$f(x) = \sqrt{x^{2}+1}-1$ (Loss of Significance) Let us say that I want to compute $f(x) = \sqrt{x^{2}+1}-1$ for small values of $x$ in a Marc-32 architecture. I can avoid loss of significance by rewriting the function
$$f(x)=\left(\sqrt{x^{2}+1}-1\right)\left(\frac{\sqrt{x^{2}+1}+1}{\sqrt{x^{2}+1}+1}\right)=\frac{x^{2}}{\sqrt{x^{2}+1}+1}$$
Even though I can solve the problem, I do not know/understand why the solution has avoid the loss of significance?
Assume that due to rounding errors, when you try to calculate the square root of $x^2 + 1$ you actually get the square root plus $\epsilon$ for some small $\epsilon$. Assume that $\epsilon = 10^{-10}$ and $x = 10^{-6}$. Perform the calculations and write down the results.