The tables-and-additions methods for accurate computation of elementary functions are fast in computation speed but require large memory. A memory-efficient method named as the integrated Add-Table Lookup-Add (iATA) is proposed in this paper. In iATA, the mathematical formulation for computing the elementary functions is derived without using the central difference formulation to save memory. Three additional techniques, specifically the carry select technique, symmetry property exploitation and unequal partitioning of input with the aid of error analysis, are integrated in iATA to further reduce the memory size. The experimental results show that the proposed method is able to achieve higher memory efficiency than the best existing tables-and-additions methods. For the reciprocal and the natural logarithm function, iATA saves 23.63 and 61.39 percent of memory when compared to the best existing results obtained, respectively, by the unified Multipartite Table Method [39] and the Symmetric Table Addition Method [37].