Fast correct rounding of elementary functions in double precision using double-extended arithmetic
Texte intégral
(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Fast correct rounding of elementary functions in double precision using double-extended arithmetic Florent de Dinechin — David Defour — Christoph Lauter. N° 5137 Mars 2004. ISSN 0249-6399. ISRN INRIA/RR--5137--FR+ENG. THÈME 2. apport de recherche.
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(244) t 53 , (80 53 J ?* 0 ( c ?)t. 4 ( ,=98+ ( ( ?4 0/(1 (2 ◦ (th ) h⊕r. 6 *
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(246) 00"#/ (rx &. ÌÌ,¯]_^0itwitwrc 0ij~bVpqZploh[wyb?|Auvb+hvp+bµlo¥ kmb+a2a¯r&q4rcv± dcrohv v±Ò± ´²b+a2a¯rv± ¶qbVkjktw9uvwµq^vrsq9q^0b2ijh0¥¦lo~a2rsqimlohºpVrs~~imbV zx imw90u0~b+kmx|uvrskmijqr&qij®Ab.Z]_^vbc T¡ p+loa20kmimrohq^vrs~4 _rs~bchvb+bV vwlohvkjx¼³Nh0l& ij¥ r imw"wQq~itp¤qkjxF[lwyrijqim®obA«huvkjkH«lo~'wyq~impqkmxºh0bVnArsqim®obo«#ql 0bVpit bD^vimp4^»lo¥ q^0b¯q^0~bVbD{lAwwyimz0kmb¯~b?wyu0kjqwql~bqu0~h¼imh z « 0b+{b+hv imh0nºrokmwlloh¼q^0b~lou0h[ ijhvn a2l b rx ±/ hq^0b¬¥¦lokmkjl&imh0n2 b¬ 0b+h0loqb¬zN= x
(247) q^0itwimh¥¦lo~a¯r&qimloh²± 64 Zl& q^vb9lA[bV~rsqimloh r ← tl ⊕ pl a2r\x¯imhN®olAkj®Ab9r"~lAu0hv imh0ncb+~~lA~V«Nwlcwloa2b9ls¥²q^0b
(248) ropVpu0~4rop+xclo¥ rn v ^ A r w [ z V b + b 2 h m k A l y w T q j i " h q 0 ^ b ro~nAu0a2b+hqw th rohv r lo¥[q^0b vhvrok 53⊕ ± ·¼bh0l& h0b+b? ql
(249) 0~l&®Abq^vr&q 1 + p h + pl q^0itwZ lNbVwhÄ q 0~b+®obVhqZplo~~bVp¤q~lou0h[ ijhvnv±· ^[r&qZ b
(250) ijkmk²v~l&®Ab9itw_rxq^vr&q q^0b'wijnAh6ls¥ r §¨^0bq^0b+~µi°q itw/huvkjkH«wyq~itp¤qkmx2[lwyijqim®obA«AlA~/wyq~impqkmx2h0b+nr&qim®ob\itwrskm_r\xNwTbV|uvrsk[qlq^vrsq lo¥¾q^0bµb©0ropq/wyuva tl + pl ± ´²bq uvw 0b vh0b δ = (1 + ph + pl ) − th = tl + pl. rohv . r = (tl + pl ) − r. .. È £
(251) 53◦ (1 + ph + pl ) = 53◦ (th + δ) & rx rx Ì,Ì # x b(vh0ijqimloh6ls¥ δ ± Zl& ^vrsq /b
(252) plAa20uqb¬itw_h0lsq 53◦ (th + δ) z0u0q z = th 53⊕ r = 53◦ (th + r) imh TT¡ plAacvkjitrshq ~louvhv imh0nv±9·Fb"q^vb+~b¥¦lo~b'hvl&0~l&®Arxb
(253) q^[r&q 53◦ (th + δ) = rx53◦ (th +rx r) ±, 0lA~Zq^0itwµijq¬imw9b+hvlou0nA^ql v~l&®AbZq^vr&q δ < 2−63 QuvwQqZrAw r «vrshv q^vr&q_qrx^vbwyimnohls¥ r itw rsrxkm r\xwq^0bwimnohls¥ δ È £
(254) δ = r + & r Ì,Ì r = tl 64⊕ pl = tl + pl − r = δ − r ± . È £
(255) , δ < 2−63 & Ì,Ì ¯]_^0itwitw"ronAroijh»rp+lohvwbV|u0bVhvpb¯lo¥´²b+a2a¯r&q4r0± ºrsh[ »0±Ò^0imp4^ wy^0l& q^[r&q'q^0b+~bDitwh0l l&®Ab+~kmroDls¥h0lAh¡H¿Vb+~l"zvi°q4w ¥¦~lAa pl rsh[ tl «vrohv ¥¦~lAa´bVa2a2r2v± 6§ |tl | < 2−63 ¤± . rn. ã ã.
(256)
(257)
(258) ! "# $%
(259) &'()*
(260) +,-. 0/(1 (2 345 6. 7 98"# :*. È £
(261) . . |r | < 2−127. ?. &. ÌÌ,# 0~loa r = (tl + pl ) − (tl 64⊕ pl) «_rshv rn §;´bVaca¯r20±Òo± È £
(262) |δ| ≥ 2−127 98+ ( 53◦ (t + δ) h rx. |tl | < 2−63. §;´bVaca¯r 0± Acrohv . |pl | < 2−63−k. 53. = ◦ (th + r) rx. ÌÌ,¯Zw itw'q^0b6wyimnoh ls¥ ± −127 §¨´²b+a2a¯rFv±j\do¤«ij¥ −127 q^0bVh£q^0bwimnoh¸lo¥ ´²b+a2a¯r&q4r"v|± d0r«| v<±Ò'2rshv 0±moµb+hvwu0~bZq^vr&q_loh0kmx2|δ|q^v≥imw_im2h¥¦lo~a¯r&qijlAhitw wimnoh0i1[p+rohq/r imhz{lsq^~lou0h[ ijhvδn lA[bV~rsqimlohvwV± È £
(263) |δ| < 2−127 98+ ( 53◦ (t + δ) = 53◦ (t + r) h h rx. rx. ÌÌ,# 0~loa |δ| < 2−127 « |r | < 2−127 §¨´²b+a2a¯r9v±j\do#rshv r = δ −r §¨´bVa2a2r¬v±j?A²/b b? uvpb q^vr&q |r| < 2−126 ±¢q'ija20kmimbVw
(264) q^vr&q tl = 0 « rAw
(265) rskmkijqw
(266) hvloh¡¢¿+bV~l6z0ijqw"rs~bclo¥a2ronoh0ijqu[ b¯no~bVrsqb+~ q^vrsh 2−63−k ijq^ k < 52 §¨´bVa2a2r¯v± N¤±]_^vimwimh6quv~h6imacvkjimbVw_q^vr&q r ← tl 64⊕ pl _row_b+©0rop¤q?«0lo~ rn «Nq^0bV~b+¥¦lo~b δ = r ± r = 0 ²´ b+a2a¯r&qr¬v±jV9rohv "v±j?µw^0l&»q^vr&qimh¯rskmk0p+rowbVwV« th 53⊕ r itw q^0bp+lo~~b?p¤q~lAu0hv imh0n9ls¥ rohv q^0bV~b+¥¦lo~b9lo¥ 1 + p(x) ±]_^0itw p+lohvp+kjuv 0bVw q^vb9v~lNls¥ rxls¥ q^0b¬q^0b+lA~bVa6± Es
(267) . ÈNÌ È. uw [-y. 1+ph +pl. x + p(x).
(268)
(269) H (. «. / . 0/(1 (2! ( * 0"#/ c$% e4 0/(1 2 34% (4
(270) #!
(271) / , (c e.
(272) 0/! 2x 34% (4 0"#/ 08798+
(273) J98+ ( G 34 *=$% %-
(274) e# 0p h8 k 98+
(275) + pl. . • ph + pe = p(x)(1 + ) 64. • p h = p h ⊕ pl rn. • ph < 2−k x. &. • 2 < k < 52. 7"#! 98+. 04/! &. • . 2−m. (8+ ( . 68+ (X G$&# 6 "# \4. . ä¡0¢£¤¥. rx. m ≤ 119. 798+ (c
(276) )2( . .(& . ( T/
(277) 1 . 80 0) 0()*
(278) N( 0/0 ( +.
(279) V. c1
(280) $4 G 8
(281) \* ) A8 % +8 H
(282) % (. th + tl ← F ast2Sum64(x, ph) 64. r ← t l ⊕ pl rn 53. z ← th ⊕ r rx.
(283) "'6% 98+ ?0.
(284) :& 6. 5 #. 0/(1 ( *
(285) = 80 ?
(286) X"# \4
(287) \*4 ez6 #6. ( $ x( + \p(x) ( 80
(288) "'6%
(289) )*
(290) w &. rx. th. Zlsqbq^vrsqDq^0buvhv b+~vl& itwp+kmroija2b+~imw¯hvlsqr 0~lozvkjbVa imh q^0bºp+lohqb+©qDlo¥
(291) ija20kmb+a2b+hqijh0n bVkjbVa2b+hqro~x¥¦u0hvp¤qijlAhvw_ kmk#q^0b
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