Asymptote A

Average Error: 14.3 → 0.1

Time: 13.9s

Precision: 64

Math

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

↓

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

C

double f(double x) {
        double r4870306 = 1.0;
        double r4870307 = x;
        double r4870308 = r4870307 + r4870306;
        double r4870309 = r4870306 / r4870308;
        double r4870310 = r4870307 - r4870306;
        double r4870311 = r4870306 / r4870310;
        double r4870312 = r4870309 - r4870311;
        return r4870312;
}

↓

double f(double x) {
        double r4870313 = 1.0;
        double r4870314 = x;
        double r4870315 = r4870314 + r4870313;
        double r4870316 = r4870313 / r4870315;
        double r4870317 = -2.0;
        double r4870318 = r4870314 - r4870313;
        double r4870319 = r4870317 / r4870318;
        double r4870320 = r4870316 * r4870319;
        return r4870320;
}

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

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

5. Taylor expanded around 0 0.4

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

7. Applied *-un-lft-identity 0.4

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

8. Applied times-frac 0.1

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

9. Final simplification 0.1

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

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (x)
  :name "Asymptote A"
  (- (/ 1 (+ x 1)) (/ 1 (- x 1))))