Goal -- Finally finish incorporating the changes of #6084, which conceptually splits the precision a polynomial is defined up to from an approximation of the coefficients.
Type -- precision handling, speed improvements
Priority -- High
Difficulty -- Annoying
Prerequisites -- None
Contributors -- David Roe
Progress - I've been almost done with this for a long time now. At this point the code has bit-rotted to some extent and there are doctests to fix. It's also going to be a big project to referee
Related Tickets --
Rescue the code from #6084 and get it in a state where it can be reviewed.