Asymptote B

Average Error: 0.0 → 0.0

Time: 2.5s

Precision: 64

Math

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

↓

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

C

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

↓

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

Error

Bits error versus x

Derivation

1. Initial program 0.0

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

3. Applied flip-+ 0.0

   \[\leadsto \color{blue}{\frac{\frac{1}{x - 1} \cdot \frac{1}{x - 1} - \frac{x}{x + 1} \cdot \frac{x}{x + 1}}{\frac{1}{x - 1} - \frac{x}{x + 1}}}\]
4. Using strategy rm

5. Applied flip-- 0.0

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

6. Applied associate-/r/ 0.0

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

7. Applied associate-*l* 0.0

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

8. Final simplification 0.0

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

Reproduce

herbie shell --seed 2020060 
(FPCore (x)
  :name "Asymptote B"
  :precision binary64
  (+ (/ 1 (- x 1)) (/ x (+ x 1))))