The elementary function approximation using piecewise quadratic polynomial interpolation requires larger area of the look-up table (LUT) and circuit. To solve the problem, this paper presents an algorithm for elementary function approximation in single-precision floating-point format, which is based on minimax piecewise cubic polynomial approximation. The algorithm can efficiently achieve the approximation of reciprocal, square root, square root reciprocal, exponentials, logarithm, and trigonometric function in single-precision floating-point format. According to the algorithm the range of parameters is narrowed first, then we get the optimal truncated bit width of coefficients through errors analyzing and dividing the interval into subsections. Meanwhile Remes algorithm is used to perform the successive optimization, so as to reduce the area of LUT and circuit. At last the intermediate parameters of the circuit are optimally truncated, and the overall framework of hardware circuit is designed. The analysis and experimental results show that comparing with piecewise quadratic polynomial approximation the circuit delay is reduced by 17.25%, the area of LUT is decreased by 53.60% and total area of circuits is reduced by 19.73%. The comparisons with the polynomial approximation of degree-1, degree-2, degree-4 are also made, the results show that the piecewise cubic polynomial interpolation algorithm proposed by this paper can achieve optimal performance and minimized cost of hardware implementation.