Floating Point Arithmetic August 31, 2009Floating Point PuzzlesIEEE Floating PointFractional Binary NumbersFrac. Binary Number ExamplesRepresentable NumbersFloating Point RepresentationFloating Point Precisions“Normalized” Numeric ValuesNormalized Encoding ExampleDenormalized ValuesSpecial ValuesSummary of Floating Point Real Number EncodingsTiny Floating Point ExampleValues Related to the ExponentDynamic RangeDistribution of ValuesDistribution of Values (close-up view)Interesting NumbersSpecial Properties of EncodingFloating Point OperationsCloser Look at Round-To-EvenRounding Binary NumbersFP MultiplicationFP AdditionMathematical Properties of FP AddMath. Properties of FP MultCreating Floating Point NumberNormalizeRoundingPostnormalizeFloating Point in CCurious Excel BehaviorSummaryFloating Point ArithmeticAugust 31, 2009Floating Point ArithmeticAugust 31, 2009TopicsTopicsIEEE Floating Point StandardRoundingFloating Point OperationsMathematical propertiesclass03.ppt15-213– 2 –15-213: Intro to Computer SystemsFall 2009 ©Floating Point PuzzlesFloating Point PuzzlesFor each of the following C expressions, either:Argue that it is true for all argument valuesExplain why not true•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 -f > -d•d * d >= 0.0•(d+f)-d == fint x = …;float f = …;double d = …;Assume neitherd nor f is NaN– 3 –15-213: Intro to Computer SystemsFall 2009 ©IEEE Floating PointIEEE Floating PointIEEE Standard 754IEEE Standard 754Established in 1985 as uniform standard for floating point arithmeticBefore that, many idiosyncratic formatsSupported by all major CPUsDriven by Numerical ConcernsDriven by Numerical ConcernsNice standards for rounding, overflow, underflowHard to make go fastNumerical analysts predominated over hardware types in defining standard– 4 –15-213: Intro to Computer SystemsFall 2009 ©Fractional Binary NumbersFractional Binary NumbersRepresentationRepresentationBits to right of “binary point” represent fractional powers of 2Represents rational number:bibi–1b2b1b0b–1b–2b–3b–j• • •• • • .1242i–12i• • •• • •1/21/41/82–jbk2kk ji– 5 –15-213: Intro to Computer SystemsFall 2009 ©Frac. Binary Number ExamplesFrac. Binary Number ExamplesValueValueRepresentationRepresentation5-3/4 101.1122-7/8 10.111263/64 0.1111112ObservationsObservationsDivide by 2 by shifting rightMultiply by 2 by shifting leftNumbers of form 0.111111…2 just below 1.01/2 + 1/4 + 1/8 + … + 1/2i + … 1.0Use notation 1.0 – – 6 –15-213: Intro to Computer SystemsFall 2009 ©Representable NumbersRepresentable NumbersLimitationLimitationCan only exactly represent numbers of the form x/2kOther numbers have repeating bit representationsValueValueRepresentationRepresentation1/3 0.0101010101[01]…21/5 0.001100110011[0011]…21/10 0.0001100110011[0011]…2– 7 –15-213: Intro to Computer SystemsFall 2009 ©Numerical FormNumerical Form–1s M 2ESign bit s determines whether number is negative or positiveSignificand M normally a fractional value in range [1.0,2.0).Exponent E weights value by power of twoEncodingEncodingMSB is sign bitexp field encodes Efrac field encodes MFloating Point RepresentationFloating Point Representations exp frac– 8 –15-213: Intro to Computer SystemsFall 2009 ©EncodingEncodingMSB is sign bitexp field encodes Efrac field encodes MSizesSizesSingle precision: 8 exp bits, 23 frac bits32 bits totalDouble precision: 11 exp bits, 52 frac bits64 bits totalExtended precision: 15 exp bits, 63 frac bitsOnly found in Intel-compatible machinesStored in 80 bits»1 bit wastedFloating Point PrecisionsFloating Point Precisionss exp frac– 9 –15-213: Intro to Computer SystemsFall 2009 ©“Normalized” Numeric Values“Normalized” Numeric ValuesConditionConditionexp 000…0 and exp 111…1Exponent coded as Exponent coded as biasedbiased value valueE = Exp – BiasExp : 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 = 2e-1 - 1, where e is number of exponent bitsSignificand coded with implied leading 1Significand coded with implied leading 1M = 1.xxx…x2xxx…x: bits of fracMinimum when 000…0 (M = 1.0)Maximum when 111…1 (M = 2.0 – )Get extra leading bit for “free”– 10 –15-213: Intro to Computer SystemsFall 2009 ©Normalized Encoding ExampleNormalized Encoding ExampleValueValueFloat F = 15213.0;1521310 = 111011011011012 = 1.11011011011012 X 213SignificandSignificandM = 1.11011011011012frac= 110110110110100000000002ExponentExponentE = 13Bias = 127Exp = 140 = 100011002Floating Point Representation:Hex: 4 6 6 D B 4 0 0 Binary: 0100 0110 0110 1101 1011 0100 0000 0000140: 100 0110 015213: 1110 1101 1011 01– 11 –15-213: Intro to Computer SystemsFall 2009 ©Denormalized ValuesDenormalized ValuesConditionConditionexp = 000…0ValueValueExponent value E = –Bias + 1Significand value M = 0.xxx…x2xxx…x: bits of fracCasesCases exp = 000…0, frac = 000…0Represents value 0Note that have distinct values +0 and –0exp = 000…0, frac 000…0Numbers very close to 0.0Lose precision as get smaller“Gradual underflow”– 12 –15-213: Intro to Computer SystemsFall 2009 ©Special ValuesSpecial ValuesConditionConditionexp = 111…1CasesCases exp = 111…1, frac = 000…0Represents value(infinity)Operation that overflowsBoth positive and negativeE.g., 1.0/0.0 = 1.0/0.0 = +, 1.0/0.0 = exp = 111…1, frac 000…0Not-a-Number (NaN)Represents case when no numeric value can be determinedE.g., sqrt(–1), – 13 –15-213: Intro to Computer SystemsFall 2009 ©Summary of Floating Point Real Number EncodingsSummary of Floating Point Real Number EncodingsNaNNaN+0+Denorm +Normalized-Denorm-Normalized+0– 14 –15-213: Intro to Computer SystemsFall 2009 ©Tiny Floating Point ExampleTiny Floating Point Example8-bit Floating Point Representation8-bit Floating Point
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