2sqrt (example 3.1)

Average Error: 29.4 → 0.2

Time: 4.8s

Precision: 64

Math

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

↓

\[\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1 + 0}{\sqrt{x + 1} + \sqrt{x}}\right)\right)\]

C

double f(double x) {
        double r158794 = x;
        double r158795 = 1.0;
        double r158796 = r158794 + r158795;
        double r158797 = sqrt(r158796);
        double r158798 = sqrt(r158794);
        double r158799 = r158797 - r158798;
        return r158799;
}

↓

double f(double x) {
        double r158800 = 1.0;
        double r158801 = 0.0;
        double r158802 = r158800 + r158801;
        double r158803 = x;
        double r158804 = r158803 + r158800;
        double r158805 = sqrt(r158804);
        double r158806 = sqrt(r158803);
        double r158807 = r158805 + r158806;
        double r158808 = r158802 / r158807;
        double r158809 = expm1(r158808);
        double r158810 = log1p(r158809);
        return r158810;
}

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-- 29.1

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

4. Simplified 0.2

   \[\leadsto \frac{\color{blue}{1 + 0}}{\sqrt{x + 1} + \sqrt{x}}\]
5. Using strategy rm

6. Applied log1p-expm1-u 0.2

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1 + 0}{\sqrt{x + 1} + \sqrt{x}}\right)\right)}\]

7. Final simplification 0.2

   \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1 + 0}{\sqrt{x + 1} + \sqrt{x}}\right)\right)\]

Reproduce

herbie shell --seed 2020062 +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)))