Average Error: 14.9 → 0.0
Time: 1.3s
Precision: binary64
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -2431087199668.861 \lor \neg \left(x \leq 3400835.701847913\right):\\ \;\;\;\;\frac{1}{{x}^{5}} + \left(\frac{1}{x} - \frac{1}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + x \cdot x}\\ \end{array}\]
\frac{x}{x \cdot x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -2431087199668.861 \lor \neg \left(x \leq 3400835.701847913\right):\\
\;\;\;\;\frac{1}{{x}^{5}} + \left(\frac{1}{x} - \frac{1}{{x}^{3}}\right)\\

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

\end{array}
(FPCore (x) :precision binary64 (/ x (+ (* x x) 1.0)))
↓
(FPCore (x)
 :precision binary64
 (if (or (<= x -2431087199668.861) (not (<= x 3400835.701847913)))
   (+ (/ 1.0 (pow x 5.0)) (- (/ 1.0 x) (/ 1.0 (pow x 3.0))))
   (/ x (+ 1.0 (* x x)))))
double code(double x) {
	return (x / ((double) (((double) (x * x)) + 1.0)));
}
↓
double code(double x) {
	double tmp;
	if (((x <= -2431087199668.861) || !(x <= 3400835.701847913))) {
		tmp = ((double) ((1.0 / ((double) pow(x, 5.0))) + ((double) ((1.0 / x) - (1.0 / ((double) pow(x, 3.0)))))));
	} else {
		tmp = (x / ((double) (1.0 + ((double) (x * x)))));
	}
	return tmp;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original14.9
Target0.1
Herbie0.0
\[\frac{1}{x + \frac{1}{x}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -2431087199668.86084 or 3400835.70184791321 < x

    1. Initial program Error: 30.9 bits

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

    2. Taylor expanded around inf Error: 0.0 bits

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

    3. SimplifiedError: 0.0 bits

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

    if -2431087199668.86084 < x < 3400835.70184791321

    1. Initial program Error: 0.0 bits

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

  4. Final simplificationError: 0.0 bits

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2431087199668.861 \lor \neg \left(x \leq 3400835.701847913\right):\\ \;\;\;\;\frac{1}{{x}^{5}} + \left(\frac{1}{x} - \frac{1}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + x \cdot x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020204 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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