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

↓

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

double f(double x) {
        double r3032882 = 1.0;
        double r3032883 = x;
        double r3032884 = r3032883 + r3032882;
        double r3032885 = r3032882 / r3032884;
        double r3032886 = 2.0;
        double r3032887 = r3032886 / r3032883;
        double r3032888 = r3032885 - r3032887;
        double r3032889 = r3032883 - r3032882;
        double r3032890 = r3032882 / r3032889;
        double r3032891 = r3032888 + r3032890;
        return r3032891;
}

↓

double f(double x) {
        double r3032892 = 1.0;
        double r3032893 = x;
        double r3032894 = r3032893 + r3032892;
        double r3032895 = r3032892 / r3032894;
        double r3032896 = 2.0;
        double r3032897 = r3032896 / r3032893;
        double r3032898 = -r3032897;
        double r3032899 = r3032893 - r3032892;
        double r3032900 = r3032892 / r3032899;
        double r3032901 = r3032898 + r3032900;
        double r3032902 = r3032895 + r3032901;
        return r3032902;
}

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

↓

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

Derivation

Initial program 1.0
\[\frac{\left(\left(\frac{\left(1\right)}{\left(\frac{x}{\left(1\right)}\right)}\right) - \left(\frac{\left(2\right)}{x}\right)\right)}{\left(\frac{\left(1\right)}{\left(x - \left(1\right)\right)}\right)}\]
Using strategy rm
Applied sub-neg1.0
\[\leadsto \frac{\color{blue}{\left(\frac{\left(\frac{\left(1\right)}{\left(\frac{x}{\left(1\right)}\right)}\right)}{\left(-\left(\frac{\left(2\right)}{x}\right)\right)}\right)}}{\left(\frac{\left(1\right)}{\left(x - \left(1\right)\right)}\right)}\]
Applied associate-+l+1.0
\[\leadsto \color{blue}{\frac{\left(\frac{\left(1\right)}{\left(\frac{x}{\left(1\right)}\right)}\right)}{\left(\frac{\left(-\left(\frac{\left(2\right)}{x}\right)\right)}{\left(\frac{\left(1\right)}{\left(x - \left(1\right)\right)}\right)}\right)}}\]
Final simplification1.0
\[\leadsto \frac{1}{x + 1} + \left(\left(-\frac{2}{x}\right) + \frac{1}{x - 1}\right)\]

Reproduce

herbie shell --seed 2019102 
(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)))))

Error

Derivation

Reproduce