Floating Point Sept 6, 2006Floating 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 PointSept 6, 2006TopicsTopics IEEE Floating Point Standard Rounding Floating Point Operations Mathematical propertiesclass03.ppt15-213“The course that gives CMU its Zip!”15-213, F’06–2–15-213, F’06Floating Point PuzzlesFloating Point Puzzles For each of the following C expressions, either:z Argue that it is true for all argument valuesz 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–3–15-213, F’06IEEE Floating PointIEEE Floating PointIEEE Standard 754IEEE Standard 754 Established in 1985 as uniform standard for floating point arithmeticz 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 fastz Numerical analysts predominated over hardware types in defining standard–4–15-213, F’06Fractional Binary NumbersFractional Binary NumbersRepresentationRepresentation Bits to right of “binary point” represent fractional powers of 2 Represents rational number:bibi–1b2b1b0b–1b–2b–3b–j••••••.1242i2i–1••••••1/21/41/82–jbk⋅2kk=−ji∑–5–15-213, F’06Frac. Binary Number ExamplesFrac. 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 just below 1.0z1/2 + 1/4 + 1/8 + … + 1/2i+ … → 1.0zUse notation 1.0 – ε–6–15-213, F’06Representable 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]…2–7–15-213, F’06Numerical FormNumerical Form –1sM 2EzSign bit s determines whether number is negative or positivezSignificand M normally a fractional value in range [1.0,2.0).zExponent 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–8–15-213, F’06EncodingEncoding MSB is sign bit exp field encodes E frac field encodes MSizesSizes Single precision: 8 exp bits, 23 frac bitsz32 bits total Double precision: 11 exp bits, 52 frac bitsz64 bits total Extended precision: 15 exp bits, 63 frac bitszOnly found in Intel-compatible machineszStored in 80 bits» 1 bit wastedFloating Point PrecisionsFloating Point Precisionss exp frac–9–15-213, F’06“Normalized” Numeric Values“Normalized” Numeric ValuesConditionCondition exp ≠ 000…0 and exp ≠ 111…1Exponent coded as Exponent coded as biasedbiasedvaluevalueE = Exp – BiaszExp : unsigned value denoted by exp zBias : 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 bitsSignificandSignificandcoded with implied leading 1coded with implied leading 1M = 1.xxx…x2z xxx…x: bits of fraczMinimum when 000…0 (M = 1.0)zMaximum when 111…1 (M = 2.0 – ε)zGet extra leading bit for “free”–10–15-213, F’06Normalized Encoding ExampleNormalized Encoding ExampleValueValueFloat F = 15213.0; 1521310= 111011011011012 = 1.11011011011012X 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, F’06Denormalized ValuesDenormalized ValuesConditionCondition exp = 000…0ValueValue Exponent value E = –Bias + 1 Significand value M = 0.xxx…x2z xxx…x: bits of fracCasesCases exp = 000…0, frac = 000…0z Represents value 0z Note that have distinct values +0 and –0 exp = 000…0, frac ≠ 000…0z Numbers very close to 0.0z Lose precision as get smallerz “Gradual underflow”–12–15-213, F’06Special ValuesSpecial ValuesConditionCondition exp = 111…1CasesCases exp = 111…1, frac = 000…0z Represents value ∞ (infinity)z Operation that overflowsz Both positive and negativez E.g., 1.0/0.0 = −1.0/−0.0 = +∞, 1.0/−0.0 = −∞ exp = 111…1, frac ≠ 000…0z Not-a-Number (NaN)z Represents case when no numeric value can be determinedz E.g., sqrt(–1), ∞ − ∞, ∞ ∗ 0–13–15-213, F’06Summary of Floating Point Real Number EncodingsSummary of Floating Point Real Number EncodingsNaNNaN+∞−∞−0+Denorm +Normalized-Denorm-Normalized+0–14–15-213, F’06Tiny Floating Point ExampleTiny Floating Point Example88--bit Floating Point Representationbit 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 fraczzSame General Form as IEEE FormatSame General Form as IEEE Format normalized, denormalized representation of 0, NaN, infinitysexpfrac02367–15–15-213, F’06Values 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 1/2701110181000+1291001+2410 1010 +3 811 1011 +4
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