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8/17/2019 Shoaib Parallel Adders (2)
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Parallel Prefx Adders
A Case Study
Muhammad Shoaib Bin Alta CS/ECE 755
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Outline
• Motivation
• ntrodu!tion
• "arious #ree adders• Com$arison
• %ayout o &o''e(Stone
• Con!lusion
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Motivation
• Addition) a undamental o$eration
– Basi! blo!* o most arithmeti! o$erations
– Address !al!ulation
• +aster, aster and aster
• -o.
– 0i$$le Carry Adder %oo* Ahead
– Carry Sele!t, !arry S*i$
– 1ood or small number o bits but2
– 3eed some !han'e or .ider adders
Square root select
Linear select
Ripple adder
20 40N
t p
( i n
u n i t d e l a y s )
600
10
0
20
30
40
50
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Pro$a'ate and 1enerate%o'i!
• +or a ull adder, defne .hat ha$$ensto !arries
– 1enerate) Cout 4 inde$endent o C
• 1 4 A 6 B
– Pro$a'ate) Cout 4 C
• P 4 A ⊕ B
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Prefx Adder Euations
• Euations oten a!tored into 1and P
• 1enerate and $ro$a'ate or 'rou$ss$annin' i)8
• Base !ase
• Sum)
: : : 1:
: : 1:
i j i k i k k j
i j i k k j
G G P G
P P P
−
−
= +
=
''
:
:
i i i i i
i i i i i
G G A B
P P A B
≡ =
≡ = ⊕
'0:0 0
0:0 0 0
inG G C
P P
≡ =
≡ =
1:0i i iS P G −= ⊕3:0 3:0 3:3 3:3 2:0G C G P G= = +
3:0 3:0 3:2 3:2 1:0G C G P G= = +
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3otations
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0i$$le Carry Adder
S1
B1
1
!1
"1
"0#0
S2
B2
!2
"2
"1#0
2
S3
B3
3
!3
"3
"2#0
S4
B4
!4
"4
"3#0
4
$in
"0
!0
1# Bit%ise !" lo&ic
2# "roup !" lo&ic
3# Su' lo&ic
$0
$1
$2
$3
$out
$4
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0i$$le Carry Adder
15:0 14:0 13:0 12:0 11:0 10:0 9:0 8:0 7:0 6:0 5:0 4:0 3:0 2:0 1:0 0:0C C C C C C C C C C C C C C C C
15:15 14:14 13:13 12:12 11:11 10:10 9:9 8:8 7:7 6:6 5:5 4:4 3:3 2:2 1:1 0:0G G G G G G G G G G G G G G G G
7:0 7:7 7:7 6:0C G P G= +
3:0 3:3 3:3 2:0C G P G= +
11:0 11:11 11:11 10:0C G P G= +
15:0 15:15 15:15 14:0C G P G= +
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%oo* Ahead Basi! idea
AN-1( BN-1 1( B1
P 1
S 1
• • •
• • • S N-1
P N-1C i ( N-1
S 0
P 0C i (0 C i (1
A
:0 1:0 1:0( , , )
k k k k k k k C f A B C G P C
− −= = +
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%oo*ahead) #o$olo'y
)*pandin& Loo+a,ead equations#
ll t,e %ay#
C o(3
C i (0
V DD
P 0
P 1
P 2
P 3
G0
G1
G2:0 1 1 2:0
( )k k k k k k
C G P G P C − − −
= + +
:0 1 1 1 0 0 0( (.... ( )))k k k k k C G P G P P G P C − −= + + + +
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%o'arithmi! %oo*aheadAdder
A7
F
A6A5A4A3A2A1
A0
A0
A1
A2
A3
A4
A5
A6
A7
F
tp∼ log2(N)
tp∼ N
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Carry loo*ahead #rees
1:0 1 1 0 1 0 0
2:0 2 2 1:0
2:0 2 2 1 2 1 0 2 1 0 0
2 2 1 2 1 0 0 0
2:1 2:1 0
( ) ( )( )
C G PG P P C
C G P G
C G P G P PG P P P C
G P G P P G P C
G P C
= + +
= +
= + + +
= + + +
= +
• #his idea !an be extended to buildhierar!hal trees
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Prefx Adder Stru!ture
• m$lement the ideao Carry %oo*aheadtree
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Brent(&un' Adder
• Sta'es
– 9:lo'3(;
• +an out
– 9• Avoids Ex$losion o
.ires
• Odd Com$utationthen even
• n any ro. at themost one $air
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Brent(&un' Adder3:2 3:3 3:3 2:2C G P G= +15:14 15:15 15:15 14:14C G P G= +
7:0 7:4 7:4 3:0C G P G= +15:8 15:12 15:12 11:8C G P G= +
4:0 4:4 4:4 3:0
C G P G= +14:0 14:14 14:14 13:0
C G P G= +
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S*lans*y Adder
• Sta'es
– %o' 3
• +an out
–
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S*lans*y Adder
3:2 3:3 3:3 2:2C G P G= +15:14 15:15 15:15 14:14C G P G= +
3:0 3:2 3:2 1:0C G P G= +
15:12 15:14 15:14 13:12C G P G= +
7:0 7:4 7:4 3:0C G P G= +15:0 15:8 15:8 7:0C G P G= +
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&o''e(Stone Adder
• Sta'es
– %o' 3
• +an out
– 9 at ea!h sta'e
• %on' .ires
• More P1 !ells
Po.er• =idely >sed
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&o''e(Stone Adder3:2 3:3 3:3 2:2C G P G= +
15:14 15:15 15:15 14:14C G P G= +
3:0 3:2 3:2 1:0C G P G= +15:12 15:14 15:14 13:12C G P G= +
7:0 7:4 7:4 3:0C G P G= +15:0 15:8 15:8 7:0C G P G= +
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-an(Carlson Adder
• Mix o &o''e(Stoneand Brent(&un'
• Sta'es
– %o' 3 ?• +an out
– 9
• #rades lo'i!al levelor .ire len'th
• n any ro. at themost one $air
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-an(Carlson Adder3:2 3:3 3:3 2:2C G P G= +15:14 15:15 15:15 14:14C G P G= +
5:0 5:2 5:2 1:0C G P G= +15:8 15:12 15:12 11:8C G P G= +
7:0 7:4 7:4 3:0C G P G= +15:0 15:8 15:8 7:0C G P G= +
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&no.les Adder
• >sin' &o''e(stoneand S*lans*y
• Sta'es
– %o' 3
• +an out
– @
• =ires
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&no.les Adder3:2 3:3 3:3 2:2C G P G= +15:14 15:15 15:15 14:14C G P G= +
5:0 5:2 5:2 1:0C G P G= +
15:8 15:12 15:12 11:8C G P G= +
7:0 7:4 7:4 3:0
C G P G= +15:0 15:8 15:8 7:0C G P G= +
4:3 4:4 4:4 3:3C G P G= +
8:0 8:1 8:1 0:0
C G P G= +
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%adner(+is!her Adder
• By Combinin'Brent(&un' andS*lans*y
• Sta'es – %o' 3 ?
• +an out
– 3/ ?
• =ires
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%adner(+is!her Adder3:2 3:3 3:3 2:2C G P G= +15:14 15:15 15:15 14:14C G P G= +
5:0 5:4 5:4 3:0C G P G= +13:8 13:12 13:12 11:8C G P G= +
7:0 7:4 7:4 3:0
C G P G= +15:0 15:8 15:8 7:0C G P G= +
8:0 8:8 8:8 7:0
C G P G= +
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Com$arison Amon' Adders
N=16 N=32 N=64 N=128
Brent(&un'
@7 D 9
S*lans*y @ 9F @D9 7D&o''e(Stone
9 7 9D
-an(Carlson
9 5 7
&no.les 7 97 7@ 95%adner(+is!her
5 D
Carryn!re
57 975 FD D@
Lo&ical e--ort o- carry propa&ate adders( .a/id arris( 2003
n ter' o- delays
- %ire capacitance ne&lected o&&eStone is est
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"alen!y o a #ree
• "alen!y
– 3umber o 'rou$s!ombine to'ether toma*e lar'er 'rou$s
– Earlier exam$les.ere o valen!y 9
– -i'h "alen!y• %ess lo'i! levels
• Ea!h sta'e has 'raterdelay
–
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S$arseness o #ree
• Com$ute Carriesor blo!*s only
• 0edu!e
– =ire !ount – 1ate !ount
– Po.er
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m$lementation o &S Adder
•
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m$lementation o &S Adder
Pro$a'ate 1enerate: : : 1:
: : 1:
i j i k i k k j
i j i k k j
G G P G
P P P
−
−
= +
=
'
'
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%ayout o &S Adder
64 bit Adder
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%ayout o &S Adder
• Area !om$letely dominated by .ires
–
•
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+e. Observations
• =ire delay ex!eeds lo'i! delay inmany !ases
• #he .ire delay in!reases .ith .idtho adder
• EHe!t o eature siIe
•@< sta!*in' !an hel$ in de!reasin'area, $o.er and delay
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Con!lusion
• +ast Adders reuired or 3J@9
• rre'ular hybrid s!hemes are $ossible
• &o''e(Stone, &no.els reuire lar'e number
o $arallel .irin' tra!*s• %ar'e .ires .ill in!rease .irin' !a$a!itan!es
• Choi!e is yours2
• #rade oH bet.een delays and Area• @< inte'ration !an hel$ in redu!in' the
delays urther
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