Global approximations to the error function of real argument for vectorized computation

The error function of a real argument can be approximated to a given uniform relative accuracy by a single closed-form expression for the whole variable range, either in terms of addition, multiplication, division, and square root operations only, or also using the exponential function. Approximations to the complementary error function have also been found, including those where either only additions, multiplications, and one division, or also the exponential, are used to reach either a uniform absolute, or a uniform relative, accuracy. The coefficients have been tabulated for up to 128-bit precision. Tests of a computer code implementation using the standard single- and double-precision floating-point arithmetic show good performance and vectorizability.