2sqrt (example 3.1)

Average Error: 29.3 → 0.2

Time: 18.4s

Precision: 64

Math

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

↓

\[\frac{\left(1\right)}{\sqrt{x + 1} + \sqrt{x}}\]

C

double f(double x) {
        double r3175327 = x;
        double r3175328 = 1.0;
        double r3175329 = r3175327 + r3175328;
        double r3175330 = sqrt(r3175329);
        double r3175331 = sqrt(r3175327);
        double r3175332 = r3175330 - r3175331;
        return r3175332;
}

↓

double f(double x) {
        double r3175333 = 1.0;
        double r3175334 = /* ERROR: no posit support in C */;
        double r3175335 = /* ERROR: no posit support in C */;
        double r3175336 = x;
        double r3175337 = r3175336 + r3175333;
        double r3175338 = sqrt(r3175337);
        double r3175339 = sqrt(r3175336);
        double r3175340 = r3175338 + r3175339;
        double r3175341 = r3175335 / r3175340;
        return r3175341;
}

Error

Bits error versus x

Target

Original	29.3
Target	0.2
Herbie	0.2

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

Derivation

1. Initial program 29.3

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

3. Applied flip-- 29.1

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

4. Simplified 28.8

   \[\leadsto \frac{\color{blue}{\left(1 + x\right) - x}}{\sqrt{x + 1} + \sqrt{x}}\]

5. Simplified 28.8

   \[\leadsto \frac{\left(1 + x\right) - x}{\color{blue}{\sqrt{x} + \sqrt{1 + x}}}\]
6. Using strategy rm

7. Applied insert-posit16 28.8

   \[\leadsto \frac{\color{blue}{\left(\left(\left(1 + x\right) - x\right)\right)}}{\sqrt{x} + \sqrt{1 + x}}\]

8. Simplified 0.2

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

9. Final simplification 0.2

   \[\leadsto \frac{\left(1\right)}{\sqrt{x + 1} + \sqrt{x}}\]

Reproduce

herbie shell --seed 2019164 +o rules:numerics
(FPCore (x)
  :name "2sqrt (example 3.1)"

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

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