Forms of Information Representation in Digital Computers.

Transcription

1 Chapter Forms of Iformatio Represetatio i Digital Computers.. Geeral Cosideratios There are distiguished the followig two fudametal modes of iformatio represetatio: (.) a) iteral; b) exteral; Iteral represetatios refers to the methods ad meas of represetatio of iformatio i iteral devices of the (.) digital computer: gates, bistables, registers, memory, etc. Exteral represetatio refers to the methods ad meas of represetatio i peripheral ad teral uits (usually (.3) magetic / optical media). I what follows it is preseted extesively the iteral represetatio based o use of the biary digits (bits). The followig cocepts are related to a sequece of bits Byte: a group of 8 bits (.4) b b. b 7 or b 7 b 6. b Word: a group 4 Bytes Byte Byte Byte 3 Byte Byte Byte Byte 3 Byte 4 Double Word: a group of 8 Bytes Byte Byte 8 or (.6) Half Word: a group of Bytes or (.7) Byte Byte or (.5) Byte Byte 8 Byte Byte - -

2 With regard to the iteral represetatio the followig two ids are idetified: a) atural represetatio correspodig to fixed poit represetatio, desigated FXP b) ormal represetatio correspodig to floatig poit represetatio, desigated FLP (.8) sig bit bs b magitude bits b... m b. Fixed-Poit Represetatio of umbers.. Itroductory cocepts Fixed-Poit represetatio of assumes that i all umber represetatios a fixed positio of the biary (..) poit is adopted. I case of the extreme positios of the biary poit there are defied: ) Iteger umbers biary poit positioed at the right. (..) b m m ) Fractioal umbers biary poit positioed at the left. (..3) b b... b b... ( m) b b m For fractioal the biary poit has a double role: ) to separate the iteger part from the fractioal part (..3) ) to separate the sig bit from the magitude b.. Aalysis of the represetatio rages for positive fractioal Assug that the module of is: b i m i m bits The positive fractioal are represeted as follows: m b b... b m sig bit m differet combiatios m Hece, positive ca be defied i the cosidered format (, m) Extreme values: (..) (..) (..3) m m... (..4) m m... (..5) Hece, the rage of positive fractioal is: m m [,( )] (..6) -3-4

3 .3. Aalysis of the represetatio rages for egative fractioal The aalysis is doe for the three codes used for egative umber represetatio : sig-magitude, two s (.3.) complemet code, oe s complemet code. A. Sig-Magitude Code Assug that the module of is: (.3.) b i m i The egative fractioal umber is represeted as follows: Two represetatios for : Positive zero m... egative zero m... Overflow (OVF) (.3.8) m b b... b m sig bit m differet combiatios m Hece, egative ca be defied i the cosidered format (, m) (.3.3) (.3.4) The overflow coditio arises wheever a attempt of m represetig a umber whose module is greater tha occurs. m Coditio: > (.3.9) OVF m positive if > OVF - m egative if < ( ) OVF sigularity case (.3.) The extreme values: m m... (.3.5) m m... ( ) (.3.6) Hece, the rage of egative is: m m [, ( )] (.3.7) Uderflow (UF) The uderflow coditio arises wheever a attempt of m represetig a umber whose module is less tha occurs. m Coditio : < (.3.) UF positive if UF - egative if < > m m UF may ot be treated as error (.3.) -5-6

4 Represetatio o the real axis : OVF Positive Overflow ( m )... B. Two s Complemet Code The discussio is restricted to the first variat (.3.4) Geeral form for - : m b b... b m positive regio Distace -m sig bit magitude (.3.5) UF Positive Uderflow UF - egative Uderflow m (.3.3) where: i i m b i bi m m The extreme values: (two s complemet represetatio) m... m (.3.6) m... sig bit magitude egative regio m m... ( ) m ( )... OVF - egative Overflow sig bit magitude (.3.7) Hece m m [, ( )] (.3.8) -7-8

5 A sigle represetatio for : m... sig bit magitude (.3.9) The dirty, correspodig to us, that is..., is used by covetio for represetatio (.3.) m of the particular value Represetatio o real axis: positive regio OVF Positive Overflow m... ( ) Distace -m... m UF Positive Uderflow UF - egative Uderflow... clea zero... m (.3.) egative regio m... ( )... (by covetio) OVF - egative Overflow -9 -

6 C. Oe s Complemet Code The discussio is restricted to the first variat (.3.) Geeral form for - : m b b... b m sig bit magitude (.3.3) Represetatio o real axis : positive regio OVF Positive Overflow... ( Distace -m m ) where bi bi The extreme values: m... sig bit magitude m... sig bit magitude m ( m (.3.4) ) (.3.5) Positive Uderflow UF egative regio zero egative Uderflow UF -... m m (.3.8) m m Hece, [, ( )] Two represetatios for : m... (.3.6) m... (.3.7)... ( m OVF - egative Overflow ) sig bit magitude - -

7 3. Floatig poit represetatio of 3.. Geeral cosideratios FXP otatios are coveiet for represetig small with bouded orders of magitude (.) For istace, if ~ ± 3 bits, the ± ( 3 ) -3 This rage is iadequate for egieerig ad scietific applicatios (.) FLP uses a two part represetatio: FLP ( m, e) (.3) where m is the matissa ad e is the expoet of FLP represetatio. Ay umber ca be represeted i the followig form e m r (.4) where: eexpoet (iteger) mmatissa (fractio) m, e are siged fixed poit (.5) rradix (self-implied ) Example: r (.6) m.95 e For the particular case r e m (.7) Example:. (.8) m. e Mai advatages of FLP otatio - drastic elargig of the rage of represetatio (.9) - costat relative error of represetatio Mai drawbacs of FLP otatio - cost of implemetatio (.) - arithmetic algorithms more complex 3.. FLP formats ormalized FLP represetatio m, r m < r for r m, m < the most sigificat bit of the matissa is always. -4 (.) (.) Example: (.3). 3

8 ormalized matissa:. Expoet: -3 represeted as iteger FXP Both matissa ad expoet are represeted i FXP (.3) otatio. U-ormalized FLP represetatio m < r (.4) for r: m < (.5) The most sigificat bits of the matissa are zero Example:.., where m. ad e - (.6)., where m. ad e - I computer desig it is preferred ormalized FLP otatio. where the most sigificat positio of the (.7) matissa cotais a o-zero digit, which is uique for the umber. ormalizig operatio cosists i shiftig the matissa ad adjustig the expoet util ormalizatio coditio (.8) is derived. Rules for ormalizatio operatio: a) for oe left shift of matissa decremet the expoet (.9) b) for oe right shift of matissa icremet the expoet -5 Example : Give the u-ormalized FLP biary umber 4. (.) After ormalizig operatio. (.) Thus, m., e Example : Give the u-ormalized FLP biary umber. (.) After ormalizig operatio:. (.3) Thus, m., e Geeral format of FLP otatio α a a m m s ms sig sig expoet matissa exp oet -6 matissa (.4) where: α s sig bit of the expoet m s sig bit of the matissa (.5) umber of bits for expoet represetatio umber of bits for matissa represetatio a i bit of the expoet m i bit of the matissa The expoet is either ubiased or biased; the biased expoet assumes that always the expoet is positive, (.6) ot requirig the sig bit α s

9 Biased expoet: C E T, such that C where r T, the biasig costat (.7) for r, T Modified format of FLP otatio Example : r 4 4 T 8 Biased expoet i the rage: [, 5 ] C C 5 (.) ms c c m sig of bit matissa biased expoet m matissa (.8) sice T 8 it results that: C E 8 C E 8 E C C E 8 E C Example : r T 5 Biased expoet i the rage: [, 99 ] C C 99 (.9) sice T 5 it results that: C E 5 C E E C C 5 E 5 E C Thus, the actual rage of expoet is E [ 5, 49] Thus, the actual rage of expoet is E [ 8, 7] 3.3. Represetatio rages for biary FLP Stadard FLP format α s sig bit of expoet ms a a m m sig bit of matissa expoet E -8 matissa m (3.) ormalized FLP case: m < (3.) m (3.3)

10 m (3.4) E (3.5) E (3.6) E ( ) (3.7) E ( ) (3.8) Hece, E m ( ) (3.9) E ( ) m (3.) E m ( ) (3.) E m ( ) U-ormalized FLP case: It is cosidered the limit case for matissa, where: m u The other coditios remai uchaged: m E u ( ) (3.) (3.4) E ( ) The: ( ) (3.5) u ( ) (3.6) u ( ) ( ) (3.7) u ( ) ( ) (3.8) u (3.3) Observatios: By comparig the above values the followig obvious relatios are observed: (3.9) < u > u u u, or (3.) u < (3.) (3.) FLP overflow (OVF): Ay attempt to represet a umber that is greater the the greatest represetable umber is called overflow Geeral coditios: (3.3) > ( ) Positive overflow ( OVF ) > ( ) (3.4) egative overflow ( OVF ) < (3.5) ( ) FLP uderflow (UF) Ay attempt to represet a umber that is smaller the the smallest represetable umber is called uderflow (3.6) Geeral coditios: ormalized: ( ) < (3.7) u-ormalized: ( ) < Positive uderflow ( UF ) Coditios: ormalized: ( ) < (3.8) -9 -

11 u-ormalized: ( ) < (3.9) egative uderflow ( UF ) Coditios: ormalized: ( ) > (3.3) u-ormalized: ( ) > (3.3) Zero represetatio: a) U-ormalized b) ormalized (3.3) Represetatio of rages for FLP o real axis: a) ormalized OVF ( ) d e FLP distace: Assug that matissa is o bits, there are represeted two cosecutive ad, for the expoet e: e m (3.33) e ( m ) The, the distace is give by the differece: e e e d m m FLP ( ) ( ) UF UF ( ) (3.34) Thus, the distace is ot costat as i case of FXP represetatio, because it depeds o the curret value (3.34) of e. The greater is e, the greater is d ad vice-versa. OVF ( ) - -

12 b) U-ormalized OVF u ( ) d e α s sig bit of expoet ms a a m m sig bit of matissa expoet matissa (4.) umber of differet expoets: biary digits, hece E, values (4.) [ ( )] [,( )] [ ( ) ], E values E values Total: (twicely tae ) (4.3) ( ) u u UF UF ( ) u u (3.35) umber of differet matissas: o biary digits o for the case of ormalized FLP (4.4) m, values matissa m - m - [ ( )] m - m (4.5) u OVF ( ) 3.4. Total umber of expressible Stadard FLP format -3 m ( ) o rages for positive ad egative matissas: m, values (4.6) [ ] [ ( ), ] m values Total: (4.7) Coclusio: total umber of expressible FLP is (4.8) ( ) -4

13 3.5. FLP represetatio ad Real umber System The FLP ca be used to model the real umber system of mathematics, although there are importat differeces The fiite ature of the umber represetatio, which is uavoidable (5.) If a umber caot be expressed i the umber represetatio set beig used the it will be used the earest ode that ca be expressed; this operatio is ow as ROUDIG operatio Highlightig the regios: I OVF II d e The followig differeces are metioed: ) Expressible appear oly i regios II, (5.3) IV, VI of real axis ) Distace betwee two cosecutive odes e d FLP Thereby d is ot costat throughout regios II (5.4) ad VI Real umbers are forg Cotiuum, that is x y x, y Z (5.5) 3) Eve i regios II,IV,VI there exists a fiite umber of odes, thereby the DESITY is fiite (5.6) Cotrol over regios II ad VI is give by the followig rules: ) By icreasig the umber of bits at expoet it results the extesio of regios II ad VI ad, (5.7) cosequetly, the rage of represetatio is elarged ) By icreasig the umber of bits at matissa it results a icrease of the umber of odes i regios II ad (5.8) VI, improvig the accuracy of represetatio. III IV V UF UF (5.) Coclusio: to comply to the eeded accuracy it will be adopted oe of differet formats of FLP represetatio: Simple, Double, Exteded Precisio (5.9) (,,EP) Example of FLP represetatio VI Give the decimal iteger 433 Assumed FLP format: 76 (6.) Biased expoet VII OVF -5-6

14 C E 64 Solutio: Coversio ito radix represetatio 433 (6.) ) ormalized represetatio Fractioal otatio of : 9. Biased expoet E 9 C (6.3) Required represetatio:. sig Biased matissa expoet Verificatio: ( ) ( ) ) U-ormalized represetatio Fractioal otatio of : Biased expoet 6 (6.4) E 6 C (6.5) Required represetatio: sig. Biased expoet Verificatio: matissa 9 6 ( ) ( ) Power of radix FLP represetatios (6.6) ecessity of zoes II ad VI extesio Simple mathematical coectio betwee radix ad (7.) radices power of two (4,8,6) represetatios. Geeral otatios: e R m R R (7.) e ( ) R m Particular otatios: ) For R 4 4 m 4 if ormalized: (7.3) m 4 4 < ) For 3 R 3 8

15 8 m 8 if ormalized: (7.4) m 8 8 < 3) For 4 R m 6 if ormalized: (7.5) m 6 6 < Geeral rules: ) m R is writte with ad, but iterpreted i radix R (7.6) ) thus m 6 is writte with ad, but iterpreted i radix 6; a group of four cosecutive bits associated (7.7) to a hexadigit 3) Expoet e is a biary iteger, but <<< 6! (7.8) Hece, the rage of expressible is sigificatly exteded. Advatages: ) a powerful mechaism of extesio the rages of represetatio ) reductio of the probability of geeratio (7.9) u-ormalized represetatios 3) reductio of ormalizig procedure duratio Example: Radix R 6 (7.) Format (,,, ) -9 α s sig bit of expoet m s sig bit of matissa a a m m expoet where ( α, m, a, m ) {,} m 6 The m s m mm3 m4 h s i h j matissa iterpreted i R6 m5 m6 m7 m8 m 3m m m (7.) h h h 6 4 ormalizatio coditio: h - correspods to the followig cases: (7.) 5 cases 3.8. Example of FLP represetatio i radix 6 Give the decimal iteger: 43 It is required FLP represetatio i radix 6 4 R 6 assumed FLP format: 7 6 (8.) Biased expoet C E 64 Solutios: ) Coversio of ito radix ad radix 6: 43 B (8.) 6-3 h 4 (7.)

16 ) Fractioal represetatios for ormalized FLP represetatio:. (8.3) ) Expoet: E 3, C (8.4) 9) Verificatio: 6964 ( B 6 ) ( 6 6 ) (8.) 4) FLP represetatio sig biased h h h 3 h 4 bit expoet ) Verificatio: ( B 4 ) ( 6 6 ) (8.5) (8.6) 6) Fractioal represetatio for u-ormalized FLP represetatio:. (8.7) ) Expoet: E 5, C (8.8) 8) FLP represetatio sig biased h h h 3 h 4 bit expoet (8.9) 3.9. Comparative aalysis of represetatio errors Maximum absolute error: ) Defiitio d i s t a ce Δ (9.) ) FXP case: FXP Δ 3) FLP case: Δ Relative error: ) Defiitio e FLP e Δ -3 (9.) (9.3) δ (9.4) A ) FXP case: FXP δ (9.5) A FXP δ (9.6) A FXP δ (9.7) A

17 3) Coclusio: a strog depedecy o the value of the umber, thereby the relative error i FXP case is ot (9.8) costat over the rage of represetatio. 4) FLP case: FLP e FLP Δ δ e A m m (9.9) FXP δ m (9.) FXP δ m (9.) 4) Coclusio: the relative error i ormalized FLP case is costat over the etire rage of (9.) represetatio. Graphical represetatio: δ 3.. Examples of FLP represetatios i differet computer families Felix C family Oly ormalized FLP ( m < ) Simple precisio ad Double precisio Biary ad hexa iterpretatios Simple precisio () requires 4 Bytes (3 bits): : Byte Byte Byte 3 Byte 4 3 bits (.) (9.3) FXP FLP format: 7 4 S C m (.) FLP where: S sig bit C biased expoet m matissa satisfyig the coditio: A Commets o δ (how to improve the FXP relative errors i case of small scalig factors). (9.4) m <

18 Double precisio () requires 8 Bytes (64 bits): : Byte Byte8 64 format: (.3) 7 56 S C m where: S sig bit C biased expoet m matissa satisfyig the coditio: m < Rages for matissa: For : matissa represeted o 4 bits, where m - is obligatory Thus, X X m m X X m m For : matissa represeted o 56 bits, where m - is obligatory Thus, (.6) (.7) For both cases the biased expoet C o 7 bits is represeted i excess 64: CE64 Rages for C ad E: C [,7] E C64 E C E C Thus, E [ 64,63] (.4) (.5) The rages of represetable : : ( ) ( ) (.8)

19 : ( ) ( ) (.9) Represetatio o the real axis: : OVF ( ) Example: Fractioal otatio of 9. Biased expoet E 9 C I represetatio matissa cosists of 4 bits Required FLP represetatio: sig Biased expoet matissa (.) Positive represetable UF UF egative represetable d 4 e (.) OVF ( )

20 : Positive represetable OVF ( ) d e Coclusio: the rages are quite idetical, but the umber of represetable is much greater i case ( d < d) Geeral represetatio i base 6: e 4 e 6 m ( ) m For ormalised case: m < 6 Matissa is iterpreted i base 6, as follows: For : (.3) (.4) UF (.5) UF (.) h h h The extreme values are: m 6 6 m 6 h h h (.6) egative represetable ( ) OVF For : h - h h (.7) The extreme values are: m 6 4 m 6 (.8) -39-4

21 Biased expoet remais the same: E [ 64, 63] Rages of represetatio: : 6 ( 6 ) ( 6 ) 6 (.9) (.) Represetatio o the real axis: : Positive represetable OVF 6 ( 6 ) 6 d 6 e : ( 6 ) ( 6 ) (.) UF (.) UF egative represetable OVF 6 ( 6 )

22 : Positive represetable OVF 6 ( 6 ) 6 d e Coclusios: ) the umber of expressible is idetical with that correspodig to iterpretatio i base, but the rages were expaded, havig i view that : e e 6 >> ) the accuracy is diished sice the distace betwee odes is greater : : d e4 e6 4e4 6 6 d d >> (.4) (.5) (.6) UF (.3) : d e56 e4 4e d d >> (.7) UF egative represetable 6 ( 6 ) OVF

23 3... P- Family Oly ormalized FLP ( m < ) Simple precisio ad Double precisio represetatios (.) Double precisio ( ) represetatio o 8 Bytes (64 bits): : Byte Byte8 Oly biary iterpretatio Simple precisio ( ) represetatio o 4 Bytes (3 bits): Byte Byte Byte 3 Byte 4 64 format: S (.3.) format: S sig bit biased expoet matissa (..) sig bit biased expoet For both cases the biased expoet C o 8 bits is represeted i excess 8 : C E 8 Rages for C ad E : C,55 [ ] E C 8 E E C C matissa (.4) (.5) Thus, E [8,7] Rages for matissa : For case matissa is represeted o 3 bits: - -3 matissa (.6)

24 But, because oly ormalized matissas are allowed, the first bit, m -, is always, so that it could be sipped, beig assumed. The, with 3 bits there are represeted matissas o 4 bits m assumed 3 (.7) (.8) m assumed 55 (.) m m assumed - 3 (.9) m m assumed 55 m 56 (.3) m 4 The rages of represetable : For case matissa is represeted o 55 bits: matissa 55 Similarly, the most sigificat bit is always,so that it is ot represeted aymore, beig assumed. Therefore, with 55 bits there are represeted matissas o 56 bits. (.) (.) : : ( ) ( ) ( ) ( ) (.4) (.5)

25 Represetatio o the real axis : : Positive represetable UF UF egative represetable OVF ( ) OVF d e ( ) (.6) : Positive represetable UF UF egative represetable OVF OVF 7 56 ( ) d 56 e ( ) (.7) -49-5

26 3.. IEEE Floatig Poit Stadard Short History Util about 98 each computer maufacturer had its ow FLP format.some of them did arithmetic operatios icorrectly ad there was o data portability. IEEE set up a committee to stadardize FLP arithmetic. Two mai advatages by stadardizatio: a) data ca be exchaged betwee differet computers b) hardware desigers have a uique set of specificatios whe desigig CPUs I 985 it was adopted IEEE stadard 754 for FLP represetatios. All big compaies producig microprocessors complied to this stadard ( Itel, Motorola, MIPS, ARC etc ) The stadard ecompasses the advaced experiece of may well-ow CPU/micro maufacturers. The stadard was developed to facilitate the portability of programs from oe processor to aother processor. 3.. Basic FLP Formats Three FLP formats : ) Sigle Precisio ( ) o 3 bits. ) Double Precisio ( ) o 64 bits. 3) Exteded Precisio ( EP ) o 8 bits. Exteded precisio is used primarily iside FLP ALUs. Its role is to reduce errors i case of the roudig operatios. Also, EP lesses the chace of a itermediate overflow to abort computatios. This format is ot available to the users. EP icludes additioal bits both at expoet ad matissa fields. EP used strictly for itermediate calculatios, because this format lesses the chace of a fial result to be cotaated by excessive roud off error. Sice EP is of iterest oly for desigers of CPUs, i what follows, it will ot be discussed. The focus of presetatio will be the Sigle Precisio () -5 (..) (..) (..) ad Double Precisio () formats. Sigle precisio format : S Biased Expoet Matissa 8 3 format is o 3 bits with a 8 bit segmet for the biased expoet. Double precisio format : S Biased Expoet Matissa format is o 64 bits with a bit segmet for the biased expoet. S sig bit of the represeted umber S for > S for < (..5) Biased expoets are differet for ad : ) I expoet uses excess 7 represetatio (..6) C E 7 ) I expoet uses excess 3 represetatio (..7) C E (..3) (..4)

27 Rage of expoets : ) I : C E 7 E C 7 (..8) C [,55], the rage for 8 bit The extreme values for C : C (..9) C 55 The extreme values for E E C (..) E C E 7, 8 Thus, [ ] ) I : C E 3 E C 3 (..) [,47] C, the rage of bit The extreme values for C : C C 47 The extreme values for E : E C E C Thus, E [ 3, 4] Matissa is always iterpreted i base ( ever a power of base) Oe of the traditioal problems with FLP is how to deal with UF,OVF or uiitialized. IEEE 754 stadard deals with these problems explicitly by addig to the traditioal ormalized other four umerical types -53 (..) (..3) (..4) (..5) Therefore the followig umerical types were desiged: a) ormalized b) Deormalized c) Zero d) Ifiity e) ot a umber ( a ) 3..3 ormalized umbers -54 (..6) The extreme values of the biased expoet are ot allowed for ormalized, beig assiged to other FLP represetatios. Therefore, the allowed biased expoet values, desigated C, are i the rages: C C (.3.) (,55) C C C (,47) C Accordig to defiitio, a ormalized umber fractio begis with a biary poit, followed by a, ad the rest of the fractio. As i case of P- computers, the leadig i the (.3.) fractio does ot have to be stored, its presece is assumed Stadard IEEE 754 uses the priciple of a implicit.

28 Sigificad is a cocept used istead of the words fractio or matissa; it is desigated S ad it is composed of: a)the implied leadig, but placed i the iteger positio. (.3.3) b)the biary poit c)3 () or 5 () arbitrary bits represetig the matissa (fractio) S. xx... x i (.3.4) 3 S. xx... x i (.3.5) 5 Coclusio: all ormalized have a sigificad S i the rage (.3.6) S < Detailed calculatio of the rage of sigificads ) for : S... (.3.7) ( ) S (.3.8) Hece, S (.3.9) S < (.3.) Rages of ormalized umber represetatios: ) for : C (,55), or < C <55, or C 54 (.3.5) E (-7,8), or -7< E <8, or -6 E 7 S S The, -3 - (- ) (- ) Represetatio o the real axis: 8 (.3.6) (.3.7) (.3.8) ) for : S... (.3.) ( ) 5 5 Hece, S (.3.) S (.3.3) 6 6 (.3.9) S < (.3.4)

29 ) for : C (,47), or < C < 47, or C 46 (.3.) E (-3,4), or -3 < E < 4,or - E 3 (.3.) S S The, 5 (- ) (- ) - Represetatio o the real axis: 4 (.3.) (.3.3) Examples of ormalized umber represetatios. Example r. : It is give a decimal fractio.5 ad it is required its FLP represetatio.. - (.) E - C E7-76 C S... 3 > Sig FLP represetatio: 3 bits (.4.) 3 F (.3.4) Thus,.5 3Fh

30 Example r. : Example r.3 : It is give the decimal iteger ad it is required its FLP represetatio. It is give the decimal umber.5 ad it is required its FLP represetatio. (.) E C E777 C S... 3 > Sig. (.) E C E777 C S... 3 > Sig FLP represetatio: 3 bits (.4.) FLP represetatio: 3 bits (.4.3) F 8 3 F C Thus, 3F8h Thus,.5 3FCh Deormalized I geeral case of FLP represetatios there were defied - - the extreme regios OVF, UF. (.5.) -59-6

31 Deormalized were itroduced to solve the case - of UF, that is whe the result of a calculatio has a magitude smaller the the smallest ormalized FLP umber that ca be represeted: UF : <, < UF : >, > - Previously, such cases were solved either by taig the result ad cotiuig calculatios, or causig a FLP uderflow trap. Sice either of these is satisfactory, IEEE Stadard 754 D iveted deormalized ( ) (.5.) (.5.3) (.5.4) Rages of deormalized : ) for : ( ) D D D D ( ) D D D D 3 5 ) for : ( ) ( ) (.5.8) (.5.9) Rules of represetatio: D a) has the biased expoet, D D D C E 7, E 3 b) Sigificad is replaced by the traditioal matissa (fractio) of 3 or 5 bits, so that the implied iteger vaishes. 3 m.... m The rages of matissas: ) for : m m < m < ) for : m - (.5.5) (.5.6) (.5.7) Represetatio o the real axis: ) for : Positive deormalized 5 egative deormalized 5 D D 7 D 5 D 5 7 ( ( 3 ) 3 ) (.5.) -6-6

32 ) for : Positive deormalized 75 egative deormalize d 75 D D 3 D 75 D 75 3 ( 5 ( ) 5 ) (.5.) Distace D ( 7 7 ) 5 ( 5 5 Distace i deormalized regio: D 7 m d D 7 D -7-5 ) for : ( m m D ) ( m 7 3 m 3 ) ) 7 m (.5.3) (.5.4) Cotact regio betwee ormalized ad Deormalized : ) for : Positive ormalized Positive deormalized D D 6 D -63 d 7 5 ( 3 ) (.5.) Positive ormalized D S Positive deormalized Distace D ( 3 ( ) 75 D d 5 75 ) 3 75 ( 5 ) (.5.5) (.5.6)

33 Distace i deormalized regio: D 3 m D 3 5 ( m ) D D ( m ) m d m m (.5.7) b) egative zero: ) for ) for (.6.3) (.6.4) Represetatio of umber There are defied two zeroes: ad -. The cofiguratios: a) Positive zero: 8 3 ) for.. (.6.) 3 Rules of represetatio of ± : C m Sig - Sig (.6.5) ) for (.6.)

34 3..7. Represetatio of ifiity The cofiguratios: It was defied a positive overflow OVF whe > (.7.) It was defied a egative overflow OVF - whe < ) : I geeral, emergece of OVF or OVF - costitutes a sigularity, which is sigalled by geeratig a error sigal. (.7.) I IEEE FLP stadard it was adopted a more flexible solutio: OVF is defied as OVF - is defied as (.7.3) (.7.6) O ± there are defied valid operatios: 5 ( ) 5 ( ) 5 ( ) ( ) (.7.4) 5 ( ) ( ) ( ) ( ) 5: ( ± ) ) : (.7.7) Rules for represetatio of ± : Biased Expoet C Matissa field (which is ot allowed for ormalized ) (.7.5) Sig Sig

35 3..8. ot a umber (a) represetatio. a represets situatios whe results of arithmetic operatios are ot defied. (.8.) a is a symbolic etity ecoded i FLP format of which there are two types: (.8.) a) sigallig b) quiet The sigallig a shows a ivalid operatio exceptio, wheever it appears as operad. (.8.3) The quiet a propagates through almost ay arithmetic operatios without sigalig a exceptio. (.8.4) Table with quite a: Operatio Example ( ) ( ) ( ) ( ) Additio/Subtractio ( ) ( ) ( ) ( ) Multiplicatio ( ± ) Divisio, Square Root x, where x < Thus, a ca be used i FLP format as a operad with predictable results. (.8.6) Rules for represetatio of a i IEEE 754 stadard: Biased Expoet: C Matissa: ay o zero bit patter (.8.5) (.8.7) Cofiguratio ±.. Ay o zero patter 8/ 3/5 3/64 Observatio: the matissa value ca be used to distiguish betwee a quiet a ad a sigalig a; also it ca specify particular exceptio coditios Recapitulatio of all IEEE FLP stadard umerical types ) ormalized : ± <EXP<Max Ay patter ) Deormalized : ± Ay ozero patter 3) Zero: ± 4) Ifiity: ± Max 5) ot a umber (a) ± Max Ay ozero patter 8/ 3/5 3/64 (.8.8) (.8.9) (.9.) -69-7

36 3.. IEEE FLP stadard represetatio o real axis A) Simple Precisio (): ) egative : 8 3 ) Positive : Positive ormalized ( ) egative deormalized D ) D ( ) egative ormalized Positive deormalized D ( ) ( ) D (..)

37 B) Double Precisio (): ) Positive umbers: Positive ormalized ( ) ) Positive deormalized D ( ) D (..3) (..4)

38 3.4 Represetatio of alphaumeric iformatio 3.4. Itroductio So far, oly represetatio of was (4..) cosidered The iformatio maipulated i a digital computer is wider tha umerical values: alphabetic characters (A,B,,Z), arithmetic operatio symbols (,,, ), (4..) puctuatio symbols (, ;.! ), memoics for differet commads ( DEL, ESC, ACK etc) a.s.o. Thus, a exteded code is required to ecompass all these characters forg the alphaumeric character (4..3) set of a digital computer, ow also as the ALPHABET of a digital computer. Ayhow these alphaumeric characters are biary ecoded iside a digital computer (4..4) Such a alphabet must comprise biary codes o bits for represetatio of decimal digits, 6 letters of the usual alphabet ( upper case ad lower case letters), (4..5) the puctuatio symbols, arithmetic operatio symbols, special character symbols, etc. If differet systems of codig are used the it must be provided a complex codig/decodig process whe (4..6) two digital computers must commuicate. Typically, such a alphaumeric set would cotai over characters. (4..7) The ey problem is how to stadardize represetatio Coditios for the Alphaumeric Character Code. The legth of the code must esure represetatio of the etire set of characters, symbols, commads. With bits there are ecoded characters; vice versa, if cosiderig characters, the the biary ecodig would require >[log ] bits. -75 (4..) I the begiig era, for such purposes, it was used Baudot code cotaiig oly 5 bits, which is completely isufficiet owadays ( 5 3 characters); this code was specific for older typewriters, allowig pritig oly uppercase letters. At preset there were adopted 8 bit alphaumeric codes, esurig represetatio of up to 56 characters.. Coveiet correlatio to the dimesio of the Addressable Iformatio Uit ( AIU). Dimesio of AIU ad memory orgaizatio. Ideally, a alphaumeric code would occupy a locatio i the mai memory, therefore dimesio of the code would comply to the dimesio of AIU. At preset, a 8 bit combiatio represets a byte, which is UIA or a divisio of UIA. Therefore, a recommeded legth for the alphaumeric code for most computers is 8. If shorter, the a iefficiet mode of memory utilizatio occurs, by wastig memory cells. 3. Coveiet correlatio to the decimal represetatio. The BCD codes are usually o 4 bits Ideally, it would be to select a 8 bit alphaumeric code to ecompass two BCD combiatios. Hece, i a memory locatio with 8 bits there ca be placed a AIU, a alphaumeric character or exactly two BCD combiatios (two decimal digits). 4. Icorporatig some umerical facilities to esure realizatio of some specific alphaumeric operatios through existig umeric operatios. -76 (4..) (4..3) (4..4) (4..5) (4..6) (4..7) (4..8) (4..9) (4..) (4..)

39 The weightig priciple of codig the alphaumeric characters, that is for a successio of related characters there is used a ascedig set of biary combiatios. For istace, assug 8, for the successio of alphabetic upper case letters, it is used the (4..) followig sequece of codes: A 4h B 4h C 43h This coditio is very useful for defiig particular maipulatios of alphaumeric iformatio, lie, for istace, comparisos (4..3) betwee words for creatig lists o ivetory, employees, payrolls, etc. Fial Coclusio: it resulted ecessity for a 8 bit legth alphaumeric code icorporatig weightig (4..4) priciple ASCII-8 ad EBCDIC codes ASCII alphaumeric code ASCII ( America Stadard Code for Iformatio Iterchage ) -the most commoly used code i digital computers ASCII is a 7 bit code used i telecommuicatios, but it was exteded to 8 bits to be adapted for digital computers alphaumeric iformatio represetatio. The ASCII code o 7 bits is give below: (4.3..) (4.3..) This code allows codig of uppercase ad lowercase letters, decimal digits, puctuatio symbols, mathematical symbols, cotrol characters for peripheral devices etc. To chage to a 8 bit code it was appeded the eighth bit i the most sigificat positio, b 7. Usually this extra bit is a parity bit for error detectio mechaisms; sometimes it is adopted a. th Alteratively, use of the 8 bit is defiig a alterate character set; with ASCII-7 there are coded 8 alphaumeric characters whereas with a exteded ASCII there ca be ecoded 56 alpha characters, by icludig also ew graphic characters. (4.3..4) (4.3..5) (4.3..6) (4.3..7)

40 The weightig priciple is respected: A a B b C c etc. etc. etc. Each ASCII combiatio is represeted i a shorthad otatio by two hexa digits (h s h i ): bbbb bbbb 3 For example: A 4h B 4h C 43h etc. hs hi (4.3..8) (4.3..9) The EBCDIC character code show i hexadecimal is preseted below: (4.3..4) EBCDIC alphaumeric code. EBCDIC (Exteded Biary Coded Decimal Iterchage Code) EBCDIC was developed by IBM, used o IBM maiframes (also i Romaia computers i family FELIX C). EBCDIC from the begiig was a 8 bit code allowig represetatio of 56 alphaumeric characters. (4.3..) (4.3..) (4.3..3) Most of computers would accept alphaumerical data i either code (ASCII or EBCDIC) ad perform coversio to the ative code. The coversio to ASCII/EBCDIC ad bac is accomplished i I/O uits. Whe a user types i a series of characters at a eyboard by meas of a ecodig chip that exist iside the eyboard iterface, each character is traslated ito its ASCII or EBCDIC equivalet o 8 bits, followed by trasmissio of the correspodig byte to CPU. (4.3..5) (4.3..6) (4.3..7) -79-8

41 The output that a CPU seds bac to the display is also a ASCII or EBCDIC code series (alpha iformatio). These codes are traslated by peripheral uit iterfaces ito a uderstadable iformatio by the user Alphaumeric iformatio (4.3..8) I the versio. of the Uicode stadard there are differet coded characters coverig the pricipal writte laguages of all cotiets. The first 56 combiatios of Uicode, show i hexadecimal are give i what follows: (4.5.4) Alphaumeric word series of alphaumeric characters. The mai characteristic of alphaumeric iformatio represetatio is its variable legth. There are upper limits specific for differet computer families (for example, 56 characters). Example: House will be represeted i ASCII as follows: F " H O U S E b/ " (4.4.) (4.4.) (4.4.3) (4.4.4) (4.5.5) 4.5. Uicode character set The ASCII ad EBCDIC codes support the historically doat (Lati) character sets i computers. But there are may character sets i the world, therefore a ew uiversal character stadard was developed that supports a great variety of the world s character sets, called Uicode. This is a 6 bit codig system ad represets a evolvig stadard. It chages as ew character sets are itroduced ito it ad as existig characters sets evolve ad their represetatio are redefied. Each Uicode biary code is represeted by a patter of 4 hexadecimal digits. (4.5.) (4.5.) (4.5.3) I defiig the Uicode there was provided a oe to oe correspodece betwee Uicode 6 bit patter ad ASCII 8 bit patters, amely betwee Uicode combiatios h up to 7Fh ad ASCII combiatios from h up to 7Fh. The 6 bit Uicode stadard is a subset of the 3 bit ISO 646 Uiversal Character Set (UCS-4). (4.5.6) (4.5.7) -8-8

42 5. Decimal iformatio represetatio The moder digital computer ca operate o decimal iformatio as well; such iformatio must be coded iside a digital computer through BCD codes. It is assumed that the mathematical operatios are implemeted with BCD arithmetic i decimal arithmetical uits. The AIU is a Byte o 8 bits. Matchig of BCD codes to the dimesio of a AIU is realized through two formats: compact ad zoed. Compact format: two BCD codes ecapsulated i a Byte, as follows: BYTE BYTE BCD BCD BCD BCD (5.) (5.) (5.3) (5.4) (5.5) Zoed format assumes each Byte cotais a group of 4 bits called zoe, followed by the BCD code:. Zoe BCD Zoe BCD. Byte Byte Zoe has differet assiged codes depedig o the alphaumeric code (ASCII, EBCDIC etc.) I EBCDIC zoe I ASCII zoe Zoed formats are used for Iput / Output operatios, whereas compact formats are used i processig uits. Automatic coversios form zoed to compact formats ad vice-versa are desig. (5.9) (5.) (5.) (5.) BCD used is the basic 84 code. Example: 754 i two cosecutive Bytes: (5.6) (5.7) Byte Byte The allowed legth L, of the decimal umber, depeds o the computer family; for istace, it could be 3 digits sig: d 3 d 9 d 8 d 7. d Byte Byte Byte 6 L group of 4 bits to ecode the sig (5.8)