# Help comparing relative error for different parenthesizations of addition

I am given two functions: $$fl(fl(x+y)+z)$$ and $$fl(x+fl(y+z))$$ and asked to derive their relative error. Then, given a set of conditions:

a) $$x < y < x$$

b) $$x > 0, y < 0, z > 0$$

c) $$x < 0, y > 0, z < 0$$

determine which parenthesization has less relative error.

After some algebra I determined the following relative errors where

$$\epsilon_1 = RE(fl(x+y)),\ \epsilon_2 = RE(fl(fl(x+y)+z)$$

$$\epsilon_3 = RE(fl(y+z)),\ \epsilon_4 = RE(fl(x+fl(y+z))$$

$$\left| \epsilon_2 + \frac{(x+y)}{(x + y + z)}\epsilon_1 \right| \ \text{and}\ \ \left| \epsilon_4 + \frac{(y+z)}{(x + y + z)}\epsilon_3 \right|$$

I determined that for the case $$x < y < z,\ fl(fl(x + y) + z$$ has less relative error. With the largest variable, $$z$$ in the denominator alone, as $$z \rightarrow \infty$$ the coefficient of $$\epsilon_1$$ would approach $$0$$.

But for $$x > 0, y < 0, z > 0$$ I'm struggling to generalize anything. I don't see how a conclusion can be drawn without an ordering between $$z$$ and $$x$$. And what about when $$y = x + z$$? Although this case is independent of the relative parenthesizations. And similarly for the case $$x < 0, y > 0, z < 0$$.