Question: > 
> I'm trying to figure out how to avoid catastrophic cancellation for the following expression
>  $$\sqrt{1+x} - 1$$
>  for $x$ being a number *very* close to $0$. 
> 
> 
> 

Of course, the answer would come to $0$ unless the expression is changed around.

Any help is appreciated! Thanks!

Answer: 
You could use $$\sqrt{1+x}-1=\frac{x}{\sqrt{1+x}+1}$$
If $x$ is seriously small, e.g. $\ x < 10^{-14}$, and you don't care too much about terms of order $O(10^{-28})$ then why not use:

$$\sqrt{1+x}\approx1+\frac{x}{2}$$