Average Error: 1.0 → 1.0
Time: 49.7s
Precision: 64
\[\frac{\left(\left(\frac{\left(1\right)}{\left(\frac{x}{\left(1\right)}\right)}\right) - \left(\frac{\left(2\right)}{x}\right)\right)}{\left(\frac{\left(1\right)}{\left(x - \left(1\right)\right)}\right)}\]
↓
\[\left(\frac{1}{x + 1} + \frac{1}{x - 1}\right) - \frac{2}{x}\]
\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)}
↓
\left(\frac{1}{x + 1} + \frac{1}{x - 1}\right) - \frac{2}{x}
double f(double x) {
        double r5426981 = 1.0;
        double r5426982 = /* ERROR: no posit support in C */;
        double r5426983 = x;
        double r5426984 = r5426983 + r5426982;
        double r5426985 = r5426982 / r5426984;
        double r5426986 = 2.0;
        double r5426987 = /* ERROR: no posit support in C */;
        double r5426988 = r5426987 / r5426983;
        double r5426989 = r5426985 - r5426988;
        double r5426990 = r5426983 - r5426982;
        double r5426991 = r5426982 / r5426990;
        double r5426992 = r5426989 + r5426991;
        return r5426992;
}
↓
double f(double x) {
        double r5426993 = 1.0;
        double r5426994 = x;
        double r5426995 = r5426994 + r5426993;
        double r5426996 = r5426993 / r5426995;
        double r5426997 = r5426994 - r5426993;
        double r5426998 = r5426993 / r5426997;
        double r5426999 = r5426996 + r5426998;
        double r5427000 = 2.0;
        double r5427001 = r5427000 / r5426994;
        double r5427002 = r5426999 - r5427001;
        return r5427002;
}

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. Using strategy rm

  7. Applied associate-+r-1.0

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

  8. Final simplification1.0

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

Reproduce

herbie shell --seed 2019133 
(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)))))