3frac (problem 3.3.3)

Average Error: 9.6 → 0.2

Time: 2.4s

Precision: binary64

Math

\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]

↓

\[\frac{2}{x \cdot \left(x \cdot x - 1 \cdot 1\right)}\]

C

double code(double x) {
	return ((double) (((double) ((1.0 / ((double) (x + 1.0))) - (2.0 / x))) + (1.0 / ((double) (x - 1.0)))));
}

↓

double code(double x) {
	return (2.0 / ((double) (x * ((double) (((double) (x * x)) - ((double) (1.0 * 1.0)))))));
}

Error

Bits error versus x

Target

Original	9.6
Target	0.2
Herbie	0.2

\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

1. Initial program 9.6

   \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
2. Using strategy rm

3. Applied frac-sub 26.1

   \[\leadsto \color{blue}{\frac{1 \cdot x - \left(x + 1\right) \cdot 2}{\left(x + 1\right) \cdot x}} + \frac{1}{x - 1}\]

4. Applied frac-add 25.6

   \[\leadsto \color{blue}{\frac{\left(1 \cdot x - \left(x + 1\right) \cdot 2\right) \cdot \left(x - 1\right) + \left(\left(x + 1\right) \cdot x\right) \cdot 1}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)}}\]

5. Simplified 25.6

   \[\leadsto \frac{\color{blue}{\left(x - 1\right) \cdot \left(1 \cdot x - \left(1 + x\right) \cdot 2\right) + 1 \cdot \left(x \cdot \left(1 + x\right)\right)}}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)}\]

6. Simplified 25.6

   \[\leadsto \frac{\left(x - 1\right) \cdot \left(1 \cdot x - \left(1 + x\right) \cdot 2\right) + 1 \cdot \left(x \cdot \left(1 + x\right)\right)}{\color{blue}{x \cdot \left(x \cdot x - 1 \cdot 1\right)}}\]

7. Taylor expanded around 0 0.2

   \[\leadsto \frac{\color{blue}{2}}{x \cdot \left(x \cdot x - 1 \cdot 1\right)}\]

8. Final simplification 0.2

   \[\leadsto \frac{2}{x \cdot \left(x \cdot x - 1 \cdot 1\right)}\]

Reproduce

herbie shell --seed 2020196 
(FPCore (x)
  :name "3frac (problem 3.3.3)"
  :precision binary64

  :herbie-target
  (/ 2.0 (* x (- (* x x) 1.0)))

  (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))))