Average Error: 0.8 → 0.2
Time: 11.8s
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 r2082644 = x;
        double r2082645 = 1.0;
        double r2082646 = /* ERROR: no posit support in C */;
        double r2082647 = r2082644 + r2082646;
        double r2082648 = sqrt(r2082647);
        double r2082649 = sqrt(r2082644);
        double r2082650 = r2082648 - r2082649;
        return r2082650;
}
↓
double f(double x) {
        double r2082651 = 1.0;
        double r2082652 = x;
        double r2082653 = r2082651 + r2082652;
        double r2082654 = sqrt(r2082653);
        double r2082655 = sqrt(r2082652);
        double r2082656 = r2082654 + r2082655;
        double r2082657 = r2082651 / r2082656;
        double r2082658 = 1.0;
        double r2082659 = r2082657 * r2082658;
        return r2082659;
}

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