2sqrt (example 3.1)

Average Error: 30.3 → 0.2

Time: 4.8s

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 r128009 = x;
        double r128010 = 1.0;
        double r128011 = r128009 + r128010;
        double r128012 = sqrt(r128011);
        double r128013 = sqrt(r128009);
        double r128014 = r128012 - r128013;
        return r128014;
}

↓

double f(double x) {
        double r128015 = 1.0;
        double r128016 = 0.0;
        double r128017 = r128015 + r128016;
        double r128018 = x;
        double r128019 = r128018 + r128015;
        double r128020 = sqrt(r128019);
        double r128021 = sqrt(r128018);
        double r128022 = r128020 + r128021;
        double r128023 = r128017 / r128022;
        double r128024 = log1p(r128023);
        double r128025 = expm1(r128024);
        return r128025;
}

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)))