Computer Arithmetic 1 Of 23 Computer Arithmetic By: Ali Bohlooli Zefreh sepahan. iut. ac. ir Professor: Dr. Shadrokh Samavi Date: 1381/ 3/ 4 Square Root Algorithm

Computer Arithmetic 2 Of 23 Overview Digit by Digit method Restoring Non Restoring SRT –Division –Square Root

Computer Arithmetic 3 Of 23 Similarities between Square Root & Division The square-root and division are inverse of multiplication, therefore expect to find some similarities between them.

Computer Arithmetic 4 Of 23 Square-Rooting methods 1) Multiplicative methods (Convergence) 2) Subtractive methods (Digit by digit) –Use the subtraction and shift operation –Restoring, NonRestoring, SRT algorithms –Adopted in most of the recent processors –Area-effective –Use the reciprocal table –Newton-Raphson, Goldschmidt’s algorithms –Relatively high-speed

Computer Arithmetic 5 Of 23 Subtractive methods

Computer Arithmetic 6 Of 23 Restoring & Non Restoring methods Assume that Q i is partial root in step I and q n-i-1 Is I+1’bit of partial root so Next partial root=r* Q i + q n-i-1 And (r Q i + q n-i-1 ) 2 =r 2 Q i 2 +2r Q i q n-i-1 + q 2 n -i-1 Therefore 2r Q i q n-i-1 + q 2 n -i-1 Must be subtract from partial reminder If radix =2 (2 Q i + q n-i-1 ) 2 =4 Q i 2 +4 Q i q n-i-1 + q 2 n -i-1 If q n-i-1 = ‘1’ then it is sufficient to concatenate ‘01’ in the right of Q i

Computer Arithmetic 7 Of 23 Restoring CS q1q1 q2q2 q3q3 q4q4 r8r8 r5r5 r6r6 r7r7 r4r4 r3r3 r2r2 r1r1 R= x8x8 x6x6 x7x7 x5x5 x4x4 x3x3 x2x2 x1x a b Mux r ou t Borrow_out Sub control Sub 0 1 Borrow_in CS 0 0 0

Computer Arithmetic 8 Of 23 Non Restoring Q= =0.q 1 q 2 …q n X=Q 2 = 0.x 1 x 2 …x 2n Non Restoring Square root algorithm step 1) R 0 = 0.x 1 x 2 ; f 0 =0.01;i=1; step 2)R i =R i-1 -f i-1 step 3)if (Ri<0) then (q i ‘0’;R i R i.x 2i+1 x 2i+2 ; F i q 1 q 2 …q i 11; R i+1 R i +F i ); Else((q i ‘1’;R i R i.x 2i+1 x 2i+2 ; F i q 1 q 2 …q i 01; R i+1 R i -F i ); step 4) i i+1; step 5)if(I<n) then (go to step 3) Else (Q=0.q 1 q 2 …q n ) End

Computer Arithmetic 9 Of 23 Proof of Non restoring Algorithm (X11) 2 =4x+3 (X01) 2 =4x+1 In restoring algorithm R j+1 R j q j+1 2q j R j+2 4R j -(4q j+1 +1)=4R j -8q j -1 In non restoring algorithm R j+1 R j -(4q j +1) q j+1 2q j R j+2 4R j+1 +(4q j+1 +3)=4R j -8q j -1 `

Computer Arithmetic 10 Of 23 Non Restoring CAS r8r8 r5r5 r6r6 r7r7 r4r4 r3r3 r2r2 r1r1 R= q4q4 q3q3 q2q2 q1q x1x1 x2x2 x4x4 x3x3 x5x5 x6x6 x7x7 x8x CAS FA  x in r out qiqi sub/add c in c out 1

Computer Arithmetic 11 Of 23 division & Square-root recursion formulas To find, set and evaluate Define, then R j+1 =rR j -q j+1 D (division recursion formula) (Square-root recursion formula) : the jth partial root : the (j+1)th bit of partial root the subtractive square root and division methods are closely related

Computer Arithmetic 12 Of 23 Assume that X is Radicand and its square root is S m = 0.s 1...s m X- S m = w m 2 -m 에서 w m = 2 m X- 2 m S m w m-1 = 2 m-1 X- 2 m-1 S m w m -2 w m-1 = -2 m (S m -S m-1 ) = -2 m ((s m 2 -m + S m-1 ) - S m-1 ) = -2 m (2s m S m-1 2 -m + s m 2 -2m ) = -s m (2S m-1 + s m 2 -m ) 22 2 w m = 2 w m-1 - s m (2S m-1 + s m 2 -m ) 2 Why

Computer Arithmetic 13 Of 23 SRT Division in Non restoring division 1 if 0< 2r i-1 <D -1 if –2D < 2r i-1 <0 r i = 2 r i-1 - q i D q i = riri D2D D 2r i-1 q i =1 -D-2D -D

Computer Arithmetic 14 Of 23 SRT Division For q i =0 implies no addition/subtraction However, comparing 2 r i-1 with D or -D takes time if D is a normalized fraction (1/2  D < 1), then we can set q i =0 for -1/2  2 r i-1 < 1/2 1 if 2 r i-1  1/2 0 if -1/2  2 r i-1 < 1/2 1 if 2 r i-1 < -1/2 r i = 2 r i-1 - q i D (SRT division)  1.|r i |  D  convergence guaranteed 2.comparing 2 r i-1 with 1/2 or -1/2 is easier if 2 r i-1 = 0.1xxx  2 r i-1  1/2 if 2 r i-1 = 1.0xxx  2 r i-1 < -1/2 if |X|=|r 0 |  1/2  |2 r 0 |  1  examine three bits of 2 r i-1 q i =

Computer Arithmetic 15 Of 23 SRT Division 1/2 1 2r i-1 q i =1 -1/2-D -1/2 q i =0 D when |2r 0 |=|2X|  1 start with a normalized divisor normalize the partial remainder by shifting over leading 0’s if positive, and over leading 1’s if negative 2 r i-1 =0.001xxx 2 r i-1 =1.110xxx riri

Computer Arithmetic 16 Of 23 SRT Division (Example) X = = 5/16 D = = 3/4 r 0 =X=5/16 2r 0 Add -D=-3/4 r 1 =2r 0 -D 2r 1  r 2 2r 2  r 3 2r 3 Add D r 4 =2r 3 +D Add D r 4 (corrected)  1/2 q 1 =  1/2 q 2 =  1/2 q 3 =  1/2 q 4 = r correction Q = – ulp( Unit in the Least Position ) = R = 1/2 * 2 -4 = 1/32

Computer Arithmetic 17 Of 23 SRT Division How to reduce the number of add/subtract operations from simulation and statistical analysis 1. Average ‘shift’ in the SRT method : 2.67 average operations : n/ Actual number of operations depends on D The smallest number is achieved when D  [17/28,3/4] with average shift of 3. Approximately D  [3/5,3/4]

Computer Arithmetic 18 Of 23 SRT Division To reduce the number of operations, modify the SRT method when D  [3/5,3/4] 1. Examine the possibility of using a multiple of D: in some of the steps during the division use 2D (one position earlier) if D is too small, or D/2 (one position later) if D is too large.

Computer Arithmetic 19 Of 23 Modifying the SRT method 2. Change the comparison constant K K= 1/2 if D  [3/5,3/4] k1=3/8 k2=7/16k3=1/2 k2=5/8k3=3/ K= D= K= 3/8 if D  [1/2,9/16] K= 7/16 if D  [9/16,10/16]

Computer Arithmetic 20 Of 23 modifying the SRT method 3- Radix  (=2 m )   # of ADD/SUB reduced # of steps =  n/m  r 0 =X; r i =  r i–1 –q i D restoring DIV: (  –1)  q i  0 Non restoring DIV: (  –1)  q i  –(  –1) High Radix SRT DIV:   q i  –   (  –1)     (  –1) /2 

Computer Arithmetic 21 Of 23 Compute Allowable Range of k (1) k|D|  |r i | (convergence condition)  r i =  r i–1 –q i D=  kD–  D  kD (1)  k =  /(  –1)

Computer Arithmetic 22 Of 23 SRT Division (4) Quotient Accumulation (QA) (1) Quotient Selection (QS) q j +1 결정 q j +1 결정 (3) Partial Remainder Formation (PRF) p j +1 = rp j - q j + 1 d p j +1 = rp j - q j + 1 d (2) Divisor Multiple Formation (DMF) q j +1 d 계산 q j +1 d 계산 Steps

Computer Arithmetic 23 Of 23 SRT Division MUX Divisor Multiple Generator(-2D,-1D,0D,1D,2D) MUX QuotientSelector(PLA) rem P Quotient Q CPA Logic Last q Dividend Divisor QS DMF PRF QA Radix-4 SRT Implementation

Computer Arithmetic 24 Of 23 Refrences 1) “AREA AND PERFORMANCE TRADEOFFS IN FLOATING-POINT DIVIDE AND SQUARE ROOT IMPLEMENTATIONS”,Miriam Leeserl, Dept. of Electrical and Computer Engineering, Northeastern UniversityBoston, Massachusetts ) “New algorithm and vlsi architecture for SRT Divesion and square root” S.E. McQuillan, J.V. McCanny, University of Belfast 4)”Computer arithmetic”,kai Hwang 5) zSzsrtlong.pdf/srt-division-architectures-models.pdf srt-division-architectures-models.pdf 6) s.pdf/harris97srt.pdf 7) g_serzSz..zSzPublicationszSzkuhlmannzSzsqrt.pdf/fast-low-power-shared.pdf 8) aperszSzSRT-thesis.pdf/alternative-implementations-of-srt.pdf 9) delmanzSzpentiumzSzpentium.pdf/the-mathematics-of-the.pdf

Computer Arithmetic 25 Of 23 References 10) _TOC.pdf/oberman97division.pdf 11) iptzSzedtc96.pdf/asynchronous-srt-dividers-the.pdf 12) Sz1995zSzlect3zSzlect3.pdf/about-these-notes-on.pdf 13) sb.mpg.dezSzpubzSzpaperszSzreportszSzMPI-I pdf/burnikel96leda.pdf 14) _KarpzSzpublicationszSzsqrtdiv.pdf/alan97high.pdf 15) vlsi.yonsei.ac.kr/seminar/ Hi-Speed%20Division%20Circuit.ppt 16) piglet.uccs.edu/~cs520/S99cha.ppt 17) sdgroup.snu.ac.kr/class/digital.2000/note/302L22.ppt