Floating Point ArithmeticSept 5, 2007Floating 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 BehaviorSummary15-213Floating Point ArithmeticSept 5, 2007TopicsTopics IEEE Floating Point Standard Rounding Floating Point Operations Mathematical propertiesclass03.ppt15-213 F’07Floating 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–2–15-213: Intro to Computer SystemsFall 2007 ©• 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 NaNIEEE Floating PointIEEE Floating Point–3–15-213: Intro to Computer SystemsFall 2007 ©IEEE 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 standardFractional Binary NumbersFractional Binary Numbersbibi–1b2b1b0b–1b–2b–3b–j••••••.1242i2i–1••••••1/21/41/82–jRepresentationRepresentation Bits to right of “binary point” represent fractional powers of 2 Represents rational number:bk⋅2kk=−ji∑–4–15-213: Intro to Computer SystemsFall 2007 ©Frac. Binary Number ExamplesFrac. Binary Number Examples–5–15-213: Intro to Computer SystemsFall 2007 ©ValueValueRepresentationRepresentation5-3/4 101.1122-7/8 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 – εRepresentable NumbersRepresentable Numbers–6–15-213: Intro to Computer SystemsFall 2007 ©LimitationLimitation 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]…2Floating Point RepresentationFloating Point Representation–7–15-213: Intro to Computer SystemsFall 2007 ©Numerical 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 Ms exp fracFloating Point PrecisionsFloating Point Precisions–8–15-213: Intro to Computer SystemsFall 2007 ©EncodingEncoding 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 wasteds exp frac“Normalized” Numeric Values“Normalized” Numeric Values–9–15-213: Intro to Computer SystemsFall 2007 ©ConditionCondition 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”Normalized Encoding ExampleNormalized Encoding Example–10–15-213: Intro to Computer SystemsFall 2007 ©ValueValueFloat 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 01Denormalized ValuesDenormalized Values–11–15-213: Intro to Computer SystemsFall 2007 ©ConditionCondition 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”Special ValuesSpecial Values–12–15-213: Intro to Computer SystemsFall 2007 ©ConditionCondition 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), ∞ − ∞, ∞ ∗ 0Summary of Floating Point Real Number EncodingsSummary of Floating Point Real Number Encodings–13–15-213: Intro to Computer SystemsFall 2007 ©NaNNaN+∞−∞-Normalized +Denorm +Normalized-Denorm−0+0Tiny Floating Point ExampleTiny Floating Point Example–14–15-213: Intro to Computer SystemsFall 2007 ©88--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
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