A Variable Latency Goldschmidt's Floating Point Number Divider
Sung-Gi Kim, Hong-Bok Song, Gyeong-Yeon Cho · The Journal of the Korean Institute of Information and Communication Engineering · 2005
The Goldschmidt iterative algorithm for a floating point divide calculates it by performing a fixed number of multiplications. In this paper, a variable latency Goldschmidt's divide algorithm is proposed, that performs multiplications a variable number of times until the error becomes smaller than a given value. To calculate a floating point divide '', multifly '' to the denominator and the nominator, then it becomes ’'. And the algorithm repeats the following operations: ’, i{0,1,...n-1}'. 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 29 for the single precision floating point, and 59 for the double precision floating point. Let ’', there is $F_{i+1}=1-e_{i+1},\;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 () with varying sizes. 1'he 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 divider. 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