x / (x^2 + 1)

Average Error: 15.6 → 0.1

Time: 2.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x}{x \cdot x + 1} \le -3.20787222115441721 \cdot 10^{-308} \lor \neg \left(\frac{x}{x \cdot x + 1} \le -0.0\right):\\ \;\;\;\;\frac{\frac{x}{\sqrt{x \cdot x + 1}}}{\sqrt{x \cdot x + 1}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{1}{{x}^{5}} - \frac{1}{{x}^{3}}\right) + \frac{1}{x}\\ \end{array}\]

C

double code(double x) {
	return ((double) (x / ((double) (((double) (x * x)) + 1.0))));
}

↓

double code(double x) {
	double VAR;
	if (((((double) (x / ((double) (((double) (x * x)) + 1.0)))) <= -3.207872221154417e-308) || !(((double) (x / ((double) (((double) (x * x)) + 1.0)))) <= -0.0))) {
		VAR = ((double) (((double) (x / ((double) sqrt(((double) (((double) (x * x)) + 1.0)))))) / ((double) sqrt(((double) (((double) (x * x)) + 1.0))))));
	} else {
		VAR = ((double) (((double) (1.0 * ((double) (((double) (1.0 / ((double) pow(x, 5.0)))) - ((double) (1.0 / ((double) pow(x, 3.0)))))))) + ((double) (1.0 / x))));
	}
	return VAR;
}

Error

Bits error versus x

Target

Original	15.6
Target	0.1
Herbie	0.1

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

Derivation

1. Split input into 2 regimes

2. if (/ x (+ (* x x) 1.0)) < -3.207872221154417e-308 or -0.0 < (/ x (+ (* x x) 1.0))

   1. Initial program 0.1

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

   3. Applied add-sqr-sqrt 0.1

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

   4. Applied associate-/r* 0.0

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

   if -3.207872221154417e-308 < (/ x (+ (* x x) 1.0)) < -0.0

   1. Initial program 59.4

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

   2. Taylor expanded around inf 0.2

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

   3. Simplified 0.2

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x \cdot x + 1} \le -3.20787222115441721 \cdot 10^{-308} \lor \neg \left(\frac{x}{x \cdot x + 1} \le -0.0\right):\\ \;\;\;\;\frac{\frac{x}{\sqrt{x \cdot x + 1}}}{\sqrt{x \cdot x + 1}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{1}{{x}^{5}} - \frac{1}{{x}^{3}}\right) + \frac{1}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020120 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

  :herbie-target
  (/ 1 (+ x (/ 1 x)))

  (/ x (+ (* x x) 1)))