2frac (problem 3.3.1)

Average Error: 14.4 → 0.1

Time: 2.9s

Precision: 64

Math

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

↓

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

C

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

↓

double code(double x) {
	return (((0.0 + sqrt(1.0)) * (1.0 / x)) / ((x + 1.0) / (sqrt(0.0) - sqrt(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 add-sqr-sqrt 0.4

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

14. Applied add-sqr-sqrt 0.4

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

15. Applied difference-of-squares 0.4

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

16. Applied times-frac 0.4

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

17. Applied associate-/r* 0.1

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

18. Simplified 0.1

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

19. Final simplification 0.1

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

Reproduce

herbie shell --seed 2020057 
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  :precision binary64
  (- (/ 1 (+ x 1)) (/ 1 x)))