Average Error: 1.0 → 1.0
Time: 19.1s
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 r3096232 = 1.0;
        double r3096233 = /* ERROR: no posit support in C */;
        double r3096234 = x;
        double r3096235 = r3096234 + r3096233;
        double r3096236 = r3096233 / r3096235;
        double r3096237 = 2.0;
        double r3096238 = /* ERROR: no posit support in C */;
        double r3096239 = r3096238 / r3096234;
        double r3096240 = r3096236 - r3096239;
        double r3096241 = r3096234 - r3096233;
        double r3096242 = r3096233 / r3096241;
        double r3096243 = r3096240 + r3096242;
        return r3096243;
}
↓
double f(double x) {
        double r3096244 = 1.0;
        double r3096245 = x;
        double r3096246 = r3096245 + r3096244;
        double r3096247 = r3096244 / r3096246;
        double r3096248 = 2.0;
        double r3096249 = r3096248 / r3096245;
        double r3096250 = r3096247 - r3096249;
        double r3096251 = r3096245 - r3096244;
        double r3096252 = r3096244 / r3096251;
        double r3096253 = r3096250 + r3096252;
        return r3096253;
}

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 2019128 +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)))))