Sung Gi Kim , Gyeong Yeon Cho

The KIPS Transactions:PartA, Vol. 12, No. 2, pp. 95-102, Apr. 2005
10.3745/KIPSTA.2005.12.2.95,   PDF Download:

Abstract

The Newton-Raphson iterative algorithm for finding a floating point reciprocal square root calculates it by performing a fixed number of multiplications. In this paper, a variable latency Newton-Raphson’s reciprocal square root algorithm is proposed, that performs multiplications a variable number of times until the error becomes smaller than a given value. To find the reciprocal square root of a floating point number F, the algorithm repeats the following operations : ´X i 1= , i∈{0, 1, 2,...n-1}´ with the initial value is ´X0=±e0´ The bits to the right of p fractional bits in intermediate multiplication results are truncated, and this truncation error is less than ´er=2-p´ The value of p is 28 for the single precision floating point, and 58 for the double precision floating point. ......is less than the smallest number which is representable by floating point number. So, Xi 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 square root tables (X0=±e0) 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 square root unit. Also, it can be used to construct optimized approximate reciprocal square root 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.

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