Computational complexity of mathematical operations

The following tables list the computational complexity of various algorithms for common mathematical operations.
Here, complexity refers to the time complexity of performing computations on a multitape Turing machine. See big O notation for an explanation of the notation used.
Note: Due to the variety of multiplication algorithms, formula_1 below stands in for the complexity of the chosen multiplication algorithm.
Special functions.
Many of the methods in this section are given in Borwein & Borwein.
Elementary functions.
The elementary functions are constructed by composing arithmetic operations, the exponential function (formula_2), the natural logarithm (formula_3), trigonometric functions (formula_4), and their inverses. The complexity of an elementary function is equivalent to that of its inverse, since all elementary functions are analytic and hence invertible by means of Newton's method. In particular, if either formula_2 or formula_3 in the complex domain can be computed with some complexity, then that complexity is attainable for all other elementary functions.
Below, the size formula_7 refers to the number of digits of precision at which the function is to be evaluated.
It is not known whether formula_8 is the optimal complexity for elementary functions. The best known lower bound is the trivial bound 
formula_9.
Mathematical constants.
This table gives the complexity of computing approximations to the given constants to formula_7 correct digits.
Number theory.
Algorithms for number theoretical calculations are studied in computational number theory.
Matrix algebra.
The following complexity figures assume that arithmetic with individual elements has complexity "O"(1), as is the case with fixed-precision floating-point arithmetic or operations on a finite field.
In 2005, Henry Cohn, Robert Kleinberg, Balázs Szegedy, and Chris Umans showed that either of two different conjectures would imply that the exponent of matrix multiplication is 2.
Transforms.
Algorithms for computing transforms of functions (particularly integral transforms) are widely used in all areas of mathematics, particularly analysis and signal processing.