Worse than Bisection???!!!!

[ttw]

Reguli Falsi with bracketing can be worse than bisection on many functions. It's like the Secant Method but keeps the root between estimates. One sets X2=(F(X1)-F(X0))/(X1-X0). If (for example) the function is sort of a curved L-shape and one point is above the X-axis near the upper leg and the other above the X-axis way out on the lower leg, the function will creep along chopping the interval a small amount at each time.

Using the Secant Method and switching to Bisection if the point falls outside works well in practice as it at worse becomes Bisection.