Asymptote A

Average Error: 14.4 → 0.1

Time: 15.2s

Precision: 64

Math

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

↓

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

C

double f(double x) {
        double r5447032 = 1.0;
        double r5447033 = x;
        double r5447034 = r5447033 + r5447032;
        double r5447035 = r5447032 / r5447034;
        double r5447036 = r5447033 - r5447032;
        double r5447037 = r5447032 / r5447036;
        double r5447038 = r5447035 - r5447037;
        return r5447038;
}

↓

double f(double x) {
        double r5447039 = -2.0;
        double r5447040 = x;
        double r5447041 = 1.0;
        double r5447042 = r5447040 + r5447041;
        double r5447043 = r5447039 / r5447042;
        double r5447044 = r5447040 - r5447041;
        double r5447045 = r5447043 / r5447044;
        return r5447045;
}

Error

Bits error versus x

Derivation

1. Initial program 14.4

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

3. Applied flip-- 28.7

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

4. Applied associate-/r/ 28.7

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

5. Applied flip-+ 14.5

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

6. Applied associate-/r/ 14.4

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

7. Applied distribute-lft-out-- 13.9

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

8. Taylor expanded around 0 0.3

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

10. Applied difference-of-squares 0.3

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

11. Applied *-un-lft-identity 0.3

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

12. Applied times-frac 0.1

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

13. Applied associate-*l* 0.1

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

14. Simplified 0.1

    \[\leadsto \frac{1}{x + 1} \cdot \color{blue}{\frac{-2}{x - 1}}\]
15. Using strategy rm

16. Applied associate-*r/ 0.1

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

17. Simplified 0.1

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

18. Final simplification 0.1

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

Reproduce

herbie shell --seed 2019164 +o rules:numerics
(FPCore (x)
  :name "Asymptote A"
  (- (/ 1 (+ x 1)) (/ 1 (- x 1))))