Average Error: 0.6 → 0.6
Time: 19.1s
Precision: 64
\[\left(\frac{\left(1\right)}{\left(\sqrt{x}\right)}\right) - \left(\frac{\left(1\right)}{\left(\sqrt{\left(\frac{x}{\left(1\right)}\right)}\right)}\right)\]
↓
\[\frac{\left(\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}\right) \cdot \left(\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\right)}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}\]
\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)
↓
\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 r3509362 = 1.0;
        double r3509363 = /* ERROR: no posit support in C */;
        double r3509364 = x;
        double r3509365 = sqrt(r3509364);
        double r3509366 = r3509363 / r3509365;
        double r3509367 = r3509364 + r3509363;
        double r3509368 = sqrt(r3509367);
        double r3509369 = r3509363 / r3509368;
        double r3509370 = r3509366 - r3509369;
        return r3509370;
}
↓
double f(double x) {
        double r3509371 = 1.0;
        double r3509372 = x;
        double r3509373 = sqrt(r3509372);
        double r3509374 = r3509371 / r3509373;
        double r3509375 = r3509372 + r3509371;
        double r3509376 = sqrt(r3509375);
        double r3509377 = r3509371 / r3509376;
        double r3509378 = r3509374 + r3509377;
        double r3509379 = r3509374 - r3509377;
        double r3509380 = r3509378 * r3509379;
        double r3509381 = r3509380 / r3509378;
        return r3509381;
}

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 2019107 
(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))))))