2frac (problem 3.3.1)

Average Error: 14.3 → 0.1

Time: 2.8s

Precision: 64

Math

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

↓

\[\frac{\frac{1}{x}}{\frac{x + 1}{0 - 1}}\]

C

double f(double x) {
        double r51991 = 1.0;
        double r51992 = x;
        double r51993 = r51992 + r51991;
        double r51994 = r51991 / r51993;
        double r51995 = r51991 / r51992;
        double r51996 = r51994 - r51995;
        return r51996;
}

↓

double f(double x) {
        double r51997 = 1.0;
        double r51998 = x;
        double r51999 = r51997 / r51998;
        double r52000 = r51998 + r51997;
        double r52001 = 0.0;
        double r52002 = r52001 - r51997;
        double r52003 = r52000 / r52002;
        double r52004 = r51999 / r52003;
        return r52004;
}

Error

Bits error versus x

Derivation

1. Initial program 14.3

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

3. Applied frac-sub 13.7

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

4. Simplified 13.7

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

6. Applied associate-/r* 13.7

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

7. Simplified 0.1

   \[\leadsto \frac{\color{blue}{\frac{1}{\frac{x + 1}{0 - 1}}}}{x}\]
8. Using strategy rm

9. Applied div-inv 0.1

   \[\leadsto \frac{\color{blue}{1 \cdot \frac{1}{\frac{x + 1}{0 - 1}}}}{x}\]

10. Applied associate-/l* 0.4

    \[\leadsto \color{blue}{\frac{1}{\frac{x}{\frac{1}{\frac{x + 1}{0 - 1}}}}}\]

11. Simplified 0.4

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

13. Applied *-un-lft-identity 0.4

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

14. Applied times-frac 0.4

    \[\leadsto \frac{1}{\color{blue}{\frac{x}{1} \cdot \frac{x + 1}{0 - 1}}}\]

15. Applied associate-/r* 0.1

    \[\leadsto \color{blue}{\frac{\frac{1}{\frac{x}{1}}}{\frac{x + 1}{0 - 1}}}\]

16. Simplified 0.1

    \[\leadsto \frac{\color{blue}{\frac{1}{x}}}{\frac{x + 1}{0 - 1}}\]

17. Final simplification 0.1

    \[\leadsto \frac{\frac{1}{x}}{\frac{x + 1}{0 - 1}}\]

Reproduce

herbie shell --seed 2020036 
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  :precision binary64
  (- (/ 1 (+ x 1)) (/ 1 x)))