Average Error: 0.8 → 0.3
Time: 9.4s
Precision: 64
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[\frac{x + \left(1 - x\right)}{\sqrt{x + 1} + \sqrt{x}}\]
double f(double x) {
        double r1812252 = x;
        double r1812253 = 1.0;
        double r1812254 = r1812252 + r1812253;
        double r1812255 = sqrt(r1812254);
        double r1812256 = sqrt(r1812252);
        double r1812257 = r1812255 - r1812256;
        return r1812257;
}
↓
double f(double x) {
        double r1812258 = x;
        double r1812259 = 1.0;
        double r1812260 = r1812259 - r1812258;
        double r1812261 = r1812258 + r1812260;
        double r1812262 = r1812258 + r1812259;
        double r1812263 = sqrt(r1812262);
        double r1812264 = sqrt(r1812258);
        double r1812265 = r1812263 + r1812264;
        double r1812266 = r1812261 / r1812265;
        return r1812266;
}


\sqrt{x + 1} - \sqrt{x}
↓
\frac{x + \left(1 - x\right)}{\sqrt{x + 1} + \sqrt{x}}

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. Using strategy rm

  5. Applied sqrt-sqrd.p160.5

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

  7. Applied sqrt-sqrd.p160.4

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

  9. Applied associate--l+0.3

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

  10. Final simplification0.3

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

Reproduce

herbie shell --seed 2019101 +o rules:numerics
(FPCore (x)
  :name "2sqrt (example 3.1)"
  (-.p16 (sqrt.p16 (+.p16 x (real->posit16 1))) (sqrt.p16 x)))