To evaluate polynomial you use Horners scheme, or if you do not know how to compute it, still straightforward naive approach works. It depends on "how big" is this polynomial...
For logarithm and exponent you could use BKM and for trigonometric functions CORDIC.
So ranking by time polynomial is the fastest, and the rest is equally time consuming.
It is hard to tell apart, as basically the same algorithm can compute all these functions.
Using floating point numbers (so inexact results are obvious) the time complexity of special functions is $O(1)$, because underneath you have constant number of rounds no matter what the input was.
Number of iterations is fixed for given precision, so it does not depend on particular input (although angle reduction is time consuming for huge input value).
Polynomial evaluation is $O(n)$ in number of given coefficients using Horner, otherwise it is $O(n^2)$.