Average Error: 0.8 → 0.3
Time: 18.7s
Precision: 64
\[\left(\sqrt{\left(\frac{x}{\left(1\right)}\right)}\right) - \left(\sqrt{x}\right)\]
↓
\[\left(\left(\frac{\left(\left(1.0\right) \cdot \left(\frac{\left(1\right)}{\left(x - \left(\left(\sqrt{x}\right) \cdot \left(\sqrt{x}\right)\right)\right)}\right)\right)}{\left(\frac{\left(\sqrt{\left(\frac{\left(1\right)}{x}\right)}\right)}{\left(\sqrt{x}\right)}\right)}\right)\right)\]
\left(\sqrt{\left(\frac{x}{\left(1\right)}\right)}\right) - \left(\sqrt{x}\right)
↓
\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)
double f(double x) {
        double r4577320 = x;
        double r4577321 = 1.0;
        double r4577322 = /* ERROR: no posit support in C */;
        double r4577323 = r4577320 + r4577322;
        double r4577324 = sqrt(r4577323);
        double r4577325 = sqrt(r4577320);
        double r4577326 = r4577324 - r4577325;
        return r4577326;
}
↓
double f(double x) {
        double r4577327 = 1.0;
        double r4577328 = /* ERROR: no posit support in C */;
        double r4577329 = 1.0;
        double r4577330 = /* ERROR: no posit support in C */;
        double r4577331 = x;
        double r4577332 = sqrt(r4577331);
        double r4577333 = r4577332 * r4577332;
        double r4577334 = r4577331 - r4577333;
        double r4577335 = r4577330 + r4577334;
        double r4577336 = r4577328 * r4577335;
        double r4577337 = r4577330 + r4577331;
        double r4577338 = sqrt(r4577337);
        double r4577339 = r4577338 + r4577332;
        double r4577340 = r4577336 / r4577339;
        double r4577341 = /*Error: no posit support in C */;
        double r4577342 = /*Error: no posit support in C */;
        return r4577342;
}

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. Simplified0.8

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

  5. Simplified0.8

    \[\leadsto \frac{\left(\left(\frac{\left(\sqrt{\left(\frac{\left(1\right)}{x}\right)}\right)}{\left(\sqrt{x}\right)}\right) \cdot \left(\left(\sqrt{\left(\frac{\left(1\right)}{x}\right)}\right) - \left(\sqrt{x}\right)\right)\right)}{\color{blue}{\left(\frac{\left(\sqrt{\left(\frac{\left(1\right)}{x}\right)}\right)}{\left(\sqrt{x}\right)}\right)}}\]
  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.p160.5

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

  13. Applied introduce-quire0.5

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

  14. Simplified0.3

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

  15. Final simplification0.3

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

Reproduce

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