Average Error: 14.4 → 0.0
Time: 1.6s
Precision: binary64
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -523.8341154073099 \lor \neg \left(x \leq 475.29682370685475\right):\\ \;\;\;\;\frac{1}{{x}^{5}} + \left(\frac{1}{x} - \frac{1}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{1 + x \cdot x}} \cdot \frac{x}{\sqrt{1 + x \cdot x}}\\ \end{array}\]
\frac{x}{x \cdot x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -523.8341154073099 \lor \neg \left(x \leq 475.29682370685475\right):\\
\;\;\;\;\frac{1}{{x}^{5}} + \left(\frac{1}{x} - \frac{1}{{x}^{3}}\right)\\

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

\end{array}
double code(double x) {
	return (x / ((double) (((double) (x * x)) + 1.0)));
}
↓
double code(double x) {
	double VAR;
	if (((x <= -523.8341154073099) || !(x <= 475.29682370685475))) {
		VAR = ((double) ((1.0 / ((double) pow(x, 5.0))) + ((double) ((1.0 / x) - (1.0 / ((double) pow(x, 3.0)))))));
	} else {
		VAR = ((double) ((1.0 / ((double) sqrt(((double) (1.0 + ((double) (x * x))))))) * (x / ((double) sqrt(((double) (1.0 + ((double) (x * x)))))))));
	}
	return VAR;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original14.4
Target0.1
Herbie0.0
\[\frac{1}{x + \frac{1}{x}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -523.834115407309923 or 475.29682370685475 < x

    1. Initial program 29.8

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

    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

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

    if -523.834115407309923 < x < 475.29682370685475

    1. Initial program 0.0

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

    3. Applied add-sqr-sqrt0.0

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

    4. Applied *-un-lft-identity0.0

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

    5. Applied times-frac0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -523.8341154073099 \lor \neg \left(x \leq 475.29682370685475\right):\\ \;\;\;\;\frac{1}{{x}^{5}} + \left(\frac{1}{x} - \frac{1}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{1 + x \cdot x}} \cdot \frac{x}{\sqrt{1 + x \cdot x}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020196 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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