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In this paper, an algorithm based on Taylor series expansion is proposed to calculate the logarithm (*log*_{2}*x*) of IEEE754 binary32 accuracy floating-point number by a multi-domain partitioning method. The general mantissa (1≤*x*<2) is multiplied by 2, 4, 8, … (or equivalently left-shifted by 1, 2, 3, … bits), the regions of (2≤*x*<4), (4≤*x*<8), (8≤*x*<16),… are considered, and Taylor-series expansion is applied. In those regions, the slope of *f*(*x*)=log_{2} *x* with respect to *x* is gentle compared to the region of (1≤*x*<2), which reduces the required number of terms. We also consider the trade-offs among the numbers of additions, subtractions, and multiplications and Look-Up Table (LUT) size in hardware to select the best algorithm for the engineer's design and build the best hardware device.

- Publication
- IEICE TRANSACTIONS on Fundamentals Vol.E105-A No.7 pp.1020-1027

- Publication Date
- 2022/07/01

- Publicized
- 2022/01/17

- Online ISSN
- 1745-1337

- DOI
- 10.1587/transfun.2021EAP1076

- Type of Manuscript
- PAPER

- Category
- Digital Signal Processing

Jianglin WEI

Gunma University

Anna KUWANA

Gunma University

Haruo KOBAYASHI

Gunma University

Kazuyoshi KUBO

Oyama National College of Technology

The copyright of the original papers published on this site belongs to IEICE. Unauthorized use of the original or translated papers is prohibited. See IEICE Provisions on Copyright for details.

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Jianglin WEI, Anna KUWANA, Haruo KOBAYASHI, Kazuyoshi KUBO, "IEEE754 Binary32 Floating-Point Logarithmic Algorithms Based on Taylor-Series Expansion with Mantissa Region Conversion and Division" in IEICE TRANSACTIONS on Fundamentals,
vol. E105-A, no. 7, pp. 1020-1027, July 2022, doi: 10.1587/transfun.2021EAP1076.

Abstract: In this paper, an algorithm based on Taylor series expansion is proposed to calculate the logarithm (*log*_{2}*x*) of IEEE754 binary32 accuracy floating-point number by a multi-domain partitioning method. The general mantissa (1≤*x*<2) is multiplied by 2, 4, 8, … (or equivalently left-shifted by 1, 2, 3, … bits), the regions of (2≤*x*<4), (4≤*x*<8), (8≤*x*<16),… are considered, and Taylor-series expansion is applied. In those regions, the slope of *f*(*x*)=log_{2} *x* with respect to *x* is gentle compared to the region of (1≤*x*<2), which reduces the required number of terms. We also consider the trade-offs among the numbers of additions, subtractions, and multiplications and Look-Up Table (LUT) size in hardware to select the best algorithm for the engineer's design and build the best hardware device.

URL: https://global.ieice.org/en_transactions/fundamentals/10.1587/transfun.2021EAP1076/_p

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@ARTICLE{e105-a_7_1020,

author={Jianglin WEI, Anna KUWANA, Haruo KOBAYASHI, Kazuyoshi KUBO, },

journal={IEICE TRANSACTIONS on Fundamentals},

title={IEEE754 Binary32 Floating-Point Logarithmic Algorithms Based on Taylor-Series Expansion with Mantissa Region Conversion and Division},

year={2022},

volume={E105-A},

number={7},

pages={1020-1027},

abstract={In this paper, an algorithm based on Taylor series expansion is proposed to calculate the logarithm (*log*_{2}*x*) of IEEE754 binary32 accuracy floating-point number by a multi-domain partitioning method. The general mantissa (1≤*x*<2) is multiplied by 2, 4, 8, … (or equivalently left-shifted by 1, 2, 3, … bits), the regions of (2≤*x*<4), (4≤*x*<8), (8≤*x*<16),… are considered, and Taylor-series expansion is applied. In those regions, the slope of *f*(*x*)=log_{2} *x* with respect to *x* is gentle compared to the region of (1≤*x*<2), which reduces the required number of terms. We also consider the trade-offs among the numbers of additions, subtractions, and multiplications and Look-Up Table (LUT) size in hardware to select the best algorithm for the engineer's design and build the best hardware device.},

keywords={},

doi={10.1587/transfun.2021EAP1076},

ISSN={1745-1337},

month={July},}

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TY - JOUR

TI - IEEE754 Binary32 Floating-Point Logarithmic Algorithms Based on Taylor-Series Expansion with Mantissa Region Conversion and Division

T2 - IEICE TRANSACTIONS on Fundamentals

SP - 1020

EP - 1027

AU - Jianglin WEI

AU - Anna KUWANA

AU - Haruo KOBAYASHI

AU - Kazuyoshi KUBO

PY - 2022

DO - 10.1587/transfun.2021EAP1076

JO - IEICE TRANSACTIONS on Fundamentals

SN - 1745-1337

VL - E105-A

IS - 7

JA - IEICE TRANSACTIONS on Fundamentals

Y1 - July 2022

AB - In this paper, an algorithm based on Taylor series expansion is proposed to calculate the logarithm (*log*_{2}*x*) of IEEE754 binary32 accuracy floating-point number by a multi-domain partitioning method. The general mantissa (1≤*x*<2) is multiplied by 2, 4, 8, … (or equivalently left-shifted by 1, 2, 3, … bits), the regions of (2≤*x*<4), (4≤*x*<8), (8≤*x*<16),… are considered, and Taylor-series expansion is applied. In those regions, the slope of *f*(*x*)=log_{2} *x* with respect to *x* is gentle compared to the region of (1≤*x*<2), which reduces the required number of terms. We also consider the trade-offs among the numbers of additions, subtractions, and multiplications and Look-Up Table (LUT) size in hardware to select the best algorithm for the engineer's design and build the best hardware device.

ER -