$f(x) = \sqrt{x^{2}+1}-1$ (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?