x / (x^2 + 1)

Average Error: 14.8 → 0.2

Time: 2.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x}{x \cdot x + 1} \le -3.5437647178871777 \cdot 10^{-305} \lor \neg \left(\frac{x}{x \cdot x + 1} \le 0.0\right):\\ \;\;\;\;\frac{1}{\sqrt{x \cdot x + 1}} \cdot \frac{x}{\sqrt{x \cdot x + 1}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{1}{{x}^{5}} - \frac{1}{{x}^{3}}\right) + \frac{1}{x}\\ \end{array}\]

C

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

↓

double code(double x) {
	double VAR;
	if ((((x / ((x * x) + 1.0)) <= -3.5437647178871777e-305) || !((x / ((x * x) + 1.0)) <= 0.0))) {
		VAR = ((1.0 / sqrt(((x * x) + 1.0))) * (x / sqrt(((x * x) + 1.0))));
	} else {
		VAR = ((1.0 * ((1.0 / pow(x, 5.0)) - (1.0 / pow(x, 3.0)))) + (1.0 / x));
	}
	return VAR;
}

Error

Bits error versus x

Target

Original	14.8
Target	0.1
Herbie	0.2

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

Derivation

1. Split input into 2 regimes

2. if (/ x (+ (* x x) 1.0)) < -3.5437647178871777e-305 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 *-un-lft-identity 0.1

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

   5. Applied times-frac 0.0

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

   if -3.5437647178871777e-305 < (/ x (+ (* x x) 1.0)) < 0.0

   1. Initial program 59.0

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

   2. Taylor expanded around inf 0.6

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

   3. Simplified 0.6

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x \cdot x + 1} \le -3.5437647178871777 \cdot 10^{-305} \lor \neg \left(\frac{x}{x \cdot x + 1} \le 0.0\right):\\ \;\;\;\;\frac{1}{\sqrt{x \cdot x + 1}} \cdot \frac{x}{\sqrt{x \cdot x + 1}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{1}{{x}^{5}} - \frac{1}{{x}^{3}}\right) + \frac{1}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020100 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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