Floating PointTopicsTopics IEEE Floating Point Standard Rounding Floating Point Operations Mathematical propertiesSystems I2IEEE Floating PointIEEE Standard 754IEEE Standard 754 Established in 1985 as uniform standard for floating pointarithmetic 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 indefining standard3Fractional 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#4Frac. 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 1.01/2 + 1/4 + 1/8 + … + 1/2i + … → 1.0Use notation 1.0 – ε5Representable 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]…26Numerical 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 Representations exp frac7EncodingEncoding 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 Precisionss exp frac8“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 bitsSignificandSignificand coded with implied leading 1 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 – ε )Get extra leading bit for “free”9Normalized Encoding ExampleValueValueFloat 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 0Binary: 0100 0110 0110 1101 1011 0100 0000 0000140: 100 0110 015213: 1110 1101 1011 0110Denormalized 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”11Special 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), ∞ − ∞12Summary of Floating PointReal Number EncodingsNaNNaN+∞−∞−0+Denorm +Normalized-Denorm-Normalized+013Tiny 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, infinitysexp frac0236714Values 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)15Dynamic 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 14/8*128 = 2240 1110 111 7 15/8*128 = 2400 1111 000 n/a infclosest to zerolargest denormsmallest normclosest to 1 belowclosest to 1 abovelargest normDenormalizednumbersNormalizednumbers16Distribution of Values6-bit IEEE-like format6-bit IEEE-like format e = 3 exponent bits f = 2 fraction bits Bias is 3Notice how the distribution gets denser toward zero.Notice how the distribution gets denser toward zero.-15 -10 -5 0 5 10 15Denormalized Normalized Infinity17Distribution of Values(close-up view)6-bit IEEE-like format6-bit IEEE-like format e = 3 exponent bits f = 2 fraction bits Bias is 3-1 -0.5 0 0.5 1Denormalized Normalized Infinity18Interesting NumbersDescriptionDescriptionexpexpfracfracNumeric ValueNumeric ValueZeroZero0000……00000000……00000.00.0Smallest Pos. Smallest Pos. DenormDenorm..0000……00000000……010122–– {23,52}{23,52} X 2 X 2–– {126,1022}{126,1022} Single ≈ 1.4 X 10–45 Double ≈ 4.9 X 10–324Largest Largest DenormalizedDenormalized0000……00001111……1111(1.0 (1.0 –– εε) X 2) X 2–– {126,1022}{126,1022} Single ≈ 1.18 X 10–38 Double ≈ 2.2 X 10–308Smallest Pos. NormalizedSmallest Pos. Normalized0000……01010000……00001.0 X 21.0 X 2–– {126,1022}{126,1022} Just larger than largest denormalizedOneOne0101……11110000……00001.01.0