I'm studying errors in functions on numerical methods. On my notes, I've written the Heron's Formula: Let $a\geq b\geq c$:$$A=\sqrt{p(p-a)(p-b)(p-c)}\ \ ,$$ where $$p=\frac{a+b+c}{2}.$$
This formula suffers cancellation if $a \sim b+c$ on the factor $(p-a)$. Looking for a reformulation, we find that $$A=\frac{1}{4}\sqrt{(a+(b+c))(c-(a-b))(c+(a-b))(a+(b-c))}.$$ It's easy to show that this formula is better analytically than the other one if we're looking to avoid cancellation errors.
How we show this? How can we see that the formula below is better than the above one? Thanks!