This has been really eye-opening, and I just wanted to drop you a note 
that I am watching this thread carefully. My first priority is to get 
the tests all passing before beta 1 on 2016-09-12, even if (as seems 
likely) that means weakening the tests, and then come back and see if we 
can tighten it up again later.

I haven't checked it in yet, but I've already managed to simplify the 
nth_root code by taking Tim's advice that more than one iteration of 
Newton's method is a waste of time. Thanks!

Can somebody comment on my reasoning here?

I start by taking an initial guess for the root by using 
r = pow(x, 1.0/n). Then I test if r**n == x, if it does I conclude that 
r is either the exact root, or as close as representable as a float, and 
just return it without bothering with even one iteration. Sensible?

Or should I just always run one iteration of Newton and trust that it 
won't make things worse?