Condition number problem

In a numerical evaluation of the difference the first term will have a relative floating point error of about $\pm 2\mu$ and the second term of about $\pm 3\mu$, where the coefficients of the machine precision are the operations counts of the terms (multiplication by 2 is exact). At $x\approx 0$ the terms evaluate to $≈1$ so that the relative are also the absolute errors.

The error of the difference can thus be as large as $\pm 5μ$. However, the exact value by algebraic simplification is $=2x^2+O(x^3)$. Thus the relative error at $x=0$ is expected to behave like $\frac{5\mu}{2x^2}$. Indeed numerical evaluation by computing the relative error for $|x|<10^{-7}$ confirms that estimate: