2frac (problem 3.3.1)

Average Error: 14.3 → 0.1

Time: 3.4s

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 r43508 = 1.0;
        double r43509 = x;
        double r43510 = r43509 + r43508;
        double r43511 = r43508 / r43510;
        double r43512 = r43508 / r43509;
        double r43513 = r43511 - r43512;
        return r43513;
}

↓

double f(double x) {
        double r43514 = 1.0;
        double r43515 = x;
        double r43516 = r43514 / r43515;
        double r43517 = r43515 + r43514;
        double r43518 = 0.0;
        double r43519 = r43518 - r43514;
        double r43520 = r43517 / r43519;
        double r43521 = r43516 / r43520;
        return r43521;
}

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 +o rules:numerics
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  :precision binary64
  (- (/ 1 (+ x 1)) (/ 1 x)))