With non-linear equations (especially) Newton-Raphson cannot always find the answer on real machines, and cannot always find a particular answer even if one starts right at the answer. The problem can occur with linear equations as well.
The difficulty has to do with floating point round-off error. {I believe that} It can be shown that for any fixed precision and any fixed round-off scheme, that when you calculate one variable in terms of another, or project new values for variables linearly based upon the old values and the evaluated value of the function, that there will be functions where it is not possible to simultaneously find the best representable values for all of the variables.
It can be the case that if (x0,y0) are the closest representable values to the root, that the calculations dance between (x0,y0*(1+eps)) and (x0*(1+eps),y0) . In other cases, the difference between what can be calculated and a theoretical root can be such that even if you start at the closest representation to the theoretical root, the routine will move off towards a different root that has a lower calculation error.
You can sometimes find the root you want by raising the tolerance away from 0 that is accepted as being 0, but there is no absolute tolerance that will work for every problem.
N-R finds answers in theory, but in practice with finite arithmetic is a different matter. :(
Re-writing the expression can help a lot. For example, Hornering a polynomial can lead to much better precision.
