2sqrt (example 3.1)

Average Error: 30.0 → 0.3

Time: 23.8s

Precision: 64

Math

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

↓

\[\frac{\sqrt{\frac{1}{\sqrt{x + 1} + \sqrt{x}}}}{\mathsf{hypot}\left(\sqrt{\sqrt{x + 1}}, \sqrt{\sqrt{x}}\right)}\]

C

double f(double x) {
        double r3923969 = x;
        double r3923970 = 1.0;
        double r3923971 = r3923969 + r3923970;
        double r3923972 = sqrt(r3923971);
        double r3923973 = sqrt(r3923969);
        double r3923974 = r3923972 - r3923973;
        return r3923974;
}

↓

double f(double x) {
        double r3923975 = 1.0;
        double r3923976 = x;
        double r3923977 = r3923976 + r3923975;
        double r3923978 = sqrt(r3923977);
        double r3923979 = sqrt(r3923976);
        double r3923980 = r3923978 + r3923979;
        double r3923981 = r3923975 / r3923980;
        double r3923982 = sqrt(r3923981);
        double r3923983 = sqrt(r3923978);
        double r3923984 = sqrt(r3923979);
        double r3923985 = hypot(r3923983, r3923984);
        double r3923986 = r3923982 / r3923985;
        return r3923986;
}

Error

Bits error versus x

Target

Original	30.0
Target	0.2
Herbie	0.3

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

Derivation

1. Initial program 30.0

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

3. Applied flip-- 29.8

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

4. Simplified 0.2

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

6. Applied add-sqr-sqrt 0.3

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

8. Applied sqrt-div 0.3

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

9. Applied associate-*l/ 0.2

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

10. Simplified 0.2

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

12. Applied add-sqr-sqrt 0.3

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

13. Applied add-sqr-sqrt 0.3

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

14. Applied sqrt-prod 0.3

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

15. Applied hypot-def 0.3

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

16. Final simplification 0.3

    \[\leadsto \frac{\sqrt{\frac{1}{\sqrt{x + 1} + \sqrt{x}}}}{\mathsf{hypot}\left(\sqrt{\sqrt{x + 1}}, \sqrt{\sqrt{x}}\right)}\]

Reproduce

herbie shell --seed 2019149 +o rules:numerics
(FPCore (x)
  :name "2sqrt (example 3.1)"

  :herbie-target
  (/ 1 (+ (sqrt (+ x 1)) (sqrt x)))

  (- (sqrt (+ x 1)) (sqrt x)))