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Unsigned 0 to 2n - 1 2 s Complement -2n-1 to 2n-1 - 1 BCD 0 to 10n/4 - 1 But, what about? very large numbers 9,349,398,989,787,762,244,859,087,678 very small number 0.0000000000000000000000045691 rationals 2/3 Irrationals 2.71828& ., 3.1415926& .., #ZMZZZ#bbjbjbjbjbb b &   2 H-Conceptual Overview: Finite-precision numbers*. f$$b$ . ORepresenting Real Numbers`  How to represent fractional parts to the right of the ``decimal'' point? A number like 0.12 (i.e., (1/2)10) not represented well by integers 0 or 1! Two ways to represent real numbers better: Fixed point Floating pointIN+I``h `h-`+` `   BFixed-Point Data Formatb(   Q Fixed Point `  Pros Add two reals just by adding the integers Much less logic than floating point Faster Often used in digital signal processing Cons The range of numbers is narrow number=400 000 000 000 000 000 000 000 000. 000 It is much more economical to represent as 4*1026} b`}`` ` ~ b  j  b &    3Recall Scientific Notation`  Issues: Arithmetic (+, -, *, / ) Representation, Normal form Range and Precision(Determined by?) Rounding and errors Exceptions (e.g., divide by zero, overflow, underflow) Properties~0FFb)bfb fgb  {Scientific Notation: Normalized ( 12.35 x 10^-9 ? 1.235 x 10^-8 ? scientific notation: has a single digit to the left of the decimal point Normalized scientific notation: such a single digit must be non-zero.T Numerical Form: ( 1)s M 2E Sign bit s determines whether number is negative or positive Significand M normally a fractional value in range [1.0,2.0). Exponent E weights value by power of two Encoding >Ua UU`dedlddl `d?`d9`d` `2Y   f  KIEEE 754 standardd  SThree formats: single/double/extended precision (32,64,80 bits). Single precision:TT` T y  the representation: ( 1)s (1+ Fraction) 2E-bias where E is the exponent representation in the exp field Sign bit s determines whether number is negative or positive Fraction is normally a fractional value in range between 0 and 1 A leading 1 added to the fraction is  implicit Exponent using a  biased notation For single precision  the bias is 127 UlUa'aUb fgfnffnn2&nn n bf>bfnbf( b {  z4Advantage of using the biased notation for exponents55$ Under the single-precision IEEE 754 standard: bias = 127 If the real exponent is +1, what is the biased exponent ? How about if the real exponent is -1 ?WNormalized Encoding Exampleb(  Value: Float F = 15213.010; 1521310 = 1.1101101101101 X 213 Significand M = 1.1101101101101 frac = 11011011011010000000000 (23 bits! With leading 1 hiding!) Exponent E = 1310, Bias = 12710 Exp = 14010 = 10001100 Z!Z ZjZ Z8ZZbcjcbbjbjb b b  b  b   b  b   b   b 5 b  bbbjbbbjbbbjbcbbL=        mDenormalized Values`   Condition exp = 000& 0 Significand value M = 0.xxx& x xxx& x: bits of frac Cases frac = 000& 0 Represents value 0 Note that have distinct values +0 and  0 frac 000& 0 Numbers very close to 0.0 Lose precision as get smaller  Gradual underflow  Z ZZZZ Z<Z ZLZ bbcbcbcbbbbccbc c  b  c  b  c  b bcbcbc<bcb bcbcLbt   #    E  U nSpecial Values`  Condition exp = 111& 1 (infinity) Operation that overflows Both positive and negative E.g., 1.0/0.0 = -1.0/-0.0 = +, 1.0/-0.0 = - Not-a-Number (NaN) Represents case when no numeric value can be determined E.g., sqrt( 1), -  Z Z ZcZZNZ bbcbcbcc  c  $c   bD b  c   b  c   b  c   b   b  c   b  c   c   b bHbc  @  @    xIEEE 754: Summary g$  oIEEE754: Summary (Cont.)b(   aSpecial Properties of Encoding`  FP Zero Same as Integer Zero All bits = 0 Can (Almost) Use Unsigned Integer Comparison Must first compare sign bits Must consider -0 = 0 NaNs problematic Will be greater than any other values Otherwise OK Denorm vs. normalized Normalized vs. infinity -Cab b-bC b ab@  @  ( | FP Addition b(  &Operands ( 1)s1 M1 2E1 ( 1)s2 M2 2E2 Assume E1 > E2 Exact Result: ( 1)s M 2E Exponent E: E1 Sign s, significand M: Result of signed align & add ZZZZDZ bgfnffngfnffnbbbbb b  g  f  n  f  f  n  b  bbbbbbbbbb&h   ! ~Decimal Number Conversion  b(  Convert Binary to Decimal (base 2 to base 10) x x x x x. d d d d d d & 2 / %0 Z -`#b(c (k (c (bb$ h !Decimal Number Conversion (Cont.)" "b$ " Convert Decimal Integer to binary Integer divide the decimal value by 2 and then write down the remainder from bottom to top (Divide 2 and Get the Remainders) 3710 = ?2 + P P)bbVbf*bbjbjbb  !Decimal Number Conversion (Cont.)" "b$ " Convert Decimal Fraction to Binary Fraction multiply the decimal value by 2 and then write down the integer number from top to bottom (Multiply 2 and Get the Integers) . 0.37510 = ?2 , P P7bQb!f*bbjbjb  !Decimal Number Conversion (Cont.)" "b$ " Convert Decimal Number to Binary Binary Put Together: Integer Part . 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Floating Point Representation g,    |  $cwawa1 ?Rectangle: Click to edit Master text styles Second level Third level Fourth level Fifth level ,$D 0 @ MSB is sign bit: S=0/1 exp field encodes E frac field encodes MJ0wnU0n<U)0n<Zc cccgcggccgcg2 +   4aYiH | 0޽h ? GGG___PPT10.+?kDD9' {= @B D' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*|%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*|Y%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*|Y%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*|%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*|%(D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*|%(D' =-o6Bdissolve*<3<*|DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* |%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* |,%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* |,A%(+p+0+|0 ++0+ |0 +0  0  X(  Xr X S ?     X 0n @ }<$ 0   X < o  *l (a   X(a ,$D  00N 0P`0  X 1(a  X Z\wawa8c?0P0  Ss" c   X Z|wawa8c? PP 0  Uexp" c    X Zwawa̙8c?` P`0  dfrac" c    X <`7   [ 1 bit 8 bits 23 bits   X <b v5 ,$  0 ^,Double precision: see page 192 in P&H book--H X 0޽h ? @Eff؂o___PPT10.+۔ҧD' = @B D' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*XA%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*XAD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*XAD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*XAT%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*XATD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*XATD' =%(D' =%(DG' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* X%(D' =-6B'barn(inHorizontal)*<3<* XD' =%(D' =%(DT' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* X%(D' =-6B'barn(inHorizontal)*<3<* X+p+0+X0 ++0+ X0 +   0L0  S(    S ~(wawa1 ?@ <$ 0  ,4aYi%  Z8۫ 8c 8c1 ?\ /IEEE 754 Standard Floating Point Representation 00g$ 0    $lwawa1 ?Rectangle: Click to edit Master text styles Second level Third level Fourth level Fifth levelD t*0wnUc  4aYiH  0޽h ? GGG___PPT10.+r<DN' {= @B D ' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*;%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*<%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*1%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*1T%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*T{%(+8+0+0 +  0 0Z(  r  S ?    r  S @ 0   #  < ,$  0 / Answer: -1+127 = 126! (Note 126-127 = -1!) 8/ wn/  <Tl,$D  0 Z(How about if the real exponent is -127 ?))  B|} *~ ,$  0 [+Answer: 1+127 = 128! (Note 128-127 = +1!),,H  0޽h ? @Eff؂o  ___PPT10 .`+B#style.visibility<*%(D' =-6B'barn(inHorizontal)*<3<*D' =%(D' =%(D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(D' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*)%(D' =-m6Bbox(in)*<3<*)+p+0+0 ++0+0 +=)  0    K(  x  c $0? w      c $PI@ <$ 0  >*4@PaYi  HԔ?F ,$D  0 FFloating Point Representation: Binary:01000110011011011011010000000000PG 2&a  aaa G  r  <sT|,$D  0r  <,$D 0  6@s,$  0 Oexp 2b     6< ,$ 0 ^frac 2b   H  0޽h ? GGG""___PPT10!.+yD ' {= @B D ' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*=%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*=I%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*Ih%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*h%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(++0+0 ++0+0 ++0+0 ++0+0 +'  0L0  (    S ~Vwawa1 ?@ <$ 0  r  S V?    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GGG.%&%___PPT10%.+D$' {= @B Da$' = @BA?%,( < +O%,( < +D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bdissolve*<3<* D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bdissolve*<3<* D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*6%(D' =-o6Bdissolve*<3<*6D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*6J%(D' =-o6Bdissolve*<3<*6JD' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*JP%(D' =-o6Bdissolve*<3<*JPD' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*P]%(D' =-o6Bdissolve*<3<*P]D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*]p%(D' =-o6Bdissolve*<3<*]pD' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*p%(D' =-o6Bdissolve*<3<*pD' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*+8+0+0 +  0L0  (    S ~uwawa1 ?@ DoW<$ 0  r  S dv?    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GGGF>___PPT10.+D' {= @B Dy' = @BA?%,( < +O%,( < +D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bdissolve*<3<* D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bdissolve*<3<* D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =-o6Bdissolve*<3<*$D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*$=%(D' =-o6Bdissolve*<3<*$=D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*=X%(D' =-o6Bdissolve*<3<*=XD' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =-o6Bdissolve*<3<*XD' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*+8+0+0 +$  0 ##9B s#(   r   S `?    -d H~   #" 6H~   <ȏ?< H~ qAny bit patternb   @`   <Ԛ?H< ~ k 0  6o ?&c `B ?  01 ?&c `B @  01 ? & c fB A  6o ?c&cc  B  04 ]  ^NaN: 2`  H   0޽h ? @Eff؂oy___PPT10Y+D=' {= @B +M  0 tl &'(  x  c $K?     dB  <D8c?==dB  <D8c?===dB  <D8c? jjM dB  <D8c?===jB  BD8c?= dB  <D8c? j    H 8c?c j#  _NaN c    dB   <D8c? M dB   <D8c? w dB   <D8c?G =G    H8c? AM  _NaN c    dB  <D8c? ==w   HP8c?\M| `+, aa      HXU8c?l.f f-2   c      HZ8c? 6  n-0.   a    dB  <D8c?=  Ht8c?  o+Denorm c  &     H(a8c?-\ Y +Normalized  c      Hdf8c?  o-Denorm c  &   dB  <D8c?=  Hk8c? Y -Normalized  c    jB  BD8c?= dB  <D8c?== = dB  <D8c?=dB  <D8c?=^B  6D8c? = ^B  6D8c?=   Nq8c? [  N+0 a    ^B  6D8c?m    <pv8c? M  `Negative underflow c    ^B ! 6D8c?   " <z8c? =  `Positive underflow c    ^B # 6D8c?  $ <48c? S  _Negative overflow c    ^B % 6D8c?  & <8c?   _Positive overflow c    H  0޽h ? GGGy___PPT10Y+D=' {= @B +  0L0  0(    S ~Xwawa1 ?@ W<$ 0  r  S Ћ?    H  0޽h ? GGGh`___PPT10@.+,KD' {= @B D' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<**%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<**W%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*Wt%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*t%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(+8+0+0 +)  0L0   P3 (    S ~8wawa1 ?@  <$ 0  l  Zwawa8c?  j ,$D 0 ( 1)s1 M1 j 8 Zgg  o  g    z       ,$D 0`B  08c? z `B  08c? z lB  <8c? " "   Tةwawa8c?  | E1 E2B c  c  c    z U      U ,$D 0t   Zxwawa8c?  ,$D 0 ( 1)s2 M2 j 8 Zgg  o  g       ZԸwawa8c?U$   O+ c    .z 6F   6F,$D 0`B   08c?6F<  Zhwawa8c? ( 1)s M j 8 Zgg  o  g    ~  s *X?    H  0޽h ? GGG___PPT10.+_/D<' = @B D' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*'%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*'6%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*6P%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*P`%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*`%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(+p+0+0 ++0+0 +6   0L0  )( (  ( (  `|xaxa ??    " (  fhxaxaG  ?@  <$ 0   ( 0 f ,$  0 ><1101.0112 = 1*23+1*22+0*21+1*20+0*2-1+1*2-2+1*2-3 = 13.37510= 2JJJJJJJJ J ( 0! ; ,$  0 R 24 23 22 21 20 2-1 2-2 2-3 2-4 2-5 2-6 & & G wn 2bjbjbjbjbjbjbjbjbjbjbjbG  ( 0   , 2B ( 0D/ 9 ,$@   0B  ( 0D{   ,$@  0B !( 0Dx$ ,$@  0B "( 0Dx ,$@  0B #( 0De ,$@  0B $( 0Dk u ,$@   0B %( 0D ,$@   0B &( 0D ,$@   0B '( 0D~ ,$@   0B (( 0D} ,$@  0B )( 0D ,$@  0H ( 0޽h ? 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" 8  f|.xaxaG  ?@  <$ 0   8 6/y~ %,$  0  Quotient reminder 372 = 18 & 1 182 = 9 & 0 92 = 4 & 1 42 = 2 & 0 22= 1 & 0 12 = 0 & 16bbb  B 8 6DԔ  ,$D 0 8 086 J P ,$  0 I1001012& 2 J ? 8 Bp= @,$   0 ;Can it always be converted into an accurate binary number? &<;b;  8 0B_ ,$   0 <YES! 2*H 8 0޽h ? 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PP 0  SE" c   X ZKwawa̙8c?` P`0  Zfraction"   c  B X 6DԔ ,$D  0B  X 6DԔSSM ,$D  0H X 0޽h ? U>=UU(d\___PPT10<..$+e D' {= @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(DG' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =-6B'barn(inHorizontal)*<3<*XD' =%(D' =%(DG' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* X%(D' =-6B'barn(inHorizontal)*<3<* X+ B 0  | ( (  ( ( T1 ?E   @  (C xmm ?#" 0e Ly   lWhen representing a real value, you have two choices: the fixed-point format or the floating-point format. Any way, to avoid nonlinear effects introduced by overflow, underflow, saturation, and wrapping during computation, control of each variable s range has to be done by appropriate scaling of operands. In the floating-point case, the exponent of a variable is part of the run-time representation and is computed automatically by the floating-point ALU. But in the fixed-point case, the exponent of each variable is implicit and determined by the programmer off-line using the predicted dynamic range of the variable. Therefore, the scaling of fixed-point operands has to be done explicitly by the programmer. This is generally a tedious and error-prone process. In fact, you may regard the fixed-point as a form of integer with an additional hypothetical binary point. 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'Times New RomanTahoma Wingdings 新細明體Arial Book AntiquaDotumTimes Courier New HelveticaSymbolBatang BlueprintMicrosoft Equation 3.0*Topic 3d Representation of Real NumbersRecap.Conceptual Overview: Finite-precision numbersRepresenting Real NumbersFixed-Point Data Format Fixed PointRecall Scientific Notation Scientific Notation: NormalizedSlide 9IEEE 754 standard Slide 115Advantage of using the biased notation for exponentsNormalized Encoding ExampleDenormalized ValuesSpecial ValuesIEEE 754: SummaryIEEE754: Summary (Cont.)Special Properties of Encoding FP AdditionDecimal Number Conversion "Decimal Number Conversion (Cont.)"Decimal Number Conversion (Cont.)"Decimal Number Conversion (Cont.)SummaryIEEE 754 Floating point Review  Fonts Used Design TemplateEmbedded OLE Servers Slide Titles_ж0HWUHWU  !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~    Root EntrydO)PicturesCurrent User SummaryInformation(9PowerPoint Document(DocumentSummaryInformation8