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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& .., ¡ö#ZMZZ¥Z#bbjbjbjbjbb¥ b ª&ë   2 ó˜HŸ¨-Conceptual Overview: Finite-precision numbers¡*. f$Éþ$b$ª . óOŸ¨Representing Real Numbers¡`ª  Ÿ¨ÜHow to represent fractional parts to the right of the ``decimal'' point? A number like 0.12 (i.e., (1/2)10) not represented well by integers 0 or 1! Two ways to represent real numbers better: Fixed point Floating point¡šIN+I``hçÿ `hçÿ-`+` ` ª Ý ó�BŸ¨Fixed-Point Data Format¡b(ª  Ÿª óQŸ¨ Fixed Point¡ `ª  Ÿ¨Pros Add two reals just by adding the integers Much less logic than floating point Faster Often used in digital signal processing Cons The range of numbers is narrow number=400 000 000 000 000 000 000 000 000. 000 It is much more economical to represent as 4*1026¡œ} b`}`` ` ~ b  j  b ª&   ÷ ó3Ÿ¨Recall Scientific Notation¡`ª  Ÿ¨¶Issues: Arithmetic (+, -, *, / ) Representation, Normal form Range and Precision(Determined by?) Rounding and errors Exceptions (e.g., divide by zero, overflow, underflow) Properties¡~0FF¯b)bfÉþb fÉþgbª · óÎ{Ÿ¨Scientific Notation: Normalized¡ (Ÿ¨± 12.35 x 10^-9 ? 1.235 x 10^-8 ? scientific notation: has a single digit to the left of the decimal point Normalized scientific notation: such a single digit must be non-zero.ó§TŸª Ÿ ”Numerical Form: ( 1)s M 2E Sign bit s determines whether number is negative or positive Significand M normally a fractional value in range [1.0,2.0). Exponent E weights value by power of two Encoding ¡>U¥a UU`dedlddl `d?`d9`d` `ª2Y   f  óœKŸ¨IEEE 754 standard¡dª  Ÿ¨SThree formats: single/double/extended precision (32,64,80 bits). Single precision:¡TT`ª T óÌyŸª Ÿ ö the representation: ( 1)s (1+ Fraction) 2E-bias where E is the exponent representation in the exp field Sign bit s determines whether number is negative or positive Fraction is normally a fractional value in range between 0 and 1 A leading 1 added to the fraction is  implicit Exponent using a  biased notation For single precision  the bias is 127 ¡²UlUÒa'aUb fgfnffnnÉ2þ&nnÈ þn bf>bfnbf( b ª{  óÍzŸ¨4Advantage of using the biased notation for exponents¡55$Ÿ¨¢ 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 ?¡£¡óªWŸ¨Normalized Encoding Example¡b(ª  Ÿ¨ôValue: 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 Z8ZZbcjçÿcbbjçÿbjb b b  b  b   b  b   b   b 5 b  bbbjçÿbbbjçÿbbbjçÿbcbbªL=      ˆ  óÀmŸ¨Denormalized Values¡`ª   Ÿ âCondition  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 ¹ð 000& 0 Numbers very close to 0.0 Lose precision as get smaller  Gradual underflow ¡Ê Z ZZZZ Z<Z ZLZ bbcbcbcbbbbccbc c  b  c  b  c  b bcbcbc<bcbâ bcbcLbªt   #    E  U óÁnŸ¨Special Values¡`ª  Ÿ ÎCondition  exp = 111& 1 ¥ð ð(infinity) Operation that overflows Both positive and negative E.g., 1.0/0.0 = -ð1.0/-ð0.0 = +¥ð, 1.0/-ð0.0 = -ð¥ð Not-a-Number (NaN) Represents case when no numeric value can be determined E.g., sqrt( 1), ¥ð ð-ð ð¥ð¡ Z Z ZcZZNZ bbcbcbcc  c  $c   bD b  c   b  c   b  c   b   b  c   b  c   c   b bHbc  ª@•  @    óËxŸ¨IEEE 754: Summary¡ g$ª  óÂoŸ¨IEEE754: Summary (Cont.)¡b(ª  Ÿª ó´aŸ¨Special Properties of Encoding¡`ª  Ÿ¨úFP 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 abª@‰  @  ( óÏ|Ÿ¨ FP Addition¡ b(ª  Ÿ &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¡Ö ZZZZDZ bgfnffngfnffnbbbbb b  g  f  n  f  f  n  b  bbbbbbbbbbª&h   ! óÒ~Ÿ¨Decimal Number Conversion ¡ b(ª  Ÿ ÎConvert Binary to Decimal (base 2 to base 10) x x x x x. d d d d d d & 2 ¡ž/ %0 Z -`#b(c (k (çÿc (bb$ª h óÓ€Ÿ¨!Decimal Number Conversion (Cont.)¡" "b$ª " Ÿ¨ÖConvert 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 ¡Ä+ P¬ P)bbVbfº*þbbjçÿbjçÿbbª × óÔ‚Ÿ¨!Decimal Number Conversion (Cont.)¡" "b$ª " Ÿ¨åConvert Decimal Fraction to Binary Fraction multiply the decimal value by 2 and then write down the integer number from top to bottom (Multiply 2 and Get the Integers) . 0.37510 = ?2 ¡¦, Pº P7bQb!fº*þbbjçÿbjçÿbª æ óØƒŸ¨!Decimal Number Conversion (Cont.)¡" "b$ª " Ÿ¨¬Convert Decimal Number to Binary Binary Put Together: Integer Part . 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ðYeö)ðà  © ð žðâ ðH “ ð6€¸“©¿�ƒ¿Àÿˆð³h¹  ð]Ÿ¨Negative overflow¡ 2`ª  ð΢ ðH “ ð6€t¥©¿�ƒ¿ÀÿˆðÀX´ ¬  ðhŸ¨Expressible negative numbers¡ 2`ª  𸢠ðH “ ð6€�©©¿�ƒ¿Àÿˆð¥D ` ì ðRŸ¨zero¡ 2bª  ðÈ¢ ðH “ ð6€\­©¿�ƒ¿ÀÿˆðBé¾ H ðbŸ¨Negative underflow¡  2dÉþª  ðdB ðH £ ð<…D¿ÀË8cÑÿˆð¼7 7 ê ðdB ðH £ ð<…D¿ÀË8cÑÿˆð~ q Ï ðâ ðH “ ð6€Ì|«¿�ƒ¿Àÿˆð Zù  ð]Ÿ¨Positive overflow¡ 2`ª  ð,ð8 ð_ü –ˆ  ð"Hð¢ _–. ðfB ðH “ ð6…D¿ÀË8cÿˆðhø ­ø ðfB ðH “ ð6…D¿ÀË8cÿˆð­• ­d ðÈ¢ ðH “ ð6€À¯©¿�ƒ¿ÀÿˆðÑV ; =  ðZŸ & & & & & & .¡ 2bª  ðfB ð H “ ð6…D¿ÀË8cÿˆð¸ ì ðfB ð H “ ð6…D¿ÀË8cÿˆð½ _ ½  ðfB ð H “ ð6…D¿ÀË8cÿˆð@ V @ v ðfB ð H “ ð6…D¿ÀË8cÿˆðŸ V Ÿ v ðÈ¢ ð H “ ð6€´©¿�ƒ¿ÀÿˆðÚ z Da  ðZŸ & & & & & & .¡ 2bª  ðfB ðH “ ð6…D¿ÀË8cÿˆðH% �% ðfB ðH “ ð6…D¿ÀË8cÿˆðQ¹ Qˆ ðrr ðH ³ ðBZ…‡�ƒ¿ÀË8cÿˆð_ü €Ë ðrr ðH ³ ðBZ…‡�ƒ¿ÀË8cÿˆðãü � Ë ðrr ðH ³ ðBZ…‡�ƒ¿ÀË8cÿˆðØ D 7  ðrr ðH ³ ðBZ…‡�ƒ¿ÀË8cÿˆð@ ; Ÿ ø ðrr ðH ³ ðBZ…‡�ƒ¿ÀË8cÿˆðuD – ðrr ðH ³ ðBZ…‡�ƒ¿ÀË8cÿˆðù  Ô ð΢ ðH “ ð6€º©¿�ƒ¿ÀÿˆðÛ› ÷Ç  ðhŸ¨Expressible positive numbers¡ 2`ª  ðdB ðH@ £ ð<…D¿ÀË8cÑÿˆðë / Ø ðÈ¢ ðH “ ð6€H¿©¿�ƒ¿Àÿˆð9¾ “? 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ðÀ€ �ðà  ½ ð žð° ðX ƒ ð0€¼n½�¿Àÿ @ ˆðÞ}ð<$ñ 0à ½ ð žð–¢ ðX £ ð<€ü‘½…‡¿�ƒ¿Àÿ𯖠o  ð*Ÿ¡ðƒðl ð¾(a  ð Xð(¾a ð,$ñD  0ð0ðN ð0P`0  ðX ðˆð1(a ðå ðX ó ðZ€\•½�wa‚¢­ƒwa„¢­…‡¿�¿ÀË8cÿ?ð0P0  ðSŸ¨s¡" cª  ðç ðX ó ðZ€|™½�wa‚¢­ƒwa„¢­…‡¿�ÿÿ™¿ÀË8cÿ?ð PP 0  ðUŸ¨exp¡" cª  ðö ð X ó ðZ€„�½�wa‚¢­ƒwa„¢­…‡¿�Ì™ÿ¿ÀË8cÿ?ð` P`0  ðdŸ¨frac¡" cª  ðÏ¢ ð X £ ð<€`¡½…‡¿�ƒ¿Àÿð¾7 Ì  ð[Ÿ¨ 1 bit 8 bits 23 bits¡ ðþ¢ ð X £ ð<€¼¦½…‡¿�ƒ¿Àÿðb v5© ð,$ñ  0 ð^Ÿ¨,Double precision: see page 192 in P&H book¡--ðH ðX ƒ ð0�ƒ“ŽŸ‹”Þ½h¿ÿ ?ð ÿÿÿ@EŒ·ÁëffìØ‚²²²o‰÷ÏÛýˆ×ŠÏº___PPT10‹¯ê.+Û”Ò§Dñ'ñ £½ÿÿÿÿ=ñ @Bñ DñÒ'ñ –ÿÿÿÿ=ñ @BñAñ?%ñ,(ñ <ñ +O%ñ,(ñ <ñ +Dñ{'ñ =ñ%ñ(ñÿÿÿÿDñ#'ñ =ñ%ñ(ñDñË'ñ =ñA@Bñ�Bñ Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*XA%ñ(ñDñ'ñ ô=ñ+ñá4ñ 8?ñdCñBñ0-#ppt_w/2BñCñèBñ#ppt_xBñ*ñY3ñ>ñBñ ppt_x<ñû*XADñ'ñ ô=ñ+ñÙ4ñ 8?ñ\CñBñ#ppt_yBñCñèBñ#ppt_yBñ*ñY3ñ>ñBñ ppt_y<ñû*XADñ{'ñ =ñ%ñ(ñÿÿÿÿDñ#'ñ =ñ%ñ(ñDñË'ñ =ñA@Bñ�Bñ Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*XAT%ñ(ñDñ'ñ ô=ñ+ñá4ñ 8?ñdCñBñ0-#ppt_w/2BñCñèBñ#ppt_xBñ*ñY3ñ>ñBñ ppt_x<ñû*XATDñ'ñ ô=ñ+ñÙ4ñ 8?ñ\CñBñ#ppt_yBñCñèBñ#ppt_yBñ*ñY3ñ>ñBñ ppt_y<ñû*XATDñ÷'ñ =ñ%ñ(ñÿÿÿÿDñŸ'ñ =ñ%ñ(ñDñG'ñ =ñ4@Bñ�Bñ Bñ°Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû* Xÿÿÿÿÿÿÿÿ%ñ(ñDñ»'ñ ô=ñ-ñƒ6ñBñ'barn(inHorizontal)*ñ<3ñ<ñû* XÿÿÿÿÿÿÿÿDñ'ñ =ñ%ñ(ñÿÿÿÿDñ¬'ñ =ñ%ñ(ñDñT'ñ =ñA@Bñ�Bñ Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû* Xÿÿÿÿÿÿÿÿ%ñ(ñDñ»'ñ ô=ñ-ñƒ6ñBñ'barn(inHorizontal)*ñ<3ñ<ñû* Xÿÿÿÿÿÿÿÿ+p+0+X0 ++0+ X0 +î ï € 0ù¿L0 Ãð» ððSð( ð ðð ð S ð~€(ë½�wa‚¢­ƒwa„¢­…†‡ˆ‰Š‹¿¿Ëœ1ÿ ?@ ˆðÝȱð<$ñ 0à ½ ð,ž¦ù4�aÕ©ÂYÓið% ð ó ðZ€8Û«� ø‚8cƒ ø„8c¿�ƒ¿ÀËœ1ÿ ?ðˆ£\ 𛟨/IEEE 754 Standard Floating Point Representation¡ 00g$ª 0 ¦ððÈ ð  ð$€lê«�wa‚¢­ƒwa„¢­¿�ƒ¿ÀËœ1ÿ ?�ÃÄRectangle: Click to edit Master text styles Second level Third level Fourth level Fifth levelðD‰ÖŽ ðtŸ¡*þ0wnUc ª ¦ù4�aÕ©ÂYÓiðH ð ƒ ð0�ƒ“ŽŸ‹”Þ½h¿ÿ ?ð ÿÿÿ‘‘‘GGGÚÚÚ‘‘‘ˆÖŠÎº___PPT10‹®ê.+ôrì<DñN'ñ {ÿÿÿÿ=ñ @Bñ Dñ 'ñ –ÿÿÿÿ=ñ @BñAñ?%ñ,(ñ <ñ +O%ñ,(ñ <ñ +DñA'ñ =ñ%ñ(ñÿÿÿÿDñé'ñ =ñ%ñ(ñDñ‘'ñ =ñA@Bñ�Bñ Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*%ñ(ñóDñA'ñ =ñ%ñ(ñÿÿÿÿDñé'ñ =ñ%ñ(ñDñ‘'ñ =ñA@Bñ�Bñ Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*;%ñ(ñóDñA'ñ =ñ%ñ(ñÿÿÿÿDñé'ñ =ñ%ñ(ñDñ‘'ñ =ñA@Bñ�Bñ Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*<‚%ñ(ñóDñA'ñ =ñ%ñ(ñÿÿÿÿDñé'ñ =ñ%ñ(ñDñ‘'ñ =ñA@Bñ�Bñ Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*‚¿%ñ(ñóDñA'ñ =ñ%ñ(ñÿÿÿÿDñé'ñ =ñ%ñ(ñDñ‘'ñ =ñA@Bñ�Bñ Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*¿%ñ(ñóDñA'ñ =ñ%ñ(ñÿÿÿÿDñé'ñ =ñ%ñ(ñDñ‘'ñ =ñA@Bñ�Bñ Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*1%ñ(ñóDñA'ñ =ñ%ñ(ñÿÿÿÿDñé'ñ =ñ%ñ(ñDñ‘'ñ =ñA@Bñ�Bñ Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*1T%ñ(ñóDñA'ñ =ñ%ñ(ñÿÿÿÿDñé'ñ =ñ%ñ(ñDñ‘'ñ =ñA@Bñ�Bñ Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*T{%ñ(ñó+8+0+0 +î×ï € 0 ÊðÂ0ððZð( ð ððr ð S ð€ó½¿ÿ? ðÀ€ �ðà  ½ ð žðr ð S ð€ðó½¿ÿ@ ð°0 ðà ½ ð žð#¢ ð £ ð<€°÷½…‡¿�ƒ¿Àÿðô ™ÆÁð,$ñ  0 ðƒŸ¨/ Answer: -1+127 = 126! 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Any way, to avoid nonlinear effects introduced by overflow, underflow, saturation, and wrapping during computation, control of each variable s range has to be done by appropriate scaling of operands. In the floating-point case, the exponent of a variable is part of the run-time representation and is computed automatically by the floating-point ALU. But in the fixed-point case, the exponent of each variable is implicit and determined by the programmer off-line using the predicted dynamic range of the variable. Therefore, the scaling of fixed-point operands has to be done explicitly by the programmer. This is generally a tedious and error-prone process. In fact, you may regard the fixed-point as a form of integer with an additional hypothetical binary point. In our research, the number of bits to the left of the binary-point excluding the sign bit is called Integer Word-Length or IWL.¡JîÒ`````ª î ¦à??__ðH ð( ƒ ð0�ƒ“’ßj”BÁ�¿ÿ ?ð ÿÿÿ‘‘‘a�ý®ü(ÎÎÎð¦ñ} 0 &ðpð$ð¶ð( ð ð$ðŽ ð$ Ó ðN¿ÀÄËœ1ÍÎ×ÿ ?$ˆð»ìMD ðà  ðè ð$  ð`€h�žg‚§°ƒžg„§°…‡ˆ¿¿ÿ ?$ˆðÚ Lì!ðà   ð@Ÿ¡`ª ðH ð$ ƒ ð0�ƒ“’ßj”BÁ�¿ÿ ?ð ÿÿÿ‘‘‘a�ý®ü(ÎÎΈ8Š0º___PPT10‹ë.$ˆÃðÕèð¦ñ~ 0 &ð�ð,ð¶ð( ð ð,ðŽ ð, Ó ðN¿ÀÄËœ1ÍÎ×ÿ ?$ˆð»ìMD ðà  ðè ð,  ð`€ú�žg‚§°ƒžg„§°…‡ˆ¿¿ÿ ?$ˆðÚ Lì!ðà   ð@Ÿ¡`ª ðH ð, ƒ ð0�ƒ“’ßj”BÁ�¿ÿ ?ð ÿÿÿ‘‘‘a�ý®ü(ÎÎΈ8Š0º___PPT10‹ë.$ˆÃðÕèð¦ñ€ 0 &ðÐð<ð¶ð( ð ð<ðŽ ð< Ó ðN¿ÀÄËœ1ÍÎ×ÿ ?$ˆð»ìMD ðà  ðè ð<  ð`€`$�žg‚§°ƒžg„§°…‡ˆ¿¿ÿ ?$ˆðÚ Lì!ðà   ð@Ÿ¡`ª ðH ð< ƒ ð0�ƒ“’ßj”BÁ�¿ÿ ?ð ÿÿÿ‘‘‘a�ý®ü(ÎÎΈ8Š0º___PPT10‹ë.$ˆÃðÕèð¦ñ‚ 0 &ððLð¶ð( ð ðLðŽ ðL Ó ðN¿ÀÄËœ1ÍÎ×ÿ ?$ˆð»ìMD ðà  ðè ðL  ð`€è*�žg‚§°ƒžg„§°…‡ˆ¿¿ÿ ?$ˆðÚ Lì!ðà   ð@Ÿ¡`ª ðH ðL ƒ ð0�ƒ“’ßj”BÁ�¿ÿ ?ð ÿÿÿ‘‘‘a�ý®ü(ÎÎΈ8Š0º___PPT10‹ë.$ˆÃðÕèð¦ñƒ 0 &ð0ðTð¶ð( ð ðTðŽ ðT Ó ðN¿ÀÄËœ1ÍÎ×ÿ ?$ˆð»ìMD ðà  ðè ðT  ð`€¼0�žg‚§°ƒžg„§°…‡ˆ¿¿ÿ ?$ˆðÚ Lì!ðà   ð@Ÿ¡`ª ðH ðT ƒ ð0�ƒ“’ßj”BÁ�¿ÿ ?ð ÿÿÿ‘‘‘a�ý®ü(ÎÎΈ8Š0º___PPT10‹ë.$ˆÃðÕèð¦ñ„ 0 &ðPð\ð¶ð( ð ð\ðŽ ð\ Ó ðN¿ÀÄËœ1ÍÎ×ÿ ?$ˆð»ìMD ðà  ›ðè ð\  ð`€\zŒ�žg‚§°ƒžg„§°…‡ˆ¿¿ÿ ?$ˆðÚ Lì!ðà  › ð@Ÿ¡`ª ðH ð\ ƒ ð0�ƒ“’ßj”BÁ�¿ÿ ?ð ÿÿÿ‘‘‘a�ý®ü(ÎÎΈ8Š0º___PPT10‹ë.$ˆÃðÕè}xœíUAkSAž™—Vóék­ЇG½bZ/"hÄ ­’z£¼`Ää5MªÄS/B/‚â]ôâUêÙCÀ‹'�xðZÿ�Oßsfw_ˆä¥¥%_ؼÙÙÙ��Ý™o;Æ·_lM}�>œ ‚0 £=:4MÁ ÓÂ0ŒÔáû ?¸�˜;L˜oðŸ÷4Ä¿C|þ5Á…Ôø»­~*ø#ŽÃH·æ…è=)}[_êµÝ¸þêé™GoÑ’ŽqÊU¸ Þ@>{aaoåpA¦èq«øl®IŠͰõ„Ù¶Û÷wþÿà…ŸòùOh«‘+'³Ù0°9 hp °   @ L Xdlä(Lecture 6 Instruction Set ArchitecturesPGaoPC:\Program Files\Microsoft Office\Templates\Presentation Designs\Blueprint.potHWU896Microsoft PowerPoint@@˜‚¨¨@° EZ´{Ã@`Õ��‚ÏÅfG<8ÿÿÿÿ; 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