计算机组成原理与接口技术2016课件lch01-floatingpoint

1、数据的表示计算机硬件能够直接识别的数据类型，如定点数、浮点数等硬件直接识别：意味着某种数据类型可用计算机硬件直接表示出来，并能由计算机指令直接调用该种数据类型。比如，数组、结构这一类数据，由于计算机不能一次对它进行存取、也不能一次处理它，它要分多次处理，所以它并不是计算机能够直接表示的数据。1.5 计算机中数据的表示Floating-PoNumbersIEEE 754 Floating-PoStandardFloating-PoAddition1.5.3 小数表示Floating PoThe World is Not JustegersProgramming languagepport num

2、bers with fractionCalled floating-ponumbersExles:3.14159265 ()2.71828 (e)0.000000001 or 1.0 109 (seconds in a nanosecond)86,400,000,000,000 or 8.64 1013 (nanoseconds in a day)last number is a largeegert cannot fit in a 32-bitegerWe use a scientific noion (科学记数法) to representFloating PoVery small num

3、bers (e.g. 1.0 109)Very large numbers (e.g. 8.64 1013)Scientific noion: d . f1f2f3f4 10Digital System II e1e2e3slide 4ExExles of floating-ponumbers in base 10 5.341103 , 0.05341105 , 2.013101 , 201.3103decimal poles of floating-ponumbers in base 2 1.00101223 , 0.0100101225 , 1.10110123 , 1101.10126b

4、inary poExponents（指数） are kept in decimal for clarityThe binary number (1101.101)2 = 23+22+20+21+23 = 13.625Floating-ponumbers should be normalized（规格化）Exactly one non-zero digit should appear before the poIn a decimal number, this digit can be from 1 to 9In a binary number, this digit should be 1No

5、rmalized FP Numbers: 5.341103and 1.10110123NOT Normalized: 0.05341105 and 1101.10126Floating PoDigital System IIslide 5Floating-PoNumbersFloating-PoRepresenionA floating-ponumber is represented by the triple（三部分）S is the Sign bit (0 isitive and 1 is negative)Represenion is called sign and magnitudeE

6、 is the Exponent field (signed)Very large numbers have largeitive exponentsVery small close-to-zero numbers have negative exponentsMore bits in exponent field increases range of valuesF is the Fraction field (fraction after binary poMore bits in fraction field improves the preci)of FP numbersFloatin

7、g PoDigital System IIslide 6Value of a floating-ponumber = (-1)S val(F) 2val(E)SExponentFractionNext . . .Floating-PoNumbersIEEE 754 Floating-PoStandardFloating-PoAdditionFloating PoDigital System IIslide 7Found in virtually every computer invented since 1980Simplified porting （移植）of floating-ponumb

8、ersUnified（） the development of floating-poalgorithmsnumbersIncreased the accuracy（准确性） of floating-poSingle PreciFloating PoNumbers (32 bits)1-bit sign + 8-bit exponent + 23-bit fractionDouble PreciFloating PoNumbers (64 bits)1-bit sign + 11-bit exponent + 52-bit fractionFloating PoDigital System I

9、Islide 8优点SExponent11Fraction52(continued)SExponent8Fraction23IEEE 754 Floating-PoStandardFor a normalized floating ponumber (S, E, F)Significand（尾数） is equal to (1.F)2 = (1.f1f2f3f4)2IEEE 754 ames hidden 1. (not stored) for normalized numbersSignificand is 1 bit longern fraction(有效位数比实际的长了1位)Value

10、of a Normalized Floating PoNumber is)(1)S is 1 when S is 0 (Floating Poitive), and 1 when S is 1 (negative)Digital System IIslide 9转换(1)S (1.F)2 2val(E)(1)S (1.f1f2f3f4 )2 2val(E)(1)S (1 + f12-1 + f22-2 + f32-3 + f42-4 )2 2val(ESEF = f1 f2 f3 f4 Normalized Floating PoNumbersFor a normalized floating

11、 ponumber (S, E, F)用 原码 表示用 移码 表示符号位：0 +1 -Floating PoDigital System IIslide 10SEF = f1 f2 f3 f4 Normalized Floating PoNumbersBiased Exponent RepresenionHow to represent a signed exponent? Choiare Sign + magnitude represenion for the exponent（符号+幅值）Twos complement represenion（补码）Biased represenion（移

12、码）IEEE 754 uses biased represenion for the exponentValue of exponent = val(E) = E Bias (Bias is a constant)Recallt exponent field is 8 bits for single preciE can behe range 0 to 255E = 0 and E = 255 are (discussed later)（保留） for spel useE = 1 to 254 are used for normalized floating poBias = 127 (hal

13、f of 254), val(E) = E 127numbersval(E=1) = 126, val(E=127) = 0, val(E=254) = 127Floating PoDigital System IIslide 11 Biased Exponent（移码） ContdFor double preci, exponent field is 11 bitsE can behe range 0 to 2047E = 0 and E = 2047 arefor spel useE = 1 to 2046 are used for normalized floating poBias =

14、 1023 (half of 2046), val(E) = E 1023numbersval(E=1) = 1022, val(E=1023) = 0, val(E=2046) = 1023Value of a Normalized Floating PoNumber isFloating PoDigital System IIslide 12转换(1)S (1.F)2 2E Bias(1)S (1.f1f2f3f4 )2 2E Bias(1)S (1 + f12-1 + f22-2 + f32-3 + f42-4 )2 2E BiasExles of Single PreciFloatWh

15、at is the decimal value of this Single Precifloat?Solution:Sign = 1 is negativeExponent = (01111100)2 = 124, E bias = 124 127 = 3Significand = (1.0100 0)2 = 1 + 2-2 = 1.25 (1. is implicit)Value in decimal = 1.25 23 = 0.15625What is the decimal value of?Solution:implicitValue in decimal = +(1.0100110

16、0 0)2 2130127=(1.01001100 0)2 23 = (1010.01100 0)2 =Digital System II10.375Floating Poslide 130100000100100110000000000000000010111110001000000000000000000000Exles of Double PreciFloatWhat is the decimal value of this Double Precifloat ?Solution:Value of exponent = (10000000101)2 Bias = 1029 1023 =

17、6Value of double float = (1.00101010 0)2 (1001010.10 0)2 = 74.5What is the decimal value of ? 26(1. is implicit) =t yourself! (answer should be 1.5 27Digital System IIFloating Po= 0.01171875)slide 1410111111100010000000000000000000000000000000000000000000000000000100000001010010101000000000000000000

18、000000000000000000000000000Converting FP Decimal to BinaryConvert 0.8125 to binary in single and double preciSolution:Fraction bits can be obtained using multiplication by 20.8125 2 = 1.6250.625 20.25 2 0.5 2= 1.25= 0.5= 1.0Stop when fractional part is 0Fraction = (0.1101)2 = (1.101)2 2 1(Normalized

19、)Exponent = 1 + Bias = 126 (single preci) and 1022 (double)Single PreciDouble PreciFloating PoDigital System IIslide 151011111111101010000000000000000000000000000000000000000000000000101000000000000000000000.8125 = (0.1101)2 = + + 1/16 = 13/Largest Normalized FloatWhat is the Largest normalized floa

20、t?Solution for Single Preci:Exponent bias = 254 127 = 127 (largest exponent for SP)Significand = (1.111 1)2 = almost 2Value in decimal 2 2127 2128: 3.4028 1038Solution for Double PreciValue in decimal 2 21023 21024 1.79769 10308Overflow: exponent is too large to fithe exponent fieldslide 16Floating

21、PoDigital System II011111111110111111111111111111111111111111111111111111111111111101111111011111111111111111111111Smallest Normalized FloatWhat is the smallest (in absolute value) normalized float?Solution for Single Preci:Exponent bias = 1 127 = 126 (smallest exponent for SP)Significand = (1.000 0

22、)2 = 1Value in decimal = 1 2126 = 1.17549 1038Solution for Double Preci:Value in decimal = 1 21022 = 2.22507 10308Underflow: exponent is too small to fit in exponent fieldFloating PoDigital System IIslide 17000000000001000000000000000000000000000000000000000000000000000000000000100000000000000000000

23、000Zero, Infinity, and NaNZeroExponent field E = 0 and fraction F = 0 +0 and 0 areInfinitysible according to sign bit SInfinity is a speFor single preciFor double precil value represented withwith 8-bit exponent: with 11-bit exponent:um E and F = 0um E = 255 um E = 2047by zeroInfinity can result fro

24、m overflow or divi+ and aresible according to sign bit SNaN (Not a Number)Floating PoNaN is a spel value represented withum E and F 0Result from exceptional situations, such as 0/0 or sqrt(negative)Operation on a NaN results is NaN: Op(X, NaN) = NaNDigital System IIslide 18 Denormalized（非规格化） Number

25、s IEEE standard uses denormalized numbers to Fill the gap betProvide graduan 0 and the smallest normalized floatderflow to zeroDenormalized: exponent field E is 0 and fraction F 0Implicit 1. before the fraction nowes 0. (not normalized)Value of denormalized number ( S, 0, F )NegativeOverflowNegative

26、UnderflowitiveitiveUnderflowOverflow-+DenormDenormNormalized (+ve)0Digital System II-2128Floating Po-212621262128slide 19Normalized (ve)Single preci:(1) S (0.F)2 2126 Double preci:(1) S (0.F)2 21022Summary of IEEE 754 EncodingFloating PoDigital System IIslide 20Double-PreciExponent = 11Fraction = 52

27、ValueNormalized Number1 to 2046Anything (1.F)2 2E 1023Denormalized Number0nonzero (0.F)2 21022Zero00 0Infinity20470 NaN2047nonzeroNaNSingle-PreciExponent = 8Fraction = 23ValueNormalized Number1 to 254Anything (1.F)2 2E 127Denormalized Number0nonzero (0.F)2 2126Zero00 0Infinity2550 NaN255nonzeroNaNCo

28、nsider Adding (Single-PreciFloating-Po):+1.1.0001020010122422Cannot add significands Why?Because exponents are not equalHow to make exponents equal?Shift the significand of the lesser exponent rightDifference betn the two exponents = 4 2 = 2So, shift right second number by 2 bits and increment expon

29、ent1.00101222=0.00001Digital System II01224Floating Poslide 21Floating PoAddition ExleNow, ADD the Significands:+1.1.000100010124221.0.000010000101 2424(shift right)(result)01 24+10.0Addition produ0011a carry bit, result is NOT normalizedNormalize Result (shift right and increment exponent):2425+10.

30、01.00001100101101=Floating PoDigital System IIslide 22Floating-PoAddition contd8位二进制数，可以代表不同的值小结：计算机有符号数de 23“Father” of the IEEE 754 standard直到80年代初，各个机器的浮点数表示格式还没有，因而相互不兼容，机器之间传送数据时，带来麻烦1970年代后期, IEEE成立着手制定浮点数标准1985年完成浮点数标准IEEE754的制定现在所有计算机都采用IEEE754来表示浮点数This standard was primarily the work of on

31、e UC Berkeley math professor William Kahan.,/wkahan/ us/754story.htmlDigital System IIieee754sFloating PoProf. William Kahanslide 2(2) IEEE 754格式单精度双精度指数采用偏移值,其中单精度为127,双精度为1023.从而所有浮点数的阶码值都可以变成非负整数,点数的比较和排序.便于浮IEEE754尾数形式为1.,其中F部分保存的的有效数字位,进一步提.这样可以保留是高数据表示的精确度.Floating PoDigital System IIslide 252

32、5S11位偏指数E52位有效尾数FS23位有效尾数F8位偏指数E与上述IEEE754格式相对应的32位浮点数的真值可表示为:N = (-1)S 2 1.FE-127随E和F取值不同，浮点数据表示具有不同的意义E=0, F =0：表示机器零；E=0, F 0：则N = (-1)S 20.F,非规格化浮点数；1.F，规格化浮点数；-1261 E 254：N = (-1)S 2E-127E=255, F =0E=255, F 0：无穷大，对应于x/0 （其中 x 0） ；：N= NaN，表示一个非数值，对应于0/0。Floating PoDigital System IIslide 26261.7C语言数据类型的含义典型的数据类型对应32位机的数据位宽Floating PoDigital System IIslide 27数据类型字节数char12short4long4char *4float（单精度浮点数）4double（双精度浮点数）81.7C语言数据类型的含义(续)例1. 6某大字节序的计算机系统内存中存放有如表1- 6所示数据，试地址 0 x1234 5678表示的，float，do