Average Error: 9.2 → 0.2
Time: 3.6s
Precision: binary64
\[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -4.34426230244491986 \cdot 10^{23} \lor \neg \left(x \le 2676405038669891\right):\\ \;\;\;\;\frac{x}{y} + \left(1 - \frac{1}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(\sqrt[3]{\frac{x}{y} + 1} \cdot \sqrt[3]{\frac{x}{y} + 1}\right)\right) \cdot \frac{\sqrt[3]{\frac{x}{y} + 1}}{x + 1}\\ \end{array}\]
\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -4.34426230244491986 \cdot 10^{23} \lor \neg \left(x \le 2676405038669891\right):\\
\;\;\;\;\frac{x}{y} + \left(1 - \frac{1}{x}\right)\\

\mathbf{else}:\\
\;\;\;\;\left(x \cdot \left(\sqrt[3]{\frac{x}{y} + 1} \cdot \sqrt[3]{\frac{x}{y} + 1}\right)\right) \cdot \frac{\sqrt[3]{\frac{x}{y} + 1}}{x + 1}\\

\end{array}
double code(double x, double y) {
	return ((double) (((double) (x * ((double) (((double) (x / y)) + 1.0)))) / ((double) (x + 1.0))));
}
↓
double code(double x, double y) {
	double VAR;
	if (((x <= -4.34426230244492e+23) || !(x <= 2676405038669891.0))) {
		VAR = ((double) (((double) (x / y)) + ((double) (1.0 - ((double) (1.0 / x))))));
	} else {
		VAR = ((double) (((double) (x * ((double) (((double) cbrt(((double) (((double) (x / y)) + 1.0)))) * ((double) cbrt(((double) (((double) (x / y)) + 1.0)))))))) * ((double) (((double) cbrt(((double) (((double) (x / y)) + 1.0)))) / ((double) (x + 1.0))))));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original9.2
Target0.1
Herbie0.2
\[\frac{x}{1} \cdot \frac{\frac{x}{y} + 1}{x + 1}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -4.34426230244491986e23 or 2676405038669891 < x

    1. Initial program 23.2

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]

    2. Simplified0.1

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

    3. Taylor expanded around inf 0.0

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

    4. Simplified0.0

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

    if -4.34426230244491986e23 < x < 2676405038669891

    1. Initial program 0.1

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\]

    2. Simplified0.1

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}}\]
    3. Using strategy rm

    4. Applied *-un-lft-identity0.1

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

    5. Applied add-cube-cbrt0.4

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

    6. Applied times-frac0.4

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

    7. Applied associate-*r*0.4

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

    8. Simplified0.4

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4.34426230244491986 \cdot 10^{23} \lor \neg \left(x \le 2676405038669891\right):\\ \;\;\;\;\frac{x}{y} + \left(1 - \frac{1}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(\sqrt[3]{\frac{x}{y} + 1} \cdot \sqrt[3]{\frac{x}{y} + 1}\right)\right) \cdot \frac{\sqrt[3]{\frac{x}{y} + 1}}{x + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020181 
(FPCore (x y)
  :name "Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1"
  :precision binary64

  :herbie-target
  (* (/ x 1.0) (/ (+ (/ x y) 1.0) (+ x 1.0)))

  (/ (* x (+ (/ x y) 1.0)) (+ x 1.0)))