\input style \chapno=4\subchno=2\subsubchno=1\chapnotrue bOLXSHINSTVO PUBLIKACIJ O DETALYAH PROGRAMM DLYA RABOTY V SISTEME S PLAVAYUSHCHEJ TOCHKOJ RASSEYANY PO "TEHNICHESKIM PAMYATNYM ZAPISKAM", RASPROSTRANYAEMYM RAZLICHNYMI PROIZVODITELYAMI evm, NO SLUCHALISX I PUBLIKACII |TIH PROGRAMM V OTKRYTOJ LITERATURE. pOMIMO PRIVEDENNYH VYSHE RABOT SM.~R.~H.~Stark, D.~v.~McMillan, {\sl Math.~Comp.,\/} {\bf 5} (1951), 86--92, GDE OPISANA PROGRAMMA DLYA REALIZACII NA PANELXNO-SHTEKERNOM USTROJSTVE; D.~McCracken, Digital Computer Programming, New York, Wiley, 1957, 121--131; J.~W.~Carr~III, {\sl CACM,\/} {\bf 2} (May, 1959), 10--15; W.~G.~Wadey, {\sl JACM,\/} {\bf 7} (1960), 129--139; D.~E.~Knuth, {\sl JACM,\/} {\bf 8} (1961), 119--128; O.~Kesner, {\sl CACM,\/} {\bf 5} (1962), 269--271; F.~r.~Brooks, K.~E.~Iverson, Automatic data processing, New York, Wiley, 1963, 184--199. oBSUZHDENIE ARIFMETICHESKIH OPERACIJ V SISTEME S PLAVAYUSHCHEJ TOCHKOJ S TOCHKI ZRENIYA INZHENERA-|LEKTRONSHCHIKA MOZHNO NAJTI V STATXE s.~g.~k|MPBELLA "Floating-point operation" V SBORNIKE "Planning a Computer System", [ed.~by~W.~Buchholz, New York, McGrow-Hill, 1962,92--121]. dOPOLNITELXNYE SSYLKI NA LITERATURU V~P.~4.2.2. \excercises \ex[10] kAK BUDUT VYGLYADETX CHISLO aVOGADRO I POSTOYANNAYA pLANKA, ESLI IH PREDSTAVITX V VIDE CHETYREHRAZRYADNYH CHISEL S PLAVAYUSHCHEJ TOCHKOJ PO OSNOVANIYU~$100$ S IZBYTKOM~$50$? (iMENNO TAKOVO BYLO BY PREDSTAVLENIE V MASHINE~\MIX{} (KAK V~\eqref[5]), ESLI BY RAZMER BAJTA RAVNYALSYA~$100$.) \ex[12] pREDPOLOZHIM, CHTO POKAZATELX~$e$ LEZHIT V INTERVALE~$0\le e \le E$; KAKOVY NAIBOLXSHEE I NAIMENXSHEE POLOZHITELXNYE ZNACHENIYA, KOTORYE MOGUT BYTX ZAPISANY KAK $p\hbox{-RAZRYADNYE}$ CHISLA S PLAVAYUSHCHEJ TOCHKOJ PO OSNOVANIYU~$b$ S IZBYTKOM~$q$? kAKOVY NAIBOLXSHEE I NAIMENXSHEE POLOZHITELXNYE ZNACHENIYA, KOTORYE MOGUT BYTX PREDSTAVLENY V VIDE \emph{NORMALIZOVANNYH} TAKIH CHISEL? \ex[20] pOKAZHITE, CHTO ESLI MY RABOTAEM S NORMALIZOVANNYMI DVOICHNYMI CHISLAMI S PLAVAYUSHCHEJ TOCHKOJ, TO SUSHCHESTVUET SPOSOB NEMNOGO UVELICHITX TOCHNOSTX BEZ UVELICHENIYA OB®EMA ISPOLXZUEMOJ PAMYATI: $p\hbox{-RAZRYADNUYU}$ DROBNUYU CHASTX MOZHNO PREDSTAVLYATX PRI POMOSHCHI VSEGO LISHX $p-1$~RAZRYADOV MASHINNOGO SLOVA, ESLI CHUTX-CHUTX UMENXSHITX INTERVAL ZNACHENIJ POKAZATELYA. \rex[15] pUSTX~$b=10$, $p=8$. kAKOJ REZULXTAT DAST ALGORITM~A DLYA OPERACII~$(50, +.98765432) \oplus (49, +.33333333)$? dLYA OPERACII~$(53, -.99987654) \oplus (54, +.10000000)$? dLYA OPERACII~$(45, -.50000001) \oplus (54, +.10000000)$? \ex[M23] dOKAZHITE, CHTO MEZHDU SHAGAMI~A5 I~A6 ALGORITMA~A MOZHNO, NE IZMENYAYA REZULXTATA, POMESTITX SLEDUYUSHCHUYU OPERACIYU: ESLI~$f_u$ I~$f_v$ IMEYUT ODINAKOVYJ ZNAK, TO ZAMENITX~$f_v$ NA~$\sign(f_v)b^{-p-2}\floor{b^{p+2}\abs{f_v}}$; ESLI ZNAKI~$f_u$ I~$f_v$ PROTIVOPOLOZHNY, TO ZAMENITX~$f_v$ NA~$\sign(f_v)b^{-p-2}\ceil{b^{p+2}\abs{f_v}}$. (eFFEKT |TOJ OPERACII SOSTOIT V "UREZANII"~$f_v$ DO $p+2$~RAZRYADOV, CHTOBY MINIMIZIROVATX DLINU REGISTRA, NEOBHODIMUYU DLYA VYPOLNENIYA SLOZHENIYA V SHAGE~A6.) \ex[22] uDACHNA LI IDEYA ZAMENITX STROKI~38--40 PROGRAMMY~A ODNOJ-EDINSTVENNOJ KOMANDOJ~"|SLC 5|"? \ex[M21] kAKIE IZMENENIYA NEOBHODIMO PROIZVESTI V ALGORITME~A, CHTOBY ON OBESPECHIVAL VYDACHU PRAVILXNO OKRUGLENNOGO NORMALIZOVANNOGO REZULXTATA, DAZHE ESLI VHODNYE DANNYE NE NORMALIZOVANY? (sLOVA "PRAVILXNO OKRUGLENNYJ" OZNACHAYUT, CHTO REZULXTAT IMEET NAIBOLXSHUYU VOZMOZHNUYU TOCHNOSTX V $p$~RAZRYADAH V PREDPOLOZHENII, CHTO VHODNYE DANNYE~$u$ I~$v$ V TOCHNOSTI RAVNY~$f_u\times b^{e_u-q}$ %% 242 I~$f_v\times b^{e_v-q}$ SOOTVETSTVENNO, HOTYA, BYTX MOZHET, I NE NORMALIZOVANY. v CHASTNOSTI, $u$ ILI~$v$ MOGUT IMETX NULEVUYU DROBNUYU CHASTX S OCHENX BOLXSHIM POKAZATELEM, I TAKOJ OPERAND V KONTEKSTE DANNOGO UPRAZHNENIYA VOSPRINIMALSYA BY KAK RAVNYJ NULYU.) \ex[25] pRIVEDITE PRIMERY VHODNYH ZNACHENIJ, DLYA KOTORYH PODPROGRAMMA~|FADD| V PROGRAMME~A NE OBESPECHIVAET POLUCHENIYA "PRAVILXNO OKRUGLENNOGO NORMALIZOVANNOGO OTVETA" V SMYSLE UPR.~7. \ex[m24] (u.~kAHAN.) pREDPOLOZHIM, CHTO ISCHEZNOVENIE POKAZATELYA PRIVODIT K PRISVOENIYU REZULXTATU ZNACHENIYA NULX BEZ KAKOGO-LIBO UKAZANIYA OB OSHIBKE. iSPOLXZUYA VOSXMIRAZRYADNYE DESYATICHNYE CHISLA S PLAVAYUSHCHEJ TOCHKOJ S IZBYTKOM NULX I POKAZATELEM~$e$ V INTERVALE~$-50\le e < 50$, NAJDITE TAKIE POLOZHITELXNYE ZNACHENIYA~$a$, $b$, $c$, $d$ I~$y$, DLYA KOTORYH VYPOLNYAYUTSYA SOOTNOSHENIYA~\eqref[11]. \ex[M15] pRIVEDITE PRIMER NORMALIZOVANNYH VOSXMIRAZRYADNYH DESYATICHNYH CHISEL S PLAVAYUSHCHEJ TOCHKOJ~$u$ I~$v$, V PROCESSE SLOZHENIYA KOTORYH PROISHODIT PEREPOLNENIE PRI OKRUGLENII. \rex[M20] dAJTE PRIMER NORMALIZOVANNYH VOSXMIRAZRYADNYH DESYATICHNYH CHISEL S PLAVAYUSHCHEJ TOCHKOJ~$u$ I~$v$, V PROCESSE UMNOZHENIYA KOTORYH PROISHODIT PEREPOLNENIE PRI OKRUGLENII. \ex[M25] dOKAZHITE, CHTO PEREPOLNENIE PRI OKRUGLENII NE MOZHET PROISHODITX V HODE VYPOLNENIYA FAZY NORMALIZACII PRI DELENII CHISEL S PLAVAYUSHCHEJ TOCHKOJ. \rex[m23] mISTER sMYSHL¸NYJ SLEDUYUSHCHIM OBRAZOM MODIFICIROVAL PRIEM, PRIVODYASHCHIJ K~\eqref[13] S TEM, CHTOBY PROIZVODITX \emph{UREZANIE} POLOZHITELXNYH CHISEL~$u$ S PLAVAYUSHCHEJ TOCHKOJ DO CELYH CHISEL~$\floor{u}$ S FIKSIROVANNOJ TOCHKOJ V PREDPOLOZHENII, CHTO~$00$, TO~$u \otimes w \le v \otimes w$, $u \oslash w \le v \oslash w$, $w \oslash u \ge w \oslash v$.}\cr } } %% 246 iTAK, ESLI OPERACII V SISTEME S PLAVAYUSHCHEJ TOCHKOJ USTANOVLENY V SOOTVETSTVII S OPREDELENNYMI USLOVIYAMI, TO, NESMOTRYA NA NETOCHNOSTX SAMIH OPERACIJ, SOHRANYAETSYA NEMALAYA REGULYARNOSTX. v PRIVEDENNOJ VYSHE KOLLEKCII TOZHDESTV, RAZUMEETSYA, VSE ZHE BROSAETSYA V GLAZA OTSUTSTVIE NESKOLXKIH IZVESTNYH ZAKONOV ALGEBRY; ZAKON ASSOCIATIVNOSTI DLYA UMNOZHENIYA V SISTEME S PLAVAYUSHCHEJ TOCHKOJ VYPOLNYAETSYA NE VPOLNE TOCHNO, KAK |TO BUDET VIDNO IZ UPR.~3, CHTO ZHE KASAETSYA ZAKONA DISTRIBUTIVNOSTI, SVYAZYVAYUSHCHEGO OPERACII~$\otimes$ I~$\oplus$), TO ON MOZHET NARUSHATXSYA, I PRI |TOM DOVOLXNO ZNACHITELXNO. pUSTX, NAPRIMER, $u=20000.000$, $v=-6.0000000$ I~$w=6.0000003$, TOGDA \EQ{ \eqalign{ (u\otimes v) \oplus (u\otimes w) &= 120000.00 \oplus 120000.01 =.010000000,\cr u \otimes (v\oplus w) &= 20000.000 \otimes .00000030000000 = .0060000000,\cr } } TAK CHTO \EQ[15]{ u\otimes (v\oplus w) \ne (u\otimes v) \oplus (u \otimes w). } aNALOGICHNO, NETRUDNO UKAZATX PRIMERY, KOGDA~$2(u^2\oplus v^2)<(u\oplus v)^2$; ZAPROGRAMMIROVANNOE VYCHISLENIE SREDNEGO KVADRATICHNOGO OTKLONENIYA DLYA RYADA NABLYUDENIJ PO FORMULE \EQ{ \sigma = {1\over n}\sqrt{n \sum_{1\le k \le n} x_k^2-\left(\sum_{1\le k \le n} x_k\right)^2} } MOZHET PRIVESTI K IZVLECHENIYU KVADRATNOGO KORNYA IZ OTRICATELXNOGO CHISLA! dAZHE ESLI ALGEBRAICHESKIE ZAKONY VYPOLNYAYUTSYA NE VPOLNE STROGO, MY MOZHEM ISPOLXZOVATX NASHI METODY DLYA OPREDELENIYA TOGO, S KAKOJ STEPENXYU TOCHNOSTI VYPOLNYAETSYA ZAKON. iZ OPREDELENIYA~$\round(x, p)$ SLEDUET, CHTO \EQ[16]{ \round(x, p)=x(1+\sigma_p(x)), } GDE \EQ[17]{ \abs{\sigma_p(x)}\le {1\over 2}b^{1-p}. } sLEDOVATELXNO, MY VSEGDA MOZHEM ZAPISATX \EQ{ \eqalignno{ a\oplus b &= (a+b)(1+\delta_p(a+b)), & (18)\cr a\ominus b &=(a-b)(1+\delta_p(a-b)), & (19)\cr a\otimes b &= (a\times a) (1+\delta_p(a\times b)), & (20)\cr a\oslash b &=(a/b)(1+\delta_p(a/b)). & (21)\cr } } zDESX DOVOLXNO PROSTYM SPOSOBOM MOZHNO OCENITX OTNOSITELXNUYU OSHIBKU NORMALIZOVANNYH VYCHISLENIJ V SISTEME S PLAVAYUSHCHEJ TOCHKOJ. fORMULY~\eqref[18]--\eqref[21] SLUZHAT GLAVNYM INSTRUMENTOM DLYA OCENKI OSHIBOK V ARIFMETIKE NORMALIZOVANNYH CHISEL S PLAVAYUSHCHEJ TOCHKOJ. %% 247 v KACHESTVE PRIMERA TIPICHNOJ PROCEDURY OCENKI OSHIBKI RASSMOTRIM ZAKON ASSOCIATIVNOSTI UMNOZHENIYA. kAK POKAZYVAET UPR.~3, $(u\times v)\otimes w$, VOOBSHCHE GOVORYA, NE RAVNO~$u\otimes (v\otimes w)$; NO SITUACIYA V DANNOM SLUCHAE NAMNOGO LUCHSHE, CHEM V SLUCHAE ZAKONA ASSOCIATIVNOSTI SLOZHENIYA~\eqref[1] I ZAKONA DISTRIBUTIVNOSTI~\eqref[15]. v SAMOM DELE, VVIDU~\eqref[17] I~\eqref[20] IMEEM \EQ{ \eqalignter{ (u\otimes v)\otimes w &= ((uv)(1+\delta_1))\otimes w &= uvw(1+\delta_1)(1+\delta_2),\cr u\otimes(v\otimes w) &= u\otimes((vw)(1+\delta_3)) &= uvw(1+\delta_3)(1+\delta_4)\cr } } DLYA NEKOTORYH~$\delta_1$, $\delta_2$, $\delta_3$, $\delta_4$ PRI USLOVII, CHTO NE PROISHODIT PEREPOLNENIYA ILI ISCHEZNOVENIYA POKAZATELYA, PRICHEM~$\abs{\delta_j}\le {1\over2}b^{1-p}$ DLYA KAZHDOGO~$j$. sLEDOVATELXNO, \EQ{ {(u\otimes v)\otimes w \over u\otimes (v\otimes w)} ={(1+\delta_1)(1+\delta_2)\over (1+\delta_3)(1+\delta_4)} =1+\delta, } GDE \EQ[22]{ \abs{\delta}\le 2b^{1-p}/\left(1-{1\over2}b^{1-p}\right)^2. } tEM SAMYM MY USTANOVILI, CHTO $(u\otimes v)\otimes w$ \emph{PRIBLIZITELXNO RAVNO}~$u\otimes (v\otimes w)$, ZA ISKLYUCHENIEM TEH SLUCHAEV, KOGDA PROISHODIT ISCHEZNOVENIE ILI PEREPOLNENIE POKAZATELYA. eTA INTUITIVNAYA IDEYA "PRIBLIZITELXNOGO RAVENSTVA" ZASLUZHIVAET BOLEE PODROBNOGO IZUCHENIYA; MOZHNO LI RAZUMNYM OBRAZOM DATX BOLEE TOCHNUYU FORMULIROVKU |TOGO UTVERZHDENIYA? pROGRAMMIST, ISPOLXZUYUSHCHIJ ARIFMETICHESKIE OPERACII V SISTEME S PLAVAYUSHCHEJ TOCHKOJ, POCHTI NIKOGDA NE ISPYTYVAET ZHELANIYA PROVERITX, NE VYPOLNYAETSYA LI RAVENSTVO~$u=v$ (ILI PO KRAJNEJ MERE ON EDVA LI KOGDA-NIBUDX PYTAETSYA |TO SDELATX), TAK KAK RAVENSTVO YAVLYAETSYA PREDELXNO MALOVEROYATNYM SOBYTIEM. nAPRIMER, ESLI ISPOLXZUETSYA REKURRENTNOE SOOTNOSHENIE \EQ{ x_{n+1}=f(x_n), } O KOTOROM TEORIYA, VZYATAYA IZ KAKOJ-TO KNIZHKI, UTVERZHDAET, CHTO $x_n$~STREMITSYA K NEKOTOROMU PREDELU PRI~$n\to\infty$, TO, KAK PRAVILO, BYLO BY OSHIBKOJ PRODOLZHATX VYCHISLENIYA, POKA DLYA NEKOTOROGO~$n$ NE OSUSHCHESTVITSYA RAVENSTVO~$x_{n+1}=x_n$, TAK KAK POSLEDOVATELXNOSTX~$x_n$ MOZHET VVIDU OKRUGLENIYA PROMEZHUTOCHNYH REZULXTATOV OKAZATXSYA PERIODICHESKOJ S BOLXSHIM PERIODOM. rAZUMNO PRODOLZHATX VYCHISLENIYA LISHX DO TEH POR, POKA DLYA NEKOTOROGO PODHODYASHCHIM OBRAZOM VYBRANNOGO~$\delta$ NE STANET SPRAVEDLIVO NERAVENSTVO~$\abs{x_{n+1}-x_n}<\delta$; NO TAK KAK MY NE ZNAEM ZARANEE PORYADKA VELICHINY~$x_n$, ESHCHE BOLEE PRAVILXNO DOZHIDATXSYA POYAVLENIYA NERAVENSTVA %%248 \EQ[23]{ \abs{x_{n+1}-x_n}\le \varepsilon\abs{x_n}; } CHISLO~$\varepsilon$ GORAZDO LEGCHE VYBRATX ZARANEE. sOOTNOSHENIE~\eqref[23]---|TO DRUGOJ SPOSOB VYRAZHENIYA TOGO FAKTA, CHTO CHISLA~$x_{n+1}$ I~$x_n$ PRIBLIZITELXNO RAVNY; I NASHE OBSUZHDENIE POKAZYVAET, CHTO PRI RASSMOTRENII VYCHISLENIJ NAD CHISLAMI S PLAVAYUSHCHEJ TOCHKOJ SOOTNOSHENIE "PRIBLIZITELXNOGO RAVENSTVA" BYLO BY BOLEE POLEZNO, CHEM TRADICIONNOE SOOTNOSHENIE RAVENSTVA, ESLI TOLXKO NAM UDASTSYA OPREDELITX PERVOE SOOTNOSHENIE NADLEZHASHCHIM OBRAZOM. dRUGIMI SLOVAMI, TOT FAKT, CHTO STROGOE RAVENSTVO VELICHIN V SISTEME S PLAVAYUSHCHEJ TOCHKOJ IGRAET OCHENX NEBOLXSHUYU ROLX, PRIVODIT K NEOBHODIMOSTI VVESTI NOVUYU OPERACIYU \emph{SRAVNENIYA VELICHIN S PLAVAYUSHCHEJ TOCHKOJ,} PREDNAZNACHENNUYU DLYA OBLEGCHENIYA OCENOK OTNOSITELXNYH ZNACHENIJ DVUH TAKIH VELICHIN. pREDSTAVLYAYUTSYA PRIGODNYMI SLEDUYUSHCHIE OPREDELENIYA DLYA CHISEL S PLAVAYUSHCHEJ TOCHKOJ~$u=(e_n, f_n)$ I~$v=(e_v, f_v)$ PO OSNOVANIYU~$b$ S IZBYTKOM~$q$: \EQ{ \eqalignno{ u &\prec v(\varepsilon) \hbox{ TOGDA I TOLXKO TOGDA, KOGDA~$v-u>\varepsilon\max(b^{e_u-q}, b^{e_v-q})$;} &(24)\cr u &\sim v(\varepsilon) \hbox{ TOGDA I TOLXKO TOGDA, KOGDA~$\abs{v-u}\le \varepsilon \max (b^{e_u-q}, b^{e_v-q})$;} &(25)\cr u &\succ v(\varepsilon) \hbox{ TOGDA I TOLXKO TOGDA, KOGDA~$u-v>\varepsilon\max(b^{e_u-q}, b^{e_v-q})$;} &(26)\cr u &\approx v(\varepsilon) \hbox{ TOGDA I TOLXKO TOGDA, KOGDA~$\abs{v-u}\le\varepsilon\min(b^{e_u-q}, b^{e_v-q})$.} &(27)\cr } } sOGLASNO |TIM OPREDELENIYAM, DLYA LYUBOJ DANNOJ PARY ZNACHENIJ~$u$,~$v$ MOZHET VYPOLNYATXSYA V TOCHNOSTI ODNO IZ SOOTNOSHENIJ~$u\prec v$ ("OPREDELENNO MENXSHE"), $u\sim v$ ("PRIBLIZITELXNO RAVNO") ILI~$u\succ v$ ("OPREDELENNO BOLXSHE"). sOOTNOSHENIE~$u\approx v$---NESKOLXKO BOLEE SILXNOE, NEZHELI~$u\sim v$, I EGO MOZHNO CHITATX TAK: "$u$ PO SUSHCHESTVU RAVNO~$v$". vSE |TI SOOTNOSHENIYA ZADAYUTSYA POSREDSTVOM POLOZHITELXNOGO CHISLA~$\varepsilon$, IZMERYAYUSHCHEGO STEPENX RASSMATRIVAEMOGO PRIBLIZHENIYA. oDNIM IZ SPOSOBOV ISTOLKOVANIYA PRIVEDENNYH OPREDELENIJ SOSTOIT V TOM, CHTOBY LYUBOMU CHISLU~$u$ S PLAVAYUSHCHEJ TOCHKOJ POSTAVITX V SOOTVETSTVIE MNOZHESTVO~$S(u)\set{x \mid \abs{x-u}\le \varepsilon b^{e_u-q}}$; MNOZHESTVO~$S(u)$ PREDSTAVLYAET SOBOJ SOVOKUPNOSTX VESHCHESTVENNYH CHISEL, RASPOLOZHENNYH VBLIZI~$u$, I OPREDELENO PRI POMOSHCHI POKAZATELYA~$u$. v TERMINAH |TIH MNOZHESTV SOOTNOSHENIE~$u\prec v$ VYPOLNYAETSYA TOGDA I TOLXKO TOGDA, KOGDA~$S(u)S(v)$ I~$S(u)>v$; $u\approx v$ TOGDA I TOLXKO TOGDA, KOGDA~$u\in S(v)$ I~$v\in S(u)$. (zDESX MY PREDPOLAGAEM, CHTO PARAMETR~$\varepsilon$, IZMERYAYUSHCHIJ STEPENX PRIBLIZHENIYA, FIKSIROVAN; V BOLEE PODROBNOJ ZAPISI MOZHNO BYLO OTRAZITX I ZAVISIMOSTX~$S(u)$ OT~$\varepsilon$.) vOT NEKOTORYE SLEDSTVIYA IZ PRIVEDENNYH OPREDELENIJ: \EQ{ \displaylinesno{ \hbox{ESLI~$u\prec v(\varepsilon)$, TO~$v\succ u(\varepsilon)$;} & (28)\cr \hbox{ESLI~$u\approx v(\varepsilon)$, TO~$u\sim v(\varepsilon)$;} & (29)\cr u\approx u(\varepsilon); & (30)\cr \hbox{ESLI~$u\prec v(\varepsilon)$, TO~$u $\sim$, sOOTNOSHENIYA~$\prec$, $\sim$, $\succ$ I~$\approx$ POLEZNY DLYA CHISLENNYH ALGORITMOV, I PO|TOMU RAZUMNA IDEYA OBESPECHENIYA evm PROGRAMMAMI SRAVNENIYA CHISEL S PLAVAYUSHCHEJ TOCHKOJ NARYADU S PROGRAMMAMI VYPOLNENIYA NAD NIMI ARIFMETICHESKIH DEJSTVIJ. %% 250 tEPERX VNOVX PEREKLYUCHIM NASHE VNIMANIE NA VOPROS O NAHOZHDENII \emph{TOCHNYH} SOOTNOSHENIJ, KOTORYM UDOVLETVORYAYUT OPERACII NAD VELICHINAMI S PLAVAYUSHCHEJ TOCHKOJ. iNTERESNO OTMETITX, CHTO SLOZHENIE I VYCHITANIE TAKIH VELICHIN NE POLNOSTXYU VYPADAYUT IZ POLYA ZRENIYA AKSIOMATIKI, TAK KAK ONI UDOVLETVORYAYUT NETRIVIALXNYM TOZHDESTVAM, FORMULIRUEMYM V TEOREMAH~A I~B. \proclaim tEOREMA~A. pUSTX~$u$ I~$v$---NORMALIZOVANNYE CHISLA S PLAVAYUSHCHEJ TOCHKOJ. tOGDA \EQ[42]{ ((u\oplus v)\ominus u)+((u\oplus v)\ominus ((u\oplus v)\ominus u))=u\oplus v, } %% PRI USLOVII CHTO => PRI USLOVII, CHTO PRI USLOVII, CHTO NE PROISHODIT PEREPOLNENIYA ILI ISCHEZNOVENIYA POKAZATELYA. \emph{zAMECHANIE.} eTO DOVOLXNO GROMOZDKOE TOZHDESTVO MOZHNO PEREPISATX V SLEDUYUSHCHEM BOLEE PROSTOM VIDE. pOLOZHIM \EQ[43]{ \twocoleqalign{ u'&=(u\oplus v)\ominus v, & v' &=(u\oplus v)\ominus u ;\cr u''&=(u\oplus v)\ominus v', & v''&=(u\oplus v)\ominus u'.\cr } } iNTUITIVNO YASNO, CHTO $u'$ I~$u''$ DOLZHNY BYTX PRIBLIZHENIYAMI K~$u$, A~$v'$ I~$v''$---PRIBLIZHENIYAMI K~$v$. tEOREMA~A UTVERZHDAET, CHTO \EQ[44]{ u\oplus v = u'+v'' = u''+v'. } eTO BOLEE SILXNOE UTVERZHDENIE, NEZHELI TOZHDESTVO \EQ[45]{ u\oplus v = u' \oplus v'' = u'' \oplus v', } YAVLYAYUSHCHEESYA ESHCHE ODNIM SLEDSTVIEM TEOREMY~A (SM.~UPR.~12). \proof vVIDU SIMMETRICHNOSTI NASHIH PREDPOLOZHENIJ DOSTATOCHNO USTANOVITX SPRAVEDLIVOSTX RAVENSTV~\eqref[44] PRI USLOVII~$u\ge\abs{v}$. v POSLEDUYUSHCHEM DOKAZATELXSTVE UDOBNO BUDET ISPOLXZOVATX SOKRASHCHENIYA \EQ[46]{ d=e_u-e_v\ge 0, \qquad w=u\oplus v \ge 0 } I RABOTATX S CELYMI CHISLAMI VMESTO DROBEJ; ESLI MALAYA LATINSKAYA BUKVA~$x$ OBOZNACHAET NORMALIZOVANNUYU VELICHINU~$(e_x, f_x)$ S PLAVAYUSHCHEJ TOCHKOJ, TO SOOTVETSTVUYUSHCHAYA ZAGLAVNAYA BUKVA~$X$ BUDET OBOZNACHATX CHISLO~$b^{p+e_x-e_v}f_x$. v CHASTNOSTI, $U=b^{p+d}f_u$, $V=b^p f_v$; |TI VELICHINY, RAVNO KAK I~$U'$, $V'$, $U''$, $V''$ I~$W$, YAVLYAYUTSYA CELYMI CHISLAMI, DESYATICHNYE TOCHKI KOTORYH VYROVNENY TAKIM OBRAZOM, CHTO~\eqref[44] |KVIVALENTNO \EQ[47]{ W=U'+V''=U''+V'. } tEPERX DOKAZATELXSTVO SVODITSYA K DOVOLXNO SKUCHNOMU RASSMOTRENIYU RYADA CHASTNYH SLUCHAEV. %% 251 \bye