Average Error: 1.0 → 1.0
Time: 50.3s
Precision: 64
\[\frac{\left(\left(\frac{\left(1\right)}{\left(\frac{x}{\left(1\right)}\right)}\right) - \left(\frac{\left(2\right)}{x}\right)\right)}{\left(\frac{\left(1\right)}{\left(x - \left(1\right)\right)}\right)}\]
↓
\[\frac{1}{x + 1} + \left(\frac{1}{x - 1} - \frac{2}{x}\right)\]
\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 r7714791 = 1.0;
        double r7714792 = /* ERROR: no posit support in C */;
        double r7714793 = x;
        double r7714794 = r7714793 + r7714792;
        double r7714795 = r7714792 / r7714794;
        double r7714796 = 2.0;
        double r7714797 = /* ERROR: no posit support in C */;
        double r7714798 = r7714797 / r7714793;
        double r7714799 = r7714795 - r7714798;
        double r7714800 = r7714793 - r7714792;
        double r7714801 = r7714792 / r7714800;
        double r7714802 = r7714799 + r7714801;
        return r7714802;
}
↓
double f(double x) {
        double r7714803 = 1.0;
        double r7714804 = x;
        double r7714805 = r7714804 + r7714803;
        double r7714806 = r7714803 / r7714805;
        double r7714807 = r7714804 - r7714803;
        double r7714808 = r7714803 / r7714807;
        double r7714809 = 2.0;
        double r7714810 = r7714809 / r7714804;
        double r7714811 = r7714808 - r7714810;
        double r7714812 = r7714806 + r7714811;
        return r7714812;
}

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 2019130 +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)))))