Average Error: 1.0 → 1.0
Time: 20.9s
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 r2948078 = 1.0;
        double r2948079 = x;
        double r2948080 = r2948079 + r2948078;
        double r2948081 = r2948078 / r2948080;
        double r2948082 = 2.0;
        double r2948083 = r2948082 / r2948079;
        double r2948084 = r2948081 - r2948083;
        double r2948085 = r2948079 - r2948078;
        double r2948086 = r2948078 / r2948085;
        double r2948087 = r2948084 + r2948086;
        return r2948087;
}
↓
double f(double x) {
        double r2948088 = 1.0;
        double r2948089 = x;
        double r2948090 = r2948089 + r2948088;
        double r2948091 = r2948088 / r2948090;
        double r2948092 = 2.0;
        double r2948093 = r2948092 / r2948089;
        double r2948094 = r2948089 - r2948088;
        double r2948095 = r2948088 / r2948094;
        double r2948096 = r2948093 - r2948095;
        double r2948097 = r2948091 - r2948096;
        return r2948097;
}


\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 2019102 +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)))))