2sqrt (example 3.1)

Average Error: 29.5 → 0.2

Time: 8.2s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

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

C

double f(double x) {
        double r125781 = x;
        double r125782 = 1.0;
        double r125783 = r125781 + r125782;
        double r125784 = sqrt(r125783);
        double r125785 = sqrt(r125781);
        double r125786 = r125784 - r125785;
        return r125786;
}

↓

double f(double x) {
        double r125787 = 1.0;
        double r125788 = x;
        double r125789 = r125788 + r125787;
        double r125790 = sqrt(r125789);
        double r125791 = sqrt(r125788);
        double r125792 = r125790 + r125791;
        double r125793 = r125787 / r125792;
        return r125793;
}

Error

Bits error versus x

Target

Original	29.5
Target	0.2
Herbie	0.2

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

Derivation

1. Initial program 29.5

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

3. Applied flip-- 29.4

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

4. Simplified 0.2

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

5. Final simplification 0.2

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

Reproduce

herbie shell --seed 2020046 
(FPCore (x)
  :name "2sqrt (example 3.1)"
  :precision binary64

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

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