2frac (problem 3.3.1)

Average Error: 14.4 → 0.4

Time: 2.9s

Precision: 64

Math

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

↓

\[\left(\sqrt[3]{\frac{1}{x + 1}} \cdot \sqrt[3]{\frac{1}{x + 1}}\right) \cdot \left(\sqrt[3]{\frac{1}{x + 1}} \cdot \frac{0 - 1}{x}\right)\]

C

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

↓

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

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 13.8

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

4. Simplified 13.8

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

6. Applied times-frac 13.8

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

7. Simplified 0.1

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

9. Applied add-cube-cbrt 0.4

   \[\leadsto \color{blue}{\left(\left(\sqrt[3]{\frac{1}{x + 1}} \cdot \sqrt[3]{\frac{1}{x + 1}}\right) \cdot \sqrt[3]{\frac{1}{x + 1}}\right)} \cdot \frac{0 - 1}{x}\]

10. Applied associate-*l* 0.4

    \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{1}{x + 1}} \cdot \sqrt[3]{\frac{1}{x + 1}}\right) \cdot \left(\sqrt[3]{\frac{1}{x + 1}} \cdot \frac{0 - 1}{x}\right)}\]

11. Final simplification 0.4

    \[\leadsto \left(\sqrt[3]{\frac{1}{x + 1}} \cdot \sqrt[3]{\frac{1}{x + 1}}\right) \cdot \left(\sqrt[3]{\frac{1}{x + 1}} \cdot \frac{0 - 1}{x}\right)\]

Reproduce

herbie shell --seed 2020053 
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  :precision binary64
  (- (/ 1 (+ x 1)) (/ 1 x)))