Asymptote A

Average Error: 14.3 → 0.1

Time: 6.4s

Precision: 64

Math

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

↓

\[\frac{-\frac{1 \cdot 2}{\left|1\right| + x}}{x - \sqrt{1 \cdot 1}}\]

C

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

↓

double code(double x) {
	return (-((1.0 * 2.0) / (fabs(1.0) + x)) / (x - sqrt((1.0 * 1.0))));
}

Error

Bits error versus x

Derivation

1. Initial program 14.3

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

3. Applied frac-sub 13.7

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

4. Simplified 13.7

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

5. Simplified 13.7

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

6. Taylor expanded around 0 0.4

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

8. Applied add-sqr-sqrt 0.4

   \[\leadsto \frac{1 \cdot \left(-2\right)}{x \cdot x - \color{blue}{\sqrt{1 \cdot 1} \cdot \sqrt{1 \cdot 1}}}\]

9. Applied difference-of-squares 0.4

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

10. Applied associate-/r* 0.1

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

11. Simplified 0.1

    \[\leadsto \frac{\color{blue}{-\frac{1 \cdot 2}{\left|1\right| + x}}}{x - \sqrt{1 \cdot 1}}\]

12. Final simplification 0.1

    \[\leadsto \frac{-\frac{1 \cdot 2}{\left|1\right| + x}}{x - \sqrt{1 \cdot 1}}\]

Reproduce

herbie shell --seed 2020066 +o rules:numerics
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1 (+ x 1)) (/ 1 (- x 1))))