Average Error: 0.8 → 0.6
Time: 16.9s
Precision: 64
\[\left(\sqrt{\left(\frac{x}{\left(1\right)}\right)}\right) - \left(\sqrt{x}\right)\]
↓
\[\frac{\left(\sqrt{1 + x} + \sqrt{x}\right) \cdot \sqrt{1 + x} + \left(\sqrt{1 + x} + \sqrt{x}\right) \cdot \left(-\sqrt{x}\right)}{\sqrt{1 + x} + \sqrt{x}}\]
\left(\sqrt{\left(\frac{x}{\left(1\right)}\right)}\right) - \left(\sqrt{x}\right)
↓
\frac{\left(\sqrt{1 + x} + \sqrt{x}\right) \cdot \sqrt{1 + x} + \left(\sqrt{1 + x} + \sqrt{x}\right) \cdot \left(-\sqrt{x}\right)}{\sqrt{1 + x} + \sqrt{x}}
double f(double x) {
        double r1859350 = x;
        double r1859351 = 1.0;
        double r1859352 = /* ERROR: no posit support in C */;
        double r1859353 = r1859350 + r1859352;
        double r1859354 = sqrt(r1859353);
        double r1859355 = sqrt(r1859350);
        double r1859356 = r1859354 - r1859355;
        return r1859356;
}
↓
double f(double x) {
        double r1859357 = 1.0;
        double r1859358 = x;
        double r1859359 = r1859357 + r1859358;
        double r1859360 = sqrt(r1859359);
        double r1859361 = sqrt(r1859358);
        double r1859362 = r1859360 + r1859361;
        double r1859363 = r1859362 * r1859360;
        double r1859364 = -r1859361;
        double r1859365 = r1859362 * r1859364;
        double r1859366 = r1859363 + r1859365;
        double r1859367 = r1859366 / r1859362;
        return r1859367;
}

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 sub-neg0.8

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

  8. Applied distribute-lft-in0.6

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

  9. Final simplification0.6

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

Reproduce

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