This paper deals with approximation circuits for sine waves, with applications to low-power direct digital frequency synthesizers (DDFSs). The goal is to determine very accurate approximations, i.e. having very low Maximum Absolute Errors (MAE). These allow for eliminating the ROM and reducing power. They rely on simple circuitry. The paper starts by presenting previous approximation techniques focusing on ROM-less piecewise linear ones. These will be followed by several new ROM-less piecewise linear approximations, and by a very thorough comparison of the newly introduced ones with respect to those presented in the literature. The solutions presented in this paper significantly reduce the MAE with a very modest increase in circuitry (one more multiplexer), and/or 1-2 bits of extra precision.