2sqrt (example 3.1)

Average Error: 29.8 → 0.3

Time: 34.2s

Precision: 64

Math

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

↓

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

C

double f(double x) {
        double r7217651 = x;
        double r7217652 = 1.0;
        double r7217653 = r7217651 + r7217652;
        double r7217654 = sqrt(r7217653);
        double r7217655 = sqrt(r7217651);
        double r7217656 = r7217654 - r7217655;
        return r7217656;
}

↓

double f(double x) {
        double r7217657 = 1.0;
        double r7217658 = x;
        double r7217659 = r7217658 + r7217657;
        double r7217660 = sqrt(r7217659);
        double r7217661 = sqrt(r7217658);
        double r7217662 = r7217660 + r7217661;
        double r7217663 = sqrt(r7217662);
        double r7217664 = r7217657 / r7217663;
        double r7217665 = r7217664 / r7217663;
        return r7217665;
}

Error

Bits error versus x

Target

Original	29.8
Target	0.2
Herbie	0.3

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

Derivation

1. Initial program 29.8

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

3. Applied flip-- 29.6

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

4. Taylor expanded around inf 0.2

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

6. Applied add-sqr-sqrt 0.4

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

7. Applied associate-/r* 0.3

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

8. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019104 +o rules:numerics
(FPCore (x)
  :name "2sqrt (example 3.1)"

  :herbie-target
  (/ 1 (+ (sqrt (+ x 1)) (sqrt x)))

  (- (sqrt (+ x 1)) (sqrt x)))