TITLE: How bad, really, is the bisection method?
QUESTION [5 upvotes]: We know that the bisection method for root finding is slow (linear convergence), but has the advantage of always working for a continuous function, if we start with a interval which brackets the root.  However, the arithmetic is very simple, no derivatives are required, and only one new function evaluation is needed each iteration.
However, in practice, and with modern computer speeds and processing power, computing, say, 30 iterates of bisection can be done in a fraction of a second.  Irrespective of the theory, solving a single root-finding problem would seem to take hardly any more computing time than a better method.
All books say: bisection - works but slow; Newton-Raphson/Secant etc: sometimes fails but much faster.  And in the days of hand computation the time differences would have been considerable.  I wonder, though, with modern computational tools, and with practical precision requirements (does anybody really need 100 decimal place accuracy?), if the bisection method is really as slow as all the texts seem to indicate?

REPLY [2 votes]: There is no reason to use the bisection method when the Regula falsi variants, the most simple one the Illinois variation, provide faster superlinear convergence with almost no more computational or coding effort and the same safety features of a bracketing method as the bisection method. 
The Dekker and Brent methods are almost as fast as, resp. in nice cases faster than a secant iteration while preserving the safety of a bracketing method, but have more code complexity.