Average Error: 1.0 → 1.0
Time: 19.3s
Precision: 64
\[\frac{\left(\left(\frac{\left(1\right)}{\left(\frac{x}{\left(1\right)}\right)}\right) - \left(\frac{\left(2\right)}{x}\right)\right)}{\left(\frac{\left(1\right)}{\left(x - \left(1\right)\right)}\right)}\]
↓
\[\frac{1}{x + 1} - \left(\frac{2}{x} - \frac{1}{x - 1}\right)\]
\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)}
↓
\frac{1}{x + 1} - \left(\frac{2}{x} - \frac{1}{x - 1}\right)
double f(double x) {
        double r1547849 = 1.0;
        double r1547850 = /* ERROR: no posit support in C */;
        double r1547851 = x;
        double r1547852 = r1547851 + r1547850;
        double r1547853 = r1547850 / r1547852;
        double r1547854 = 2.0;
        double r1547855 = /* ERROR: no posit support in C */;
        double r1547856 = r1547855 / r1547851;
        double r1547857 = r1547853 - r1547856;
        double r1547858 = r1547851 - r1547850;
        double r1547859 = r1547850 / r1547858;
        double r1547860 = r1547857 + r1547859;
        return r1547860;
}
↓
double f(double x) {
        double r1547861 = 1.0;
        double r1547862 = x;
        double r1547863 = r1547862 + r1547861;
        double r1547864 = r1547861 / r1547863;
        double r1547865 = 2.0;
        double r1547866 = r1547865 / r1547862;
        double r1547867 = r1547862 - r1547861;
        double r1547868 = r1547861 / r1547867;
        double r1547869 = r1547866 - r1547868;
        double r1547870 = r1547864 - r1547869;
        return r1547870;
}

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 2019107 
(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)))))