Asymptote A

Average Error: 14.7 → 0.1

Time: 21.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -232.32776769692475 \lor \neg \left(x \le 283.61304447385839\right):\\ \;\;\;\;\left(\frac{-2}{{x}^{6}} - \frac{\frac{2}{x}}{x}\right) - \frac{2}{{x}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{x \cdot x - 1 \cdot 1}, x - 1, -\left(x + 1\right) \cdot \frac{1}{x \cdot x - 1 \cdot 1}\right) + \frac{1}{x \cdot x - 1 \cdot 1} \cdot \left(\left(-\left(x + 1\right)\right) + \left(x + 1\right)\right)\\ \end{array}\]

C

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

↓

double code(double x) {
	double VAR;
	if (((x <= -232.32776769692475) || !(x <= 283.6130444738584))) {
		VAR = (((-2.0 / pow(x, 6.0)) - ((2.0 / x) / x)) - (2.0 / pow(x, 4.0)));
	} else {
		VAR = (fma((1.0 / ((x * x) - (1.0 * 1.0))), (x - 1.0), -((x + 1.0) * (1.0 / ((x * x) - (1.0 * 1.0))))) + ((1.0 / ((x * x) - (1.0 * 1.0))) * (-(x + 1.0) + (x + 1.0))));
	}
	return VAR;
}

Error

Bits error versus x

Derivation

1. Split input into 2 regimes

2. if x < -232.32776769692475 or 283.6130444738584 < x

   1. Initial program 29.0

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

   3. Applied flip-+ 58.5

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

   4. Applied associate-/r/ 58.6

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

   5. Applied fma-neg 58.7

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

   6. Taylor expanded around inf 0.8

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

   7. Simplified 0.1

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

   if -232.32776769692475 < x < 283.6130444738584

   1. Initial program 0.0

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

   3. Applied flip-- 0.0

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

   4. Applied associate-/r/ 0.0

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

   5. Applied flip-+ 0.0

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

   6. Applied associate-/r/ 0.0

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

   7. Applied prod-diff 0.0

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

   8. Simplified 0.0

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -232.32776769692475 \lor \neg \left(x \le 283.61304447385839\right):\\ \;\;\;\;\left(\frac{-2}{{x}^{6}} - \frac{\frac{2}{x}}{x}\right) - \frac{2}{{x}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{x \cdot x - 1 \cdot 1}, x - 1, -\left(x + 1\right) \cdot \frac{1}{x \cdot x - 1 \cdot 1}\right) + \frac{1}{x \cdot x - 1 \cdot 1} \cdot \left(\left(-\left(x + 1\right)\right) + \left(x + 1\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020102 +o rules:numerics
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1 (+ x 1)) (/ 1 (- x 1))))