Average Error: 15.3 → 0.2
Time: 2.6s
Precision: binary64
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x}{x \cdot x + 1} \leq -1.6172421973863297 \cdot 10^{-304} \lor \neg \left(\frac{x}{x \cdot x + 1} \leq 0\right):\\ \;\;\;\;\frac{1}{\sqrt{x \cdot x + 1}} \cdot \frac{x}{\sqrt{x \cdot x + 1}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x} + \frac{1}{{x}^{5}}\right) - {\left(\frac{1}{x}\right)}^{3}\\ \end{array}\]
\frac{x}{x \cdot x + 1}
↓
\begin{array}{l}
\mathbf{if}\;\frac{x}{x \cdot x + 1} \leq -1.6172421973863297 \cdot 10^{-304} \lor \neg \left(\frac{x}{x \cdot x + 1} \leq 0\right):\\
\;\;\;\;\frac{1}{\sqrt{x \cdot x + 1}} \cdot \frac{x}{\sqrt{x \cdot x + 1}}\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{1}{x} + \frac{1}{{x}^{5}}\right) - {\left(\frac{1}{x}\right)}^{3}\\

\end{array}
(FPCore (x) :precision binary64 (/ x (+ (* x x) 1.0)))
↓
(FPCore (x)
 :precision binary64
 (if (or (<= (/ x (+ (* x x) 1.0)) -1.6172421973863297e-304)
         (not (<= (/ x (+ (* x x) 1.0)) 0.0)))
   (* (/ 1.0 (sqrt (+ (* x x) 1.0))) (/ x (sqrt (+ (* x x) 1.0))))
   (- (+ (/ 1.0 x) (/ 1.0 (pow x 5.0))) (pow (/ 1.0 x) 3.0))))
double code(double x) {
	return x / ((x * x) + 1.0);
}
↓
double code(double x) {
	double tmp;
	if (((x / ((x * x) + 1.0)) <= -1.6172421973863297e-304) || !((x / ((x * x) + 1.0)) <= 0.0)) {
		tmp = (1.0 / sqrt((x * x) + 1.0)) * (x / sqrt((x * x) + 1.0));
	} else {
		tmp = ((1.0 / x) + (1.0 / pow(x, 5.0))) - pow((1.0 / x), 3.0);
	}
	return tmp;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original15.3
Target0.1
Herbie0.2
\[\frac{1}{x + \frac{1}{x}}\]

Derivation

  1. Split input into 2 regimes

  2. if (/.f64 x (+.f64 (*.f64 x x) 1)) < -1.6172421973863297e-304 or -0.0 < (/.f64 x (+.f64 (*.f64 x x) 1))

    1. Initial program 0.1

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

    3. Applied add-sqr-sqrt_binary64_11330.1

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

    4. Applied *-un-lft-identity_binary64_11110.1

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

    5. Applied times-frac_binary64_11170.0

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

    if -1.6172421973863297e-304 < (/.f64 x (+.f64 (*.f64 x x) 1)) < -0.0

    1. Initial program 58.9

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

    2. Taylor expanded around inf 0.7

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

    3. Simplified0.7

      \[\leadsto \color{blue}{\left(\frac{1}{x} + \frac{1}{{x}^{5}}\right) - {\left(\frac{1}{x}\right)}^{3}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x \cdot x + 1} \leq -1.6172421973863297 \cdot 10^{-304} \lor \neg \left(\frac{x}{x \cdot x + 1} \leq 0\right):\\ \;\;\;\;\frac{1}{\sqrt{x \cdot x + 1}} \cdot \frac{x}{\sqrt{x \cdot x + 1}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x} + \frac{1}{{x}^{5}}\right) - {\left(\frac{1}{x}\right)}^{3}\\ \end{array}\]

Reproduce

herbie shell --seed 2020280 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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