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http://dx.doi.org/10.3745/KIPSTA.2005.12A.2.095

A Variable Latency Newton-Raphson's Floating Point Number Reciprocal Computation  

Kim Sung-Gi (부경대학교 대학원 컴퓨터공학과)
Cho Gyeong-Yeon (부경대학교 전자컴퓨터정보통신공학부)
Publication Information
The KIPS Transactions:PartA / v.12A, no.2, 2005 , pp. 95-102 More about this Journal
Abstract
The Newton-Raphson iterative algorithm for finding a floating point reciprocal which is widely used for a floating point division, calculates the reciprocal by performing a fixed number of multiplications. In this paper, a variable latency Newton-Raphson's reciprocal algorithm is proposed that performs multiplications a variable number of times until the error becomes smaller than a given value. To find the reciprocal of a floating point number F, the algorithm repeats the following operations: '$ with the initial value $. The bits to the right of p fractional bits in intermediate multiplication results are truncated, and this truncation error is less than $. The value of p is 27 for the single precision floating point, and 57 for the double precision floating point. Let $, these is $'X_{i+1}=\frac{1}{F}-e_{i+1},\;where\;{'}e_{i+1} is approximate to $. Since the number of multiplications performed by the proposed algorithm is dependent on the input values, the average number of multiplications per an operation is derived from many reciprocal tables $(X_0=\frac{1}{F}{\pm}e_0)$ with varying sizes. The superiority of this algorithm is proved by comparing this average number with the fixed number of multiplications of the conventional algorithm. Since the proposed algorithm only performs the multiplications until the error gets smaller than a given value, it can be used to improve the performance of a reciprocal unit. Also, it can be used to construct optimized approximate reciprocal tables. The results of this paper can be applied to many areas that utilize floating point numbers, such as digital signal processing, computer graphics, multimedia scientific computing, etc.
Keywords
부동소수점;뉴톤-랍손;역수;가변시간;
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