Small Multiplier-based Multiplication and Division ... fileJean-Luc Beuchat, Arnaud Tisserand. Small...
Transcript of Small Multiplier-based Multiplication and Division ... fileJean-Luc Beuchat, Arnaud Tisserand. Small...
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Small Multiplier-based Multiplication and DivisionOperators for Virtex-II Devices
Jean-Luc Beuchat, Arnaud Tisserand
To cite this version:Jean-Luc Beuchat, Arnaud Tisserand. Small Multiplier-based Multiplication and Division Operatorsfor Virtex-II Devices. [Research Report] RR-4494, INRIA. 2002. �inria-00072094�
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INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE
Small Multiplier-based Multiplication andDivision Operators for Virtex-II Devices
Jean-Luc Beuchat, Arnaud Tisserand
No 4494
July 2002
THEME 2
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MULT18x18S
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(a) Virtex−II CLB
(b) Embedded multiplier
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Critical path
Critical path
library virtex2;use virtex2.components.all
mult: MULT18x18S port map ( P => XY, A => X, B => Y, C => Clk, CE => Ce, R => Clr);
. . .
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X0²X1
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Y1§
&
&
X0
Y0X1
Y1X0
Y1X1
Y0
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5:0
3:0
7:4
X0 * Y0
X1 * Y0 * k
X0 * Y1 * k
X1 * Y1 * k * k
X1 X0
Y1 Y0
Optional pipeline level
n bits
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(X1k +X0)(Y1k + Y0) = X1Y1(k2 − k) + (X1 +X0)(Y1 + Y0)k +X0Y0(1− k).
�Ðs�l{kQ��pSo{��d3w�s�am~�d3d¼cgvmo{s�l{�mo{lnd_~�wu²Ëk�tCced_o{yX1Y1
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(X1 + X0)tCk��
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l{c��m~�p»��d_ced_kQs8zQy 41~�l{s�l{kmq
(X1k +X0)(Y1k + Y0) = X1Y1(k2 + k)− (X1 −X0)(Y1 − Y0)k +X0Y0(1 + k).
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&
&
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X1
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X0
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−X1 * Y1 * k
Sign extension
(X0+X1) * (Y0+Y1) * k
−X0 * Y0 * k
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itC~�d�w�vmc�ced3��v�w�l{kmq½tKs�~�d3d�pC£ËrutC~�~�y��Ò�m~�pS��tCqQtCs�detS�m�md_~�we°¤¡$l{q:vm~�d��:z"´]§-À%tC~�~�y�� w�tu��d�tS�m�md_~�wtC~�d�k�pSs=v�w�d]£¤vmo�l{k��8l{~�s�d]��¡�}Ëi-�Lw=zMd3rutCv�w�d�pC£Ës�a�d�£ÒtSw�s8rutC~�~�y�onpSq:lnr:§g¡�pS~�c©tCo{o{y�²³}H}
ilnw�t
(2m− i− 1)�Òzml{sLl{kQs�d_q�d_~(�md]·�k�d3�zQy
}H}i =
m−1−i∑
j=0
xiyj2i+j +
m−1∑
j=i+1
xjym−1−i2j+m−i−1.
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max (}®}
i) =
m−1−i∑
j=0
2i+j +
m−1∑
j=i+1
2j+m−i−1
=m−1−i∑
j=0
2i+j +2m−2i−2∑
j=m−i2i+j = 2i
(22m−2i−1 − 1
).
`%a�d_~�d]£pS~�d:²
max (}H}
i +}®}
i+1) = 22m−i−1 + 22m−i−2 − 2i+1 − 2i°��»´
= 10 11 · · · 11︸ ︷︷ ︸2m−2i−4×
01 0 · · · 0︸ ︷︷ ︸i×
°f�´
tCk���}®}i +
}H}i+1
lnw�t(2m − i) �Òzml{s�l{k�s�d_q�d_~u§��Lw�w�vmced�s�a�tCs m = 21
tCk���r3pSk�w�ln�md_~-s�a�dw�vmc
(}®}
0 +}H}
1) + (}H}
2 +}H}
3)§H¡�~�pSc °f�´]²04Ëdg�md3��v�r3d
max(
(}H}
0 +}H}
1)︸ ︷︷ ︸42 ��� ���
+ (}H}
2 +}H}
3)︸ ︷︷ ︸40 ��� ���
)
= 10 11 · · · 11︸ ︷︷ ︸38×
01 + 10 11 · · · 11︸ ︷︷ ︸34×
0100.
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��� * ��: ";=��W� � UV�7U#�=� ���!�5? � ;U#?� d�a�t3��d:41~�l{s�s�d_k0t�À o{l{zm~�tC~�y�41amlnr�a0q�d_k�d_~�tCs�d3w®�=¸L¹=º �md3w�r_~�l{�ms�lnpSk�w%pC£Ms�a�d8tC~�r�aml{s�d3r_s�vm~�d3w�md3w�r_~�l{z�d3� tCzMpu��d:§ `%a�d¨�=¸L¹=º6r3pA�md 4¢tSw¼w�yAkQs�a�d3w�lA@ud3� 41l{s�a ; y�km�mo{l�£¤y;}H~�p VA§Ý��§ 2 tCk��l{c��mond_ced_k�s�d3�YpSk tK�Ll{~�s�d]�A� ���1¥LÀ%fC�XD:�:�^� D½�md_�Alnr3d�v�w�l{kmq½¥Ll{o{l{k��¨�(o{o{l¬tCk�r3d ; d_~�lnd3w���§{�:§Ý� 2 lÒ§�Ðk�s�a�d8£¤pSo{onp*41l{kmq�²
mtCk��
n~�d3w���d3r_s�l{��d_o{y��md_k�pSs�d=s�a�d=s�pSs�tCo³w�lA@ud-pC£$s�a�d-pS��d_~�tCk��mw(tCk���s�a�d
w�lA@ud-pC£$s�a�d_l{~�onp*4Ëd_~L��tC~�su§`%a�d�·�~�w�sgd]����d_~�l{ced_k�s�r3pSc���tC~�d3w�s�a�d�s�am~�d3d©w�pSo{vms�lnpSk�wK°¤¡$l{q:vm~�d D�´]§K`%a�de��l{��ln�md]�ÐtCk��A�
r3pSk�|�v�d_~=w�s�~�tCs�d_q:y½ondutS�mw(s�p©s�a�d�w�c©tCo{ond3w�sLr_l{~�r_vml{s�w8l{ks�d_~�cew=pC£Hw�o{lnr3d3wLkQvmc�z�d_~u²Md]�mr3d_�ms(£pS~
i/i ñkj,l,l�mnl
ª � ���%$ ��� <!��7�,0!����� ��:��$�%<���/���n�����g�%$�
. . .. . .
n = 17
X0
Y0
X1
Y1
. . .. . .
. . .
n = 17
. . .. . .
. . .
. . .. .
.. .
.
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
PP3PP2 PP0
PP1
. . .
. . .
. . .
0
. . .
. . .
0000 0
000
0
0000 0
0
PP0PP1PP2PP3
n = 17
n = 17
(a) Proposed multiplier
m bits
MULT18x18
m−n part
ial pr
oduc
ts
PP0PP1
PP2PP3
(c) Adding the partial products
(b) Adding the partial products (tree structure)
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m = 18tCk��
m = 1941a�d_~�d-s�a�d�w�l{kmq:ond É P8º�`-�uª^�O�uªetC~�r�aml{s�d3r_s�vm~�d�lnw�w�c©tCo{ond_~u§®`%a�d-sO4¢pC�
4%t3y�ced_s�a�pA��lnw ��lnw�tC�m�MpSl{kQs�l{kmq �s�a�pSvmq:ael{sHw���tC~�d3wËt-w�c©tCo{omcgvmo{s�l{�mo{lnd_~u²�l{s lnwHw�onp*4Ëd_~%tCk���s�a�dtS�m��l{s�lnpSk�tCo®onpSq:lnr�lnwLsR41lnr3d�zml{q:q�d_~-s�a�tCk�s�a�d���l{�Aln�md]�ÐtCk��A� r3pSk�|Qv�d_~�tC�m�m~�p�tSr�a³§�¡>l{k�tCo{o{y�²³s�a�dtC~�r�aml{s�d3r_s�vm~�d��md3w�r_~�l{zMd3�l{k ��tC~�tCq:~�tC�ma 2 § 2 lnw=teq�pAp��s�~�tS�md]� pC¯YzMd_sO4Ëd3d_k�w�lA@ud�tCk�� ��d_~�lnpA�41a�d_k
mlnw(r_onp:w�d=s�p
17§
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m = 23²C41amlnr�a;lnw©s�a�d c©tCkQs�lnw�wVtYw�lA@udl{k � M M M w�s�tCk���tC~�� B�p�tCs�l{kmqS�Ò��pSl{k�s
k�vmcgzMd_~�w�°t�w�vm~���d_y©pSk B�p�tCs�l{kmqS�Ò��pSl{k�s(rutCk0z�dL£pSvmk��©l{k�BÝfGF\´]§ � d8a�t3��d8l{c��mond_ced_kQs�d3�¼sO4Ëp��d_~�w�lnpSk�wxpC£�s�a�d®f 2 × f 2 vmk�w�l{q:k�d3��cgvmo{s�l{�mo{lnd_~$l{kQ��pSo{��d3��l{k�s�a�d`B�p�tCs�l{kmqS�Ò�MpSl{kQs$cgvmo{s�l{�mo{lnrutCs�lnpSk41l{s�aYsO4Ëp�ml{��d_o{l{k�d©w�s�tCq�d3w�tCk���s�a�d É P8º�`-�uª^�O�uª ; �m~�l{c�l{s�l{��d:§K`%a�d©��l{�Aln�md]�ÐtCk��A� r3pSk�|Qv�d_~tC�m�m~�p�tSr�al{c��mo{lnd3w
50w�o{lnr3d3wu²A£pSvm~ É PLºA`-�uª^�M�uª ; zmonp�r�«Aw»²�tCk��½a�tSwLt��Md_~�lnp��
τd3|�v�tCo³s�pYD
k�w»§ 9Lvm~�tCo{q�pS~�l{s�amc k�d3d3�mw182
w�o{lnr3d3w»²>t�w�l{kmq:ond É PLºA`-�uª^�O�uª ; zmonpAr�« tCk��¨lnwgt0o{l{s�s�ond�zml{sw�onp54Ëd_~�°
τ = 8k�wV´]§ � d=�mo¬tCk¼s�pew�s�v���y B�p�tCs�l{kmqS�Ò�MpSl{kQs=tC~�l{s�amced_s�lnrgpSk¡�}Ëi=�8wu§
º³d_sgv�w�k�p54¿w�s�v���y�s�a�del{c���tSr_s�pC£npSkYs�a�d©��l{�Aln�md]�ÐtCk��A� r3pSk�|�v�d_~�ced_s�a�pA�O§ � d©pSz��
s�tCl{k s�a�d¼w�c©tCo{ond3w�s©r_l{~�r_vml{s�w 41a�d_kn = 17
°¤¡$l{q:vm~�d D�´]§ `%amlnwe~�d3w�vmo{selnwek�pSs©w�vm~��m~�lnw�l{kmq ~�d_ced_c�z�d_~=s�a�tCs8s�a�d
nondutSw�s8w�l{q:kml�·�rutCk�s=zml{s�wLr3pSced���l{~�d3r_s�o{y½£¤~�pSc¦t É P8ºA`-�uª^�M�uª©zmonp�r�«�§
À¢pSk�w�d3|Qv�d_k�s�o{y�²xs�a�d�o¬tC~�q�d_~nlnwu²³s�a�dew�c©tCo{ond_~�s�a�d�o¬tSw�s�tS�m�md_~gpC£¢¡$l{q:vm~�d©f�zMd3r3pSced3wu§�¡�pS~
tKr3pScgzml{k�tCs�pS~�l¬tCo¢r_l{~�r_vml{su²Or�a�pAp:w�l{kmqn = m/2
ondutS�mw=a�p*4Ëd_��d_~�s�p�t©£tSw�s�d_~8r_l{~�r_vml{su§ �;a�d_k4¢d-l{kQs�~�pA��v�r3dgt��ml{�Md_o{l{k�d�w�s�tCq�d=z�yK~�d_�mo¬tSr_l{kmq�tCo{o É P8º�`-�uª^�O�uª�zmonpAr�«�w1zQy É PLºA`-�uª^�M�uª ; ²pSvm~8d]�A�Md_~�l{ced_kQs�w8w�a�p*4 s�a�tCs
n = 17lnw�s�a�dgz�d3w�s8w�pSo{vms�lnpSk ¢s�a�dgw�lA@ud�lnwLd]��tSr_s�o{y�s�a�d�w�tCced
pSk�d=s�a�tCs%l{k½s�a�d-r3pSc�zml{k�tCs�pS~�l¬tCo>rutSw�d�°\s�a�tCs5Uíw 41a�yKl{s%lnw%k�pSs1�monpSs�s�d3��l{k¼¡$l{q:vm~�d D�´%tCk��0s�a�d�Md_~�lnpA�½�mp�d3w�k U s��md_�Md_k��tCk�y�cepS~�d�pSk½s�a�d-�StCo{v�dgpC£
n§
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050
100150200250300350400450500550
18 20 22 24 26 28 30 32
slic
enu
mb
er
operand size
3 3 3 3 3 3 3 3r rr rr rrr
e e e e e e e e13
14
15
16
17
18
19
20
21
22
18 20 22 24 26 28 30 32τ
[ns]
operand size
rrr rrr r r
e e ee e e e e
3 3
3
33 3 3
3
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Unit e de recherche INRIA Lorraine, Technopole de Nancy-Brabois, Campus scientifique,615 rue du Jardin Botanique, BP 101, 54600 VILLERS LES NANCY
Unit e de recherche INRIA Rennes, Irisa, Campus universitaire de Beaulieu, 35042 RENNES CedexUnit e de recherche INRIA Rhone-Alpes, 655, avenue de l’Europe, 38330 MONTBONNOT ST MARTIN
Unit e de recherche INRIA Rocquencourt, Domaine de Voluceau, Rocquencourt, BP 105, 78153 LE CHESNAY CedexUnit e de recherche INRIA Sophia-Antipolis, 2004 route des Lucioles, BP 93, 06902 SOPHIA-ANTIPOLIS Cedex
EditeurINRIA, Domaine de Voluceau, Rocquencourt, BP 105, 78153 LE CHESNAY Cedex (France)
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ISSN 0249-6399