Average Error: 0.8 → 0.2
Time: 10.9s
Precision: 64
\[\left(\sqrt{\left(\frac{x}{\left(1\right)}\right)}\right) - \left(\sqrt{x}\right)\]
↓
\[\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot 1.0\]
\left(\sqrt{\left(\frac{x}{\left(1\right)}\right)}\right) - \left(\sqrt{x}\right)
↓
\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot 1.0
double f(double x) {
        double r1759833 = x;
        double r1759834 = 1.0;
        double r1759835 = /* ERROR: no posit support in C */;
        double r1759836 = r1759833 + r1759835;
        double r1759837 = sqrt(r1759836);
        double r1759838 = sqrt(r1759833);
        double r1759839 = r1759837 - r1759838;
        return r1759839;
}
↓
double f(double x) {
        double r1759840 = 1.0;
        double r1759841 = x;
        double r1759842 = r1759840 + r1759841;
        double r1759843 = sqrt(r1759842);
        double r1759844 = sqrt(r1759841);
        double r1759845 = r1759843 + r1759844;
        double r1759846 = r1759840 / r1759845;
        double r1759847 = 1.0;
        double r1759848 = r1759846 * r1759847;
        return r1759848;
}

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. Using strategy rm

  5. Applied p16-flip--1.0

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

  6. Applied associate-/l/1.0

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

  7. Simplified0.5

    \[\leadsto \frac{\color{blue}{\left(\left(\frac{\left(0.0\right)}{\left(1\right)}\right) \cdot \left(\frac{\left(\frac{x}{x}\right)}{\left(1\right)}\right)\right)}}{\left(\left(\frac{\left(\sqrt{\left(\frac{x}{\left(1\right)}\right)}\right)}{\left(\sqrt{x}\right)}\right) \cdot \left(\frac{\left(\left(\sqrt{\left(\frac{x}{\left(1\right)}\right)}\right) \cdot \left(\sqrt{\left(\frac{x}{\left(1\right)}\right)}\right)\right)}{\left(\left(\sqrt{x}\right) \cdot \left(\sqrt{x}\right)\right)}\right)\right)}\]

  8. Simplified0.2

    \[\leadsto \color{blue}{\left(\frac{\left(1\right)}{\left(\frac{\left(\sqrt{\left(\frac{\left(1\right)}{x}\right)}\right)}{\left(\sqrt{x}\right)}\right)}\right) \cdot \left(1.0\right)}\]

  9. Final simplification0.2

    \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot 1.0\]

Reproduce

herbie shell --seed 2019124 
(FPCore (x)
  :name "2sqrt (example 3.1)"
  (-.p16 (sqrt.p16 (+.p16 x (real->posit16 1))) (sqrt.p16 x)))