2isqrt (example 3.6)

Average Error: 0.6 → 0.6

Time: 50.1s

Precision: 64

Math

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

↓

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

C

double f(double x) {
        double r2038153 = 1.0;
        double r2038154 = /* ERROR: no posit support in C */;
        double r2038155 = x;
        double r2038156 = sqrt(r2038155);
        double r2038157 = r2038154 / r2038156;
        double r2038158 = r2038155 + r2038154;
        double r2038159 = sqrt(r2038158);
        double r2038160 = r2038154 / r2038159;
        double r2038161 = r2038157 - r2038160;
        return r2038161;
}

↓

double f(double x) {
        double r2038162 = 1.0;
        double r2038163 = x;
        double r2038164 = r2038162 + r2038163;
        double r2038165 = sqrt(r2038164);
        double r2038166 = r2038162 / r2038165;
        double r2038167 = sqrt(r2038163);
        double r2038168 = r2038162 / r2038167;
        double r2038169 = r2038166 + r2038168;
        double r2038170 = r2038168 - r2038166;
        double r2038171 = r2038170 / r2038169;
        double r2038172 = r2038169 * r2038171;
        return r2038172;
}

Error

Bits error versus x

Derivation

1. Initial program 0.6

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

3. Applied p16-flip-- 0.7

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

5. Applied p16-*-un-lft-identity 0.7

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

6. Applied p16-*-un-lft-identity 0.7

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

7. Applied p16-times-frac 0.7

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

8. Applied *p16-lft-identity-expand 0.7

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

9. Applied p16-*-un-lft-identity 0.7

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

10. Applied p16-times-frac 0.7

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

11. Applied distribute-lft-out 0.7

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

12. Applied difference-of-squares 0.6

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

13. Applied p16-times-frac 0.6

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

14. Simplified 0.6

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

15. Simplified 0.6

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

16. Final simplification 0.6

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

Reproduce

herbie shell --seed 2019153 
(FPCore (x)
  :name "2isqrt (example 3.6)"
  (-.p16 (/.p16 (real->posit16 1) (sqrt.p16 x)) (/.p16 (real->posit16 1) (sqrt.p16 (+.p16 x (real->posit16 1))))))