For practical matters, smooth functions (exp, sin, atan...) are often computed numerically using polynomials of high order.

Calculators compute the exponential function in the range [0,1] by evaluating a 12 degree polynomial whose coefficients are chosen so as to get 8 or 10 digits correct. These approximating polynomials can be found in formulas handbooks (e.g. "Methods and Programs for Mathematical Functions". Also the EDM lists a few in its appendices). They do a better job, at the given precision 10^(-8), than the standard Taylor polynomials.

Also have a look at your favorite language mathematical library, to see how standard functions (sin, exp, Atan...) are implemented (someone may suggest a link ?).