Binary Search Root Error: Estimating Accuracy

[c299792458]

I have written a computer program that uses binary search to find the root to f(x) = 0, where f is an arbitrary (user-defined) function.

If the rounding error $\leq \epsilon$ and the truncation error $\leq \delta$, what is the estimated accuracy of the output?

Would the following reasoning be correct?
Let $r$ be s.t. $f(r) = 0$. Then $f(x) = f(r) + (x-r)f'(r) + O((x-r)^2)$. So the estimated error is $|x-r| \approx |\frac{f(x)}{f'(r)}| = \frac{\epsilon+\delta}{|f'(r)|}$?

Thanks.

Edited: I meant expanded till the divisor is NOT 0.

 

[Future Bruno]

Yes, that reasoning is correct. In fact, since the binary search method only requires two evaluations of the function f(x) (one at the lower bound and one at the upper bound), the accuracy of the output can be estimated as $\frac{\epsilon+\delta}{|f'(r)|}$, where $r$ is the root found using the binary search method.