Asymptote A

Average Error: 14.4 → 0.1

Time: 3.4s

Precision: 64

Math

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

↓

\[1 \cdot \frac{\frac{-2}{x + 1}}{x - 1}\]

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 / (x + 1.0)) / (x - 1.0)));
}

Error

Bits error versus x

Derivation

1. Initial program 14.4

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

3. Applied flip-- 29.0

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

4. Applied associate-/r/ 29.0

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

5. Applied flip-+ 14.5

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

6. Applied associate-/r/ 14.4

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

7. Applied distribute-lft-out-- 13.8

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

8. Taylor expanded around 0 0.4

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

10. Applied div-inv 0.4

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

11. Applied associate-*l* 0.4

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

12. Simplified 0.1

    \[\leadsto 1 \cdot \color{blue}{\frac{\frac{-2}{x + 1}}{x - 1}}\]

13. Final simplification 0.1

    \[\leadsto 1 \cdot \frac{\frac{-2}{x + 1}}{x - 1}\]

Reproduce

herbie shell --seed 2020106 +o rules:numerics
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1 (+ x 1)) (/ 1 (- x 1))))