The Newton mode efficiency is relatively low, Is it can be improved ?
Of course it is slower because it uses an iterative calculation. It is also inaccurate and it fails most of the times
However, the differences in your benchmark test doesn't comes from newton's method but rather from my algorithm.
Newton's method needs an initial guess in order to find a root, so I tried to isolate roots by solving f'(x)=0 (between 2 consecutive roots of first derivate can be at most 1 root of function). So my algorithm solves n-1 equations and this is the cause of it's slowness. Of course, it is useful for higher degree polynomial equations, where you can not find an exact solution.
Updated the code in my previous post... It is not faster (au contraire) but it didn't fail so often
In some cases, that initial guess is not good so newton's method is not convergent. I'll try to solve this, but
my initial guess is that I can not make it "full proof".