浮点运算单元公开课一等奖省优质课大赛获奖课件

浮点运算单元浮点运算Floating-PointNumbersIEEE754Floating-PointStandardFloating-PointAdditionandSubtractionFloating-PointMultiplication浮点数在计算机内格式浮点数:X=MS

ESEm-1...E2E1

M-1M-2...M-n

符号位

阶码位

尾数数码位

总位数短浮点数:

1

8

23

32长浮点数:

1

11

52

64

暂时浮点数:1

15

64

80IEEE标准：阶码用移码，基为2；尾数用原码X=MX*2EX浮点数阶码位数决定数表示范围，

尾数位数决定数有效精度。浮点数在计算机内格式浮点数:X=M

EE...EE

MM...M

ssm-110-1-2-nIEEE标准：尾数用原码X=MX

*2EX浮点数是数学中实数子集合，由一个纯小数乘上一个指数值来组成。在计算机内，其纯小数部分被称为浮点数尾数，对非0值浮点数，要求尾数绝对值必须>=1/2，称满足这种表示要求浮点数为规格化表示；把不满足这一表示要求尾数，变成满足这一要求尾数操作过程，叫作浮点数规格化处理，经过尾数移位和修改阶码实现。浮点数在计算机内格式浮点数:X=M

EE...EE

MM...M

ssm-110-1-2-nIEEE标准：尾数用原码X=MX*2EX按国际电子电气工程师协会要求标准，浮点数尾数要用原码表示，即符号位Ms:0表示正，1表示负，且非0值尾数数值最高位M-1必为1,才能满足浮点数规格化表示要求；既然非0值浮点数尾数数值最高位必定为1，则在保留浮点数到内存前，经过尾数右移,强行把该位去掉,用一样多尾数位就能多存一位二进制数，有利于提升数据表示精度，称这种处理方案使用了隐藏位技术。当然，在取回这么浮点数到运算器执行运算时，必须先恢复该隐藏位。FloatingPoint浮点数在计算机内格式X=Ms

EsEm-1...E1E0

M-1M-2...M-n

IEEE标准：阶码用移码，基为2X=MX*2EX按国际电子电气工程师协会要求国际通用标准，浮点数阶码用整数给出，而且要用移码表示，用作为以2为底指数幂。既然该指数底一定为2，能够无须在浮点数格式中明确表示出来，只需给出阶码幂值即可。

移码表示只用于表示整数，只用在浮点数阶码部分，其定义类似于整数补码定义，差异在符号位。

移码符号位是0表示负，1表示正，与补码符号位恰好相反，移码是指机器数在数轴上有个移位关系；

移码数值位则与补码数值位完全相同。浮点数格式：关于移码知识浮点数:X=M

EE...EE

MM...M

ssm-110-1-2-nX=MX*2EX移码表示整数，用在浮点数阶码部分。一位符号位和n位数值位组成移码,其定义为；[E]移=2n+E-2n<=E<2n表示范围：00000000111111110负数正数机器数[X]补=X0X<2n

2n+1+X-2nX0浮点数格式：关于移码知识一位符号位和n位数值位组成移码,其定义为；[E]移=2n+E-2n<=E<2n表示范围：00000000~11111111

负数

正数机器数0移码只执行二数加减运算与增1、减1操作。加减运算时，符号位计算结果求反后,才是加减运算正确符号位值。注意:当用双符号位时，00代表负，01代表正，而不是11代表正8位阶码能表示-128~+127，当阶码为-128时，其补码表示为00000000，该浮点数绝对值<2-128,人们要求此浮点数值为零，若尾数不为0就清其为0，并特称此值为机器零。8位移码表示机器数为数真值在数轴上向右平移了128个位置-128+127BiasedExponentValueofexponent=val(E)=E–Bias(Biasisaconstant)8bitsforsingleprecisionEcanbeintherange0to255E=0andE=255arereservedforspecialuseE=1to254areusedfornormalizedfloatingpointnumbersBias=127(halfof254),val(E)=E–127val(E=1)=–126,val(E=127)=0,val(E=254)=127ExampleofExponentExponent(E)Adjusted

Binary(E+127)

+5

132

100001000

127

1111111-10

117

1110101+128

255

11111111-127

0

0-1

126

1111110ExampleofNormalizedMantissaBinaryValueNormalizedAsExponent1101.1011.10110130.001011.01-31.00011.00010100000111.00000117BiasedExponentExampleofFloatingPointLargestNormalizedFloatSmallestNormalizedFloatZeroInfinityNaNDenormalizednumbersZero&InfinityThevalueNaN(NotaNumber)isusedtorepresentavaluethatdoesnotrepresentarealnumber.NaNisaspecialvaluerepresentedwithmaximumEandF≠0Resultfromexceptionalsituations,suchas0/0orsqrt(negative)OperationonaNaNresultsisNaN:Op(X,NaN)=NaNQNaNdenoteindeterminateoperations,SNaNdenoteinvalidoperationsNaNSignExponent(e)Fraction(f)Value000..0000..00+0000..0000..01PositiveDenormalizedReal:0.f×2(-b+1)11..11

000..01XX..XXPositiveNormalizedReal:1.f×2(e-b)11..10

011..1100..00+Infinity011..1100..01SNaN:01..11011..1110..00QNaN:11..11SignExponent(e)Fraction(f)Value100..0000..00-0100..0000..01NegativeDenormalizedReal:-0.f×2(-b+1)11..11

100..01XX..XXNegativeNormalizedReal:-1.f×2(e-b)11..10

111..1100..00-Infinity111..1100..01SNaN:01..11111..1110..00QNaN:11.11OperationResultn÷±Infinity0±Infinity×±Infinity±Infinity±nonzero÷0±InfinityInfinity+InfinityInfinity±0÷±0NaNInfinity-InfinityNaN±Infinity÷±InfinityNaN±Infinity×0NaNFPAddFPAddFloatingPointSubtractionExampleFloatingPointSubtractionExampleExtrabitsGuardbitExtrabitRoundingModenearestInthismode,theinexactresultsareroundedtothenearerofthetwopossibleresultvalues.Iftheneitherpossibilityisnearer,thentheevenalternativeischosen.Thisformofroundingisalsocalled``roundtoeven''。“Even”whenleastsignificantbitis0Value Binary Rounded Action RoundedValue23/32 10.000112 10.002 (<1/2—down)223/16 10.001102 10.012 (>1/2—up) 21/427/8 10.111002 11.002 (1/2—up) 325/8 10.101002 10.102 (1/2—down) 21/2RoundingModeStepsinAddition/SubtractionofFloating-PointNumbersStep1:Calculatedifferencedofthetwoexponents-d=|E1-E2|Step2:Shiftsignificandofsmallernumberbyd-base

positionstotherightStep3:AddalignedsignificandsandsetexponentofresulttoexponentoflargeroperandStep4:NormalizeresultantsignificandandadjustexponentifnecessaryStep5:RoundresultantsignificandandadjustexponentifnecessaryAddition/SubtractionStructureAddition/SubtractionE1E2-

Exponentoflargernumbernotdecreased-thiswillresultinalargersignificandadderrequired.

Addition-resultantsignificandM(sumoftwoalignedsignificands)isinrange1/

M<2

IfM>1-apostnormalizationstep-shiftingsignificandtotherighttoyieldM3andincreasingexponentbyone-isrequired(anexponentoverflowmayoccur)Addition/SubtractionNormalizationSubtraction-ResultantsignificandMisinrange0|M|<1-postnormalizationstep-shiftingsignificandtoleftanddecreasingexponent-isrequiredifM<1/

(anexponentunderflowmayoccur)Inextremecases,thepostnormalizationstepmayrequireashiftleftoperationoverallbitsinsignificand,yieldingazeroresult.EffectiveAddition/SubtractionDistinguishbetweeneffectiveadditionandeffectivesubtractionDependsonsignbitsofoperandsandinstructionexecutedEffectiveaddition:CalculateexponentdifferencetodeterminealignmentshiftShiftsignificandofsmalleroperand,addalignedsignificandsTheresultcanoverflowbyatmostonebitpositionLongpost-normalizationshiftnotneededSinglebitoverflowcanbedetectedand,iffound,a1-bitnormalizationisperformedusingamultiplexorEliminateIncrementinRoundingSignificandadderdesignedtoproducetwosimultaneousresults-sumandsum+1Calledcompoundadder;canbeimplementedinvariousways(e.g.,carry-look-aheadorconditionalsum)Round-to-nearest-even-useroundingbitstodeterminewhichofthetwoshouldbeselectedThesetwoaresufficientevenifasinglebitoverflowoccursIncaseofoverflow,1isaddedinRposition(insteadofLSBposition),andsinceR=1ifroundingneeded,acarrywillpropagatetoLSBtogeneratecorrectsum+1

Directedroundings-Rnotnecessarily1-sum+2maybeneededEffectiveSubtractionMassivecancellationofmostsignificantbitsmayoccur-resultinginlengthypostnormalizationHappensonlywhenexponentsofoperandsareclose(difference1)-pre-alignmentcanbeeliminatedTwoseparateprocedures-(1)exponentsareclose(difference1)-onlyapostnormalizationshiftmaybeneeded(2)exponentsarefar

(difference>1)-onlyapre-alignmentshiftmaybeneededCLOSECaseExponentdifferencepredictedbasedontwoleastsignificantbitsofoperands-allowssubtractionofsignificandstostartassoonaspossibleIf0-subtractexecutedwithnoalignmentIf1-significandofsmalleroperandisshiftedoncetotheright(usingamultiplexor)andthensubtractedfromothersignificandInparallel-trueexponentdifferencecalculatedIf>1-procedureabortedandFARprocedurefollowedIf

1-

CLOSEprocedurecontinuedInparallelwithsubtraction-numberofleadingzerospredictedtodeterminenumberofshiftpositionsinpostnormalizationCLOSECase-NormalizationandRoundingNext-normalizationofsignificandandcorrespondingexponentadjustmentLast-rounding-precomputingsum,sum+1-selectingtheonewhichisproperlyrounded-negationofresultmaybenecessaryResultofsubtractionusuallypositive-negationnotrequiredOnlywhenexponentsequal-resultofsignificandsubtractionmaybenegative(intwo'scomplement)-requiringanegationstepNegationandroundingsteps-mutuallyexclusiveFARCaseFirst-exponentdifferencecalculatedNext-significandofsmalleroperandshiftedtorightforalignmentShifted-outbitsusedtosetstickybitSmallersignificandsubtractedfromlarger-resulteithernormalized.Laststep-roundingLeadingZerosPredictionCircuitPredictpositionofleadingnon-zerobitinresultofsubtractbeforesubtractioniscompletedAllowingtoexecutepostnormalizationshiftimmediatelyfollowingsubtractionExaminebitsofoperands(ofsubtract)inaserialfashion,startingwithmostsignificantbitstodeterminepositionoffirst1Thisserialoperationcanbeacceleratedusingaparallelschemesimilartocarry-look-aheadLeadingZerosPredictionCircuitPredictpositionofleadingnon-zerobitinresultofsubtractbeforesubtractioniscompletedAllowingtoexecutepostnormalizationshifti