2sqrt (example 3.1)

Average Error: 29.4 → 0.2

Time: 14.2s

Precision: 64

Math

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

↓

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

C

double f(double x) {
        double r2278061 = x;
        double r2278062 = 1.0;
        double r2278063 = r2278061 + r2278062;
        double r2278064 = sqrt(r2278063);
        double r2278065 = sqrt(r2278061);
        double r2278066 = r2278064 - r2278065;
        return r2278066;
}

↓

double f(double x) {
        double r2278067 = 1.0;
        double r2278068 = x;
        double r2278069 = sqrt(r2278068);
        double r2278070 = hypot(r2278069, r2278067);
        double r2278071 = r2278070 + r2278069;
        double r2278072 = r2278067 / r2278071;
        return r2278072;
}

Error

Bits error versus x

Target

Original	29.4
Target	0.2
Herbie	0.2

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

Derivation

1. Initial program 29.4

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

3. Applied flip-- 29.2

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

4. Simplified 28.8

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

5. Taylor expanded around 0 0.2

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

7. Applied *-un-lft-identity 0.2

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

8. Applied add-sqr-sqrt 0.2

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

9. Applied hypot-def 0.2

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

10. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019153 +o rules:numerics
(FPCore (x)
  :name "2sqrt (example 3.1)"

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

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