Fast Algorithm of A 64-bit Decimal Logarithmic Converter

Abstract

<!--[if gte mso 9]><xml> <w:WordDocument> <w:View>Normal</w:View> <w:Zoom>0</w:Zoom> <w:PunctuationKerning /> <w:ValidateAgainstSchemas /> <w:SaveIfXMLInvalid>false</w:SaveIfXMLInvalid> <w:IgnoreMixedContent>false</w:IgnoreMixedContent> <w:AlwaysShowPlaceholderText>false</w:AlwaysShowPlaceholderText> <w:Compatibility> <w:BreakWrappedTables /> <w:SnapToGridInCell /> <w:WrapTextWithPunct /> <w:UseAsianBreakRules /> <w:DontGrowAutofit /> <w:UseFELayout /> </w:Compatibility> <w:BrowserLevel>MicrosoftInternetExplorer4</w:BrowserLevel> </w:WordDocument> </xml><![endif]--><!--[if gte mso 9]><xml> <w:LatentStyles DefLockedState="false" LatentStyleCount="156"> </w:LatentStyles> </xml><![endif]--> <!-- /* Style Definitions */ p.MsoNormal, li.MsoNormal, div.MsoNormal {mso-style-parent:""; margin:0in; margin-bottom:.0001pt; text-align:center; mso-pagination:widow-orphan; font-size:10.0pt; font-family:"Times New Roman"; mso-fareast-font-family:"Times New Roman"; mso-fareast-language:EN-US;} @page Section1 {size:8.5in 11.0in; margin:1.0in 1.25in 1.0in 1.25in; mso-header-margin:.5in; mso-footer-margin:.5in; mso-paper-source:0;} div.Section1 {page:Section1;} --> <!--[if gte mso 10]> <mce:style><! /* Style Definitions */ table.MsoNormalTable {mso-style-name:"Table Normal"; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-parent:""; mso-padding-alt:0in 5.4pt 0in 5.4pt; mso-para-margin:0in; mso-para-margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:10.0pt; font-family:"Times New Roman"; mso-fareast-font-family:"Times New Roman"; mso-ansi-language:#0400; mso-fareast-language:#0400; mso-bidi-language:#0400;} --> <!--[endif]--><span style="font-size: 10pt; font-family: ">The paper presents an efficient algorithm to compute base-10 logarithm of a decimal number. The algorithm uses a 64-bit floating-point arithmetic, and is based on a digit-by-digit iterative computation that does not require look-up tables, curve fitting, decimal-binary conversion, or division operations. It is the first FPGA prototype of its kind that uses a 64-bit (decimal 16-digit) precision. Two numerical examples have been presented for the purpose of illustration. The algorithm produces very accurate result with a maximum absolute error of 3.53x10<sup>-14</sup>. The architecture is pipelined and implemented on to the Xilinx Virtex2p FPGA. It costs 6,752 logic cells, outputs at a minimum rate of 51 mega-samples/sec, and consumes 125.7 mW of power. The scheme is very suitable for timing and accuracy critical applications and compliant with the IEEE754-2008 standard (<em>decimal</em>64 format).</span&gt

Non-linear partitioning for decimal logarithm approximation

“…Also in this method of 16-region partitions, 32 4-digit values are to be stored in ROM. x3 = n.x, and x4 = n.x2 (8) III. PROPOSED LINEAR APPROXIMATION OF DECIMAL LOGARITHM (9)…”

“…This algorithm gives accurate results but has longer latency. Recently a 64-bit decimal logarithmic converted was proposed in [8]. This algorithm results in reduced number of clock cycles when compared to [7], but the implementation includes a power 10 block, which results in increased hardware complexity.…”

FPGA Based Efficient Multiplier for Image Processing Applications Using Recursive Error Free Mitchell Log Multiplier and KOM Architecture

“…Ramin Tajallipour et al, [7] proposed an efficient algorithm for computing decimal logarithm using 64 bit floating point arithmetic. The algorithm is based on digit by digit iterative computation that does not require lookup tables, curve fitting, decimal to binary conversion and division algorithms.…”

Implementation of DWT Integrated Log Based FPU with SPIHT Coders on FPGA

“…Overflow condition occurs, when the addition of two numbers of same bit size resulted with the increase in bit width. [11].In this work, log based 32 bit floating point multiplication was computed based on look up table method. [15].128 values of single precision floating point numbers is called as a function and accessed as 7 bit integer.…”