Average Error: 1.0 → 1.0
Time: 16.3s
Precision: 64
\[\frac{\left(\left(\frac{\left(1\right)}{\left(\frac{x}{\left(1\right)}\right)}\right) - \left(\frac{\left(2\right)}{x}\right)\right)}{\left(\frac{\left(1\right)}{\left(x - \left(1\right)\right)}\right)}\]
↓
\[\frac{1}{x + 1} + \left(\left(-\frac{2}{x}\right) + \frac{1}{x - 1}\right)\]
\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 r1203846 = 1.0;
        double r1203847 = /* ERROR: no posit support in C */;
        double r1203848 = x;
        double r1203849 = r1203848 + r1203847;
        double r1203850 = r1203847 / r1203849;
        double r1203851 = 2.0;
        double r1203852 = /* ERROR: no posit support in C */;
        double r1203853 = r1203852 / r1203848;
        double r1203854 = r1203850 - r1203853;
        double r1203855 = r1203848 - r1203847;
        double r1203856 = r1203847 / r1203855;
        double r1203857 = r1203854 + r1203856;
        return r1203857;
}
↓
double f(double x) {
        double r1203858 = 1.0;
        double r1203859 = x;
        double r1203860 = r1203859 + r1203858;
        double r1203861 = r1203858 / r1203860;
        double r1203862 = 2.0;
        double r1203863 = r1203862 / r1203859;
        double r1203864 = -r1203863;
        double r1203865 = r1203859 - r1203858;
        double r1203866 = r1203858 / r1203865;
        double r1203867 = r1203864 + r1203866;
        double r1203868 = r1203861 + r1203867;
        return r1203868;
}

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 2019121 +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)))))