Average Error: 14.8 → 0.0
Time: 1.3s
Precision: binary64
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -111091.15615886822 \lor \neg \left(x \leq 413.7446332704056\right):\\ \;\;\;\;\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - {\left(\frac{1}{x}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{1 + x \cdot x}} \cdot \frac{x}{\sqrt{1 + x \cdot x}}\\ \end{array}\]
\frac{x}{x \cdot x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -111091.15615886822 \lor \neg \left(x \leq 413.7446332704056\right):\\
\;\;\;\;\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - {\left(\frac{1}{x}\right)}^{3}\\

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

\end{array}
(FPCore (x) :precision binary64 (/ x (+ (* x x) 1.0)))
↓
(FPCore (x)
 :precision binary64
 (if (or (<= x -111091.15615886822) (not (<= x 413.7446332704056)))
   (- (+ (/ 1.0 (pow x 5.0)) (/ 1.0 x)) (pow (/ 1.0 x) 3.0))
   (* (/ 1.0 (sqrt (+ 1.0 (* x x)))) (/ x (sqrt (+ 1.0 (* x x)))))))
double code(double x) {
	return x / ((x * x) + 1.0);
}
↓
double code(double x) {
	double tmp;
	if ((x <= -111091.15615886822) || !(x <= 413.7446332704056)) {
		tmp = ((1.0 / pow(x, 5.0)) + (1.0 / x)) - pow((1.0 / x), 3.0);
	} else {
		tmp = (1.0 / sqrt(1.0 + (x * x))) * (x / sqrt(1.0 + (x * x)));
	}
	return tmp;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original14.8
Target0.1
Herbie0.0
\[\frac{1}{x + \frac{1}{x}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -111091.1561588682 or 413.744633270405586 < x

    1. Initial program 31.0

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

    2. Taylor expanded around inf 0.0

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

    3. Simplified0.0

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

    if -111091.1561588682 < x < 413.744633270405586

    1. Initial program 0.0

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

    3. Applied add-sqr-sqrt_binary640.0

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

    4. Applied *-un-lft-identity_binary640.0

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

    5. Applied times-frac_binary640.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -111091.15615886822 \lor \neg \left(x \leq 413.7446332704056\right):\\ \;\;\;\;\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - {\left(\frac{1}{x}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{1 + x \cdot x}} \cdot \frac{x}{\sqrt{1 + x \cdot x}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020219 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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