Given two floating point representations of x and y as $x'=x(1+\delta_x)$ and $y'=y(1+\delta_y)$ with $|\delta_x|,|\delta_y|\leq \epsilon_M<<1$. How do I find the relative error in $x'\cdot y'$ and $x'/y'$. Is it just straight forward $$x(1+\delta_x)\cdot y(1+\delta_y)=(x+x\delta_x)\cdot(y + y\delta_y)...$$ That seems too simple, also I would not understand how to show that way that multiplication and division do not result in a serious loss of significance.
Thank you in advance