Average Error: 0.8 → 0.6
Time: 21.0s
Precision: 64
\[\left(\sqrt{\left(\frac{x}{\left(1\right)}\right)}\right) - \left(\sqrt{x}\right)\]
↓
\[\frac{\sqrt{1 + x} \cdot \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \sqrt{x}\right) + \sqrt{x} \cdot \left(-\sqrt{x}\right)}{\sqrt{1 + x} + \sqrt{x}}\]
\left(\sqrt{\left(\frac{x}{\left(1\right)}\right)}\right) - \left(\sqrt{x}\right)
↓
\frac{\sqrt{1 + x} \cdot \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \sqrt{x}\right) + \sqrt{x} \cdot \left(-\sqrt{x}\right)}{\sqrt{1 + x} + \sqrt{x}}
double f(double x) {
        double r1927353 = x;
        double r1927354 = 1.0;
        double r1927355 = /* ERROR: no posit support in C */;
        double r1927356 = r1927353 + r1927355;
        double r1927357 = sqrt(r1927356);
        double r1927358 = sqrt(r1927353);
        double r1927359 = r1927357 - r1927358;
        return r1927359;
}
↓
double f(double x) {
        double r1927360 = 1.0;
        double r1927361 = x;
        double r1927362 = r1927360 + r1927361;
        double r1927363 = sqrt(r1927362);
        double r1927364 = sqrt(r1927361);
        double r1927365 = r1927363 - r1927364;
        double r1927366 = r1927365 + r1927364;
        double r1927367 = r1927363 * r1927366;
        double r1927368 = -r1927364;
        double r1927369 = r1927364 * r1927368;
        double r1927370 = r1927367 + r1927369;
        double r1927371 = r1927363 + r1927364;
        double r1927372 = r1927370 / r1927371;
        return r1927372;
}

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-rgt-identity-expand0.8

    \[\leadsto \frac{\left(\left(\frac{\left(\sqrt{\left(\frac{\left(1\right)}{x}\right)}\right)}{\left(\sqrt{x}\right)}\right) \cdot \left(\color{blue}{\left(\frac{\left(\sqrt{\left(\frac{\left(1\right)}{x}\right)}\right)}{\left(0.0\right)}\right)} - \left(\sqrt{x}\right)\right)\right)}{\left(\frac{\left(\sqrt{\left(\frac{\left(1\right)}{x}\right)}\right)}{\left(\sqrt{x}\right)}\right)}\]

  8. Applied associate--l+0.8

    \[\leadsto \frac{\left(\left(\frac{\left(\sqrt{\left(\frac{\left(1\right)}{x}\right)}\right)}{\left(\sqrt{x}\right)}\right) \cdot \color{blue}{\left(\frac{\left(\sqrt{\left(\frac{\left(1\right)}{x}\right)}\right)}{\left(\left(0.0\right) - \left(\sqrt{x}\right)\right)}\right)}\right)}{\left(\frac{\left(\sqrt{\left(\frac{\left(1\right)}{x}\right)}\right)}{\left(\sqrt{x}\right)}\right)}\]

  9. Applied distribute-rgt-in0.6

    \[\leadsto \frac{\color{blue}{\left(\frac{\left(\left(\sqrt{\left(\frac{\left(1\right)}{x}\right)}\right) \cdot \left(\frac{\left(\sqrt{\left(\frac{\left(1\right)}{x}\right)}\right)}{\left(\sqrt{x}\right)}\right)\right)}{\left(\left(\left(0.0\right) - \left(\sqrt{x}\right)\right) \cdot \left(\frac{\left(\sqrt{\left(\frac{\left(1\right)}{x}\right)}\right)}{\left(\sqrt{x}\right)}\right)\right)}\right)}}{\left(\frac{\left(\sqrt{\left(\frac{\left(1\right)}{x}\right)}\right)}{\left(\sqrt{x}\right)}\right)}\]

  10. Simplified0.6

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

  12. Applied distribute-rgt-in0.7

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

  13. Applied associate-+r+0.9

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

  14. Simplified0.6

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

  15. Final simplification0.6

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

Reproduce

herbie shell --seed 2019154 
(FPCore (x)
  :name "2sqrt (example 3.1)"
  (-.p16 (sqrt.p16 (+.p16 x (real->posit16 1))) (sqrt.p16 x)))