Average Error: 14.9 → 0.0
Time: 1.4s
Precision: binary64
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -2796513144.817297 \lor \neg \left(x \leq 767.2537428637866\right):\\ \;\;\;\;\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{1 + x \cdot x}\\ \end{array}\]
\frac{x}{x \cdot x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -2796513144.817297 \lor \neg \left(x \leq 767.2537428637866\right):\\
\;\;\;\;\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{{x}^{3}}\\

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

\end{array}
(FPCore (x) :precision binary64 (/ x (+ (* x x) 1.0)))
↓
(FPCore (x)
 :precision binary64
 (if (or (<= x -2796513144.817297) (not (<= x 767.2537428637866)))
   (- (+ (/ 1.0 (pow x 5.0)) (/ 1.0 x)) (/ 1.0 (pow x 3.0)))
   (* x (/ 1.0 (+ 1.0 (* x x))))))
double code(double x) {
	return (x / ((double) (((double) (x * x)) + 1.0)));
}
↓
double code(double x) {
	double tmp;
	if (((x <= -2796513144.817297) || !(x <= 767.2537428637866))) {
		tmp = ((double) (((double) ((1.0 / ((double) pow(x, 5.0))) + (1.0 / x))) - (1.0 / ((double) pow(x, 3.0)))));
	} else {
		tmp = ((double) (x * (1.0 / ((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 < -2796513144.817297 or 767.25374286378656 < x

    1. Initial program 30.6

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

    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

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

    if -2796513144.817297 < x < 767.25374286378656

    1. Initial program 0.0

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

    3. Applied div-inv_binary640.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2796513144.817297 \lor \neg \left(x \leq 767.2537428637866\right):\\ \;\;\;\;\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{1 + x \cdot x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020205 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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