Evaluation Of Elementary Functions Research Articles

Modular design and implementation of field-programmable-gate-array-based Gaussian noise generator

ABSTRACTThe modular design of a Gaussian noise generator (GNG) based on field-programmable gate array (FPGA) technology was studied. A new range reduction architecture was included in a series of elementary function evaluation modules and was integrated into the GNG system. The approximation and quantisation errors for the square root module with a first polynomial approximation were high; therefore, we used the central limit theorem (CLT) to improve the noise quality. This resulted in an output rate of one sample per clock cycle. We subsequently applied Newton's method for the square root module, thus eliminating the need for the use of the CLT because applying the CLT resulted in an output rate of two samples per clock cycle (>200 million samples per second). Two statistical tests confirmed that our GNG is of high quality. Furthermore, the range reduction, which is used to solve a limited interval of the function approximation algorithms of the System Generator platform using Xilinx FPGAs, appeared to have a higher numerical accuracy, was operated at >350 MHz, and can be suitably applied for any function evaluation.

A Domain-Specific Architecture for Elementary Function Evaluation

We propose a Domain-Specific Architecture for elementary function computation to improve throughput while reducing power consumption as a model for more general applications: support fine-grained parallelism by eliminating branches, and eliminate the duplication required by coprocessors by decomposing computation into instructions which fit existing pipelined execution models and standard register files. Our example instruction architecture (ISA) extension supports scalar and vector/SIMD implementations of table-based methods of calculating all common special functions, with the aim of improving throughput by (1) eliminating the need for tables in memory, (2) eliminating all branches for special cases, and (3) reducing the total number of instructions. Two new instructions are required, a table lookup instruction and an extended-precision floating-point multiply-add instruction with special treatment for exceptional inputs. To estimate the performance impact of these instructions, we implemented them in a modified Cell/B.E. SPU simulator and observed an average throughput improvement of 2.5 times for optimized loops mapping single functions over long vectors.

Unified Tables for Exponential and Logarithm Families

Accurate table methods allow for very accurate and efficient evaluation of elementary functions. We present new single-table approaches to logarithm and exponential evaluation, by which we mean that a single table of values works for both log( x ) and log(1 + x ), and a single table for e x and e x − 1. This approach eliminates special cases normally required to evaluate log(1 + x ) and e x − 1 accurately near zero, which will significantly improve performance on architectures which use SIMD parallelism, or on which data-dependent branching is expensive. We have implemented it on the Cell/B.E. SPU (SIMD compute engine) and found the resulting functions to be up to twice as fast as the conventional implementations distributed in the IBM Mathematical Acceleration Subsystem (MASS). We include the literate code used to generate all the variants of exponential and log functions in the article, and discuss relevant language and hardware features.

Space-efficient evaluation of hypergeometric series

We consider the evaluation of the truncated hypergeometric series [EQUATION] to high precision, where <i>a, b, p</i>, and <i>q</i> are polynomials with integer coefficients, and <i>a</i>(<i>n</i>), <i>b</i>(<i>n</i>), <i>p</i>(<i>n</i>), <i>q</i>(<i>n</i>) have bit length <i>O</i>(log <i>n</i>). We also assume that the series is linearly convergent, so that the <i>n</i>th term of (1) is <i>O</i>(<i>c<sup>-n</sup></i>) with <i>c</i> > 1. These series are commonly used in the high precision evaluation of elementary functions and other constants, including the exponential function, logarithms, trigonometric functions, and constants such as the Apéry's constant ζ(3) [9, 10]. "Binary splitting" is an approach that has been independently discovered and used by many authors in the computation of (1) [2, 3, 4, 5, 8, 10, 12]. Binary splitting computes the numerator and denominator of the rational number <i>S</i>(<i>N</i>). The decimal representation of <i>S</i>(<i>N</i>) is then computed by fixed-point division of the numerator by the denominator. The binary splitting approach takes advantage of the special form of the series (1) to obtain a denominator that is relatively small (of size <i>O</i>(<i>N</i> log <i>N</i>)). It also takes advantage of fast integer multiplication to obtain a time complexity of <i>O</i>((log <i>N</i>)<sup>2</sup>M(<i>N</i>)), where M(<i>N</i>) = <i>O</i>(<i>N</i> log <i>N</i> log log <i>N</i>) is the complexity of integer multiplication of two <i>N</i>-bit integers [16]. The space complexity of the algorithm is <i>O</i>(<i>N</i> log <i>N</i>), the size of the computed numerator and denominator. Typically, the numerator and denominator computed by binary splitting have large common factors. For example, in the computation of 640000 digits of ζ(3), as much as 86% of the size of the computed numerator and denominator can be attributed to their common factor [7]. Empirically, we have observed that the size of the reduced numerator and denominator is <i>O</i>(<i>N</i>) instead of <i>O</i>(<i>N</i> log <i>N</i>) as computed by binary splitting. The additional digits computed not only slow down the final division but also require more memory to be used during the computation. For computing a large number of decimal digits, either the computation cannot be done at all or some data would have to be swapped out of memory, increasing the computation time dramatically. In this poster, we study the application of well-known techniques in computer algebra to the evaluation of (1). If a bound on the size of the <i>reduced</i> numerator and denominator is known, we can compute the image of <i>S</i>(<i>N</i>) in (1) under an appropriately chosen modulus. Fast rational number reconstruction can then be applied to recover the reduced numerator and denominator [13, 14, 15, 17]. We show how to apply our techniques to the computation of ζ(3), including the prediction of the size of the reduced numerator and denominator. In particular, we obtain the desired <i>O</i>(<i>N</i>) bound on the size of reduced numerator and denominator, which is an interesting result by itself. The techniques used in the analysis can be applied to similar hypergeometric series.

On-the-Fly Range Reduction

In several cases, the input argument of an elementary function evaluation is given bit-serially, most significant bit first. We suggest a solution for performing the first step of the evaluation (namely, the range reduction) on the fly: the computation is overlapped with the reception of the input bits. This algorithm can be used for the trigonometric functions sin, cos, tan as well as for the exponential function.

Evaluating Elementary Functions with Guaranteed Precision

Because of changes in the computer markets, the problem of efficient evaluation of elementary functions (which seemed to be already solved) becomes important again. In this paper, after a brief review of current approaches to this problem, algorithms for finding guaranteed bounds for values of elementary functions are suggested. The evaluation time is reduced through increasing the amount of memory used.

Approximate Euclidean Shortest Paths in 3-Space

Papadimitriou's approximation approach to the Euclidean shortest path (ESP) in 3-space is revisited. As this problem is NP-hard, his approach represents an important step towards practical algorithms. However, there are several gaps in the original description. Besides giving a complete treatment in the framework of bit complexity, we also improve on his subdivision method. Among the tools needed are root-separation bounds and nontrivial applications of Brent's complexity bounds on evaluation of elementary functions using floating point numbers.

A new addition scheme and fast scaling factor compensation methods for CORDIC algorithms

As CORDIC algorithms attract more and more attention in elementary function evaluation and signal processing applications, the problem of their VLSI realization has drawn considerable interest. In this work we present speed enhancement techniques for these types of algorithms, covering algorithmic and implementation issues. We discuss speed limiting issues, namely addition techniques, appropriate number systems and scaling factor compensation with special emphasis on low latency time. A new carry-select adder structure will be presented that offers an excellent tradeoff between speed and VLSI chip area. With two scaling factor compensation schemes the fastest or most area economical CORDIC implementations are feasible. Also, the close relationship between SRT division and the new redundant CORDIC algorithms will be demonstrated.

Evaluating elementary functions in a numerical coprocessor based on rational approximations

A different approach to hardware evaluation of elementary functions for high-precision floating-point numbers (in particular, the extended double precision format of the IEEE standard P754) is examined. The evaluation is based on rational approximations of the elementary functions, a method which is commonly used in scientific software packages. A hardware model is presented of a floating-point numeric coprocessor consisting of a fast adder and a fast multiplier, and the minimum hardware required for evaluation of the elementary functions is added to it. Next, rational approximations for evaluating the elementary functions and testing the accuracy of the results are derived. The calculation time of these approximations in the proposed numeric processor is then estimated. It is concluded that rational approximations can successfully complete with previously used methods when execution time and silicon area are considered.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Efficient Multiple-Precision Evaluation of Elementary Functions

Let $M(t)$ denote the time required to multiply two t-digit numbers using base b arithmetic. Methods are presented for computing the elementary functions in $O({t^{1/3}}M(t))$ time.

Efficient multiple-precision evaluation of elementary functions

Let M ( t ) M(t) denote the time required to multiply two t-digit numbers using base b arithmetic. Methods are presented for computing the elementary functions in O ( t 1 / 3 M ( t ) ) O({t^{1/3}}M(t)) time.

An economic version of numerical Laplace inversion

A modified form of Papoulis's trigonometric algorithm for Laplace inversion is described; it amounts to summation of a finite number of values of the transform with previously computed coefficients, and avoids computer evaluation of elementary functions, thus minimizing the number of operations when finding. the original approximately. The rounding errors are analyzed.

A pipelined computer architecture for unified elementary function evaluation

A pipelined computer architecture for rapid consecutive evaluation of several elementary functions ( x/ y, √ x, sin x, cos, x, e x , ln x, …) using basic CORDIC algorithms is proposed. Continued products iterations of the form (1 + σ i m 2 − k ) allow linking n-identical ALU structures to permit n different function evaluations. New algorithms for sin −1, cos −1, cot −1, sinh −1, cosh −1 and x v are developed. Lastly, a new functional efficiency is defined for pipeline architectures which compares favorably to iterative arrays. Index terms—Digital Arithmetic, Pipeline, Unified Elementary Functions, Iterative Algorithms, CORDIC

Fast Multiple-Precision Evaluation of Elementary Functions

Let ƒ( x ) be one of the usual elementary functions (exp, log, artan, sin, cosh, etc.), and let M ( n ) be the number of single-precision operations required to multiply n -bit integers. It is shown that ƒ( x ) can be evaluated, with relative error Ο (2 - n ), in Ο ( M ( n )log ( n )) operations as n → ∞, for any floating-point number x (with an n -bit fraction) in a suitable finite interval. From the Schönhage-Strassen bound on M ( n ), it follows that an n -bit approximation to ƒ( x ) may be evaluated in Ο ( n log 2 ( n ) log log( n )) operations. Special cases include the evaluation of constants such as π, e , and e π . The algorithms depend on the theory of elliptic integrals, using the arithmetic-geometric mean iteration and ascending Landen transformations.