I've done some work with fast approximate math libraries,and went down this path. Taylor was a dead-end, but Chebyshev polynomials work great. They have the nice property of having (close to) the minimum maximum (minimax) error over the entire range you are approximating.

well, from memory (this was > 20y ago)

  - Taylor gives you polynomial optimized for the evaluation of one point.
  - naive Newton gives you a polynomial with 0 error in the interpolation points [but the system has a bad condition]
  - Chebychev gives you a polynomial with minimal error over the interval of choice.

So there is no real reason to ever use the Taylor series to approximate a function over an interval. The high school math teachers are lying.

not lying, probably just mistaken