6.004 Computation Structures Spring 2009

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1 MIT OpenCourseWre Computtion Structures Spring 009 For informtion out citing these mterils or our Terms of Use, visit:

2 Cost/Performnce Trdeoffs: cse study Binry Multipliction Digitl Systems Architecture 1.01 x n its n its n its since ( n -1) < n EASY PROBLEM: design comintionl circuit to multiply tiny (1-, -, 3-it) opernds... HARD PROBLEM: design circuit to multiply BIG (3-it, 64-it) numers We cn mke ig multipliers out of little ones! Engineering Principle: Exploit STRUCTURE in prolem. L #3 due tonight! modified /3/09 10:44 L09 - Multipliers 1 L09 - Multipliers Given n-it multipliers: x = n its n its n its Mking n-it multiplier using n-it multipliers x H H L L Our Bsis: n=1: minimlist strting point Multiplying two 1-it numers is pretty simple: x = 0 Synthesize n-it multipliers: x 4n its n its n its L L L H H L H H L09 - Multipliers 3 Of course, we could strt with optimized comintionl multipliers for lrger opernds; e.g it Multiplier 4 c 3 c c 1 c 0 the logic gets more complex, ut some optimiztions re possile... L09 - Multipliers 4

3 n-it y n-it multipliction: Our induction step: 1. Divide multiplicnds into n-it pieces. Form n-it prtil products, using n-it y n-it multipliers. 3. Align ppropritely 4. Add. H L x H H H L L L H L Induction: we cn use the sme structuring principle to uild 4n-it multiplier from our newly-constructed n-it ones... L H REGROUP prtil products - dditions rther thn 3! Brick Wll view of prtil products Mking 4n-it multipliers from n-it ones: induction steps x L09 - Multipliers 5 L09 - Multipliers 6 Given prolem: Multiplier Cookook: Chpter x Sussemlies: Prtil Products Adders MULT ADD Step : Sum Step 1: Form (& rrnge) Prtil Products: "Order Of" nottion: (...) implies oth inequlities; O(...) implies only the second. Performnce/Cost Anlysis "g(n) is of order f(n)" g(n) = (f(n)) if there exist C C 1 > 0, such tht for ll ut finitely mny integrl n 0 c 1 f(n) g(n) c f(n) g(n) = O(f(n)) g(n) = (f(n)) Exmple: n +n+3 = (n ) since n (n +n+3) n "lmost lwys" Prtil Products: n = (n ) Things to Add: n - = (n) Adder Width: n = (n) Hrdwre Cost:? = (n ) Ltency: (n )?? L09 - Multipliers 7 L09 - Multipliers 8

4 Oservtions: Repckging Function MULT ADD (n ) prtil products. (n ) full dders. Hmmm. Engineering Principle #: Put the Solution where the Prolem is. MULT ADD (n ) prtil products. (n ) full dders How out n locks, ech doing little multipliction nd little ddition? L09 - Multipliers 9 L09 - Multipliers C i+1 A i FA B i (A+B) i Gol: Arry of Identicl Multiplier Cells C k+ C i S k+1 S k Single "rick" of rick-wll rry... Forms prtil product Adds to ccumulting sum long with crry S k+1 S k Necessry Component: Full Adder Tkes ddend its plus crry it. Produces sum nd crry output its. CASCADE to form n n-it dder. i j C k L09 - Multipliers 11 Design of 1-it multiplier "Brick": Brick design: AND gte forms 1x1 product -it sum propgtes from top to ottom Crry propgtes to left Wstes some gtes ut consider (sy) optimized 4x4-it rick! Arry Lyout: opernd its used digonlly Crry its propgte right-to-left Sum its propgte down C k+ S k+1 0 FA FA S k+1 S k S k i j C k L09 - Multipliers 1

5 Ltency revisited Here s our comintionl multiplier: Comintionl Multiplier: Multiplier Cookook: Chpter Wht s its propgtion dely? Nive (ut vlid) ound: O(n) dditions O(n) time for ech ddition Hence O(n ) time required On closer inspection: Propgtion only towrd left, ottom Hence longest pth ounded y length + width of rry: O(n+n) = O(n)! Hrdwre for n y n its: Ltency: Throughput: (n ) (n) (1/n) Note: lots of tricks re ville to mke fster comintionl multiplier L09 - Multipliers 13 L09 - Multipliers 14 Comintionl Multiplier: est ng for the uck? The Pipelining Bndwgon... where do I get on? Suppose we hve LOTS of multiplictions. Cn we do etter from cost/performnce stndpoint? PIPELINING WE HAVE: Pipeline rules - "well formed pipelines" Plenty of registers Demnd for higher throughput. Wht do we do? Where do we define stges? L09 - Multipliers 15 L09 - Multipliers 16

6 Stupid Pipeline Tricks Even Stupider Pipeline Tricks gott rek tht long crry chin! Stges: (n) Clock Period: (n) Hrdwre cost for n y n its: (n ) Ltency: (n ) Throughput: ( 1/n ) WORSE ide: Doesn t rek long comintionl pths NOT well-formed pipeline different register counts on lterntive pths... dt crosses stge oundries in oth directions! Bck to sics: wht s the point of pipelining, nyhow? L09 - Multipliers 17 L09 - Multipliers 18 Breking O(n) comintionl pths Multiplier Cookook: Chpter 3 LONG PATHS go down, to left: Brek rry into digonl slices Segment every long comintionl pth Stges: (n) Clock Period: (1) 0 3 Hrdwre cost for n y n its: (n ) Ltency: (n) Throughput: (1) Well-formed pipeline 1 0 (creful!) Constnt (high!) 3 0 throughput, independently of 3 1 opernd size. 1 0 GOAL: (n) stges; (1) clock period! L09 - Multipliers ut suppose we don t need the throughput? L09 - Multipliers 0

7 Moving down the cost curve... Multiplier Cookook: Chpter 4 Suppose we hve INFREQUENT multiplictions... pipelining doesn t help us. Cn we do etter from cost/ performnce stndpoint? Hmmm, do I relly need ll these extrs? i 3 1 i 3 i i 1 i 0 0 Sequentil Multiplier: Re-uses single n-it slice to emulte ech pipeline stge opernd entered serilly Lots of detils to e filled in... Stges: 1 Clock Period: ( 1 ) ( constnt!) Hrdwre cost for n y n its: (n ) Ltency: Throughput: (n ) (1/n) L09 - Multipliers 1 L09 - Multipliers (Ridiculous?) Extremes Dept... Multiplier Cookook: Chpter 5 Cost minimiztion: how fr cn we go? 3 i 1 i 3 i i 1 i 0 0 Suppose we wnt to minimize hrdwre (t ny cost) Consider it-seril! Form nd dd 1-it prtil product per clock Reuse single rick for ech it j of slice; Re-use slice for ech it of opernd Bit Seril multiplier: Re-uses single rick to emulte n n-it slice oth opernds entered serilly O(n ) clock cycles required Needs dditionl storge (typiclly from existing registers) Stges: Clock Period: Hrdwre cost for n y n its: i 3 1 i 3 i ( 1 n ) (1) (constnt) (1) +? i 1 i 0 0 Ltency: (n ) Throughput: (1/n ) L09 - Multipliers 3 L09 - Multipliers 4

8 Summry: Scheme: Comintionl $ (n ) Ltency (n) Thruput (1/n) N-pipe Slice-seril Bit-seril (n ) (n) (1) * (n) (n) (n ) (1) (1/n) (1/n ) Lots more multiplier technology: fst dders, Booth Encoding, column compression,... L09 - Multipliers 5