sqrt sqr

Average Error: 32.3 → 0

Time: 2.3s

Precision: 64

Math

\[\frac{x}{x} - \frac{1}{x} \cdot \sqrt{x \cdot x}\]

↓

\[1 - \frac{\left|x\right| \cdot 1}{x}\]

C

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

↓

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

Error

Bits error versus x

Target

Original	32.3
Target	0
Herbie	0

\[\begin{array}{l} \mathbf{if}\;x \lt 0.0:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;0.0\\ \end{array}\]

Derivation

1. Initial program 32.3

   \[\frac{x}{x} - \frac{1}{x} \cdot \sqrt{x \cdot x}\]
2. Using strategy rm

3. Applied *-un-lft-identity 32.3

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

4. Applied *-un-lft-identity 32.3

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

5. Applied times-frac 32.3

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

6. Applied associate-*l* 32.3

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

7. Applied *-un-lft-identity 32.3

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

8. Applied *-un-lft-identity 32.3

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

9. Applied times-frac 32.3

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

10. Applied distribute-lft-out-- 32.3

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

11. Simplified 0

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

12. Final simplification 0

    \[\leadsto 1 - \frac{\left|x\right| \cdot 1}{x}\]

Reproduce

herbie shell --seed 2020066 +o rules:numerics
(FPCore (x)
  :name "sqrt sqr"
  :precision binary64

  :herbie-target
  (if (< x 0.0) 2 0.0)

  (- (/ x x) (* (/ 1 x) (sqrt (* x x)))))