Average Error: 1.0 → 1.0
Time: 17.8s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\frac{1}{x + 1} - \left(\frac{2}{x} - \frac{1}{x - 1}\right)\]
double f(double x) {
        double r2765844 = 1.0;
        double r2765845 = x;
        double r2765846 = r2765845 + r2765844;
        double r2765847 = r2765844 / r2765846;
        double r2765848 = 2.0;
        double r2765849 = r2765848 / r2765845;
        double r2765850 = r2765847 - r2765849;
        double r2765851 = r2765845 - r2765844;
        double r2765852 = r2765844 / r2765851;
        double r2765853 = r2765850 + r2765852;
        return r2765853;
}
↓
double f(double x) {
        double r2765854 = 1.0;
        double r2765855 = x;
        double r2765856 = r2765855 + r2765854;
        double r2765857 = r2765854 / r2765856;
        double r2765858 = 2.0;
        double r2765859 = r2765858 / r2765855;
        double r2765860 = r2765855 - r2765854;
        double r2765861 = r2765854 / r2765860;
        double r2765862 = r2765859 - r2765861;
        double r2765863 = r2765857 - r2765862;
        return r2765863;
}


\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\frac{1}{x + 1} - \left(\frac{2}{x} - \frac{1}{x - 1}\right)

Error

Bits error versus x

Derivation

  1. Initial program 1.0

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

  3. Applied associate-+l-1.0

    \[\leadsto \color{blue}{\left(\frac{\left(1\right)}{\left(\frac{x}{\left(1\right)}\right)}\right) - \left(\left(\frac{\left(2\right)}{x}\right) - \left(\frac{\left(1\right)}{\left(x - \left(1\right)\right)}\right)\right)}\]

  4. Final simplification1.0

    \[\leadsto \frac{1}{x + 1} - \left(\frac{2}{x} - \frac{1}{x - 1}\right)\]

Reproduce

herbie shell --seed 2019101 +o rules:numerics
(FPCore (x)
  :name "3frac (problem 3.3.3)"
  (+.p16 (-.p16 (/.p16 (real->posit16 1) (+.p16 x (real->posit16 1))) (/.p16 (real->posit16 2) x)) (/.p16 (real->posit16 1) (-.p16 x (real->posit16 1)))))