small multiplier-based multiplication and division ... filejean-luc beuchat, arnaud tisserand. small...

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HAL Id: inria-00072094 https://hal.inria.fr/inria-00072094 Submitted on 23 May 2006 HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. Small Multiplier-based Multiplication and Division Operators for Virtex-II Devices Jean-Luc Beuchat, Arnaud Tisserand To cite this version: Jean-Luc Beuchat, Arnaud Tisserand. Small Multiplier-based Multiplication and Division Operators for Virtex-II Devices. [Research Report] RR-4494, INRIA. 2002. inria-00072094

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HAL Id: inria-00072094https://hal.inria.fr/inria-00072094

Submitted on 23 May 2006

HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.

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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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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XY°¤¡$l{q:vm~�d¢f�´OtCk��-lnwOz�tSw�d3�

pSk½s�a�d-£¤pSo{onp*41l{kmqKd3|�v�tCs�lnpSk

(X1k +X0)(Y1k + Y0) = X1Y1k2 + (X1Y0 +X0Y1)k +X0Y0,

41a�d_~�dk = 2n

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7:4

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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).

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(X1k +X0)(Y1k + Y0) = X1Y1(k2 + k)− (X1 −X0)(Y1 − Y0)k +X0Y0(1 + k).

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Sign extension

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max (}®}

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).

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max (}H}

i +}®}

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= 10 11 · · · 11︸ ︷︷ ︸2m−2i−4×

01 0 · · · 0︸ ︷︷ ︸i×

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(}®}

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1) + (}H}

2 +}H}

3)§H¡�~�pSc °­f�´]²04Ëdg�md3��v�r3d

max(

(}H}

0 +}H}

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+ (}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��

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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~

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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�

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mlnw(r_onp:w�d=s�p

17§

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τd3|�v�tCo³s�pYD

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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

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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

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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)

http://www.inria.fr

ISSN 0249-6399