Asymptote A

Average Error: 14.6 → 0.0

Time: 3.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -2101387514886478 \lor \neg \left(x \le 1434.76562703285413\right):\\ \;\;\;\;\frac{-2}{{x}^{6}} - 2 \cdot \left({x}^{\left(-2\right)} + \frac{1}{{x}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot \left(\left(x - 1\right) - \left(x + 1\right)\right)}{x \cdot x - 1 \cdot 1}\\ \end{array}\]

C

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

↓

double code(double x) {
	double VAR;
	if (((x <= -2101387514886478.0) || !(x <= 1434.7656270328541))) {
		VAR = ((-2.0 / pow(x, 6.0)) - (2.0 * (pow(x, -2.0) + (1.0 / pow(x, 4.0)))));
	} else {
		VAR = ((1.0 * ((x - 1.0) - (x + 1.0))) / ((x * x) - (1.0 * 1.0)));
	}
	return VAR;
}

Error

Bits error versus x

Derivation

1. Split input into 2 regimes

2. if x < -2101387514886478.0 or 1434.7656270328541 < x

   1. Initial program 29.7

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

   2. Taylor expanded around inf 0.7

      \[\leadsto \color{blue}{-\left(2 \cdot \frac{1}{{x}^{6}} + \left(2 \cdot \frac{1}{{x}^{2}} + 2 \cdot \frac{1}{{x}^{4}}\right)\right)}\]

   3. Simplified 0.7

      \[\leadsto \color{blue}{\frac{-2}{{x}^{6}} - 2 \cdot \left(\frac{1}{{x}^{2}} + \frac{1}{{x}^{4}}\right)}\]
   4. Using strategy rm

   5. Applied pow-flip 0.0

      \[\leadsto \frac{-2}{{x}^{6}} - 2 \cdot \left(\color{blue}{{x}^{\left(-2\right)}} + \frac{1}{{x}^{4}}\right)\]

   if -2101387514886478.0 < x < 1434.7656270328541

   1. Initial program 0.6

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

   3. Applied frac-sub 0.0

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

   4. Simplified 0.0

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

   5. Simplified 0.0

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

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2101387514886478 \lor \neg \left(x \le 1434.76562703285413\right):\\ \;\;\;\;\frac{-2}{{x}^{6}} - 2 \cdot \left({x}^{\left(-2\right)} + \frac{1}{{x}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot \left(\left(x - 1\right) - \left(x + 1\right)\right)}{x \cdot x - 1 \cdot 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020092 
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1 (+ x 1)) (/ 1 (- x 1))))