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Exponent E weights value by power of two Encoding MSB is sign bit exp field encodes E (note: encode != is) frac field encodes M N     ,_ FhFloating point precisionsEncoding Sign bit; exp (encodes E): k-bit; frac (encodes M): n-bit Sizes Single precision: k = 8 exp bits, n= 23 frac bits 32 bits total Double precision: k = 11 exp bits, n = 52 frac bits 64 bits total Extended precision: k = 15 exp bits, n = 63 frac bits Only found in Intel-compatible machines Stored in 80 bits 1 bit wasted Value encoded  three different cases, depending on value of expT Z;ZZ2ZZ4ZZ6Z:Z ZAZ  2    46: =      P-B>@GiNormalized numeric values Condition exp 000& 0 and exp 111& 1 Exponent coded as biased value E = Exp  Bias Exp : unsigned value denoted by exp Bias : Bias value Single precision: 127 (Exp: 1& 254, E: -126& 127) Double precision: 1023 (Exp: 1& 2046, E: -1022& 1023) in general: Bias = 2k-1 - 1, where k is number of exponent bits Significand coded with implied leading 1 M = 1.xxx& x2 (1+f & f = 0.xxx2) xxx& x: bits of frac Minimum when 000& 0 (M = 1.0) Maximum when 111& 1 (M = 2.0  ) Get extra leading bit for  free  7)#t    !x ))       ;  $"$#((B2 Q<#HjNormalized encoding exampleValue Float F = 15213.0; 1521310 = 111011011011012 = 1.11011011011012 X 213 Significand M = 1.11011011011012 frac = 11011011011010000000000 Exponent E = 13 Bias = 127 exp = 140 =100011002 &I 4 +           ' ,O QIkDenormalized values >Condition exp = 000& 0 Value Exponent value E = 1 - Bias Note: not simply E=  Bias Significand value M = 0.xxx& x2 (0.f) xxx& x: bits of frac Cases exp = 000& 0, frac = 000& 0 Represents value 0 Note that have distinct values +0 and  0 exp = 000& 0, frac 000& 0 Numbers very close to 0.0` &<        <     $$PU *R#JlSpecial valuesnCondition exp = 111& 1 Cases exp = 111& 1, frac = 000& 0 Represents value "(infinity) Operation that overflows Both positive and negative E.g., 1.0/0.0 = -1.0/-0.0 = +", 1.0/-0.0 = -" exp = 111& 1, frac 000& 0 Not-a-Number (NaN) Represents case when no numeric value can be determined E.g., sqrt(-1), - ("-") d   "\" " $$$$$$$$^$$$$( (>+SKm#Summary of FP real number encodingsLnTiny floating point example8-bit Floating Point Representation the sign bit is in the most significant bit. the next four (k) bits are the exponent, with a bias of 7 (2k-1-1) the last three (n) bits are the frac Same General Form as IEEE Format normalized, denormalized representation of 0, NaN, infinityp$!<$i )!<  >.  MoValues related to the exponentNp Dynamic range OqDistribution of values6-bit IEEE-like format e = 3 exponent bits f = 2 fraction bits Bias is 3 Notice how the distribution gets denser toward zero. J3634Pr&Distribution of values (close-up view)6-bit IEEE-like format e = 3 exponent bits f = 2 fraction bits Bias is 3 Note: Smooth transition between normalized and de-normalized numbers due to definition E = 1 - Bias for denormalized values J22}  QsInteresting numbersDescription exp frac Numeric Value Zero 00& 00 00& 00 0.0 Smallest Pos. Denorm. 00& 00 00& 01 2 {23,52} X 2 {126,1022} Single ~ 1.4 X 10 45 Double ~ 4.9 X 10 324 Largest Denormalized 00& 00 11& 11 (1.0  ) X 2 {126,1022} Single ~ 1.18 X 10 38 Double ~ 2.2 X 10 308 Smallest Pos. Normalized 00& 01 00& 00 1.0 X 2 {126,1022} Just larger than largest denormalized One 01& 11 00& 00 1.0 Largest Normalized 11& 10 11& 11 (2.0  ) X 2 {127,1023} Single ~ 3.4 X 1038 Double ~ 1.8 X 10308z+;,9&N)%;    ("    ,  b "     \  =:RtFloating point operationsConceptual view First compute exact result Make it fit into desired precision Possibly overflow if exponent too large Possibly round to fit into frac Rounding modes (illustrate with $ rounding) $1.40 $1.60 $1.50 $2.50  $1.50 Zero $1 $1 $1 $2  $1 Round down (-") $1 $1 $1 $2  $2 Round up (+") $2 $2 $2 $3  $1 Nearest Even (default) $1 $2 $2 $2  $2>H,#}>H, #$" " :$$SuCloser look at round-to-evenDefault rounding mode All others are statistically biased Sum of set of positive numbers will consistently be over- or under- estimated Applying to other decimal places / bit positions When exactly halfway between two possible values Round so that least significant digit is even E.g., round to nearest hundredth 1.2349999 1.23 (Less than half way) 1.2350001 1.24 (Greater than half way) 1.2350000 1.24 (Half way round up) 1.2450000 1.24 (Half way round down)$N11.!$N1 1.!TvRounding binary numbersrBinary fractional numbers  Even when least significant bit is 0 Half way when bits to right of rounding position = 100& 2 Examples Round to nearest 1/4 (2 bits right of binary point) Value Binary Rounded Action Rounded Value 2 3/32 10.000112 10.002 (<1/2 down) 2 2 3/16 10.001102 10.012 (>1/2 up) 2 1/4 2 7/8 10.111002 11.002 (1/2 up) 3 2 5/8 10.101002 10.102 (1/2 down) 2 1/2 l` 4^  4  E      )      +      &           UwFP multiplication@Operands ( 1)s1 M1 2E1 * ( 1)s2 M2 2E2 Exact result ( 1)s M 2E Sign s: s1 ^s2 Significand M: M1 *M2 Exponent E: E1 +E2 Fixing If M e" 2, shift M right, increment E If E out of range, overflow Round M to fit frac precision Implementation Biggest chore is multiplying significandsr  Ja*              >  a*>S {7 Vx FP addition\Operands ( 1)s1 M1 2E1 ( 1)s2 M2 2E2 Assume E1 > E2 Exact Result ( 1)s M 2E Sign s, significand M: Result of signed align & add Exponent E: E1 Fixing If M e" 2, shift M right, increment E if M < 1, shift M left k positions, decrement E by k Overflow if E out of range Round M to fit frac precision - $ - $    ,W  Wy!Mathematical properties of FP addCompare to those of Abelian Group Closed under addition? YES But may generate infinity or NaN Commutative? YES Associative? NO Overflow and inexactness of rounding (3.14+1e10)-1e10=0 (rounding) 3.14+(1e10-1e10)=3.14 0 is additive identity? YES Every element has additive inverse ALMOST Except for infinities & NaNs Monotonicity a e" b a+c e" b+c? ALMOST Except for NaNs "!!%4F "!!%  4F      $$tA Xz%Math. properties of FP multiplication&Compare to commutative ring Closed under multiplication? YES But may generate infinity or NaN Multiplication Commutative? YES Multiplication is Associative? NO Possibility of overflow, inexactness of rounding 1 is multiplicative identity? YES Multiplication distributes over addition? NO Possibility of overflow, inexactness of rounding Monotonicity a e" b & c e" 0 a *c e" b *c? ALMOST Except for NaNs!!B1O1 &!!B1O  1 >Z 2Y{Floating point in C<C guarantees two levels float single precision double double precision Conversions int ! float : maybe rounded int ! double : exact value preserved (double has greater range and higher precision) float ! double : exact value preserved (double has greater range and higher precision) double ! float : may overflow or be rounded double ! int : truncated toward zero (-1.999 ! -1) float ! int : truncated toward zero/ L/   "", " $$("(PS0@bFloating point puzzlessFor each of the following C expressions, either: Argue that it is true for all argument values Explain why not true*1C1CZ|SummaryIEEE Floating point has clear mathematical properties Represents numbers of form M X 2E Not the same as real arithmetic Violates associativity/distributivity Makes life difficult for compilers & serious numerical applications programmers:7Bv7BvQ/[\]^_`abc d e f g hijklmnopqrstuO   0` .T3f` T3f3f` 999MMM` lff3f3޲` eoHff33Ҷ` ff!` T3f3fffq` T3f3fff[>?" dd@ ?nPd@ d " @ ` n?" dd@   @@``PR   @ ` `$p>>  C; ,(  , , s *ʼn "`0  T Click to edit Master title style! ! , c $ȉ "@0  RClick to edit Master text styles Second level Third level Fourth level Fifth level!     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GGGy___PPT10Y+D='  = @B +  0 @%,( =(  ( ~ ( s *,`0   ~ ( s *d,@0  W L P* ( #   9  (  B8c?P Xbi:0   (  BXŤ8c?P hbi 1D0    (  Bhˤ8c?Pz Tb260   (  BȤ8c?zPj  Tb160    (  Bdդ8c?j PZ  Tb060    (  BӤ8c? Pz  \b 1:0     (  Bߤ8c?z Pj  Nb 2,0    (  Bݤ8c?j PZ  Nb 3,0    (  B8c?:P* hb jF0     (  B 8c?Z P: D " " " 0  (  B8c?P D " " " 0  (  B8c?Z P  G.*0  ( H8c?9 F   =1 0  ( H\E8c?yF  `  =2 0  ( HE8c?F   =4 0  ( HxE8c?IF } 0 v2i 1R0     ( HE8c?yF  ` `2iB0    (  BC`DE F8c?`@ P )  (  BC`DE F8c?`@ F )  (  BC`DE F8c?`@Y< )  (  BC`DE F8c?`@F )  (  BC`DE F8c?`@ F )  ( BpE8c?i  D " " " 0  (  LBC`DE F8c?`@ E  (  LBC`DE F8c?`@ O 8  (  LBC`DE F8c?`@ Y  (  LBC`DE F8c?`@ O x  ( HLxE8c?  (  D " " " 0  !( H0E8c?  B  A1/2"0  "( H!E8c?  F  A1/4"0  #( H%E8c?  F  A1/8"0  $( H)E8c?K R  f2 jD0   ` %(  c $A ??`( H ( 0޽h ? GGGy___PPT10Y+D='  = @B +  0 P , ~(  , r , S ,`0   r , S w,@0    ,  fAE1?"`V`pP,$D 0 >????0(2  ,  fFE1?"`F`p@,$D 0 >????0(2  ,  fIE1?"`6`p0,$D 0 >????0(2H , 0޽h ? GGG___PPT10+aWD'  = @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<* , %(++0+ , 0 ++0+ , 0 ++0+ , 0 +m  0 `0 $(  0 r 0 S xbE,`0   r 0 S   =s 0  P Be8c?@ >  Kexp,0  P B8j8c?@   Zfrac 0  P H\8c? `?  =0 0  P Hm8c? @ @  =2 0   P H@u8c? @  =3 0   P Hlx8c? @  =6 0   P HD|8c? >@  =7 0 H P 0޽h ? GGGy___PPT10Y+D='  = @B +5  0 \TT (  T r T S ,`0   B T H8c?@4 Exp exp E 2E 0 0000 -6 1/64 (denorms) 1 0001 -6 1/64 2 0010 -5 1/32 3 0011 -4 1/16 4 0100 -3 1/8 5 0101 -2 1/4 6 0110 -1 1/2 7 0111 0 1 8 1000 +1 2 9 1001 +2 4 10 1010 +3 8 11 1011 +4 16 12 1100 +5 32 13 1101 +6 64 14 1110 +7 128 15 1111 n/a (inf, NaN)80   P8X T  ft1?`N PNormalized E = e - Bias0 2 T  f1?`  lDenormalized E = 1 - Bias0 2 H T 0޽h ? GGGy___PPT10Y+D='  = @B +   0 0 (  X  (  X r X S ,`0    X H8c?@ $ .s exp frac E Value 0 0000 000 -6 0 0 0000 001 -6 1/8*1/64 = 1/512 0 0000 010 -6 2/8*1/64 = 2/512 & 0 0000 110 -6 6/8*1/64 = 6/512 0 0000 111 -6 7/8*1/64 = 7/512 0 0001 000 -6 8/8*1/64 = 8/512 0 0001 001 -6 9/8*1/64 = 9/512 & 0 0110 110 -1 14/8*1/2 = 14/16 0 0110 111 -1 15/8*1/2 = 15/16 0 0111 000 0 8/8*1 = 1 0 0111 001 0 9/8*1 = 9/8 0 0111 010 0 10/8*1 = 10/8 & 0 1110 110 7 14/8*128 = 224 0 1110 111 7 15/8*128 = 240 0 1111 000 n/a infX0  , X H츫8c?*> Gclosest to zero0  X H(ɫ8c?B T) hlargest denorm0  X Hͫ8c?8PD E smallest norm0  X HЫ8c?x lX_  Jclosest to 1 below0  X HHԫ8c? `\j  Jclosest to 1 above0  X Hdث8c? D  D largest norm 0   X Hܫ8c?$t jDenormalized numbers 0    X H08c? UD  NNormalized numbers 0 jB X BD8c?2 2jB X BD8c? t H X 0޽h ? GGGy___PPT10Y+D='  = @B +  0 \ (  \ r \ S ,`0   r \ S ,@0  ~ \  BA ?8c? m  H \ 0޽h ? GGGy___PPT10Y+D='  = @B +  0  ` (  ` r ` S ,`0   r ` S T,@0  ~ `  BA ?8c?^  H ` 0޽h ? GGGy___PPT10Y+D='  = @B +  0L0 0d $(  d r d S  ,`0   r d S ,@0  H d 0޽h ? GGGy___PPT10Y+D='  = @B +  0L0 @h w(  h r h S 5,`0   r h S 6,@0  K h H78c? P Note: 1. Round down: rounded result is close to but no greater than true result. 2. Round up: rounded result is close to but no less than true result. 0 H h 0޽h ? GGGy___PPT10Y+D='  = @B +m  0 Pl $(  l r l S ?,`0   r l S (K,@0  H l 0޽h ? GGGy___PPT10Y+D='  = @B +m  0 `p $(  p r p S xV,`0   r p S [,@0  H p 0޽h ? GGGy___PPT10Y+D='  = @B +  0L0 pt $(  t r t S $,`0   r t S Lo,@0  H t 0޽h ? GGGy___PPT10Y+D='  = @B +   0L0  x I(  x r x S (v,`0   r x S X,@0  F X oh8 x  oX h8 x  ZLwawa8c? X ( 1)s1 M1 V 08 Z  x  Z9wawa8c?88X ( 1)s2 M2 V 08 Z `B x  08c?`B x  08c?@@lB  x  <8c?8  x  Twawa8c?7o- ` E1 E2:0   x  Zwawa8c?w =  9+0 `B  x  08c?X h   x  Z褭wawa8c? 88 ( 1)s M V 08 Z H x 0޽h ? GGGy___PPT10Y+D='  = @B +  0L0 | $(  | r | S ,`0   r | S ,@0  H | 0޽h ? GGGy___PPT10Y+D='  = @B +  0L0  $(   r  S ϭ,`0   r  S Э,@0  H  0޽h ? GGGy___PPT10Y+D='  = @B +  0L0  $(   r  S L,`0   r  S $,@0  H  0޽h ? GGGy___PPT10Y+D='  = @B +X  0L0 g_  (   r  S ,`0   r  S ,@0  F  TXwawa8c??@ x == (int)(float) x x == (int)(double) x f == (float)(double) f d == (float) d f == -(-f); 2/3 == 2/3.0 d < 0.0 ((d*2) < 0.0) (d+f)-d == f:3 2p,  j B  Nwawa8c?"`Iv  Jint x = & ; float f = & ; double d = & ; &0 2&#`3  Z wawa8c? M4  Assume neither d nor f is NaNN0 H  0޽h ? GGGy___PPT10Y+D='  = @B +m  0  $(   r  S  ,`0   r  S ,@0  H  0޽h ? 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