2frac (problem 3.3.1)

Average Error: 14.4 → 0.1

Time: 3.1s

Precision: 64

Math

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

↓

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

C

double code(double x) {
	return ((1.0 / (x + 1.0)) - (1.0 / x));
}

↓

double code(double x) {
	return ((1.0 / x) / ((x + 1.0) / (0.0 - 1.0)));
}

Error

Bits error versus x

Derivation

1. Initial program 14.4

   \[\frac{1}{x + 1} - \frac{1}{x}\]
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 2020057 +o rules:numerics
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  :precision binary64
  (- (/ 1 (+ x 1)) (/ 1 x)))