2isqrt (example 3.6)

Average Error: 19.5 → 0.3

Time: 3.5s

Precision: binary64

Math

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

↓

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

C

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

↓

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

Error

Bits error versus x

Target

Original	19.5
Target	0.6
Herbie	0.3

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

Derivation

1. Initial program 19.5

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

3. Applied frac-sub 19.5

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

4. Simplified 19.5

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

5. Simplified 19.5

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

7. Applied flip-- 19.3

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

8. Simplified 0.4

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

10. Applied associate-/r* 0.4

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

11. Simplified 0.3

    \[\leadsto \frac{\color{blue}{\frac{1 \cdot 1}{x + \sqrt{x} \cdot \sqrt{x + 1}}}}{\sqrt{1 + x}}\]
12. Using strategy rm

13. Applied add-sqr-sqrt 0.4

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

14. Applied distribute-lft-out 0.4

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

15. Applied times-frac 0.4

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

16. Applied associate-/l* 0.4

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

17. Simplified 0.3

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

18. Final simplification 0.3

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

Reproduce

herbie shell --seed 2020185 
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64

  :herbie-target
  (/ 1.0 (+ (* (+ x 1.0) (sqrt x)) (* x (sqrt (+ x 1.0)))))

  (- (/ 1.0 (sqrt x)) (/ 1.0 (sqrt (+ x 1.0)))))