Asymptote C

Average Error: 29.3 → 0.0

Time: 21.5s

Precision: 64

Math

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

↓

\[\frac{-\left(-\left(1 \cdot \frac{1}{x} + 3\right)\right)}{\frac{x + 1}{x} \cdot \left(-\left(x - 1\right)\right)}\]

C

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

↓

double code(double x) {
	return ((double) (((double) -(((double) -(((double) (((double) (1.0 * ((double) (1.0 / x)))) + 3.0)))))) / ((double) (((double) (((double) (x + 1.0)) / x)) * ((double) -(((double) (x - 1.0))))))));
}

Error

Bits error versus x

Derivation

1. Initial program 29.3

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

3. Applied frac-2neg 29.3

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

4. Applied clear-num 29.3

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

5. Applied frac-sub 29.1

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

6. Simplified 29.1

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

7. Taylor expanded around 0 0.0

   \[\leadsto \frac{-\color{blue}{\left(-\left(1 \cdot \frac{1}{x} + 3\right)\right)}}{\frac{x + 1}{x} \cdot \left(-\left(x - 1\right)\right)}\]

8. Final simplification 0.0

   \[\leadsto \frac{-\left(-\left(1 \cdot \frac{1}{x} + 3\right)\right)}{\frac{x + 1}{x} \cdot \left(-\left(x - 1\right)\right)}\]

Reproduce

herbie shell --seed 2020114 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))