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A new scheme for table-based evaluation of functions

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(1)A new scheme for table-based evaluation of functions David Defour, Florent de Dinechin, Jean-Michel Muller. To cite this version: David Defour, Florent de Dinechin, Jean-Michel Muller. A new scheme for table-based evaluation of functions. [Research Report] RR-4637, LIP RR-2002-45, INRIA, LIP. 2002. �inria-00071948�. HAL Id: inria-00071948 https://hal.inria.fr/inria-00071948 Submitted on 23 May 2006. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published 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..

(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. A new scheme for table-based evaluation of functions David Defour, LIP Florent de Dinechin, LIP Jean-Michel Muller, CNRS. N° 4637 Novembre 2002. ISSN 0249-6399. ISRN INRIA/RR--4637--FR+ENG. THÈME 2. apport de recherche.

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(28) $, (')  f (x). n. = f (x0 ) + [x − x0 ]f 0 (x0 ) + 12 [x − x0 ]2 f 00 (x0 ) + 16 [x − x0 ]3 f 000 (x0 ) 1 [x − x0 ]4 f (4) (x0 ) + 24 1 + 120 [x − x0 ]5 f (5) (x0 ) + 1. (T0 ) (T1 ) (T2 ) (T3 ) (T4 ) (T5 ). \9”‹]. ] ^b bS~~’m7~qm7`c`Zjkrzrba„ uj x  = ([x 2 + x 2 + x 2 + x 2 2 c¹iba|Umax vwstr’jnmp|fi \9”|]¤ªž¢b(b ¦ ƒwts iP„Hr^b(r’ba~’` (T ) = [x − x ]f (x ) spx¢ ¡mplnlnmž7j 1 720. 1. 1 −6k 720. (6). −k. 1. 2. 1. −2k. 0. 3. 0. −3k. 4. −4k 6. ] )f (6) (ζ1 ) <. 0. (T1 ) = [x − x0 ]f 0 (x0 ) = x1 2−k f 0 (x0 )+ x2 2−2k f 0 (x0 )+ x3 2−3k f 0 (x0 )+ x4 2−4k f 0 (x0 ). 1 24. ŽVmž lnbr&vwx ¡m0qSvwxm7i rbS~`Zx (T ) = [x − x ] f (x ) ª (T ) = [x − x ] f (x ) ª spiw„ (T ) = [x − x ] f (x )  ¡~mp` Db |7vPsr’jnmpi \9”]¤¬ f b^ws£pbj [x − x ] f (x ) 0. 4 (4). ž"jkr’^ f b%mp{ r†st<jni©r’2^b 2. s #Ò Ž  ’‘ “. 0. 1 −5k 2. 5. 1 2. 2 1 120.  (T2 )                            (T ) 3                 (T4 )            (T5 ). 0. 0 5 (5). 2 00. 3. 0. 1 6. 0. 3 000. 0. (T4 ) =. 0. =. 1 2 −2k 00 f (x0 )+ 2 x1 2 x1 x2 2−3k f 00 (x0 )+ x1 x3 2−4k f 00 (x0 )+ 1 2 −4k 00 f (x0 )+ 2 x2 2 x2 (1/2)2−5k f 00 (x0 )+ 1 −5k 00 f (x0 ) + 2 2 x1 (1/2)2. =. 1 3 −3k 000 f (x0 )+ 6 x1 2 1 2 −4k 000 x x 2 f (x0 )+ 2 2 1 1 2 −5k 000 x x 2 f (x0 )+ 3 2 1 1 2 −5k 000 x x 2 f (x0 ) + 3 1 2 2. =. 1 4 −4k (4) f (x0 )+ 24 x1 2 1 3 −5k (4) x x 2 f (x0 ) + 4 6 1 2. =. 1 5 −5k (5) f (x0 ) 120 x1 2. + 5. ª stiw„ ¬ max |f |  < 2 ʙ3œpª› ™ ° ™a› – °7³ max –I›˜ ° |f¶ |µ š  < ʛ 0{ 2yZ~’baž"max ~’jkr’jni|fo‘º,||Uvwsrjkm7iL<\9”],s7x?2 ¡m7lklnmmax ž xj |f | 00. 3. 1 −6k 3. 000. 4. 1 −6k 24. (4). 5. 1 −6k 120. (5).

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(32)  P$, ` Š. . f (x). ž"^wbS~b. . = A(x0 , x1 ) + B(x0 , x2 )+ C(x0 , x3 ) + D(x0 , x4 )+ x2 × E(x0 , x1 )+ x3 2−k × E(x0 , x1 ) + f. . . \3dG]. = f (x0 ) + x1 2−k f 0 (x0 )+ 1 2 −2k 00 f (x0 )+ 2 x1 2 1 3 −3k 000 x 2 f (x0 )+ 6 1 1 4 −4k (4) f (x0 )+ 24 x1 2 1 5 −5k (5) x 2 f (x0 )+ 1 120 1 −5k 00 x (1/2)2 f (x0 )+ 1 2 B(x0 , x2 ) = x2 2−2k f 0 (x0 )+ 1 2 −4k 00 f (x0 )+ 2 x2 2 (1/2)x2 2−5k f 00 (x0 ) C(x0 , x3 ) = x3 2−3k f 0 (x0 ) D(x0 , x4 ) = x4 2−4k f 0 (x0 ) E(x0 , x1 ) = x1 2−3k f 00 (x0 )+ 1 2 −4k 000 f (x0 )+ 2 x1 2 1 3 −5k (4) x 2 f (x0 )+ 6 1 1 −5k 000 x (1/2)2 f (x0 ) 1 2 f ≤ 1 + 2 + 3 + 4 + 5 1 −6k 2 max |f (6) | + 21 2−5k max |f 00 |+ ≤ 720 1 −6k 1 −6k max |f 000 | + 24 2 max |f (4) | 32 1 −6k (5) | 120 2  max |f ≤ 2−5k 21 max |f 00 | + 13 max |f 000 |+ 1 1 (4) | + 120 max |f (5) |+ 24 max |f 1 (6) |+ 720 max |f A(x0 , x1 ). VbSiwqSbpª qSsti {Pb(m7{ rspjkibD„—{0y©ƒObS~’ ¡mp~`Zjkio©d ` vlkr’jnƒlnjnqasr’jnmpiPx"stiw„ sp„„ jnio©x’j ¦ rbS~`Zx¢ž"jkr’^ ps i ba~’~mp~lkbDx’fx(x) r’^Psti  ¬ ]^b¢x’jiSb¢mt wr^brst{wlkbDx ž"^bS~bsplkl0r^bax’b,rbS~`©xIsp~’b¢lnmUm7®pbD„0§Kvwƒ ¡~’m7`ªt„ bSƒObSiw„ ^wbas£0jklny©mt Ir’^wb( ¡viwq¤rjkm7ižbqm7iwx’jn„ ba~a¬f bž"jklnl„ juxqvwxx"st{Ompv r"jkr jkir^b k0baqr’jnmpi‰w¬¿d ¬ c¨r` vwxzr{Pb(impr’juqba„©r’^wstr¢r^b%ƒ~’m7ƒPmUxzbD„Z`Zbr’^wm0„ b ¦ ƒlnmpjkrx¢„ ba~’jnsr’jnpb­ƒ~mpƒObS~’r’jnbaxmt G ¡viPq¤r’jnmpiPxS¬ ]^wb(x’vwqaqbaxx’jk7b(„bS~jksrjk7bax¢` vPx9rV„ bDq~bas7xzbD„H~†stƒju„ lnyHbSim7vop^ r’mZ`©st®7b(~’ba`cm7ba„Hr’bS~`©xjkiPxzjnopij ¥ § qastiUr qm7`cƒPst~ba„©rmZr’^bm7ib(žb(®7bSƒ rD¬ f. V

(33). 

(34) "‘(I)& Vi st{Px9r~s7q¤rVst~†q†^jkr’bDq¤r’vw~’b( ¡m7~ r’^wb ` vwl«rjkƒlnjuqSsrjk7b&`Zbr^m „±jnx­ƒ~bax’bSiUr’bD„mpi Ijnopv~bH”p¬Mc¨r%jnxVpba~’y x’jn`cjnlust~r’mcr^b` vlkr’jnƒwsp~zrj«rb&st~†q†^j«rbaqr’v~b­mp  W ‹#[¨ªž"jkr’^r’^bb ¦ qbSƒr’jnmpimt r^b(r9ž¢mc`‘vlkr’jnƒlkjnbS~†xa¬ %. Ñ,+JGÑ.-.

(35) 

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(39) $, (')  Input word. Tables. Œ. x0. x1. x2. x3. x4. k. k. k. k. p. E A. B. C. D. k+p+g. Small multipliers. 4k+ p+g. 2k+ p+g. k+ p+g. k+ p+g. p+g. p+g. Multi−operand adder Rounding. jnopvw~’bZ”Gj?]^wb(sp{wxzr’~†spq¤r sp~q†^jkr’bDq¤rv~’b­ ¡mp~r^bƒ~mpƒOm7x’ba„ x’q†^ba`Zb 4k+p. Cuv  †x C‡_} „‚ € €&†+€ .  . . c¹i sp„„ jkr’jnmpiHr’m&r^b(stƒƒ~m ¦ jk`©str’jnmpiZbS~~’m7~x,qmpiPxzju„ bS~ba„Zjni©r^b­ƒw~’baUjnmpvPx,x’baq¤rjkm7iGª7r’^juxsp~q†^wj«rbaq¤rv~b jni0pm7lk7bax¢xzbapbS~†stl x’mpv~†qbDx¢mt  ~’m7viw„ jnioZbS~~’m7~ j.  ]^b r†st{lnbax ^Ps£pb r’m"{Ob ¥ lnlnba„(ž"jkr’^ ¥¦ ba„ §KƒOmpjniUrstlnvbax ž"^wjnq†^st~b r’^wb?ƒ~bS0jkm7vwx;`©sr^bS`©srjnqastl. stlnvbax~mpviP„ ba„ r’m©x’mp`Zb(ƒ~baqjux’jkm7iG¬  k0jn`cjnlust~lky7ª0r’^bm7v r’ƒvr mt Ir’^b`‘vl«rjkƒwlkjnbS~†x^ws£pb%rmc{Ob~mpviw„ba„—r’mZr’^wb(rst~opbSr¢ƒw~’bDqjuxzjnmpiG¬  ]^bax’b­spƒƒ~m ¦ jk`©srjkm7i©bS~~’m7~xaªpspiw„Zr’^wb%spƒƒ~m ¦ jn`©srjkm7i©bS~~’m7~aªt~ba|Uvjn~bVvwx,r’mcqmp`Zƒvr’bVž"jkr’^ stijniUr’ba~’iwsplƒ~baqSjnx’jnmpiž"^juq†^jnxV^jnop^ba~ r’^Psti±r^b ¥ iPstlI~’bD|7vwjk~ba„ƒ~baqSjnx’jkm7iG5¬ f b‘qSlns7x’x’jnqastlnlky vwx’b g sp„„ jkr’jnmpiPstl {jkrx"mp ƒ~baqSjnx’jnmpi \¡mp~  $&# ˆ' ) ]¬ i \¡ž"jkr’^  ]^bSir’^bc~’bDxzvwl«rV^PspxVr’m—{Ob ~mpvwiw„ ba„rmHr^b ¥ iPstlIƒ~’bDqjuxzjnmpist ¡bS~Vr’^b ¥ iwsplsp„w„ j«rjkm7l stibS~~mp~sr ž¢mp~†x9rmt  sti^wstlk  !Jk · ]¤¬ ]^ba~’bS ¡mp~bpª0r^b(rsp{lkbx’`j iabaxibaba„ bD„st~b%x’vwq†^r’^wstr j Size(A[x0 , x1 ]) Size(B[x0 , x2 ]) Size(C[x0 , x3 ]) Size(D[x0 , x4 ]) Size(E[x0 , x5 ]). ≤ ≤ ≤ ≤ ≤. 22∗k (4k + p + g) 22∗k (2k + p + g) 22∗k (k + p + g) 2p+k (p + g) 22∗k (k + p + g). ]^wbV` vwl«rjkƒlnjnbS~vwx’ba„©rm `‘vl«rjkƒwlky ž"j«r^ juxs ` v lkr’jnƒlnjkba~spiw„©r^b%mpib­vwxzbD„ rmZ` vlkr’jnƒlny E ž"j«r^ x jux s (k) ×E(p + g)xmpiwbp¬ (k) × (k + p + g) ]^b‘i0v` {ObS~­mt  opvwsp~„{jkrxVjnx%„ bSr’bS~`Zjkiwba„±spx"r’^bcxz`©stlnlnbaxzrVstlnvb ’x vwq†^r’^wstr r’^wb‘x’v`¸mp  r’^b spƒƒ~m ¦ jk`©srjkm7iJba~’~mp~&spiw„ stlnl?r^b—~mpvwiw„ jnio±bS~~’m7~x(juxcxz`©stlnlnbS~r^wsti s !Jk · mt r’^b ¥ iwspl,~bax’vl«r \¡ Êstjkr’^  ¡vwl~mpviw„jkio ]¤¬ 2. 3. s #Ò Ž  ’‘ “.

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(44) ]ª?j« %qmp~~bax’ƒPm7iw„ jnioqm7~’~baqr’jnio±r’ba~’`©x ^Ps£pb%{PbabSi±sp„„ba„ r’mcr’^wb£splkvwb%vPxzbD„—rm ¥ lklGr’^b A r†st{lnbp¬. Cu(t 0‚ C‚ €  zI€ ‚ xCz { *. *. . ]^wb `Zbr^m „ ƒ~bax’bSiUrba„ sp{Pm7bH^ws7x‘{ObSbai jn`Zƒlkba`ZbSiUr’bD„ jni s  ƒ~mpo7~sp`¬ ]^juxcƒ~’m7op~†st` ¥ lnlnx r^bcrsp{lkbZstiP„ „ bSr’bS~`Zjkiwbax­r’^bcrmtrspl bS~~’m7~T\¡{0ybai0v`ZbS~†sr’jnmpi] ¡mp~(jniwq~bas7xzjnio £splkvwbax%mt  g ªGviUrjkl  Êspj«r^  ¡vlG~mpvwiw„ jnioZjnx"s7q†^jkbapbD„ ¬ ]^b,rsp{lkbxzjiSbDx  ¡mp~st~jkm7vwx; ¡viwqr’jnmpiwxopbaibS~†srba„%{0yVr^b,ƒw~’m7op~†st`Tsp~’b ƒw~’bDxzbai7rba„(jnirsp{lnb ”p¬sc¹i s7„„ jkr’jnmpiGª7žb qSmp`Zƒwsp~’bD„ m7v~?~bax’vl«r†x ž"jkr’^Zr’^wb"{PbDx9r,®0imž"icr†st{lnb xzj iSbVmt ;opbSiwbS~†stln`j iaba„‘` vlkr’jnƒwst~’r’jkr’b `ZbSr’^m „7x W ‹ [¨¬ ƒ ]Ix’st`j ia{b lnb qŽ mpvw~~` baq¤{OrbS~ {jkmpr x xz~j biS  b f k ‹ d dUŒŠ ”a‰ ‹0Œ0”Dd sin ‰ ‹ ” ‹ p n t “ U ‰ “ ” ‹ Y d  Yp‰p‰7“ [0, π/4[ n ‹ Œt“ npd dp‹ ”a‹ pŠ7dt‰ ‹ d ‹Udt‹7d ”a‰ Š7dUŒtd exp ‰ ‹ a ” U Š d G p n Š ” ‹ Y t n Š7‹7dt“ [0, 1[ n ‰ Ud‰7‹Ud dt‰ ‹pŠ7Šp“ 7“ ‹ d ‹7G‹ Y7d ”a‰ Œ0”DŠ 2 −1 ‰ ‹ ” 7“ t ‰ ” ‹ Y t n Š7‹7dt“ [0, 1[ n ‰ YpŠp“7“ dt‰ Gd nGYn p‰ ]Ist{lnbZG” j ]sp{lnb&xzj iSbDx"stiw„spqaqv~†spqSyZ ¡mp~‹” n§K{wj«rDªwdt“§¨{jkr stiw„Gd n§K{jkr"jkiwƒv r mpƒObS~†stiP„x x. a i—r’baxzr’bD„Z ¡viwqr’jnmpiwx¢ž¢bV~bS`©st~®‘r’^wstrr^bVr†st{lnb­x’j iSbV~ba|Uvjn~ba„©{0yZ` vlkr’jnƒlnjnqasrjk7b rsp{lnb­{ws7xzbD„ Z` bSr’^m „jnxlnbaxx¢r’^Psti r’^bm7ib(~’bD|7vwjk~ba„ {0yHopbaibS~†stlnj`iaba„—` vlkr’jnƒwst~’r’jkr’b(`Zbr^m „ ¬?]^b(r†st{lnbax A ª B ª ª ^ws£7b jniƒvr¢{jkrxaª0ž"^bS~bas7x r†st{lnb ^ws£pb jniƒv r{jkrxa¬ m7~¢splklOmt  r^bS` r^bVi0v`‘{Pba~ C E mp  mpv rƒv rV{2kjkrxVjnxVst~jnsp{lkb&spiw„±„ baƒPbaiw„x m7Dir’^b&rstp{w+lkb7kªP{vr­stluxzm©m7ir’^b& ¡viwq¤rjkm7iG¬ ]^bDxzb(r†st{lnbax sp~’bHr’^0vwxcxz`©stlnlnbS~&r^wsti r’^mUxzb jki r’^b` vlkr’jnƒwsp~zrj«rb—`Zbr^m F „ \ 2 žm7~„wx&pba~x’vwx 2 ]ª ž"^juq†^ qSmp`ZƒObSiwxsrbax¢ ¡mp~r^b` vwl«rjkƒlnjnbS~†xS¬ 2n/5. Cu. . I€ +‚ { { }  . ( . n/2. . ]^wjnx¢`Zbr^m „H~bSƒ~bax’bSiUrxsti jk`ZƒOmp~’rstiUrjk`Zƒ~mpba`ZbSiUr,mpba~?r^b­`‘vlkr’jnƒwst~’r’jkr’b%`Zbr’^wm0„ ž"^baiHrsp~z§ o7br’r’jnio©lns7x9r’§Ko7bSiba~str’jnmpi }¢f(Vx \ Vjn~zrb ¦ cc"stiw„ ­jk~’r’b ¦ }?~m ]ªž"^juq†^ba` {Oba„±sclust~opb(i0v` {ObS~ mp  x’`©stln+l \ 18 × 18 → 35 {jkrx

(45) ]¢` vlkr’jnƒlnjkba~xa¬?]^bSjn~"ƒ~baqSjnx’jkm7i—jux"`Zmp~bVr^wstibaimpvo7^— ¡m7~r’^jux"xq†^bS`Zbpª spiw„ r’^bay splklnmžTs‘r’~bSbS§3 ¡m7ln„~ba„ vPq¤r’jnmpimpir^brsp{lnb&xzjiSbsr iwmHqm7xzr xzjniwqSb(r’^baysp~’bs£stjnlnsp{lnb­m7i r^b(q†^jkƒ¬ wmp~xz`©splkl k jkr¢`©s£yZbS7bSiH{Ob(ƒPmUx’x’jn{lkbVr’m‘vwxzb%mpiwlkyZmpib%` vwl«rjkƒlnjnbS~r’mZqmp`Zƒvr’b­r’^b­r9žm ƒw~’m „ vwqrxa¬ Ñ,+JGÑ.-.

(46) 

(47)    

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(76) 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).   

(77).   . ISSN 0249-6399.

(78)

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