*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*Background*--*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*--

## Discussion

## Tasks

Rescue the code from #6084 and get it in a state where it can be reviewed.