Average Error: 0.8 → 0.4
Time: 16.2s
Precision: 64
\[\left(\sqrt{\left(\frac{x}{\left(1\right)}\right)}\right) - \left(\sqrt{x}\right)\]
↓
\[\frac{\left(\sqrt{1 + x} + \sqrt{x}\right) \cdot \left(\mathsf{qms}\left(\left(\mathsf{qma}\left(\left(1\right), x, 1.0\right)\right), \left(\sqrt{x} \cdot \sqrt{x}\right), 1.0\right)\right)}{\left(\sqrt{1 + x} + \sqrt{x}\right) \cdot \left(\sqrt{1 + x} + \sqrt{x}\right)}\]
\left(\sqrt{\left(\frac{x}{\left(1\right)}\right)}\right) - \left(\sqrt{x}\right)
↓
\frac{\left(\sqrt{1 + x} + \sqrt{x}\right) \cdot \left(\mathsf{qms}\left(\left(\mathsf{qma}\left(\left(1\right), x, 1.0\right)\right), \left(\sqrt{x} \cdot \sqrt{x}\right), 1.0\right)\right)}{\left(\sqrt{1 + x} + \sqrt{x}\right) \cdot \left(\sqrt{1 + x} + \sqrt{x}\right)}
double f(double x) {
        double r894514 = x;
        double r894515 = 1.0;
        double r894516 = /* ERROR: no posit support in C */;
        double r894517 = r894514 + r894516;
        double r894518 = sqrt(r894517);
        double r894519 = sqrt(r894514);
        double r894520 = r894518 - r894519;
        return r894520;
}
↓
double f(double x) {
        double r894521 = 1.0;
        double r894522 = x;
        double r894523 = r894521 + r894522;
        double r894524 = sqrt(r894523);
        double r894525 = sqrt(r894522);
        double r894526 = r894524 + r894525;
        double r894527 = /*Error: no posit support in C */;
        double r894528 = 1.0;
        double r894529 = /*Error: no posit support in C */;
        double r894530 = r894525 * r894525;
        double r894531 = /*Error: no posit support in C */;
        double r894532 = /*Error: no posit support in C */;
        double r894533 = r894526 * r894532;
        double r894534 = r894526 * r894526;
        double r894535 = r894533 / r894534;
        return r894535;
}

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

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

  14. Applied insert-quire-add0.5

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

  15. Applied insert-quire-sub0.4

    \[\leadsto \frac{\left(\left(\frac{\left(\sqrt{\left(\frac{\left(1\right)}{x}\right)}\right)}{\left(\sqrt{x}\right)}\right) \cdot \color{blue}{\left(\left(\mathsf{qms}\left(\left(\mathsf{qma}\left(\left(\left(1\right)\right), x, \left(1.0\right)\right)\right), \left(\left(\sqrt{x}\right) \cdot \left(\sqrt{x}\right)\right), \left(1.0\right)\right)\right)\right)}\right)}{\left(\left(\frac{\left(\sqrt{\left(\frac{\left(1\right)}{x}\right)}\right)}{\left(\sqrt{x}\right)}\right) \cdot \left(\frac{\left(\sqrt{\left(\frac{\left(1\right)}{x}\right)}\right)}{\left(\sqrt{x}\right)}\right)\right)}\]

  16. Final simplification0.4

    \[\leadsto \frac{\left(\sqrt{1 + x} + \sqrt{x}\right) \cdot \left(\mathsf{qms}\left(\left(\mathsf{qma}\left(\left(1\right), x, 1.0\right)\right), \left(\sqrt{x} \cdot \sqrt{x}\right), 1.0\right)\right)}{\left(\sqrt{1 + x} + \sqrt{x}\right) \cdot \left(\sqrt{1 + x} + \sqrt{x}\right)}\]

Reproduce

herbie shell --seed 2019153 +o rules:numerics
(FPCore (x)
  :name "2sqrt (example 3.1)"
  (-.p16 (sqrt.p16 (+.p16 x (real->posit16 1))) (sqrt.p16 x)))