After reducing the interval, you don't want to use the Taylor series as you're building an approximation that's really good in 0 but not so good moving away fro

After reducing the interval, you don't want to use the Taylor series as you're building an approximation that's really good in 0 but not so good moving away from 0. It's better to use an interpolating polynomial (Chebychev comes to mind) over the whole target interval.