Harvey Mudd CS 105 - Floating Point

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Floating PointFloating Point PuzzlesIEEE Floating PointFractional Binary NumbersFractional 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 MultFloating Point in CAnswers to Floating Point PuzzlesAriane 5SummaryFloating PointTopicsTopicsIEEE Floating-Point StandardRoundingFloating-Point OperationsMathematical PropertiesCS 105“Tour of the Black Holes of Computing!”– 2 –CS 105Floating Point PuzzlesFloating Point PuzzlesFor each of the following C expressions, either:Argue that it is true for all argument valuesExplain 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 –CS 105IEEE Floating PointIEEE Floating PointIEEE Standard 754IEEE Standard 754Established in 1985 as uniform standard for floating point arithmeticBefore that, many idiosyncratic formatsSupported by all major CPUsDriven by numerical concernsDriven by numerical concernsNice standards for rounding, overflow, underflowHard to make go fastNumerical analysts predominated over hardware types in defining standard– 4 –CS 105Fractional Binary NumbersFractional Binary NumbersRepresentationRepresentationBits to right of “binary point” represent fractional powers of 2Represents rational number:bibi–1b2b1b0b–1b–2b–3b–j• • •• • • .1242i–12i• • •• • •1/21/41/82–jbk2kk  ji– 5 –CS 105Fractional Binary Number ExamplesFractional Binary Number ExamplesValueValueRepresentationRepresentation5-3/4 101.1122-7/8 10.111263/64 0.1111112ObservationsObservationsDivide by 2 by shifting rightMultiply by 2 by shifting leftNumbers of form 0.111111…2 are just below 1.01/2 + 1/4 + 1/8 + … + 1/2i + …  1.0Use notation 1.0 – – 6 –CS 105Representable NumbersRepresentable NumbersLimitationLimitationCan only exactly represent numbers of the form x/2kOther numbers have repeating bit representationsValueValueRepresentationRepresentation1/3 0.0101010101[01]…21/5 0.001100110011[0011]…21/10 0.0001100110011[0011]…2– 7 –CS 105Numerical FormNumerical Form–1s 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 twoEncodingEncodingMSB is sign bitexp field encodes Efrac field encodes MFloating-Point RepresentationFloating-Point Representations exp frac– 8 –CS 105EncodingEncodingMSB is sign bitexp field encodes Efrac field encodes MSizesSizesSingle precision: 8 exp bits, 23 frac bits32 bits totalDouble precision: 11 exp bits, 52 frac bits64 bits totalExtended precision: 15 exp bits, 63 frac bitsOnly found in Intel-compatible machinesStored in 80 bits»1 bit wastedFloating-Point PrecisionsFloating-Point Precisionss exp frac– 9 –CS 105“Normalized” Numeric Values“Normalized” Numeric ValuesConditionConditionexp  000…0 and exp  111…1Exponent coded as Exponent coded as biasedbiased value 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 = 2e-1 - 1, where e is number of exponent bitsSignificand coded with implied leading 1Significand coded with implied leading 1M = 1.xxx…x2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”– 10 –CS 105Normalized Encoding Example Normalized Encoding Example ValueValueFloat F = 15213.0;1521310 = 111011011011012 = 1.11011011011012 X 213SignificandSignificandM = 1.11011011011012frac= 110110110110100000000002ExponentExponentE = 13Bias = 127Exp = 140 = 100011002Floating-Point Representation (Class 02):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 –CS 105Denormalized ValuesDenormalized ValuesConditionConditionexp = 000…0ValueValueExponent value E = –Bias + 1Significand value M = 0.xxx…x2xxx…x: bits of fracCasesCases 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Lose precision as get smaller“Gradual underflow”– 12 –CS 105Special ValuesSpecial ValuesConditionConditionexp = 111…1CasesCases 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), – 13 –CS 105Summary of Floating-Point Real Number EncodingsSummary of Floating-Point Real Number EncodingsNaNNaN+0+Denorm +Normalized-Denorm-Normalized+0– 14 –CS 105Tiny Floating Point ExampleTiny Floating Point Example8-bit Floating Point Representation8-bit Floating Point RepresentationThe sign bit is in the most significant bit.The next four bits are the exponent, with a bias of 7.The last three bits are the fracSame General Form as IEEE FormatSame General Form as IEEE FormatNormalized, denormalizedRepresentation of 0, NaN, infinitys exp frac02367– 15 –CS 105Values Related to the ExponentValues Related to the ExponentExp exp E 2E0 0000 -6 1/64 (denorms)1 0001 -6 1/642 0010 -5 1/323 0011 -4 1/164 0100 -3 1/85 0101 -2 1/46 0110 -1 1/27 0111 0 18 1000 +1 29 1001 +2 410 1010 +3 811 1011 +4 1612 1100 +5 3213 1101 +6 6414 1110 +7 12815 1111 n/a (inf,


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Harvey Mudd CS 105 - Floating Point

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