Elementary Functions and Approximate Computing

Abstract

We review some of the classical methods used for quickly obtaining low-precision approximations to the elementary functions. Then, for each of the three main classes of elementary function algorithms (shift-and-add algorithms, polynomial or rational approximations, table-based methods) and for the additional, specific to approximate computing, "bit-manipulation" techniques, we examine what can be done for obtaining very fast estimates of a function, at the cost of a (controlled) loss in terms of accuracy.

Modified Fast Inverse Square Root and Square Root Approximation Algorithms: The Method of Switching Magic Constants

“…This also applies to the square root calculation algorithms, which we denote as Sqrt3. The proposed method can be thought of as a relatively accurate and fast initial guess 1 y -the DC initial approximation-for other iterative algorithms (see Template 1, lines [14][15][16]. In this paper, we consider modified NR iterations written in a special form, with combined multiply-add operations.…”

“…These elementary function algorithms-and especially those for calculating x and x / 1 -can be divided into several classes [2,[15][16][17]: digit-recurrence (shift-and-add) methods [2,16,18], iterative methods [16,19], polynomial methods [4,16,17,20], rational methods [16], table-based methods [2,[21][22][23], bit-manipulation techniques [24][25][26], and their combinations.…”

“…The use of direct lookup tables (LUTs) is less practical than combining several smaller LUTs with addition and multiplication operations. Bit-manipulation techniques are fast in both HW and SW, but have very limited accuracy-about 5-6 bits [15]. They are based on the peculiarities of the in-memory binary representation formats of integer and FP numbers.…”

“…Among the better-known methods of calculating the square root and reciprocal square root [1,2,15,20], the FISR method [20,[25][26][27][28][29] has recently gained increasing popularity in SW [8,20,27,[29][30][31][32] and HW [3,[33][34][35][36][37] applications. The algorithm was proposed for the first time in [24] but gained wider popularity through its use in the computer game Quake III Arena [27].…”

Fast Calculation of Cube and Inverse Cube Roots Using a Magic Constant and Its Implementation on Microcontrollers

“…Most modern low-cost microcontrollers that support floating-point computing only use slow (but accurate) C math.h library functions such as cbrt and 1.0f/cbrt, see [18]. In this article, to increase the speed of computing the inverse cube root, we propose algorithms using the bit manipulation technique [19]. These are based on the fast inverse cube root of the method [3,5,6,13] in modified versions; these are quite accurate and relatively fast.…”