A correctly rounded implementationof the exponential function on the Intel Itanium architecture
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(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. A correctly rounded implementation of the exponential function on the Intel Itanium architecture Christoph Quirin Lauter , Technische Universität München / École Normale Supérieure de Lyon. N° 5024 Decembre 2003. ISSN 0249-6399. ISRN INRIA/RR--5024--FR+ENG. THÈME 2. apport de recherche.
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(12) ü. Unité de recherche INRIA Rhône-Alpes 655, avenue de l’Europe, 38330 Montbonnot-St-Martin (France) Téléphone : +33 4 76 61 52 00 — Télécopie +33 4 76 61 52 52.
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(140) " m a . c. |d| < |c| · 2−64. a*
(141) # U N^& U Q # a+b ¤ ØØ*}wzsjl*{~zs+l>~MOtxs®Ê zÌGÉÈ|¨hjt{ wM¨wMvuv~tx~|¨tx¼tx~zvŨl: jvu|wº|¨hjl*jl¼txwz3 Èlk«k*~Fm¸ ~zs3 ~*{ wl%¨l>Rjl>s3{ ljl |¨l>¨k«txsjtxsjy«|¨hjl%k7txsjtxk%jk ~sWk*~¯txk%3k¿wz
(142) |hjl%~W wMvu|l¼-~vxjlw|¹w jwzjjvxl«l ¯|¨l>s3l:_sjk!Wl>rOk«l>~zs3#wº|¨hjljlrk O~sWNjlrkml txs3 |¨3{|tuwMs¸Ê Ì.3¨l:{ t l:}|h3~-| |hjlªwMl>wf|hjl4~yz3k7l>sR|#wf|hjljlrk Ok!3 |%WlwzjBwM¨tu|¨l«|¨wO|¨h3~|%wzsjl4tus,¹hjt{hT|¨h3lT~l yMtu¼Mls4|¨w«|¨h3lj lrkml «tus3 |¨3{ |¨txwzsÉ|hjt 3|¨txvutu|4hW~zfBlls l>¨Wl:{|l> ¸ c. Ô ¾¥¤ ŧ gfhjljl>{ l>l>txsjy«txs3|¨W{|¨txwzsF¨l>Rjl>s3{ ltlv~-|¨tx¼zl>vu4vuwMsjy3¸ gfh3wzjyMhhW~@¼Rtxsjy7|¨h3l%¨~zk«l v~-|ls3{_|h3~sT|hjl43s3{ wMs3tu|¨txwzs,¼zl>¨txwzsTwf|¨h3l4 l:Rjls3{lzÉtu|7¨hjwzjvTBl4~@¼zwMtxl:T~ztu|%k*~»Ml>~ h3tuyMhjl3 l!wÅ|hjl%jw{ l:¨¨wz ¨l:¨¨wzj{ l:f|¨h3Bl¼zlsR|tusjy~*l>~M ¨{h3l>jvxtxsjy4wzÅ|¨hjl!lsjyMvuwMjtus3y {wlz¸³µ|{>~sªBl}l>vu3jl>*«~s3~zvu¨tusjy~|{wzk«jtxvul|tuk«l}|¨hjlk«~zyzsjtu|¨Wl>ºw2|¨h3l~yz3k7l>sR| a ~s3 ¸ BwM¨tÁ|tu¼Ml tlrwÈ|hjl l:Rjls3{lt|¨hW~-|f|¨hjl#hjtxyzh3~z |fw
(143) |¨hjl#l>¨jvÁ|:É Étxf~@¼-~txv~jvxlr~ò|¨l>z°´ b{{ vxl>>ÉM|h3º|¨hjl#v~-|¨l>s3{ *wz|¨h3l#¨l>{wzs3ª¨l: jvu|{~s3¨3~vxvu*Bl hjtjl>s ctxs|¨hjlrvx~|¨l>s3{ *wz
(144) w|hjl {>~v{ jv~-|tuwMs3>¸ Âl¹txvuv{~zvuv |hjt}~vxyzwM¨tu|¨h3k |¨h3 l
(145)
(146) Ã l:M3ls3{l%~s3sjwz|¨ltu|~zvx¨w c + d ←
(147) ¸ (a, b) !#" ÔØ ¤ Ô
(148) %$ . ah4. =. al4. =. a5. =. 12297829382473034411 73786976294838206464 −3098113767111307095 680564733841876926926749214863536422912 12297829382473034411 295147905179352825856. ðpw uð .
(149)
(150)
(151) "! #
(152) ! $ %&. a6. =. a7. =. ln 2 ln 2 0
(153) . x ∈
(154) m&I− 2
(155) ; 2 15. m ? %& # . M. 4919131753080926207 590295810358705651712 13117684675687318145 9444732965739290427392. 15. 1 p(x) = x + x2 + x3 · 2 ex − 1. ah4 + al4 + (a5 + (a6 + a7 · x) · x) · x. 0 . %m!-?
(156) % . |ex − 1 − p(x)| < 2−123.
(157) '1r
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