Average Error: 0.8 → 0.2
Time: 5.1s
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 r2754462 = x;
        double r2754463 = 1.0;
        double r2754464 = /* ERROR: no posit support in C */;
        double r2754465 = r2754462 + r2754464;
        double r2754466 = sqrt(r2754465);
        double r2754467 = sqrt(r2754462);
        double r2754468 = r2754466 - r2754467;
        return r2754468;
}
↓
double f(double x) {
        double r2754469 = 1.0;
        double r2754470 = x;
        double r2754471 = r2754470 + r2754469;
        double r2754472 = sqrt(r2754471);
        double r2754473 = sqrt(r2754470);
        double r2754474 = r2754472 + r2754473;
        double r2754475 = r2754469 / r2754474;
        return r2754475;
}

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 2019128 
(FPCore (x)
  :name "2sqrt (example 3.1)"
  (-.p16 (sqrt.p16 (+.p16 x (real->posit16 1))) (sqrt.p16 x)))