I am unsure if numerical analysis questions are suitable for this forum, but I have nowhere else to ask, so if this question is inappropriate, tell me and I will delete it.
If $x_k$ are the Chebyshev coefficients, that is for $n \in \mathbb{N}^*$, we have $x_k = \cos(\pi \frac{2k + 1}{n})$. Now suppose you have approximated values of $x_k$ and $x_{k'}$ for $k,k' \leq n$. In the worst case, what is the number of significant values we may loose when we calculate $x_{k'} - x_k$?
Now, taking into account the fact that each real is represented as $x \approx m \times b^e$, we then have a formula that for $x'$ the approximated value of $x$ and $y'$ approximated value of $y$, $\Delta (x' + y') \leq \epsilon(\Delta| x| + \Delta |y| + |x| + |y|)$ where $\epsilon = b^{1-N}$, where $N$ is there number of significant numbers after the comma.
So for a fixed $N \in \mathbb{N}^*$ we have $\Delta (x_{k'}' - x_k') \leq \epsilon (\Delta |x_{k'}| + \Delta |x_k| + |x_{k'}| + |x_k| ) \leq 4 \epsilon $
Thus in worst case we have a loss of precision of $\frac{4}{b^{N-1}}$. Thus we loose one significant term?
I am sorry if this doesn't make much sense. I am pretty lost in this subject.