Average Error: 1.0 → 1.0
Time: 1.5m
Precision: 64
\[\frac{\left(\left(\frac{\left(1\right)}{\left(\frac{x}{\left(1\right)}\right)}\right) - \left(\frac{\left(2\right)}{x}\right)\right)}{\left(\frac{\left(1\right)}{\left(x - \left(1\right)\right)}\right)}\]
↓
\[\frac{1}{x + 1} + \left(\frac{1}{x - 1} - \frac{2}{x}\right)\]
\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{1}{x - 1} - \frac{2}{x}\right)
double f(double x) {
        double r1325705 = 1.0;
        double r1325706 = /* ERROR: no posit support in C */;
        double r1325707 = x;
        double r1325708 = r1325707 + r1325706;
        double r1325709 = r1325706 / r1325708;
        double r1325710 = 2.0;
        double r1325711 = /* ERROR: no posit support in C */;
        double r1325712 = r1325711 / r1325707;
        double r1325713 = r1325709 - r1325712;
        double r1325714 = r1325707 - r1325706;
        double r1325715 = r1325706 / r1325714;
        double r1325716 = r1325713 + r1325715;
        return r1325716;
}
↓
double f(double x) {
        double r1325717 = 1.0;
        double r1325718 = x;
        double r1325719 = r1325718 + r1325717;
        double r1325720 = r1325717 / r1325719;
        double r1325721 = r1325718 - r1325717;
        double r1325722 = r1325717 / r1325721;
        double r1325723 = 2.0;
        double r1325724 = r1325723 / r1325718;
        double r1325725 = r1325722 - r1325724;
        double r1325726 = r1325720 + r1325725;
        return r1325726;
}

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. Simplified1.0

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

  6. Final simplification1.0

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

Reproduce

herbie shell --seed 2019153 +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)))))