Average Error: 0.8 → 0.2
Time: 29.5s
Precision: 64
\[\left(\sqrt{\left(\frac{x}{\left(1\right)}\right)}\right) - \left(\sqrt{x}\right)\]
↓
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]
\left(\sqrt{\left(\frac{x}{\left(1\right)}\right)}\right) - \left(\sqrt{x}\right)
↓
\frac{1}{\sqrt{x + 1} + \sqrt{x}}
double f(double x) {
        double r5151064 = x;
        double r5151065 = 1.0;
        double r5151066 = /* ERROR: no posit support in C */;
        double r5151067 = r5151064 + r5151066;
        double r5151068 = sqrt(r5151067);
        double r5151069 = sqrt(r5151064);
        double r5151070 = r5151068 - r5151069;
        return r5151070;
}
↓
double f(double x) {
        double r5151071 = 1.0;
        double r5151072 = x;
        double r5151073 = r5151072 + r5151071;
        double r5151074 = sqrt(r5151073);
        double r5151075 = sqrt(r5151072);
        double r5151076 = r5151074 + r5151075;
        double r5151077 = r5151071 / r5151076;
        return r5151077;
}

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. Simplified0.2

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

  5. Final simplification0.2

    \[\leadsto \frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Reproduce

herbie shell --seed 2019135 +o rules:numerics
(FPCore (x)
  :name "2sqrt (example 3.1)"
  (-.p16 (sqrt.p16 (+.p16 x (real->posit16 1))) (sqrt.p16 x)))