Floating PointSept 4, 2003TopicsTopics IEEE Floating Point Standard Rounding Floating Point Operations Mathematical propertiesclass04.ppt15-213“The course that gives CMU its Zip!”– 2 –15-213, F’03Floating 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 –15-213, F’03IEEE 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 –15-213, F’03Fractional 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 –15-213, F’03Frac. 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 1.01/2 + 1/4 + 1/8 + … + 1/2i+ … →→→→ 1.0Use notation 1.0 – εεεε– 6 –15-213, F’03Representable 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’03Numerical FormNumerical Form –1sM 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 twoEncodingEncoding MSB is sign bit exp field encodes E frac field encodes MFloating Point RepresentationFloating Point Representations exp frac– 8 –15-213, F’03EncodingEncoding MSB is sign bit exp field encodes E frac field encodes MSizesSizes Single precision: 8 exp bits, 23 frac bits32 bits total Double precision: 11 exp bits, 52 frac bits64 bits total Extended 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, F’03“Normalized” Numeric Values“Normalized” Numeric ValuesConditionCondition exp ≠≠≠≠ 000…0 and exp ≠≠≠≠ 111…1Exponent coded as Exponent coded as biasedbiasedvaluevalueE = 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 bitsSignificandSignificandcoded with implied leading 1coded with implied leading 1M = 1.xxx…x2 xxx…x: bits of fracMinimum when 000…0 (M = 1.0)Maximum when 111…1 (M = 2.0 – εεεε)Get extra leading bit for “free”– 10 –15-213, F’03Normalized Encoding ExampleNormalized Encoding ExampleValueValueFloat F = 15213.0; 1521310= 111011011011012 = 1.11011011011012X 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 –15-213, F’03Denormalized 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 –15-213, F’03Special 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 –15-213, F’03Summary of Floating Point Real Number EncodingsSummary of Floating Point Real Number EncodingsNaNNaN+∞−∞−0+Denorm +Normalized-Denorm-Normalized+0– 14 –15-213, F’03Tiny 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 fracSame General Form as IEEE FormatSame General Form as IEEE Format normalized, denormalized representation of 0, NaN, infinitysexpfrac02367– 15 –15-213, F’03Values 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, NaN)– 16 –15-213, F’03Dynamic RangeDynamic Ranges exp frac E Value0 0000 000 -6 00 0000 001 -6 1/8*1/64 = 1/5120 0000 010 -6 2/8*1/64 = 2/512…0 0000 110 -6 6/8*1/64 = 6/5120 0000 111 -6 7/8*1/64 = 7/5120 0001 000 -6 8/8*1/64 = 8/5120 0001 001 -6 9/8*1/64 = 9/512…0 0110 110 -1 14/8*1/2 = 14/160 0110 111 -1 15/8*1/2 = 15/160 0111 000 0 8/8*1 = 10 0111 001 0 9/8*1 = 9/80 0111 010 0 10/8*1 = 10/8…0 1110 110 7
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