Average Error: 0.6 → 0.6
Time: 19.3s
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{\left(\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}\right) \cdot \left(\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\right)}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}\]
double f(double x) {
        double r4347124 = 1.0;
        double r4347125 = x;
        double r4347126 = sqrt(r4347125);
        double r4347127 = r4347124 / r4347126;
        double r4347128 = r4347125 + r4347124;
        double r4347129 = sqrt(r4347128);
        double r4347130 = r4347124 / r4347129;
        double r4347131 = r4347127 - r4347130;
        return r4347131;
}
↓
double f(double x) {
        double r4347132 = 1.0;
        double r4347133 = x;
        double r4347134 = sqrt(r4347133);
        double r4347135 = r4347132 / r4347134;
        double r4347136 = r4347133 + r4347132;
        double r4347137 = sqrt(r4347136);
        double r4347138 = r4347132 / r4347137;
        double r4347139 = r4347135 + r4347138;
        double r4347140 = r4347135 - r4347138;
        double r4347141 = r4347139 * r4347140;
        double r4347142 = r4347141 / r4347139;
        return r4347142;
}


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

Error

Bits error versus x

Derivation

  1. Initial program 0.6

    \[\left(\frac{\left(1\right)}{\left(\sqrt{x}\right)}\right) - \left(\frac{\left(1\right)}{\left(\sqrt{\left(\frac{x}{\left(1\right)}\right)}\right)}\right)\]
  2. Using strategy rm

  3. Applied p16-flip--0.7

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

  5. Applied difference-of-squares0.6

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

  6. Final simplification0.6

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

Reproduce

herbie shell --seed 2019102 +o rules:numerics
(FPCore (x)
  :name "2isqrt (example 3.6)"
  (-.p16 (/.p16 (real->posit16 1) (sqrt.p16 x)) (/.p16 (real->posit16 1) (sqrt.p16 (+.p16 x (real->posit16 1))))))