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Floating-Point Division and Square Root Implementation using a Taylor-Series Expansion Algorithm with Reduced Look-up Tables @Taek-Jun Kwon, Jeff Draper, University of Southern California / Information Sciences Institute Marina del Rey, CA 90292, USA {tjkwon, draper}@ISI.EDU, VOL. 27, NO. 12, DECEMBER 2008 ©2008 IEEE Vishesh Kalra EE800 11089943 Vishesh Kalra EE800 11089943

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I NTRODUCTION Division and Square root are open considered as infrequent in general purpose applications. But in modern applications like CAD tools and 3D Graphics DIV/SQRT become performance bottlenecks. A fused Floating point multiply/square root/divide unit has been presented. Algorithm used- >Taylor Series Expansion with reduced lookup tables

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DIVISION ALGORITHIM

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STAGE DIAGRAM

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PROPOSED METHOD FUSED MUL/DIV/SQRT ALGORITHM

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STAGE DIAGRAM

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STEPS INVOLVED

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PIPELINE DIAGRAM

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REDUCED LOOK UP TABLES For Yo, generally two look up tables are used based upon exponent value of input operand-even or odd. To generate IEEE 754 standard 8 bit seed is used when exponent is even and 9 bit if odd. In addition to this Yo2 values are used via a small multiplier(9b*9b)

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REDUCED LOOK UP TABLES As a Result, three look up tables and a small Multiplier >Xo for division =1/b; >Yo_even and Yo_odd for 1/sqrt(b); However since all entries in each look-up table start with same constants (eg.Xo=0.1xxxxxxx, Yo_even=0.1xxxxxxx and Yo_odd=0.10xxxxxxx) Therefore Table sizes can be reduced by storing them. Table Sizes are >Xo=128*7b; >Yo_even=128*7b; >Yo_odd=256*7b;

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PARALLEL POWERING UNITS(PPU) Lets Recall the Algorithm 3/8=1/8 + 2/8, that means 2 bit shifted and 3 bit shifted values of (1-bYo2)2 are accumulated and in case of other term 5/16 =2/8 + 1/8,that means 2 bit shifted and 4 bit shifted. So 2 bit shifted is common so accumulated by PPU.

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STAGE DIAGRAM

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RESULTS-Area Comparison Without using a Multiplier to generate constants for reduced lookup tables.

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Conclusion As shown in Results, Additional Look up tables contribute to major area overhead. Also Pipelined Accumulators(Mux) contribute to area increase. To incorporate a Square root function with a modest 20 % increase is desirable and to incorporate it in entire FPU(Floating Point Unit) is a mere 10 %. Multiplier Approach for generating constants was done and resulted in area savings of 2.4 %.

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