Average Error: 1.0 → 1.0
Time: 24.0s
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 r4614420 = 1.0;
        double r4614421 = /* ERROR: no posit support in C */;
        double r4614422 = x;
        double r4614423 = r4614422 + r4614421;
        double r4614424 = r4614421 / r4614423;
        double r4614425 = 2.0;
        double r4614426 = /* ERROR: no posit support in C */;
        double r4614427 = r4614426 / r4614422;
        double r4614428 = r4614424 - r4614427;
        double r4614429 = r4614422 - r4614421;
        double r4614430 = r4614421 / r4614429;
        double r4614431 = r4614428 + r4614430;
        return r4614431;
}
↓
double f(double x) {
        double r4614432 = 1.0;
        double r4614433 = x;
        double r4614434 = r4614433 + r4614432;
        double r4614435 = r4614432 / r4614434;
        double r4614436 = r4614433 - r4614432;
        double r4614437 = r4614432 / r4614436;
        double r4614438 = 2.0;
        double r4614439 = r4614438 / r4614433;
        double r4614440 = r4614437 - r4614439;
        double r4614441 = r4614435 + r4614440;
        return r4614441;
}

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 2019134 
(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)))))