Average Error: 0.8 → 0.2
Time: 30.2s
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 r6681537 = x;
        double r6681538 = 1.0;
        double r6681539 = /* ERROR: no posit support in C */;
        double r6681540 = r6681537 + r6681539;
        double r6681541 = sqrt(r6681540);
        double r6681542 = sqrt(r6681537);
        double r6681543 = r6681541 - r6681542;
        return r6681543;
}
↓
double f(double x) {
        double r6681544 = 1.0;
        double r6681545 = x;
        double r6681546 = r6681545 + r6681544;
        double r6681547 = sqrt(r6681546);
        double r6681548 = sqrt(r6681545);
        double r6681549 = r6681547 + r6681548;
        double r6681550 = r6681544 / r6681549;
        return r6681550;
}

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 2019133 +o rules:numerics
(FPCore (x)
  :name "2sqrt (example 3.1)"
  (-.p16 (sqrt.p16 (+.p16 x (real->posit16 1))) (sqrt.p16 x)))