It is important to note two things, though, for those following along: 1. Approximations using orthogonal polynomial bases (the least squares method, and more

It is important to note two things, though, for those following along: 1. Approximations using orthogonal polynomial bases (the least squares method, and more generally the Chebyshev methods) are, for transcendental functions, typically as accurate as Remez exchange (including with range reduction) to 16 or so digits of precision. Remez exchange is more difficult to implement correctly than the simple linear algebra methods, the latter of which rely "only" on doing a comparatively simple change of basis. Practically speaking you gain nothing from using Remez exchange instead of Chebyshev to approximate exp(x), for example. 2. The Remez exchange (and more generally, the MiniMax methods) does not guarantee anything beyond minimizing the worst case error. For many practical applications you don't care about worst case error, you care about average case relative error. It's not uncommon for the Remez exchange to produce polynomials which actually have worse average case relative error. This is also covered in considerable depth in Trefethen's Approximation Theory and Approximation Practice.