2sqrt (example 3.1)

Average Error: 29.9 → 0.2

Time: 20.9s

Precision: 64

Math

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

↓

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

C

double f(double x) {
        double r5228725 = x;
        double r5228726 = 1.0;
        double r5228727 = r5228725 + r5228726;
        double r5228728 = sqrt(r5228727);
        double r5228729 = sqrt(r5228725);
        double r5228730 = r5228728 - r5228729;
        return r5228730;
}

↓

double f(double x) {
        double r5228731 = 1.0;
        double r5228732 = /* ERROR: no posit support in C */;
        double r5228733 = /* ERROR: no posit support in C */;
        double r5228734 = x;
        double r5228735 = r5228734 + r5228731;
        double r5228736 = sqrt(r5228735);
        double r5228737 = sqrt(r5228734);
        double r5228738 = r5228736 + r5228737;
        double r5228739 = r5228733 / r5228738;
        return r5228739;
}

Error

Bits error versus x

Target

Original	29.9
Target	0.2
Herbie	0.2

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

Derivation

1. Initial program 29.9

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

3. Applied flip-- 29.6

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

4. Simplified 29.2

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

5. Simplified 29.2

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

7. Applied insert-posit16 29.2

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

8. Simplified 0.2

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

9. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (x)
  :name "2sqrt (example 3.1)"

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

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