= Solution
At a root of size between $1$ and $2$, the spacing of <floating-point numbers> is $\epsilon=2^{-40}$. With ordinary stable <function> evaluations, a sensible generic absolute position tolerance has scale \b[$2^{-40}\approx9.09\times10^{-13}$]; sub-spacing progress cannot generally be represented. This is a resolution estimate, not a lower bound excluding an accidentally exactly representable root.
Define one <bisection method> step as replacing the current bracket by the half retaining the sign change. Starting from $[1,2]$, the width after $N$ such updates is $2^{-N}$. Therefore
$$
\boxed{N=40\quad\text{for bracket width at most }2^{-40}.}
$$
The midpoint of a bracket has an exact-arithmetic error bound of half its width. If tolerance is defined instead as that midpoint error bound, $39$ updates suffice to make it at most $2^{-40}$. Stating the convention avoids an off-by-one difference; once the endpoints are adjacent <floating-point numbers>, their exact midpoint need not itself be representable.
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