Average Error: 1.0 → 1.0
Time: 5.9m
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 r7851764 = 1.0;
        double r7851765 = /* ERROR: no posit support in C */;
        double r7851766 = x;
        double r7851767 = r7851766 + r7851765;
        double r7851768 = r7851765 / r7851767;
        double r7851769 = 2.0;
        double r7851770 = /* ERROR: no posit support in C */;
        double r7851771 = r7851770 / r7851766;
        double r7851772 = r7851768 - r7851771;
        double r7851773 = r7851766 - r7851765;
        double r7851774 = r7851765 / r7851773;
        double r7851775 = r7851772 + r7851774;
        return r7851775;
}
↓
double f(double x) {
        double r7851776 = 1.0;
        double r7851777 = x;
        double r7851778 = r7851777 + r7851776;
        double r7851779 = r7851776 / r7851778;
        double r7851780 = 2.0;
        double r7851781 = r7851780 / r7851777;
        double r7851782 = r7851779 - r7851781;
        double r7851783 = r7851777 - r7851776;
        double r7851784 = r7851776 / r7851783;
        double r7851785 = r7851782 + r7851784;
        return r7851785;
}

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 2019162 +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)))))