Second Order Function Approximation with a Single Small Multiplication
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Texte intégral
(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Second Order Function Approximation with a Single Small Multiplication Jérémie Detrey, Florent de Dinechin. N° 5140 Mars 2004. ISSN 0249-6399. ISRN INRIA/RR--5140--FR+ENG. THÈME 2. apport de recherche.
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(21) &(4 #. }pxzY"Wklxz§CY8Nutr ®hTtnVc¥izYkl` t® iaf`VYaf`uqbXraf` iToYh"k:psYeCizTVYWnVc¥izabklxiabiYWYizTVeLRiTVYikQ© ou cbY8poxY8psYX`:idpeCWYZprtWWZY"ixzrC¤ Y Tok£CY TO1(A1 , B1) = 2−α−β K1(A1 ) × B1 LTVamhT h"kl`¡uY xzY"xzabisiYX`q« 0. óó Ð sQ.
(22) . }. TO1 (A1 , B1 ) = 2−α−β0 K 1 (A1 ) × B1. 1−2−β1 2. = 2−α−β0 K1 (A1 ) × (B1 −. B ./ Q+ 0! (-!. ) + K1 (A1 ) ×. 1−2−β1 2. . TVYXxY iTVYizY"xzW −α−β h"kl` u5Yk:VYX¡ize iTVYQklcfnVYeliToY ¤ STVamp 1−2 kCcbcfeQ~p2nop2ienopYHiTVYd2pY"g:WZYX`7Ki1prt(AW1WZ) Y"×ixzr kC2pYXVamh(iY8af` Rabg¤o[Vpzk£7af`VgZkuoa¥i~af` k:VTIV xzYXpzpsaf`VgiTVY kliyiToY~Y¢5Y"`opY~elLk%YX ,H g7kQiY8p`VYXYXY8ZiedxzYXhe:`opsixznoh(iizTVY~elizTVY"xToklcbLel5iTVYHpsYXgCWY"`7iX¤ TO1 STVY%QklcfnVY8peC TO2 kCcfpeZVxY8psYX`7i2pr7WWYizxr:tTVafhT klcfcbeQ~pieaftafYHabizppab¦XYjutri|ekCp2Y"cfc>¤ wÀ`NiTVampHh"kCpYizTVYZeCnizVni%elyizTVYikluVcfYpTVe:nVcfN`VeliHu Y ,HY8LLkCp TO2 TVe:cfopkC`@Y":Y"` nV`h(iafeC` >psYXY afgo¤9 (¤. . −β1. 0. . f. Ä(Á¼Ô8¾s»¨»>½>Ã(ÇȾsÍ º¼Ð~½J¹£¾ TO. TIV(A). 1. Rabg:nVxzYd[V« ¢VklWVcfY%elpsYXgCWY"`7i~psrtWWYizxr:¤ . # "(
(23) +#& . . . }j`NY"¢kCWVcbYel®¡®/, eC5Y"xkQizeCxHklxhTVa¥izYXhinVxzYampjg:ab:Y"` Rafgo¤EV¤d}~cfcRiTVYizkCuVcbYcbete:§7noopkCxYYXxs© e:xWY8af`1oklxklcfcbYXc>¤ ,`VYdpTVeCnocf klcmpse`VeCiamhYHiTkQii|2eZeC¨izTVY%iTVxzY"Yk:VabiafeC`opeC¨izTVYkCVVY"xixzY"Y hXkl`/u5Y5Y"xeCxzWYX af`/okCxzkCcbcfY"cieNiTVY WnVcbiafVcfafhXkQizabe:`q¤¡STVYXxY"eCxzYiToY h"xabiamh"kCcokliT/ampizTVY1S~® ikluVcfY%cbete:§7noqtiTVYdWnVcbiafVcbamh"kliafeC` kl` izTVYdcfk:p|ijk:VabiafeC`q¤ }j`abW5eCxizkC`:i¨k:Qkl`7izkCgCYeltizTVafpphToY"WYafpqiTokliqiToYWnoc¥izabVcfafY"xqampq§:Y"ipWklcfcCkl`%xY8h(izkC`VgCnocfkCx VnVY@ize iTVY psocbabisizab`Vg<el B ¤ STVamp afcbcHcbY8kC ize Y h"abYX`:iabWVcfY"WY"`7ikQiafeC` eC` hnoxxzY"`7i Rw]%} Tklx2kCxYa¥izTZÈkCpsihXklxzxrh"abxhnoa¥izxr >kl`oY":Y"`ZWeCxzY2Y hafY"`7ia¥EuVcfeth§WnVcbiafVcfabYXxzpykCxYk£QklafcfkCuVcf1Y (¤ STVYdxzY"WkCab`oVY"xeliTVamp~klxiamhcfY%psTVeQ~p2TVeQ iehTVete:pYiTVYd`tnVWY"xzeCnpklxklWYizY"xpaf`7ixzetVnohY8 ToY"xzYizeY"`opnVxzY%kZg:ab:Y"`1kCh"h"nVxkChru5eCno`oL¤. . . . .
(24) . X!$ ¨#%R . d R#%Rd wÀ`/iTVYeCcfcbeQaf`VgR2Ykl`7ik¡®N® ,klxhTVa¥izYXhinVxzYie Èh"cfk:ppamh"klcfcfrdVxeQtamY :Q"!!(:. kChXhnVxkCh"rE« SToYxzYXpnVcbiHxzYiznVx`oYX@Wnopsi%u5YeC`VYZeliTVYZi|2e`tnVWuYXxzp%pnVxxzeCno`oaf`Vg izTVYY¢VkChiWkQizTVY"WkQizafhXklc xzYXpnVcbiX:eCxyaf`eliToY"xizY"xzWpX:iTVYizeliklcVYXxxzeCxelEizTVYjpzhTVY"WY~pTVeCnVcmklcf2k£rpu5YjpsixzafhicfrpWkCcbcfY"xyiTokC` ¤ jeQYXCY"x2klcfc5iTVYeCcfcbeQaf`Vgafp2YXk:psafcfrk:VklViYX ieZelizTVY"x2Y"xzxe:xu5eCnV`Vp"¤STVYHu5eCnV`oeC`iTVY 2−w g:cbe:uoklcEYXxxzeCx WkCVYHutriTVY®¡/® , e:YXxzklieCxamp2iTVYpnVW elpsYXCY"xklciYXxWpX« O. Ï ìRó ÏÑõ.
(25)
(26) !#"$&%'!)(
(27) "*+!-,./0 #".!.12.3$ 4.3". α. β. xor. TIV. y. TS. TO2. xor. TO1 xor. RafgCnVxzY o«2}jxzhTVabkCi`oY8h( iznVxYeliToYZ®N®. β0 = 5 β 1 = 3. ,. round. eC5Y"xkQie:xeCx. . . . . α = 4 β = 8 α 0 = α = 4 α 1 = α2 = 2. β2 = 3. TVYXxY:«. = poly + tab + rt + rm + rf ,. amp2iTVYdYXxxzeCxnoY%ieZiTVYd5eCcfrt`VeCWamklcqkCVVxze£¢tafWkQizabe:`qp|iznoafYX ab`1V¤f ? 5 poly 5 afpizTVYZkCVVxze£¢abWkQizabe:` Y"xzxe:xnoYizexzY"WeQtab`oguVabizpxzeCW iToYikluVcfYaf`VVnVizp%kCp%pTVeQ`. V tab xzY"tabe:nopcbr ? wªi~ampp|iznoafYX ab` o¤ [ ? 5 kC`o kCxYHxzeCnV`oVab`VgZY"xzxzeCxp"7TVY"`ocfcbaf`VgZiTVYHikluVcfYXpXtiTVY%VxzenohikC`oizTVYHo`okCcqpsnVW ? SrtTVY"r klxzYHrfpsinabY8 ab` V¤ ? wÀ`¡iTVYe:cbcfeQaf`VgoL2YZpTVeQToeQiToYXpYizY"xzWpHhXkl` uYh"eCWVnizYXL¨VY"5Y"`oaf`Vg1eC`¡iToYVYXpabg:` klxklWYizY"xp"¤}~` TVY"noxamp|izafhjeCxe:iafWafpab`ogZk®N® ,¯eC5Y"xkQizeCxiTVYX` h"eC`opafpsizp2af`YX`tnVWY"xkQiaf`VgiTVY klxklWYizY"x~pokCh"YCh"eCWVnizab`ogiTVYY"xzxe:xeCx~Y8kChT1QklcfnVYelizTVYokCxzkCWZY"iYXxzpXV§CYXY"Vaf`Vge:`VcbriTVe7psY ToafhT Y"`opnVxzY ÈklabiTtnVcjk:h"hnoxzk:hrCkl`o pY"cfYXh(izab`og kCWeC`Vg¡iTVYXW izTVY eCizabWkCc~Y"abiTVYXxab` izY"xzWpeC p5Y"YXeCxeCkCxY8kV¤. . ' + (A%'. D C(A . . poly. S ToY hetY hafY"`7ip kl` klxzY he:WVniY8 eC` YXk:hT el2iTVY α af`7iY"xzQklcmpdkCpk Waf`VafWkl¢1kCVVxze£¢abKW0kQ(A) izabe:`1KuokC1p(A) YX1e:`1iTVYZK2(A) Y"WY"¦klcfgCeCxzabiTVWOqfCrª¤jSTVamp~WYizTVe 2VxzeQ7amY8pnop~abiT izTVY1NhetY h"abYX`7izpZklcfeC`Vg¡a¥izT/iToYQklcfnVYel poly ¤ S¨e hnViizTVYY"¢Vcbe:xzkliafeC` eCiTVY1okCxzkCWZY"iYXx óó Ð sQ.
(28) . . B ./ Q+ 0! (-!. pok:hYC2Y@Wk£r zx Y"Wklxz§/izTokQiiTVampYXxxzeCxamp e:ut7afeCnpscfr u5eCnV`YX utr<iTVY psY8heC`<e:xzVY"xSRk£rtcbe:x kCVVxze£¢abWkQizabe:` Y"zx xzeCx8« poly ≤ 16 2−3α−3 maxX∈[0;1[ |f 000 (X)|. . !*. C(4. %'A ( (C . tab. ~ Y"WeQtab`Vg uVabizp%xzeCW izTVYaf`VVnieCeC`oYikluVcfYWYXkl`p%abW5e:pab`Vg k1he:`opsizkl`7i%izkCuVcfYZQkCcbnVY eQ:Y"xkl` af`:izαY"xzQ−klcLαeli psaf¦"Y 2α−α ¤y}p2iTVYh"eC`7iYX`7ieCizTVY%izkCuVcfYHampnopnokCcbcfr WeC`VeCie:`VeCnopX7iTVYdQkCcbnVY izTokQidWaf`VabWampsY8piTVYY"xzxzeCx%nVYie iTVampdkloVxe£¢afWkQiafeC` ampHiToYZWY8kl`NeCyiToYZY"¢tixzY"WklcQklcfnVYXpHe:` izTVamp~af`:izY"xzQklcJokl`o izTVYYXxxzeCxaf`onoh"YX1ampiToY"`1iToYToklcbelRiTVYafpsizkC`ohYuY"i|YXY"` iTVY8psYY¢tixzY"Wklc QkCcbnVY8p"VpnVabizkCuVcbrphXklcfYX kCh"h"eCxaf`Vgize v5¤qC¤ STVY prtWWZY"ixzr xY8nohiafeC` YXpzhxzabu5YX¡af` ®tY8h(izabe:` [V¤ [ie1TokCcb:YizTVY pab¦XYZeCizTVY TOi p%YX`:iklafcfp `oek:VabiafeC`okCcqklVoxe£¢afWkliafeC`Y"xzxe:xX¤ i. . "* ( . . (4*+ $#/% &(. . j`eCxino`okQizY"cfrC:iTVYikluVcfYXph"kC`V`VeliuYjocbcfYX. rt. % *. rf. a¥izTxY8psnVcbizpxe:nV`oY8ZiedizTVY~iklxzgCY"iyVxY8hampsafeC`¨« k:hT ikluVcfY2eCnVcm1Y"`7iklafckWkQ¢afWnVWxzeCnV`oVab`VgY"xzxe:xel 2−w −1 Y"¢VhY"Y8af`VgiToYdieCizklcY"xzxe:x~uVnoVgCYi eC 2−w ¤ YizTVY"xzYe:xYocfciTVY TIV kl`o@iTVY TOi pabiT k oxY8hampsafeC`¡gCxzYXkQizY"xjiTokC`@iToYiklxzgCY"i oxY8hampsafeC`utr g0 uoa¥ip"g:noklxZuVabizp (¤ySTtnopxzeCnV`ab`ogdY"xzxe:xzpaf`ocfcbaf`Vge:`VYizkCuVcfYafp`VeQ 2−w −g −1 kC`ohXkl`uYWkCYkCppWklcfctk:pYXpabxzYXutrdab`oh"xY8kCpab`og ¤ Ve:xheC`psamp|izY"`oh"rdelVizTVY2o`okCctpnVWWkQizabe:` 2YhTVe:pY2iedxzeCnV`izTVYeCnViVniel5iTVYWnVc¥izabocbafY"xize gg00 uVabizpk:pYXcbcJCutrixznV`oh"kliaf`Vg%a¥i2kl`oZkCVVab`Vg Tklcbk uVa¥iieizTVYQklcfnVYaf` izTVY u5Ye:xYxe:nV`oaf`Vg ¤~STtnopizTVYizeliklcqYXxxzeCxjVnVYdie iTVY8psYde:nVx xzeCno`oaf`Vg:p2ampue:nV`oY8 u7r 4 × 2TIV −w −g −1 −w −g +1 ¤ STVYeCnizVniZeliTVY TS izkCuVcfYamp`VeCiZhe:=`ohYX2x`oYX utrNizTVYVxzY"tafeCnopamph"noppafeC`qkl`o/Y Wk£r h"eC`7ixzeCctabizpxzeCnV`ab`ogYXxxzeCxutrkl`oeliToY"x`tnVWu5Y"xeCgCnklxuVabizp ¤RSToafpYX`:iklafcfpkC`VelizTVY"xxzeCnV`ab`og YXxxzeCxiTkQijkCoVpnV1a¥izT iTVYpnVWWkQiafeC`1YXxxzeCxp"¤ Rab`okCcbcfr2Y%gTo1k£CY:« O. . O. O. O. 0. O. 0. 0.
(29) . rt = 2−wO −g0 +1 + 2−wO −g1 −1. .. S TVY`oklcypnVWWkQiafeC` amp%`VeQ kCcfpe5Y"xeCxzWYXNeC` g0 WZe:xYZuVabizp%izTokl`¡iToYizklxzgCY"iHoxY8hampsafeC`q¤ ~eCnV`oVab`VgNiTVYo`okCcpsnVW izeNizTVYiklxzgCY"iVxzYXhampabe:`/`VeQ Y"`7iklafcfpZk@xzeCnV`ab`ogNY"xzxzeCxnV ie rf = ¤ } h m c C k z p p f a X h l k R c i z x f a h @ § V V n Y i e Hk C p ® C k x W. k l k o ` @ Q k z i nVcmk&q¼[srkCcbcfeQ~pie1afWZoxeQ:Yabi%ie rf = 2−w −1 2−w −1 (1 − 2−g ). tjeliYiTkQiiTVampamph"noppafeC` TkCpZkCVY8 kC`VelizTVY"xi|e okCxzkCWZY"iYXxzp kC`o ie¡iTVY@®¡®/, kCxzhToa¥izYXh(iznVxzYC¤y¬nVxxzY"`7izcbr:tiTVYdQklcfnVY8pel g0 kl`o g1 klxzYHh"eCWVnizYX utrixzgam0klcb©ªkC`ot©ªgY"1xzxe:xX:ab`oh"xY8kCpab`og izTVY"W TVafcbY%izTVYkChXhnVxkCh"rZu5eCno`o afp`oelixY8kChTVY8L¤ STVYXxYafpkl`1abWVcfamhabiafWZocbYXWZYX`7izkQizabe:`@hTVeCamhYdaf`1iToYVxzY"tabe:nopYXxxzeCx~kC`oklcfrpsamp"oTVafhT ampizTokQi 2YnpsY2kl`Y"¢k:h(i8Xnocbcb©JoxY8hampsafeC`dWnVc¥izabocbafY"x8¤R}j`VelizTVY"xReCViafeC`2eCnVcmdu5Yyize~ixznV`ohXkQizYyiToYWnVc¥izabocbafY"x Tklx2kCxY2afxY8h(icfrC¤ ,nVxyhTVe:afh"Yafpe:ut7afeCnpToY"`ZizkCxg:Yisizab`og Rw]%}pabiTpsWkCcbcVWnVcbiafVcbafY"xpXlcbaf§CY izTVY {jafxsizY¢t© w w( ¤ wªiklcmpeWkl§CY8ppY"`psY~af` izTVYHeliToY"xh"kCpYXpXtkCpa¥iklcfcbeQ~pyieZhcfYXkl`ocbrY¢VxzYXpzpizTVYHY"xzxe:x k:pkZnV`ohiafeC` elRiToYoklxklWY"iY"xpX¤ Y8psamY8p"tizTVYY¢5YXhiYX g7klaf` af`1nopab`Vg kZizxno`oh"kliYX WnVc¥izabocbafY"x ampdcfYXpzpHiTokC` Toklcb2iTVY pab¦XYel2iTVYWnVcbiafVcfabYXxXTVafhT ampdabizpY"cbpWklcfche:WoklxzYX@ie1iToYizkCuVcbY8pkCp ®YXh(izabe:`afcbcpsTVeQd¤STVYXxY"eCxzYjizTVamp~hTVeCamhYdpY"YXWpR{|np|iza¥oY81kQ©ªe7p|izY"xzabe:xaJ¤ O. O. 0. . Ï ìRó ÏÑõ.
(30)
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(32) "*+!-,./0 #".!.12.3$ 4.3". -( (C %. " . #& $%. % *. *+ !' %. . &( % &( . wÀ`iToafppYXhiafeC`2YgCafCY2YXpsiafWkQiafeC`peC5kCxY8kjkl`h"xabiamh"kCc7kQiTYXcfk£r%e:xRQkCxrtaf`VgoxY8hampsafeC`op abiT e:xiTVY%e:cbcfeQab`ogizTVxYXY%nV`ohiafeC`opX«. wI = w O. 5 5 5. S TVYd`oklinVxklcLcfeCg7klxza¥izTVW1« log(1 + x) : [0; 1[→ [0; 1[ ? STVYd5eQYXx2eC[V« 2x − 1 : [0; 1[→ [0; 1[ ? STVYpab`VY:« sin( π x) : [0; 1[→ [0; 1[ ¤ 4. %'&)(* + "! # #$ 50000. log(1 + x) 2x − 1 sin x. 45000 40000 35000 30000 25000 20000 15000 10000 5000 0. &)
(33) 0 12#1 1 " 3 #$. 8.
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(43) 8. . VnV`h(iafeC` wyxzYXh"afpafeC`. uVabizp nVcbiafVcfabYXx~uVabijpsaf¦"Y 6×9 8 × 14 10 × 16 6 × 12 8 × 14 `VeCi~nopsaf`Vg kCxY8k.>pscfafh"YXp 162 406 1400 133 372 uocbeh§ WnVc¥izabocbafY"xp YXcfk£r `op 19 20 24 18 20 nopab`Vg kCxY8. k >pscfafh"YXp 135 349 1318 98 315 uocbeh§ WnVc¥izabocbafY"xp YXcfk£r `op 16 18 20 16 18 SRkluVcfYC|« wÀWokChi~elRnopsaf`VgizTVY~{jafxsizY¢t© w w 18 × 18 uVcfeh§ WnVcbiafVcfabYXxzpX¤ . wI = w O. . 16. uoa¥ip. uVabizp. B ./ Q+ 0! (-!. log(1 + x) 20 24. uVabizp. 16. uoa¥ip. sin x 20. 24. uVabizp. 10 × 15 1293 24 1218 20. e:xZh"eCWuVab`kQie:xamklceC5Y"xkQizeCxpklxzYVYizkCabcfYXqkCpiTVY YXpsiafWkQizabe:`ope:xVab5Y"cfaf`VYX<hafxzh"nVa¥ipZVxY8psYX`7i e:`VcfrpscfafgCT7i~a EY"xzY"`hYXpX¤ Rafgo¤R Èk pTVeQ~pdizTokQiiTVY1he:WuVaf`VY8/psaf¦"Yel~izTVY e:nVxikluVcfYXp TS kC`o TOi p gCxzeQ~p Y"¢e:`VY"`7iamklcfcfrNabiT iToY VxzYXhampabe:`qRk:pdY¢5YXhiYX e:xk ikluVcfY©ªuokCpYX TIV WYizTVeL¤ Rabg¤¨ Èu dh"cbe7psYXcbr xzYXpY"WuVcfYXp afgo¤ È9k TVamhT af`oamh"kliY8pizTokQiizTVY k:VYXxzpkl`/Wnoc¥izabVcfafY"xhe:`:izxafuVnizYe:`Vcfr u7r/k pWklcfckCWZe:nV`7i~ieiTVYeQCYXxzkCcbc¨kCxY8keliToYe:YXxzklieCxpX¤SToafpjÈkChijafpHklcmpe nV`oYXxcfaf`VYX1utr SkCuVcfY C ToafhT psinoVabY8pyizTVYjafWokChieC`klxzYXkdkl`Y"cmk£reCqnopab`VguVcbeh§ZWnVcbiafVcfabYXxzpX«RiTVYHa 5YXxYX`ohY~af`kCxY8k h"eCxzxY8ps5eC`VpxzeCnogCTVcfrieZizTVYklxzYXkelRiTVYdWnVcbiafVcbafY"xafWVcfY"WY"`7iY8 ab` pcbamhY8p"¤ Rafgo¤ >
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(61) Unité de recherche INRIA Rhône-Alpes 655, avenue de l’Europe - 38330 Montbonnot-St-Martin (France) Unité de recherche INRIA Lorraine : LORIA, Technopôle de Nancy-Brabois - Campus scientifique 615, rue du Jardin Botanique - BP 101 - 54602 Villers-lès-Nancy Cedex (France) Unité de recherche INRIA Rennes : IRISA, Campus universitaire de Beaulieu - 35042 Rennes Cedex (France) Unité de recherche INRIA Rocquencourt : Domaine de Voluceau - Rocquencourt - BP 105 - 78153 Le Chesnay Cedex (France) Unité de recherche INRIA Sophia Antipolis : 2004, route des Lucioles - BP 93 - 06902 Sophia Antipolis Cedex (France). Éditeur INRIA - Domaine de Voluceau - Rocquencourt, BP 105 - 78153 Le Chesnay Cedex (France).
(62). . ISSN 0249-6399.
(63)
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