Suppose you have two measured (independent) physical quantities $x$ and $y$ with relative errors $r_x := \frac{\delta x}{x}$ and $r_y := \frac{\delta y}{y}$, where $\delta x$ and $\delta y$ are the corresponding absolute errors. Now you want to calculate $z = xy$ (or $z = \frac{x}{y}$).
Usually the relative error $r_z$ is calculated as
$$ r_z = \sqrt{r_x^2 + r_y^2} $$
Sometimes (usually in lower level or high school courses) it is said that you have to take just the sum of the relative errors, i.e.
$$ r_z = r_x + r_y $$
For example for the product this seems easy to derive: $$ \begin{align} (x + \delta x)\cdot (y + \delta y) = xy + x \delta y + y \delta x + \delta x \delta y \\ (x - \delta x)\cdot (y - \delta y) = xy - x \delta y - y \delta x + \delta x \delta y \\ \end{align} $$
Subtracting both equations you get for the right side: $2(x\delta y + y \delta x)$ and half of it seems to be a good measure for the absolute value of $\delta z$. So you get for the relative error $r_z$
$$ r_z = \frac{\delta z}{z} = \frac{x\delta y + y \delta}{xy} = \frac{\delta y}{y} + \frac{\delta x}{x} = r_y + r_x $$
What's wrong with this reasoning?
What is the correct formula and why? Are there different domains of application of both formulas?
