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It is clear that multiply is composed of iterative Shift and addPSbjQbb b DMultiply hardware (V1)`   EMultiply Algorithm (V1)b$   F"Observations on Multiply Version 1##b( # 1/2 bits in multiplicand always 0 => 64-bit adder is wasted 0 s inserted in right of of multiplicand as shifted => least significant bits of product never changed once formed How many cycles it takes to multiply two 32-bit numbers ? &`=d  (X C Instead of shifting multiplicand to left, shift product to right? .DBdB GMultiply hardware (V2)`   HMultiply Algorithm V2b$   I What s going on?`  bMultiplicand stay s still and product moves right22b 2 L"Observations on Multiply Version 2##b( # yProduct register wastes space that exactly matches size of multiplier => combine Multiplier register and Product registerzzb( z MMultiply hardware V3`   NMultiply Algorithm V3b$   OSign Multiplicationb(  Easiest solution is to make both positive remember whether to complement product when done That is: calculate the sign of the product, convert the operands into positive numbers, leave out the sign bit, run for 31 steps, then fix the result. `I`I`` `   'W#Faster Algorithms or Multiplication$$b$ $  Booth s Algorithm multiply signed numbers using same hardware as before and save cycles can handle multiple bits at a time Using an array of adders Observation: whether to add or not add a particular shifted multiplicand  a decision can be made all in parallel ..LZiZZ`i``  "TDivide: Paper & Pencil`  t 1001 Quotient Divisor 1000 1001010 Dividend  1000 10 101 1010  1000 10 Remainder Dividend = Quotient * Divisor + Remainder cgcgcg (cgg c gb*c   #UHow to do Division ?`   )YDivision process B step 0: Remainder is initialized with the divident step 1: The divisor is shift to the right at each step step 2: And we test if remainder  divisor < 0 ? 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O0 d   x  f<8Sxaxa1?t O0 d   x  f x  Z 64-bit ALU  f  XB | 08c?Z XB | 08c?D |  fXSxaxa1?s  X Shift Left  d   |  fSxaxa1?^ :2|  Y Shift Right  d   |  fSxaxa1?{  SWrite d  p" | HG(8c?k W`  |  fHSxaxa1? q %  WControl  f   |  fPSxaxa1?  W32 bits  f   |  fSxaxa1? " O W64 bits  f   |  f4Sxaxa1?   W64 bits  f   | s BCDEF3f8c? @g`Hu  | s BCvDEF3f8c?u @@ e  | s BvCDE F3f8c?uu@@ n n  | s BCDEF8c?@d @ e  | s BC!DE F(8c?  j ` @  | s BCDEF8c?@   | s BCVDEF 8c?+UU    @   | s BClDEF8c?k@DKLXB "|@ 01?  &| <S0  _Note: The multiplicand R, Product R and ALU are all 64-bits, while the Multilier R is 32-bits.``H H | 0޽h ? X(=^y___PPT10Y+D='  = @B +q  0L0 x'+(     ZȺS 8c 8c1 ?? 2.?   S 8 ~A +Sf  6?a     fSxaxa8c?"   }-3. Shift the Multiplier register right 1 bit. . .f . l  <8c?\/ 8 r"  BG$8c? vD    fHSxaxa8c?D I A TDone  f      fxSxaxa8c?   SYes  f  f   68c?{ f     `Sxaxa8c?< 4  ~.2. Shift the Multiplicand register left 1 bit. / /f / l   <8c?7 1     ftSxaxa8c? ~  dNo: < 32 repetitions  f  f  68c?; I nT  ;; # ]  s   `Sxaxa8c?  R1.  f  IN  ;;   ;;   `Sxaxa8c?{ VTest" f     `Sxaxa8c? z;; ] Multiplier0"   f     f8Sxaxa8c?* _Multiplier0 = 0  f     fSxaxa8c?  ] Multiplier0=1  f  f  68c?Q %;   f,Sxaxa8c?  M1a. Add multiplicand to product & place the result in Product register N Nf N l  <8c?  s BC0DEF?/ @ f  68c? F    f@Sxaxa8c?  c32nd repetition?" f  r"  BG$8c? j   f`Sxaxa8c?S   UStart  f  rB  B8c? xcrB  B8c?~    s B6C6DE F8c?55@U  ! s B CDEF8c?   @kw F  " s BkCDEF8c?jj @  <  # s BCDEF8c?@ %   $ s BCDEF8c?@: 2 ;  % s BCDEF8c?@% & s & s BsC DEF8c? s s @ y  ' s BC DEF8c? @ p |H  0޽h ? X(=^y___PPT10Y+D='  = @B +  0L0 $(     ZS 8c 8c1 ?? 1  S    ZM 8c 8c1 ?@  S $88XH  0޽h ? X(=^y___PPT10Y+D='  = @B +}  0 X$(  Xr X S  M?   M r X S d M@ 0 M H X 0޽h ? @Eff؂o___PPT10i.}+D='  = @B +  0L0  D(     ZxM 8c 8c1 ?? _$%  M X # 01? /& ^ $ 68c? 7-  %  `Mxaxa1?   WProduct  f  X & 01?K^ ' 68c? (  `>/ xaxa1?% Z Multiplier  f  X ) 08c?9/& 1^ * 68c?A7- 7 +  ` Mxaxa1?i ] \ Multiplicand  f   ,  `D$Mxaxa1?3> x  Z 32-bit ALU  f  RB . s *8c?D 0  `)Mxaxa1?^ :2|  Y Shift Right  d   1  `'Mxaxa1? $  Y Shift right  d  j" 2 BG(8c?k  W`  3  `h1Mxaxa1? q %  WControl  f   4  `@0Mxaxa1?  W32 bits  f   5  `t,Mxaxa1? " O W32 bits  f   6  `P=Mxaxa1?   W64 bits  f   7 s BCDEF3f8c? @g`Hu  9 s BvCDE F3f8c?uu@@ n  : s BCDEF8c?@ @  ; s BC!DE F(8c?  j ` @  < s BCDEF8c?@   = s BCVDEF 8c?+UU    @   > s BClDEF8c?k@DKLRB ?@ s *1? ^B @ 6D8c  RB A s *8c?t  t  B  `XCMxaxa1?S q  SWrite d   C s BCDEF8c?@  . H  0޽h ? X(=^y___PPT10Y+D='  = @B +t  0L0 {'+(     ZKM 8c 8c1 ?? ! *  M X  0? 9     `OMxaxa8c? ]  {-3. Shift the Multiplier register right 1 bit.. .d . ^  68c? k d"  <G$8c? K    `OMxaxa8c?@   TDone  f      `TMxaxa8c?)- ! cYes: 32 repetitions  f  X   08c?(  ,    `XMxaxa8c?   *2. Shift the Product register right 1 bit.B+  ddd + ^   68c? v     `^Mxaxa8c?  dNo: < 32 repetitions  f  X  08c?T F | |C  RP k)   `bMxaxa8c?  P1. d  RN | |C  | |C   `hMxaxa8c?MA a Test  d     `\iMxaxa8c?| y|C [ Multiplier0   d     `kMxaxa8c?jya [ Multiplier0=0 d     `,oMxaxa8c?c{  [ Multiplier0=1 d  8  +wZ`  08c?" m    `LxMxaxa8c? i1a. Add multiplicand to the left half of product & place the result in the left half of Product registerxj dddddd j l  <8c?  c BC0DEF?/ @]  X  08c?E _  ZPMxaxa8c?d Y 0 c32nd repetition?" f  d"  <G$8c?R )   `ԄMxaxa8c?7$ 6U SStart d  dB  <8c? 'dB   <8c?e a  ! c B6C6DE F8c?55@hn  " c B CDEF8c?   @  # s BCDEF8c? @h   $ c BCDEF8c?@ D E  % c BCDEF8c?@ & ' n  & c BCDEF8c?@g, -  ' c BC DEF8c?   @ ~ ( c BCDEF8c?@(  H  0޽h ? X(=^y___PPT10Y+D='  = @B +R  0L0 QQ >Q(     ZT/ 8c 8c1 ?? _I  M    ZM 8c 8c1 ?@ J /  $88XO8  <    `Mxaxa1?`l hB06 fn  `B B 08c? ZB  s *8c?  `B B 08c? ZB  s *8c?p p ZB   s *8c?  ZB   s *8c?pP pQ ZB   s *8c? ZB   s *8c?p p     `Mxaxa1?0< hB16 fn     `,Mxaxa1?  hB26 fn     `ȧMxaxa1?   hB36 fn     `4?/ xaxa1?  `P0. bj     `Mxaxa1?g F `P1. bj     `Mxaxa1?  `P2. bj     `ԹMxaxa1?  `P3. bj     `Mxaxa1? g  `P4. bj     `0Mxaxa1?  `P5. bj     `Mxaxa1?W 6 `P6. bj     `Mxaxa1?  `P7. bj  ZB  s *8c?p p    `Mxaxa1?  Q0  f     `Mxaxa1?1 Q0  f     `Mxaxa1?= Q0  f     `Mxaxa1? 2  Q0  f  TB  c $8c? p @ N x @  x @~ N Gx   Gx N x    x  !  `Mxaxa1?  `A0. bj  l " <8c? x HTB # c $8c?  TB $ c $8c? P N G % G &  `Mxaxa1?G6 `A1. bj  l ' <8c?HTB ( c $8c?ppTB ) c $8c?@P@N x * x +  `|Mxaxa1? `A2. bj  l , <8c?xHTB - c $8c?TB . c $8c?PN G / G 0  `Mxaxa1?G5 `A3. bj  l 1 <8c?HTB 2 c $8c?ppTB 3 c $8c?@P@TB 4B c $8c? TB 5 c $8c?TB 6 c $8c?@ @TB 7 c $8c?p@TB 8 c $8c?@@TB 9 c $8c?p@ N @x  : @x ~ N G@x P ; G@x PN @x P < @x P =  `tMxaxa1?o { `A0. bj  l > <8c? 8x TB ? c $8c? @ 0TB @ c $8c?  PN G@P A G@P B  `Ixaxa1?Go6{ `A1. bj  l C <8c?8TB D c $8c?p@p0TB E c $8c?@@PN @xP F @xP G  `Ixaxa1?o{ `A2. bj  l H <8c?8xTB I c $8c?@0TB J c $8c?PN G@P K G@P L  ` Ixaxa1?Go5{ `A3. bj  l M <8c?8TB N c $8c?p@p0TB O c $8c?@@PTB PB c $8c? TB Q c $8c? PTB R c $8c?@P TB S c $8c?PpTB T c $8c?@PTB U c $8c?PpTB V c $8c? Pp TB W c $8c?p  TB X c $8c?p @p `B YB 08c? 0 0  N x  Z x ~ N Gx  [ Gx N x  \ x  ]  `lIxaxa1?   `A0. bj  l ^ <8c? x TB _ c $8c?  TB ` c $8c? N G  a G  b  `0Ixaxa1?G6  `A1. bj  l c <8c? TB d c $8c?ppTB e c $8c?@ @ N x  f x  g  `Ixaxa1?  `A2. bj  l h <8c?x TB i c $8c?TB j c $8c?  N G  k G  l  `L%Ixaxa1?G5  `A3. bj  l m <8c? TB n c $8c?ppTB o c $8c?@ @ TB pB c $8c?`  TB q c $8c?  TB r c $8c?@ TB s c $8c? p TB t c $8c?@  TB u c $8c? p TB v c $8c? p TB w c $8c?p P