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 you want accurate results without computing any square root, you could use $[n,n]$ Padé approximants.

These could be
$$\sqrt{1+x}-1\sim \frac{2 x}{x+4}$$
$$\sqrt{1+x}-1\sim \frac{4 x (x+2)}{x (x+12)+16}$$