Average Error: 0.6 → 0.6
Time: 37.0s
Precision: 64
\[\left(\frac{\left(1\right)}{\left(\frac{x}{\left(1\right)}\right)}\right) - \left(\frac{\left(1\right)}{x}\right)\]
↓
\[\frac{\frac{1}{x + 1} - \frac{1}{x}}{\frac{1}{x} + \frac{1}{x + 1}} \cdot \frac{\frac{1}{x} + \frac{1}{x + 1}}{1.0}\]
\left(\frac{\left(1\right)}{\left(\frac{x}{\left(1\right)}\right)}\right) - \left(\frac{\left(1\right)}{x}\right)
↓
\frac{\frac{1}{x + 1} - \frac{1}{x}}{\frac{1}{x} + \frac{1}{x + 1}} \cdot \frac{\frac{1}{x} + \frac{1}{x + 1}}{1.0}
double f(double x) {
        double r2271787 = 1.0;
        double r2271788 = /* ERROR: no posit support in C */;
        double r2271789 = x;
        double r2271790 = r2271789 + r2271788;
        double r2271791 = r2271788 / r2271790;
        double r2271792 = r2271788 / r2271789;
        double r2271793 = r2271791 - r2271792;
        return r2271793;
}
↓
double f(double x) {
        double r2271794 = 1.0;
        double r2271795 = x;
        double r2271796 = r2271795 + r2271794;
        double r2271797 = r2271794 / r2271796;
        double r2271798 = r2271794 / r2271795;
        double r2271799 = r2271797 - r2271798;
        double r2271800 = r2271798 + r2271797;
        double r2271801 = r2271799 / r2271800;
        double r2271802 = 1.0;
        double r2271803 = r2271800 / r2271802;
        double r2271804 = r2271801 * r2271803;
        return r2271804;
}

Error

Bits error versus x

Derivation

  1. Initial program 0.6

    \[\left(\frac{\left(1\right)}{\left(\frac{x}{\left(1\right)}\right)}\right) - \left(\frac{\left(1\right)}{x}\right)\]
  2. Using strategy rm

  3. Applied p16-flip--1.3

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

  4. Simplified1.0

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

  5. Simplified1.0

    \[\leadsto \frac{\left(\left(\left(\frac{\left(1\right)}{\left(\frac{x}{\left(1\right)}\right)}\right) - \left(\frac{\left(1\right)}{x}\right)\right) \cdot \left(\frac{\left(\frac{\left(1\right)}{x}\right)}{\left(\frac{\left(1\right)}{\left(\frac{x}{\left(1\right)}\right)}\right)}\right)\right)}{\color{blue}{\left(\frac{\left(\frac{\left(1\right)}{x}\right)}{\left(\frac{\left(1\right)}{\left(\frac{x}{\left(1\right)}\right)}\right)}\right)}}\]
  6. Using strategy rm

  7. Applied *p16-rgt-identity-expand1.0

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

  8. Applied p16-times-frac0.6

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

  9. Final simplification0.6

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

Reproduce

herbie shell --seed 2019153 +o rules:numerics
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  (-.p16 (/.p16 (real->posit16 1) (+.p16 x (real->posit16 1))) (/.p16 (real->posit16 1) x)))