Synthesis of Arithmetic Circuits: FPGA, ASIC and Embedded Systems

7.3 LOGARITHMIC, EXPONENTIAL, AND TRIGONOMETRIC FUNCTIONS

Most often the computation of functions such as logarithms and exponential or trigonometric functions are made through software-implemented algorithms applied to floating-point representations. Hardware or microprogrammed systems are mainly justified for special-purpose computing devices such as ASIC or embedded systems. As it is generally not possible to get an exact result, approximation methods have to be used together with error estimation techniques. Newton–Raphson, Taylor–MacLaurin series, or polynomial approximations are the most common approaches to compute these functions. For trigonometric functions, CORDIC (linear convergence) algorithms are well suited. Arguments included in the range [1, 2[ (floating-point IEEE standard) are suitable for most approximation methods that need to limit the range of the argument. Whenever a specific range is imposed on the operands, a prescaling operation may be necessary: so an initial step may be included in the algorithmic procedure. Crucial questions for approximation methods are error estimation and effective rounding techniques; these problems start from table design (first approximation LUT) up to the final result. Numerous methods, algorithms, and implementations are proposed in the literature ([SPE1965], [CHE1972], [ERC1977], [COD1980], [KOS1991], [MAR1990], [TAN1991], [ERC1995], [PAL2000], [CAO2001]). As for the basic operations, the choice will depend on the speed/cost ...