Asymptote A

Average Error: 14.1 → 0.1

Time: 2.1s

Precision: 64

Math

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

↓

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

C

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

↓

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

Error

Bits error versus x

Derivation

1. Initial program 14.1

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

3. Applied clear-num 14.1

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

4. Applied frac-sub 13.6

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

5. Simplified 13.6

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

6. Taylor expanded around 0 0.4

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

8. Applied *-commutative 0.4

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

9. Applied associate-/r* 0.1

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

10. Final simplification 0.1

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

Reproduce

herbie shell --seed 2020071 +o rules:numerics
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1 (+ x 1)) (/ 1 (- x 1))))