The original paper is in English. Non-English content has been machine-translated and may contain typographical errors or mistranslations. ex. Some numerals are expressed as "XNUMX".
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The original paper is in English. Non-English content has been machine-translated and may contain typographical errors or mistranslations. Copyrights notice
Três algoritmos multiplicativos para a operação de divisão de ponto flutuante são comparados: o método de Newton-Raphson, o algoritmo de Goldschmidt e um método ingênuo que simplesmente calcula uma forma de expansão em série de Taylor de um recíproco. A série também fornece uma base teórica para o algoritmo de Goldschmidt. É bem sabido que, do método de Newton-Raphson e do algoritmo de Goldschmidt, o primeiro é o mais preciso, enquanto o último é o mais rápido em uma unidade em pipeline. Porém, pouco se relata sobre o método ingênuo. Neste relatório, analisamos a velocidade e precisão de cada método e apresentamos os resultados dos testes numéricos, que realizamos para confirmar a validade da análise de precisão. Basicamente, as comparações são feitas no contexto de implementação de software (por exemplo, uma biblioteca de macros) e a conformidade com o arredondamento da norma IEEE 754 não é considerada. É mostrado que o método ingênuo é útil em um cenário realista onde o número de iterações é pequeno e o método é implementado em uma unidade de ponto flutuante em pipeline com uma configuração de acumulação múltipla. Em tal situação, o método ingênuo fornece um resultado mais preciso com uma latência ligeiramente menor, em comparação com o algoritmo de Goldschmidt, e é muito mais rápido, mas ligeiramente inferior em precisão ao método de Newton-Raphson.
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Takashi AMISAKI, Umpei NAGASHIMA, Kazutoshi TANABE, "Floating-Point Divide Operation without Special Hardware Supports" in IEICE TRANSACTIONS on Fundamentals,
vol. E82-A, no. 1, pp. 173-177, January 1999, doi: .
Abstract: Three multiplicative algorithms for the floating-point divide operation are compared: the Newton-Raphson method, Goldschmidt's algorithm, and a naive method that simply calculates a form of the Taylor series expansion of a reciprocal. The series also provides a theoretical basis for Goldschmidt's algorithm. It is well known that, of the Newton-Raphson method and Goldschmidt's algorithm, the former is the more accurate while the latter is the faster on a pipelined unit. However, little is reported about the naive method. In this report, we analyze the speed and accuracy of each method and present the results of numerical tests, which we conducted to confirm the validity of the accuracy analysis. Basically, the comparison are made in the context of software implementation (e. g. , a macro library) and compliance with the IEEE Standard 754 rounding is not considered. It is shown that the naive method is useful in a realistic setting where the number of iterations is small and the method is implemented on a pipelined floating-point unit with a multiply-accumulate configuration. In such a situation, the naive method gives a more accurate result with a slightly lower latency, as compared with Goldschmidt's algorithm, and is much faster than but slightly inferior in accuracy to the Newton-Raphson method.
URL: https://global.ieice.org/en_transactions/fundamentals/10.1587/e82-a_1_173/_p
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@ARTICLE{e82-a_1_173,
author={Takashi AMISAKI, Umpei NAGASHIMA, Kazutoshi TANABE, },
journal={IEICE TRANSACTIONS on Fundamentals},
title={Floating-Point Divide Operation without Special Hardware Supports},
year={1999},
volume={E82-A},
number={1},
pages={173-177},
abstract={Three multiplicative algorithms for the floating-point divide operation are compared: the Newton-Raphson method, Goldschmidt's algorithm, and a naive method that simply calculates a form of the Taylor series expansion of a reciprocal. The series also provides a theoretical basis for Goldschmidt's algorithm. It is well known that, of the Newton-Raphson method and Goldschmidt's algorithm, the former is the more accurate while the latter is the faster on a pipelined unit. However, little is reported about the naive method. In this report, we analyze the speed and accuracy of each method and present the results of numerical tests, which we conducted to confirm the validity of the accuracy analysis. Basically, the comparison are made in the context of software implementation (e. g. , a macro library) and compliance with the IEEE Standard 754 rounding is not considered. It is shown that the naive method is useful in a realistic setting where the number of iterations is small and the method is implemented on a pipelined floating-point unit with a multiply-accumulate configuration. In such a situation, the naive method gives a more accurate result with a slightly lower latency, as compared with Goldschmidt's algorithm, and is much faster than but slightly inferior in accuracy to the Newton-Raphson method.},
keywords={},
doi={},
ISSN={},
month={January},}
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TY - JOUR
TI - Floating-Point Divide Operation without Special Hardware Supports
T2 - IEICE TRANSACTIONS on Fundamentals
SP - 173
EP - 177
AU - Takashi AMISAKI
AU - Umpei NAGASHIMA
AU - Kazutoshi TANABE
PY - 1999
DO -
JO - IEICE TRANSACTIONS on Fundamentals
SN -
VL - E82-A
IS - 1
JA - IEICE TRANSACTIONS on Fundamentals
Y1 - January 1999
AB - Three multiplicative algorithms for the floating-point divide operation are compared: the Newton-Raphson method, Goldschmidt's algorithm, and a naive method that simply calculates a form of the Taylor series expansion of a reciprocal. The series also provides a theoretical basis for Goldschmidt's algorithm. It is well known that, of the Newton-Raphson method and Goldschmidt's algorithm, the former is the more accurate while the latter is the faster on a pipelined unit. However, little is reported about the naive method. In this report, we analyze the speed and accuracy of each method and present the results of numerical tests, which we conducted to confirm the validity of the accuracy analysis. Basically, the comparison are made in the context of software implementation (e. g. , a macro library) and compliance with the IEEE Standard 754 rounding is not considered. It is shown that the naive method is useful in a realistic setting where the number of iterations is small and the method is implemented on a pipelined floating-point unit with a multiply-accumulate configuration. In such a situation, the naive method gives a more accurate result with a slightly lower latency, as compared with Goldschmidt's algorithm, and is much faster than but slightly inferior in accuracy to the Newton-Raphson method.
ER -