Asymptote A

Average Error: 14.6 → 0.1

Time: 7.5s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\frac{\frac{1 \cdot \left(-4 \cdot \left(1 \cdot 1\right)\right)}{1 + x}}{\left(x - 1\right) \cdot \left(1 + 1\right)}\]

C

double f(double x) {
        double r148250 = 1.0;
        double r148251 = x;
        double r148252 = r148251 + r148250;
        double r148253 = r148250 / r148252;
        double r148254 = r148251 - r148250;
        double r148255 = r148250 / r148254;
        double r148256 = r148253 - r148255;
        return r148256;
}

↓

double f(double x) {
        double r148257 = 1.0;
        double r148258 = 4.0;
        double r148259 = r148257 * r148257;
        double r148260 = r148258 * r148259;
        double r148261 = -r148260;
        double r148262 = r148257 * r148261;
        double r148263 = x;
        double r148264 = r148257 + r148263;
        double r148265 = r148262 / r148264;
        double r148266 = r148263 - r148257;
        double r148267 = r148257 + r148257;
        double r148268 = r148266 * r148267;
        double r148269 = r148265 / r148268;
        return r148269;
}

Error

Bits error versus x

Derivation

1. Initial program 14.6

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

3. Applied flip-- 28.8

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

4. Applied associate-/r/ 28.9

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

5. Applied flip-+ 14.6

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

6. Applied associate-/r/ 14.6

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

7. Applied distribute-lft-out-- 14.0

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

8. Simplified 0.4

   \[\leadsto \frac{1}{x \cdot x - 1 \cdot 1} \cdot \color{blue}{\left(0 - \left(1 + 1\right)\right)}\]
9. Using strategy rm

10. Applied difference-of-squares 0.4

    \[\leadsto \frac{1}{\color{blue}{\left(x + 1\right) \cdot \left(x - 1\right)}} \cdot \left(0 - \left(1 + 1\right)\right)\]

11. Applied associate-/r* 0.1

    \[\leadsto \color{blue}{\frac{\frac{1}{x + 1}}{x - 1}} \cdot \left(0 - \left(1 + 1\right)\right)\]
12. Using strategy rm

13. Applied flip-- 0.1

    \[\leadsto \frac{\frac{1}{x + 1}}{x - 1} \cdot \color{blue}{\frac{0 \cdot 0 - \left(1 + 1\right) \cdot \left(1 + 1\right)}{0 + \left(1 + 1\right)}}\]

14. Applied frac-times 0.1

    \[\leadsto \color{blue}{\frac{\frac{1}{x + 1} \cdot \left(0 \cdot 0 - \left(1 + 1\right) \cdot \left(1 + 1\right)\right)}{\left(x - 1\right) \cdot \left(0 + \left(1 + 1\right)\right)}}\]

15. Simplified 0.1

    \[\leadsto \frac{\color{blue}{\frac{1 \cdot \left(-4 \cdot \left(1 \cdot 1\right)\right)}{1 + x}}}{\left(x - 1\right) \cdot \left(0 + \left(1 + 1\right)\right)}\]

16. Simplified 0.1

    \[\leadsto \frac{\frac{1 \cdot \left(-4 \cdot \left(1 \cdot 1\right)\right)}{1 + x}}{\color{blue}{\left(x - 1\right) \cdot \left(1 + 1\right)}}\]

17. Final simplification 0.1

    \[\leadsto \frac{\frac{1 \cdot \left(-4 \cdot \left(1 \cdot 1\right)\right)}{1 + x}}{\left(x - 1\right) \cdot \left(1 + 1\right)}\]

Reproduce

herbie shell --seed 2019351 
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1 (+ x 1)) (/ 1 (- x 1))))