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Fast correct rounding of elementary functions in double precision using double-extended arithmetic

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(1)Fast correct rounding of elementary functions in double precision using double-extended arithmetic Florent de Dinechin, David Defour, Christoph Lauter. To cite this version: Florent de Dinechin, David Defour, Christoph Lauter. Fast correct rounding of elementary functions in double precision using double-extended arithmetic. [Research Report] RR-5137, LIP RR-2004-10, INRIA, LIP. 2004. �inria-00071446�. HAL Id: inria-00071446 https://hal.inria.fr/inria-00071446 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. 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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(183) q^0bDž ŸTŸŸŠ¡¢p+loa2ƒ0kmimrohŽq"a¯rop4^0imh0bDrA 0 †ijq„imloh»imh£ †lou0zvkjb+¡Hb+©Žqb+hv 0bV ¾« ~louvhv ¬q„lµh0bVro~„b?wQq a2lN 0rnbo«srsh[ 53⊕ †bVh0lsqbVw²q„^0b a¯rop4^0imh0blAƒ[bV~rsq„imloh

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(186) ~„b?wyu0kjq¬q„l‘rohFž ŸTŸŸ †lou0zvkjb2uvw„ijhvn6q„^0b¯uvw„b+~„¡ wybVkjb?p¤q„b? ‘~lou0h[ †ijhvn a2l† †bA± ]_^0bVw„b6loƒ{b+~4r&q„imloh[wcro~„ba¯rop4^0imh0b6loƒ{b+~4r&qijlAhvw2lohrs~4p4^0ijq„b?p¤q„uv~„b?w"^vimp4^wyuvƒ0ƒ[lA~yqD †lAu0z0kmb¡ b+©Nq„b+h[ †bV ƒ0~bVp+imw„ijlAh± . . . 

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(197) 1 . J98+ G+) 0(:* N( 0/0 ( +. th + tl ← F ast2Sum64(1, ph) 64. r ← t l ⊕ pl rn 53. z ← th ⊕ r rx. 

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(199) X"# \4. ]_^0b¼rowwyuvacƒ0q„imloh ph = ph 64⊕ pl imwh0lsq~„b?wQq~„itp¤qij®Ab‘r&qrokjkH«imh ƒvrs~„q„itpuvkmro~Di°qimww„rsq„itw ˜[bV ij¥ ro~„bµq„^0b¬~bVw„u0kjq_lo¥ rX vrAwQq¤rndoÆNuvalo~' vrAwQq¤dsuvk°q rskmnolA~„ijq„^va«†rAw/itw q„^vb¬pVrow„b¬ijh6rcƒ[lAkjxNh0lAacitrsk ph + p l bV®&rskmuvr&qijlAhimh Zlo~h0b+~_¥¦lA~„a6±ž¢q itw rsktwylcbVrAwyxDq„l¯w„rsq„itwQ¥¦xi°¥ ph + pl ro~„bµqroz0u0ktr&q„b? ®\rokjuvbVwV± ™ Ì,Ì 2·¼b'³Nh0l&Y¥¦~„lAa(] rsz0kmb"dcq^vr&q 53◦ (1 + p(x)) = 53◦ (1 + ph + pl) ±·¼b

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(201) h0bVbV q„l rx ƒv~„l&®AbZq^vr&q z = 53◦ (1 + ph + pl) rx   imnou0~bVw_‹rsh[rx cd9nAij®Ab roh"l&®Ab+~®Žimb+¸ls¥vq^0ba¯rshŽq„itww„rAw#imhN®olAkj®AbV imh2ijh[wQq~„uvpq„imloh2wyb?|Žu0b+hvp+brsz{l&®obo± µh q^0bVw„X b ˜vnouv~„b?w+«²q„^0bDa2b?rsh0imh0nls¥ q„^vbD®obV~yqimpVrskŠkmijh0b?witw¬q„^vb¯¥¦lokmkjl&imh0$n œhNu0a'z{b+~'wyq„~itp¤q„kmx w„a¯rskmkjbV~¬ijh a¯rsnAh0i°quv †bcq„^[rsh −n ^vrow

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(205) kmijh0bA± ]_^0b  vrAwQq¤doÆNuvaObVhvw„u0~„b?w¾q^vr&q th = 1 64⊕ ph rsh[ th +tl = 1+ph «?q„^0bV~„b+¥¦lo~b bT^vr\®ob 1+ph +pl = rn ± th + t l + p l ´²bq k z{b q^0bZktrs~nob?wQqTimhŽq„bVnob+~ wyu[p4^2q„^vrsq ph < 2−k ±Š]_^0itwTa2bVrohvwŠq^vr&q/q„^0bµa2rohŽq„itw„wr¬ls¥ ph rsh[ ro~„b¬imhv †b+b? rokjimnoh6row_lAhq^0?b ˜vnAu0~„bA± 1 

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(213)  2 −k. ph + p l. −52. È £š 

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(215)  &. row„b. −63. ph ≥ 0 2−63−k. . 2−127.  

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(223)    imnou0~b2‹.œ. 1. 2−63−k. . r. th + t l. −63. ‹A‹.   imnou0~bdœ 1/2 < |th | < 2.   98+. row„b. ph < 0. "#$) (™ . ph. . 1. 

(224)  4  ™

(225) – .  N  98+. ™ ÌÌ,# 0~loa ph < 2−k < 2−2 ±  È £š 

(226)  |t | < 2−63 & l ™ Ì,Ì  Zw„imh0n¯´b+a2a¯r¯Œ0±Òd†«Žq^0imw imwbVhvw„u0~„b? zNx¯q^0b vrAwQq¤doÆNuva,(o“"ƒv~„l† †uvp+ijhvn th + tl ±  È £š 

(227)  +98+ /)  t E"X\

(228) $)04 E: *:–&?"#1 ( '98+ 2−63−k 

(229)  \ (  ‘   98  ( l •.•0  ›  80  

(230)   &. ™ Ì,Ì ¯]_^0Gb  [rowyq4dAÆNu0a (s“2ƒ0~l† †uvpimh0n bVhvw„u0~„b?w_q„^[r&q th + tl = 1 + ph b©0rAp¤q„kmxo±_]_^0bV~„b+¥¦lo~b l or kjk=q^0b

(231) z0ijqwlo¥ tl ls¥ a¯rsnohvi°quv †bw„a2rokjkmb+t~ h q+^vrsth q„^[r&q ls¥#q„^0b

(232) ktrowyqz0ijqZls¥ ro~„b9¿+bV~„lNbVwV±ŠZw itw_q„^0b ktrs~nob?wQq ijhŽqb+nobV~wyuvp4^q„^vrsq ph < 2−k q„^vb9ktrowyq_z0i°q lo¥²q^0b¬a2rohŽq„itw„wrlo¥ pphh ^[row a¯rsnohvi°quv †b 2k−63−k ±  È £š 

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(236)   l l. 2−63−k. &. ™ ÌÌ,# 0~loa. È £š 

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(243) |r| < 2−63 & ™ Ì,Ì ¯]_^0itwitwrc 0ij~bVpqZploh[wyb?|Auvb+hvp+bµlo¥ kmb+a2a¯r&q4rcŒv± Œ2rohv Œv± (v±  È £š 

(244)  t 53  , (80  53 J  ?* 0 (  c  ?)t•. 4  (  ,=98+ ( ( ?4 0/(1 (2 ◦ (th ) h⊕r. •6   *

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(247) ‘q„^0itwimh†¥¦lo~a¯r&q„imloh²± 64 Zl& q„^vb9lAƒ[bV~rsq„imloh r ← tl ⊕ pl a2r\x¯imhN®olAkj®Ab9r"~„lAu0hv †imh0ncb+~~„lA~V«Nw„lcw„loa2b9ls¥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+hŽqw th rohv r lo¥[q^0b ˜vhvrok 53⊕ ± ·¼bh0l& h0b+b? ql

(249) ƒ0~„l&®Abq^vr&q 1 + p h + pl q^0itwZ †lNbVw„hÄ q ƒ0~b+®obVhŽqZplo~~bVp¤q~lou0h[ †ijhvnv±· ^[r&qZ b

(250) ijkmk²ƒv~„l&®Ab9itw_rxq^vr&q q^0b'w„ijnAh6ls¥ r §¨^0bq^0b+~µi°q itw/hŽuvkjkH«Žwyq„~itp¤q„kmx2ƒ[lŽwyijq„im®obA«AlA~/wyq„~impq„kmx2h0b+nŽr&q„im®ob\­itwrskm_r\xNwTbV|Žuvrsk[q„lq„^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 £š 

(251)  53◦ (1 + ph + pl ) = 53◦ (th + δ) & rx rx ™ Ì,Ì # x †b(˜vh0ijq„imloh6ls¥ δ ± Zl& ^vrsq /b

(252) plAa2ƒ0u†q„b¬itw_h0lsq 53◦ (th + δ) z0u0q z = th 53⊕ r = 53◦ (th + r) imhž ŸTŸTŸŠ¡ plAacƒvkjitrshŽq ~louvhv †imh0nv±9·Fb"q„^vb+~b¥¦lo~b'hvl&‡ƒ0~„l&®Arxb

(253) q„^[r&q 53◦ (th + δ) = rx53◦ (th +rx r) ±, 0lA~Zq^0itwµijq¬imw9b+hvlou0nA^q„l ƒv~„l&®AbZq^vr&q δ < 2−63 €QuvwQqZrAw r «vrshv q^vr&q_q„rx^vbwyimnohls¥ r itw rsrxkm r\x†wq„^0bw„imnohls¥ δ  È £š 

(254)  δ = r +  & r ™ Ì,Ì  r = tl 64⊕ pl = tl + pl − r = δ − r ± . È £š 

(255) , δ < 2−63 & ™ Ì,Ì ¯]_^0itwitw"ronAroijh»r‘p+lohvw„bV|Žu0bVhvpb¯lo¥´²b+a2a¯r&q4r‘Œ0± “ºrsh[ »Œ0±Ò‰^0imp4^ wy^0l& q„^[r&q'q^0b+~bDitwh0l l&®Ab+~kmroƒDls¥h0lAh†¡H¿Vb+~l"zvi°q4w ¥¦~„lAa pl rsh[ tl «vrohv ¥¦~„lAa´bVa2a2r2Œv± Œ6§ |tl | < 2−63 ­¤± . rn. ã  ã.

(256) 

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(261) . . |r | < 2−127. ‹?Œ. &. ™ ÌÌ,# 0~loa r = (tl + pl ) − (tl 64⊕ pl) «_rshv rn §;´bVaca¯r2Œ0±Ò‰o­±  È £š 

(262)   |δ| ≥ 2−127 98+ ( 53◦ (t + δ) h rx. |tl | < 2−63. §;´bVaca¯r Œ0± ŒA­crohv . |pl | < 2−63−k. 53. = ◦ (th + r) rx. ™ ÌÌ,¯Zw itw'q^0b6wyimnoh ls¥ ± −127 §¨´²b+a2a¯rFŒv±j‹\do­¤«Šij¥ −127 q„^0bVh£q^0bw„imnoh¸lo¥ ´²b+a2a¯r&q4r"Œv|± d0r«Ž| Œv<±Ò'2rshv Œ0±m‹o‹µb+hvw„u0~bZq^vr&q_loh0kmx2|δ|q„^v≥imw_im2h†¥¦lo~a¯r&qijlAhitw w„imnoh0i1˜[p+rohŽq/r imhz{lsq^~lou0h[ †ijhvδn lAƒ[bV~rsq„imlohvwV±  È £š 

(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¯r9Œv±j‹\do­#rshv r = δ −r §¨´bVa2a2r¬Œv±j‹?’A­²/b †b? †uvpb q^vr&q |r| < 2−126 ±ž¢q'ija2ƒ0kmimbVw

(264) q^vr&q tl = 0 « rAw

(265) rskmkijqw

(266) hvloh†¡¢¿+bV~„l6z0ijqw"rs~bclo¥a2ronoh0ijq„u[ †b¯no~bVrsq„b+~ q^vrsh 2−63−k ijq„^ k < 52 §¨´bVa2a2r¯Œv± “N­¤±]_^vimwimh6q„uv~„h6imacƒvkjimbVw_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¬Œv±j‹V“9rohv "Œv±j‹?Œµw„^0l&»q^vr&qŠimh¯rskmk0p+row„bVwV« 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„^vb9ƒv~„lNls¥ rxls¥ q„^0b¬q^0b+lA~„bVa6± Es

(267) . ÈNÌ ™ È. uw [‹-y. 1+ph +pl. x + p(x). 

(268) 

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(279) ‹V“. c1

(280)  $4 G 8›

(281) ž\*› )  A8  % •+8 H

(282) % (. th + tl ← F ast2Sum64(x, ph) 64. r ← t l ⊕ pl rn 53. z ← th ⊕ r rx. 

(283) "'•6%  98+ ?ž–0. 

(284)   :–&š  6. š5 #. 0/(1 •  ( *

(285) = 80 ? 

(286) X"# \4 Š• 

(287) ž\*4 ez6 #6. ( $ x( + \p(x) (    80 ›

(288) "'•6%

(289) )*

(290) w  &. rx. th. Zlsq„b‘q„^vrsqDq„^0buvhv †b+~—vl& †itw„p+kmroija2b+~imw¯hvlsqr ƒ0~lozvkjbVa imh q^0bºp+lohŽq„b+©ŽqDlo¥

(291) ija2ƒ0kmb+a2b+hŽqijh0n bVkjbVa2b+hŽqro~„x¥¦u0hvp¤qijlAhvwœ_ kmk#q„^0b

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