2sqrt (example 3.1)

Average Error: 29.5 → 0.2

Time: 5.1s

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 r126985 = x;
        double r126986 = 1.0;
        double r126987 = r126985 + r126986;
        double r126988 = sqrt(r126987);
        double r126989 = sqrt(r126985);
        double r126990 = r126988 - r126989;
        return r126990;
}

↓

double f(double x) {
        double r126991 = 1.0;
        double r126992 = 0.0;
        double r126993 = r126991 + r126992;
        double r126994 = x;
        double r126995 = r126994 + r126991;
        double r126996 = sqrt(r126995);
        double r126997 = sqrt(r126994);
        double r126998 = r126996 + r126997;
        double r126999 = r126993 / r126998;
        double r127000 = log1p(r126999);
        double r127001 = expm1(r127000);
        return r127001;
}

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-- 29.2

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

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 2020027 +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)))