Average Error: 1.0 → 1.0
Time: 17.5s
Precision: 64
\[\frac{\left(\left(\frac{\left(1\right)}{\left(\frac{x}{\left(1\right)}\right)}\right) - \left(\frac{\left(2\right)}{x}\right)\right)}{\left(\frac{\left(1\right)}{\left(x - \left(1\right)\right)}\right)}\]
↓
\[\frac{1}{x + 1} + \left(\left(-\frac{2}{x}\right) + \frac{1}{x - 1}\right)\]
\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(\left(-\frac{2}{x}\right) + \frac{1}{x - 1}\right)
double f(double x) {
        double r2120990 = 1.0;
        double r2120991 = /* ERROR: no posit support in C */;
        double r2120992 = x;
        double r2120993 = r2120992 + r2120991;
        double r2120994 = r2120991 / r2120993;
        double r2120995 = 2.0;
        double r2120996 = /* ERROR: no posit support in C */;
        double r2120997 = r2120996 / r2120992;
        double r2120998 = r2120994 - r2120997;
        double r2120999 = r2120992 - r2120991;
        double r2121000 = r2120991 / r2120999;
        double r2121001 = r2120998 + r2121000;
        return r2121001;
}
↓
double f(double x) {
        double r2121002 = 1.0;
        double r2121003 = x;
        double r2121004 = r2121003 + r2121002;
        double r2121005 = r2121002 / r2121004;
        double r2121006 = 2.0;
        double r2121007 = r2121006 / r2121003;
        double r2121008 = -r2121007;
        double r2121009 = r2121003 - r2121002;
        double r2121010 = r2121002 / r2121009;
        double r2121011 = r2121008 + r2121010;
        double r2121012 = r2121005 + r2121011;
        return r2121012;
}

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 sub-neg1.0

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

  4. Applied associate-+l+1.0

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

  5. Final simplification1.0

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

Reproduce

herbie shell --seed 2019107 +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)))))