 / x);
	} else if (x <= 0.28438769866631414) {
		tmp = exp(x * (log(cbrt(x) / cbrt(x + y)) + (2.0 * log(cbrt(x) / cbrt(x + y))))) / x;
	} else {
		tmp = exp(-y) * (1.0 / 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

Original11.7
Target8.2
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;y < -3.7311844206647956 \cdot 10^{+94}:\\ \;\;\;\;\frac{e^{\frac{-1}{y}}}{x}\\ \mathbf{elif}\;y < 2.817959242728288 \cdot 10^{+37}:\\ \;\;\;\;\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}\\ \mathbf{elif}\;y < 2.347387415166998 \cdot 10^{+178}:\\ \;\;\;\;\log \left(e^{\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{-1}{y}}}{x}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -12.73578745103003 or 0.284387698666314137 < x

    1. Initial program 11.2

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

    2. Taylor expanded around inf 0.1

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

    3. Simplified0.1

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

    5. Applied div-inv_binary640.1

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

    if -12.73578745103003 < x < 0.284387698666314137

    1. Initial program 12.4

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x}\]
    2. Using strategy rm

    3. Applied add-cube-cbrt_binary6412.4

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

    4. Applied add-cube-cbrt_binary6412.4

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

    5. Applied times-frac_binary6412.4

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

    6. Applied log-prod_binary642.7

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

    7. Simplified0.0

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -12.735787451030035:\\ \;\;\;\;e^{-y} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq 0.28438769866631414:\\ \;\;\;\;\frac{e^{x \cdot \left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right) + 2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;e^{-y} \cdot \frac{1}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020273 
(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))