Asymptote A

Average Error: 14.3 → 0.1

Time: 9.6s

Precision: 64

Math

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

↓

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

C

double f(double x) {
        double r114615 = 1.0;
        double r114616 = x;
        double r114617 = r114616 + r114615;
        double r114618 = r114615 / r114617;
        double r114619 = r114616 - r114615;
        double r114620 = r114615 / r114619;
        double r114621 = r114618 - r114620;
        return r114621;
}

↓

double f(double x) {
        double r114622 = 1.0;
        double r114623 = x;
        double r114624 = r114623 - r114622;
        double r114625 = r114622 / r114624;
        double r114626 = 2.0;
        double r114627 = -r114626;
        double r114628 = r114625 * r114627;
        double r114629 = r114623 + r114622;
        double r114630 = r114628 / r114629;
        return r114630;
}

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 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/ 28.8

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

5. Applied flip-+ 14.4

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

6. Applied associate-/r/ 14.3

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

7. Applied distribute-lft-out-- 13.7

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

8. Taylor expanded around 0 0.4

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

10. Applied difference-of-squares 0.4

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

11. Applied add-sqr-sqrt 0.4

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

12. Applied times-frac 0.1

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

13. Applied associate-*l* 0.1

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

15. Applied associate-*l/ 0.1

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

16. Simplified 0.1

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

17. Final simplification 0.1

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

Reproduce

herbie shell --seed 2019235 
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1 (+ x 1)) (/ 1 (- x 1))))