Average Error: 0.8 → 0.4
Time: 15.9s
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 r940053 = x;
        double r940054 = 1.0;
        double r940055 = /* ERROR: no posit support in C */;
        double r940056 = r940053 + r940055;
        double r940057 = sqrt(r940056);
        double r940058 = sqrt(r940053);
        double r940059 = r940057 - r940058;
        return r940059;
}
↓
double f(double x) {
        double r940060 = 1.0;
        double r940061 = x;
        double r940062 = r940060 + r940061;
        double r940063 = sqrt(r940062);
        double r940064 = sqrt(r940061);
        double r940065 = r940063 + r940064;
        double r940066 = /*Error: no posit support in C */;
        double r940067 = 1.0;
        double r940068 = /*Error: no posit support in C */;
        double r940069 = r940064 * r940064;
        double r940070 = /*Error: no posit support in C */;
        double r940071 = /*Error: no posit support in C */;
        double r940072 = r940065 * r940071;
        double r940073 = r940065 * r940065;
        double r940074 = r940072 / r940073;
        return r940074;
}

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