What Secrets the Bisection Method Hides?

[ttw]

There several demonstrations of the bisection method's optimality (over a suitable set of functions.)

https://cs.stackexchange.com/questions/7...ion-method

Two intuitive observations also indicate this: first, if the function f(x) and -f(x) are equally likely, then going either way is equivalent. Second, when dividing an interval into two parts, the longer part is "more likely" (in some sense) to contain the root or any other part of interest.

If you know something about the function, it's possible to do better. It's not obvious if there is a learning procedure that works often enough to be useful. The "no free lunch" theorem points out that methods good for one particular subset of functions will always have a set that they fail badly on.