Multipliers. Overview. Introduction. Reading. Computer Systems Laboratory. Stanford University. Copyright 2001 by Mark Horowitz
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1 Lctr : ltplrs ptr Systs Laratry Stafrd Uvrsty hrwtz@stafrd.d pyrght 00 y ark Hrwtz H/JZ EE 7 Lctr Ovrvw adg Itrdct Thr ar ts f paprs wrtt ltplcat. Ufrtatly t s rar that th papr talks at th lgc ad crct sss tgthr. Thr ar ay lgc paprs wth dscr clvr ways f addg th partal prdcts, t thy rarly addrss th cst f th wrg. Othr crct paprs talk at fast ways t ld th asc addr clls. W wll start wth th ltplr rgazat, ad lk at th th lgc ad crct sss. H/JZ EE 7 Lctr
2 ltplcat asc da s prtty spl * = D y gratg a r f partal prdcts, ad addg th tgthr: w f dts s a partal prdct: slt f ltplyg y t f ( s th ltplcad) slt s ts Fr tgr prats wat LS ts, fr F wat S F.XXXX *.XXX H/JZ EE 7 Lctr Spl ltplr Taks cycls, ad grats ad adds partal prdct ach cycls: ltplcad artal rdct Grat ltplr ddr Shft rght ach cycl slt Shft Shft rght ach cycl H/JZ EE 7 Lctr 4
3 . rat th partal prdcts. S th tgthr a Isss Fr a fast ltplr dc th r f partal prdcts Fast ddr clls Tr addrs Talk at ths stff t H/JZ EE 7 Lctr 5 artal rdct Grat I th spl fr t s vry asy: 0 * = 0 * = Ths s a spl ND gat at ach t t grat a fr ach ltplr t W ca grat a fr vry tw ltplr ts * 0 * (ths s asy t grat y a shft) * (ych -- t ca ak asy 4 -, ad dlay th 4 t th t grp H/JZ EE 7 Lctr 6
4 dfd th dg ak t asy Wh th S f th ts s add 4 t t grp p ( = 4 -) th rcdr lks at ts Th tw ts t s rcdg, ad th S f prvs grp Tal S LS rv Ot H/JZ EE 7 Lctr 7 th Ecdr Fr d crcts Otpts ar ht a d gat Nd larg tpt ffrs t drv lg wr t ctrl all th that ar cctd t t. lk Zr Dr 4 lck Dlay.4 6 Tw_ Dr lk 4/ 4/ 4/ 4/4 O_ OTw Dr Dr Dr _ Otpt Drvr (Dr) 9/ /0 4/5 0/60 00/0 lk 4 lk 5 _ ac ZrOO_TwTw_ Ipt Ivs 9/ _ H/JZ EE 7 Lctr 8
5 .4 0/4 th lk _ 8 Zr 6 O 6 Tw O_ Tw lk 0 ltplcad s cvrtd t dal ral partal prdcts aga y sg lcal vrtrs, ad s st dw t sgfcac hghr th, gratg. Th addrs gt tc dal ral partal prdcts fr th th. H/JZ EE 7 Lctr 9 t th a rcd ltplr ts at a t Grats / f th partal prdcts t y d p wth dg *ltplcad Ths taks a addr S ths s grally t d Tal S LS Ot Nd t lk at S f prvs grp ad add f t s. Ths stll kps th rs tw -4 ad 4. Th ly hard t grat s H/JZ EE 7 Lctr 0
6 ddg artal rdcts If y wat t add r partal prdcts a cycl y cld s ltpl addrs Ths slt has tw prls: Ev fast addrs ar prtty slw 7 FO4 dlays Fast addrs ar prtty larg Epsv ara artal rdct Grat ddr artal rdct Grat ddr artal rdct Grat ddr H/JZ EE 7 Lctr arry Sav ddr T gt a fastr addr w wll s a trck: prst th rslt f addr (s) a rddat fr S = S S That s wh y add S ad S tgthr y gt th rght rslt. Ths s c sc w w d t d t prpagat th carry th addr. t Σ Istad f rpplg th carry, shft t lft y, ad s t as th thr part f th s. Each stag f th addr taks pts, ad prdcs tw tpts. H/JZ EE 7 Lctr
7 arry Sav ddr Each addr ca add partal prdct t a rddat s ddat s taks tw pts, partal prdct taks th thrd Each addr s sall ad fast D S s ard.5 FO S s prtty spl, t s jst a fll addr Th prl s th tpt s rally a sal fr t th d f th array y d t add tw parts f rddat r tgthr Ths tak a fast addr, t y ly d at th d f ltplr, t fr ach partal prdct H/JZ EE 7 Lctr ltplr Ovrvw lck dagra f ltplr: N/N N ltplr ltplcad OOTH UXES Σ pp Σ N t N Ecdrs Th Σ array s carry sav addrs, ad fal s s a ral addr H/JZ EE 7 Lctr 4
8 rray ltplr Fr flatg pt rs y hav 5 t rs, whch gv 7 = S 4 S 7 S 5 5 Ss datapath, all crtcal path (Frst addr adds ) 56 ts tall rglar datapath Shrt wrs H/JZ EE 7 Lctr 5 Tr ltplrs a ld fastr Σ arrays t sg tr addr Nd t carfl sc: Trs hav lgr wrs (ach addr s a rw f addrs) Draw a ary tr, t S ar pt tpt dvcs a ld a 4: cr sg S H/JZ EE 7 Lctr 6
9 4- ddr t Σ If y gr th sd rachs y gt 4 pts ach t pst, ad tw tpt at ach t pst. Ths allws y t ld ary trs. ca ld : trs, Wallac Trs, (ad t s a lttl fastr) t th wrg s ch r cpl H/JZ EE 7 Lctr 7 Tr Layt Fll (4:) Trs: t-slc: 4 Datapath Hght: Ss datapath, 8 crtcal path Icrasd datapath hght Lg wrs, ay crss-vrs, wrg chals H/JZ EE 7 Lctr 8
10 prs slt Itrlavd rrays 4: r d d v 4: Fll-array crtcal path rdcd y half fr sall ara vrhad Lgr wrs wth th array a w d v ttr? H/JZ EE 7 Lctr 9 rray-f-rrays Gral Ida: p p p p r partal arrays: hghr parallls, wrg cplty, largr cg twrk hs dpdg prl sz H/JZ EE 7 Lctr 0
11 EE 7 Lctr H/JZ Tw artal rrays glar wrg, sall r f crss-vrs a ffctv wr ladg ~.5S dlays Fr ptal parallls, p= p=4. ~4.5 S dlays crtcal path 4: 4: r EE 7 Lctr H/JZ Fr artal rrays Tr cr: Wrg, ara vrhad atchg rray Dlays: 5570 pp.0 Ss Optal arallls: 448 pp. Ss r artal arrays?: argal rtrs 4 4: r
12 Dtals Thr ar tw dtals that w stll d t cvr Dalg wth th ts that fall ff th d f th array I th array rgazat th datapath wdth ra cstat Th tra (LS) ts fall ff th tp f th array t ths ts ar stll rddat fr Fr rdg ths ts d t addd t Dalg wth gatv partal prdcts Th prl s wth sg ts Nd t td th sg ts th fll N ts Ths s N addtal ts fr th frst Lk at ths sss t H/JZ EE 7 Lctr LSs Fallg Off th rray Spd thrgh th array s S dlay = S 4 S 7 S 5 t sc th ar th cdd, tw t psts fall ff ach stag Nd t d a tw t carry prpagat add S dlay ld a sall carry slct addr arry t carry t s ly a dlay H/JZ EE 7 Lctr 4
13 Fastr Strctrs D r adds paralll, akg dalg wth th d ts hardr I tr addr, all th ts fall ff th d at rghly th sa t rray f arrays has a slar prl Nd t ld a cplt carry prpagat addr fr ths ts Nral Fast ddr 4: r H/JZ EE 7 Lctr 5 4 rtsy D. Fty Sg Ets Wh th s gatv, d t sg td Ths s ad sc th fat ca larg Us a trck stad r-add th tragl f s 000 T clar t s add t th rw S H/JZ EE 7 Lctr 6
14 Sg Ets S y ly d t add fw ts: S SS S S S 000 ddg ths ts tgthr s th sa as cplt sg ts H/JZ EE 7 Lctr 7
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