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A Pipelined LNS ALU Mark G Arnold marnold uwyo edu University of Wyoming University of Manchester Institute of Science and Technology Outline What is LNS and Why Bother Three Step Addition Algorithm Prior research Reduce ROM for step 2 Reduce latency here for steps 1 and 3 Novel interpolator to keep same ROM Logarithmic Number System LNS Convert to base b logarithms once x logb X Keep in LNS throughout all computation Convert back only when done X bx upper case values of real numbers lower case logarithmic representations Why bother with LNS 1 Natural for some applications Speech law CODEC HMMs ex Neural Nets 1 1 e x 2 Over one hundred papers see www xlnsresearch com 3 Lower Power Consumption 4 Can tune precision range to problem 5 Easy Multiplication Division Square Root LNS Power Consumption LNS compresses information High order bits of LNS change less frequently than equivalent fixed point LNS can take fewer bits Can tune b to the application 1 Only precision needed by app 2 Less wasted dynamic range LNS takes less power from Paliouras and Stouraitis Easy Multiplication and Division Given LNS representations x logb X y logb Y LNS multiplication algorithm only 1 step Let p x y X x 1 2 3 4 5 6 78 91 2 3 4 5 6 7 8 91 P p 1 2 Justification logb P logb X Y logb X logb Y Hardware Fixed point adder or a slide rule Division square square root similar1 3 4 5 6 78 91 2 3 4 5 6 78 91 y Y Addition is more difficult LNS addition algorithm 1 Let z1 y1 x1 2 Compute sb z1 where sb z1 logb 1 bz1 3 Let y2 x1 sb z1 Hardware required 1 Fixed point subtractor 2 ROM lookup and possibly interpolation 3 Fixed point adder Addition Justification LNS addition algorithm 1 Let z1 y1 x1 Justification 1 logb Z1 logb Y1 X1 logb Y1 logb X1 Addition Justification LNS addition algorithm 1 Let z1 y1 x1 2 Compute sb z1 logb 1 bz1 Justification 1 logb Z1 logb Y1 X1 logb Y1 logb X1 2 logb 1 Z1 logb 1 Y1 X1 Addition Justification LNS addition algorithm 1 Let z1 y1 x1 2 Compute sb z1 logb 1 bz1 3 Let y2



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