x / (x^2 + 1)

Average Error: 15.2 → 0.0

Time: 2.0s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -406.611233561352321 \lor \neg \left(x \le 415.65688547362504\right):\\ \;\;\;\;\frac{1}{x} - \left(\frac{1}{{x}^{3}} - 1 \cdot \frac{1}{{x}^{5}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x \cdot x + 1}} \cdot \frac{x}{\sqrt{\sqrt{x \cdot x + 1}} \cdot \sqrt{\sqrt{x \cdot x + 1}}}\\ \end{array}\]

C

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

↓

double code(double x) {
	double VAR;
	if (((x <= -406.6112335613523) || !(x <= 415.65688547362504))) {
		VAR = ((double) (((double) (1.0 / x)) - ((double) (((double) (1.0 / ((double) pow(x, 3.0)))) - ((double) (1.0 * ((double) (1.0 / ((double) pow(x, 5.0))))))))));
	} else {
		VAR = ((double) (((double) (1.0 / ((double) sqrt(((double) (((double) (x * x)) + 1.0)))))) * ((double) (x / ((double) (((double) sqrt(((double) sqrt(((double) (((double) (x * x)) + 1.0)))))) * ((double) sqrt(((double) sqrt(((double) (((double) (x * x)) + 1.0))))))))))));
	}
	return VAR;
}

Error

Bits error versus x

Target

Original	15.2
Target	0.1
Herbie	0.0

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

Derivation

1. Split input into 2 regimes

2. if x < -406.611233561352321 or 415.65688547362504 < x

   1. Initial program 30.3

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

   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. Simplified 0.0

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

   if -406.611233561352321 < x < 415.65688547362504

   1. Initial program 0.0

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

   3. Applied add-sqr-sqrt 0.0

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

   4. Applied *-un-lft-identity 0.0

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

   5. Applied times-frac 0.0

      \[\leadsto \color{blue}{\frac{1}{\sqrt{x \cdot x + 1}} \cdot \frac{x}{\sqrt{x \cdot x + 1}}}\]
   6. Using strategy rm

   7. Applied add-sqr-sqrt 0.0

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

   8. Applied sqrt-prod 0.0

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

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -406.611233561352321 \lor \neg \left(x \le 415.65688547362504\right):\\ \;\;\;\;\frac{1}{x} - \left(\frac{1}{{x}^{3}} - 1 \cdot \frac{1}{{x}^{5}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x \cdot x + 1}} \cdot \frac{x}{\sqrt{\sqrt{x \cdot x + 1}} \cdot \sqrt{\sqrt{x \cdot x + 1}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020156 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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