This thesis describes methods and algorithms for the approximation of the elementary functions. It begins by providing some fundamental concepts and principles of floating-point arithmetic. The main focus is on the construction of various types of polynomial and rational approximations. We start with two types of polynomial approximations: least squares approximation and minimax approximation. An iterative algorithm (Remez algorithm) is demonstrated, which constructs a minimax approximation using Chebyshev’s characterization theorem of the minimax approximation of a function. Subsequently, rational approximations are discussed, which are essentially extensions of the polynomial approximations (a polynomial is a rational function with a denominator of 1). The thesis then explores methods for polynomial evaluation and the errors that arise during approximation, with Horner’s scheme being commonly used. Finally, it delves into table-based methods that are employed when approximating a function over a large interval. Two classes of methods are considered, one using a standard table and a polynomial or rational approximation, and the other utilizing accurate tables.