2sqrt (example 3.1)

Average Error: 29.5 → 0.2

Time: 12.4s

Precision: 64

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

Math

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

↓

\[\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{x} + \sqrt{x + 1}}\right)\right)\]

C

double f(double x) {
        double r143284 = x;
        double r143285 = 1.0;
        double r143286 = r143284 + r143285;
        double r143287 = sqrt(r143286);
        double r143288 = sqrt(r143284);
        double r143289 = r143287 - r143288;
        return r143289;
}

↓

double f(double x) {
        double r143290 = 1.0;
        double r143291 = x;
        double r143292 = sqrt(r143291);
        double r143293 = r143291 + r143290;
        double r143294 = sqrt(r143293);
        double r143295 = r143292 + r143294;
        double r143296 = r143290 / r143295;
        double r143297 = log1p(r143296);
        double r143298 = expm1(r143297);
        return r143298;
}

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-- 29.3

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

4. Simplified 0.2

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

5. Simplified 0.2

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

7. Applied expm1-log1p-u 0.2

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{x} + \sqrt{x + 1}}\right)\right)}\]

8. Final simplification 0.2

   \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{x} + \sqrt{x + 1}}\right)\right)\]

Reproduce

herbie shell --seed 2019351 +o rules:numerics
(FPCore (x)
  :name "2sqrt (example 3.1)"
  :precision binary64

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

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