Average Error: 0.8 → 0.2
Time: 4.5s
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 r3617423 = x;
        double r3617424 = 1.0;
        double r3617425 = /* ERROR: no posit support in C */;
        double r3617426 = r3617423 + r3617425;
        double r3617427 = sqrt(r3617426);
        double r3617428 = sqrt(r3617423);
        double r3617429 = r3617427 - r3617428;
        return r3617429;
}
↓
double f(double x) {
        double r3617430 = 1.0;
        double r3617431 = x;
        double r3617432 = r3617431 + r3617430;
        double r3617433 = sqrt(r3617432);
        double r3617434 = sqrt(r3617431);
        double r3617435 = r3617433 + r3617434;
        double r3617436 = r3617430 / r3617435;
        return r3617436;
}

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 2019125 
(FPCore (x)
  :name "2sqrt (example 3.1)"
  (-.p16 (sqrt.p16 (+.p16 x (real->posit16 1))) (sqrt.p16 x)))