Average Error: 14.6 → 0.0
Time: 2.7s
Precision: 64
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -8878.5441343882085 \lor \neg \left(x \le 1226.01187072725065\right):\\ \;\;\;\;\frac{1}{x} - \left(\frac{1}{{x}^{3}} - 1 \cdot \frac{1}{{x}^{5}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{x \cdot x + 1}\\ \end{array}\]
\frac{x}{x \cdot x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -8878.5441343882085 \lor \neg \left(x \le 1226.01187072725065\right):\\
\;\;\;\;\frac{1}{x} - \left(\frac{1}{{x}^{3}} - 1 \cdot \frac{1}{{x}^{5}}\right)\\

\mathbf{else}:\\
\;\;\;\;x \cdot \frac{1}{x \cdot x + 1}\\

\end{array}
double code(double x) {
	return ((double) (x / ((double) (((double) (x * x)) + 1.0))));
}
↓
double code(double x) {
	double VAR;
	if (((x <= -8878.544134388208) || !(x <= 1226.0118707272507))) {
		VAR = ((double) (((double) (1.0 / x)) - ((double) (((double) (1.0 / ((double) pow(x, 3.0)))) - ((double) (1.0 * ((double) (1.0 / ((double) pow(x, 5.0))))))))));
	} else {
		VAR = ((double) (x * ((double) (1.0 / ((double) (((double) (x * x)) + 1.0))))));
	}
	return VAR;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original14.6
Target0.1
Herbie0.0
\[\frac{1}{x + \frac{1}{x}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -8878.544134388208 or 1226.0118707272507 < x

    1. Initial program 29.5

      \[\frac{x}{x \cdot x + 1}\]

    2. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{\left(1 \cdot \frac{1}{{x}^{5}} + \frac{1}{x}\right) - 1 \cdot \frac{1}{{x}^{3}}}\]

    3. Simplified0.0

      \[\leadsto \color{blue}{\frac{1}{x} - \left(\frac{1}{{x}^{3}} - 1 \cdot \frac{1}{{x}^{5}}\right)}\]

    if -8878.544134388208 < x < 1226.0118707272507

    1. Initial program 0.0

      \[\frac{x}{x \cdot x + 1}\]
    2. Using strategy rm

    3. Applied div-inv0.0

      \[\leadsto \color{blue}{x \cdot \frac{1}{x \cdot x + 1}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -8878.5441343882085 \lor \neg \left(x \le 1226.01187072725065\right):\\ \;\;\;\;\frac{1}{x} - \left(\frac{1}{{x}^{3}} - 1 \cdot \frac{1}{{x}^{5}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{x \cdot x + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020130 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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