Average Error: 1.0 → 1.0
Time: 21.3s
Precision: 64
\[\frac{\left(\left(\frac{\left(1\right)}{\left(\frac{x}{\left(1\right)}\right)}\right) - \left(\frac{\left(2\right)}{x}\right)\right)}{\left(\frac{\left(1\right)}{\left(x - \left(1\right)\right)}\right)}\]
↓
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
\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{2}{x}\right) + \frac{1}{x - 1}
double f(double x) {
        double r1543022 = 1.0;
        double r1543023 = /* ERROR: no posit support in C */;
        double r1543024 = x;
        double r1543025 = r1543024 + r1543023;
        double r1543026 = r1543023 / r1543025;
        double r1543027 = 2.0;
        double r1543028 = /* ERROR: no posit support in C */;
        double r1543029 = r1543028 / r1543024;
        double r1543030 = r1543026 - r1543029;
        double r1543031 = r1543024 - r1543023;
        double r1543032 = r1543023 / r1543031;
        double r1543033 = r1543030 + r1543032;
        return r1543033;
}
↓
double f(double x) {
        double r1543034 = 1.0;
        double r1543035 = x;
        double r1543036 = r1543035 + r1543034;
        double r1543037 = r1543034 / r1543036;
        double r1543038 = 2.0;
        double r1543039 = r1543038 / r1543035;
        double r1543040 = r1543037 - r1543039;
        double r1543041 = r1543035 - r1543034;
        double r1543042 = r1543034 / r1543041;
        double r1543043 = r1543040 + r1543042;
        return r1543043;
}

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. Final simplification1.0

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

Reproduce

herbie shell --seed 2019119 +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)))))