2sqrt (example 3.1)

Average Error: 30.3 → 0.2

Time: 5.2s

Precision: 64

Math

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

↓

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

C

double f(double x) {
        double r120578 = x;
        double r120579 = 1.0;
        double r120580 = r120578 + r120579;
        double r120581 = sqrt(r120580);
        double r120582 = sqrt(r120578);
        double r120583 = r120581 - r120582;
        return r120583;
}

↓

double f(double x) {
        double r120584 = 1.0;
        double r120585 = 0.0;
        double r120586 = r120584 + r120585;
        double r120587 = x;
        double r120588 = r120587 + r120584;
        double r120589 = sqrt(r120588);
        double r120590 = sqrt(r120587);
        double r120591 = r120589 + r120590;
        double r120592 = r120586 / r120591;
        double r120593 = log1p(r120592);
        double r120594 = expm1(r120593);
        return r120594;
}

Error

Bits error versus x

Target

Original	30.3
Target	0.2
Herbie	0.2

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

Derivation

1. Initial program 30.3

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

3. Applied flip-- 30.1

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

4. Simplified 0.2

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

6. Applied expm1-log1p-u 0.2

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

7. Final simplification 0.2

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

Reproduce

herbie shell --seed 2020001 +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)))