2sqrt (example 3.1)

Average Error: 0.8 → 0.3

Time: 17.3s

Precision: 64

Math

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

↓

\[\left(\left(\frac{1.0 \cdot \left(1 + \left(x - \sqrt{x} \cdot \sqrt{x}\right)\right)}{\sqrt{1 + x} + \sqrt{x}}\right)\right)\]

C

double f(double x) {
        double r6756234 = x;
        double r6756235 = 1.0;
        double r6756236 = /* ERROR: no posit support in C */;
        double r6756237 = r6756234 + r6756236;
        double r6756238 = sqrt(r6756237);
        double r6756239 = sqrt(r6756234);
        double r6756240 = r6756238 - r6756239;
        return r6756240;
}

↓

double f(double x) {
        double r6756241 = 1.0;
        double r6756242 = 1.0;
        double r6756243 = x;
        double r6756244 = sqrt(r6756243);
        double r6756245 = r6756244 * r6756244;
        double r6756246 = r6756243 - r6756245;
        double r6756247 = r6756242 + r6756246;
        double r6756248 = r6756241 * r6756247;
        double r6756249 = r6756242 + r6756243;
        double r6756250 = sqrt(r6756249);
        double r6756251 = r6756250 + r6756244;
        double r6756252 = r6756248 / r6756251;
        double r6756253 = /*Error: no posit support in C */;
        double r6756254 = /*Error: no posit support in C */;
        return r6756254;
}

Error

Bits error versus x

Derivation

1. Initial program 0.8

   \[\left(\sqrt{\left(\frac{x}{\left(1\right)}\right)}\right) - \left(\sqrt{x}\right)\]
2. Using strategy rm

3. Applied p16-flip-- 0.6

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

4. Simplified 0.8

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

5. Simplified 0.8

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

7. Applied p16-flip-- 0.6

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

8. Applied associate-*r/ 0.6

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

9. Applied associate-/l/ 0.6

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

11. Applied sqrt-sqrd.p16 0.5

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

13. Applied introduce-quire 0.5

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

14. Simplified 0.3

    \[\leadsto \color{blue}{\left(\left(\frac{\left(\left(1.0\right) \cdot \left(\frac{\left(1\right)}{\left(x - \left(\left(\sqrt{x}\right) \cdot \left(\sqrt{x}\right)\right)\right)}\right)\right)}{\left(\frac{\left(\sqrt{\left(\frac{\left(1\right)}{x}\right)}\right)}{\left(\sqrt{x}\right)}\right)}\right)\right)}\]

15. Final simplification 0.3

    \[\leadsto \left(\left(\frac{1.0 \cdot \left(1 + \left(x - \sqrt{x} \cdot \sqrt{x}\right)\right)}{\sqrt{1 + x} + \sqrt{x}}\right)\right)\]

Reproduce

herbie shell --seed 2019163 
(FPCore (x)
  :name "2sqrt (example 3.1)"
  (-.p16 (sqrt.p16 (+.p16 x (real->posit16 1))) (sqrt.p16 x)))