2sqrt (example 3.1)

Average Error: 29.5 → 0.3

Time: 6.1s

Precision: 64

Math

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

↓

\[\sqrt{\frac{1 + 0}{\sqrt{x + 1} + \sqrt{x}}} \cdot \sqrt{\frac{1 + 0}{\sqrt{x + 1} + \sqrt{x}}}\]

C

double f(double x) {
        double r95069 = x;
        double r95070 = 1.0;
        double r95071 = r95069 + r95070;
        double r95072 = sqrt(r95071);
        double r95073 = sqrt(r95069);
        double r95074 = r95072 - r95073;
        return r95074;
}

↓

double f(double x) {
        double r95075 = 1.0;
        double r95076 = 0.0;
        double r95077 = r95075 + r95076;
        double r95078 = x;
        double r95079 = r95078 + r95075;
        double r95080 = sqrt(r95079);
        double r95081 = sqrt(r95078);
        double r95082 = r95080 + r95081;
        double r95083 = r95077 / r95082;
        double r95084 = sqrt(r95083);
        double r95085 = r95084 * r95084;
        return r95085;
}

Error

Bits error versus x

Target

Original	29.5
Target	0.2
Herbie	0.3

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

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 add-sqr-sqrt 0.3

   \[\leadsto \color{blue}{\sqrt{\frac{1 + 0}{\sqrt{x + 1} + \sqrt{x}}} \cdot \sqrt{\frac{1 + 0}{\sqrt{x + 1} + \sqrt{x}}}}\]

7. Final simplification 0.3

   \[\leadsto \sqrt{\frac{1 + 0}{\sqrt{x + 1} + \sqrt{x}}} \cdot \sqrt{\frac{1 + 0}{\sqrt{x + 1} + \sqrt{x}}}\]

Reproduce

herbie shell --seed 2020089 +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)))