Floating Point Arithmetic Feb 17, 2000Floating Point PuzzlesIEEE Floating PointFractional Binary NumbersFractional Binary Number ExamplesFloating Point Representation“Normalized” Numeric ValuesNormalized Encoding ExampleDenormalized ValuesInteresting NumbersSpecial ValuesSummary of Floating Point Real Number EncodingsTiny floating point exampleValues related to the exponentDynamic RangeSpecial Properties of EncodingFloating Point OperationsA Closer Look at Round-To-EvenRounding Binary NumbersFP MultiplicationFP AdditionMathematical Properties of FP AddAlgebraic Properties of FP MultFloating Point in CAnswers to Floating Point Puzzlesx86 Floating PointFPU Data Register StackFPU instructionsFloating Point Code ExampleInner product stack traceFloating Point ArithmeticFeb 17, 2000Topics•IEEE Floating Point Standard•Rounding•Floating Point Operations•Mathematical properties•IA32 floating pointclass10.ppt15-213“The course that gives CMU its Zip!”CS 213 S’00– 2 –class10.pptFloating Point Puzzles•For each of the following C expressions, either:–Argue that 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 NANCS 213 S’00– 3 –class10.pptIEEE Floating PointIEEE Standard 754•Estabilished in 1985 as uniform standard for floating point arithmetic–Before that, many idiosyncratic formats•Supported by all major CPUsDriven by Numerical Concerns•Nice standards for rounding, overflow, underflow•Hard to make go fast–Numerical analysts predominated over hardware types in defining standardCS 213 S’00– 4 –class10.pptFractional Binary NumbersRepresentation•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–jbk2kk jiCS 213 S’00– 5 –class10.pptFractional Binary Number ExamplesValue Representation5-3/4 101.1122-7/8 10.111263/64 0.1111112Observation•Divide by 2 by shifting right•Numbers of form 0.111111…2 just below 1.0–Use notation 1.0 – Limitation•Can only exactly represent numbers of the form x/2k•Other numbers have repeating bit representationsValue Representation1/3 0.0101010101[01]…21/5 0.001100110011[0011]…21/10 0.0001100110011[0011]…2CS 213 S’00– 6 –class10.pptNumerical 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 twoEncoding•MSB is sign bit•exp field encodes E•frac field encodes MSizes•Single precision: 8 exp bits, 23 frac bits–32 bits total•Double precision: 11 exp bits, 52 frac bits–64 bits totalFloating Point Representations exp fracCS 213 S’00– 7 –class10.ppt“Normalized” Numeric ValuesCondition•exp 000…0 and exp 111…1Exponent coded as biased valueE = 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 = 2m-1 - 1, where m is the number of exponent bitsSignificand coded with implied leading 1m = 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”CS 213 S’00– 8 –class10.pptNormalized Encoding ExampleValueFloat F = 15213.0;•1521310 = 111011011011012 = 1.11011011011012 X 213SignificandM = 1.11011011011012frac= 110110110110100000000002ExponentE = 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 01CS 213 S’00– 9 –class10.pptDenormalized ValuesCondition•exp = 000…0Value•Exponent value E = –Bias + 1•Significand value m = 0.xxx…x2–xxx…x: bits of fracCases• 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”CS 213 S’00– 10 –class10.pptInteresting NumbersDescription exp frac Numeric ValueZero 00…00 00…00 0.0Smallest Pos. Denorm. 00…00 00…01 2– {23,52} X 2– {126,1022}•Single 1.4 X 10–45•Double 4.9 X 10–324Largest Denormalized 00…00 11…11 (1.0 – ) X 2– {126,1022}•Single 1.18 X 10–38•Double 2.2 X 10–308Smallest Pos. Normalized 00…01 00…00 1.0 X 2– {126,1022}•Just larger than largest denormalizedOne 01…11 00…00 1.0 Largest Normalized 11…10 11…11 (2.0 – ) X 2{127,1023}•Single 3.4 X 1038•Double 1.8 X 10308CS 213 S’00– 11 –class10.pptSpecial ValuesCondition•exp = 111…1Cases• 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), CS 213 S’00– 12 –class10.pptSummary of Floating Point Real Number EncodingsNaNNaN+--0 +0+Denorm +Normalized-Denorm-NormalizedCS 213 S’00– 13 –class10.pptTiny floating point exampleAssume an 8-bit floating point representation where•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•Otherwise, the rules are the same as IEEE floating point format (normalized, denormalized, representation of 0, NaN, infinity)s exp frac02367CS 213 S’00– 14 –class10.pptValues related to the exponentExp exp E0 0000 -6 (denorms)1 0001 -62 0010 -53 0011 -44 0100 -35 0101 -26 0110 -17 0111 08 1000 19 1001 210 1010 311 1011 412 1100 513 1101 614 1110 715 1111 (inf, Nan)CS 213 S’00– 15 –class10.pptDynamic Rangeexp E value0 0000 000 n/a 00 0000 001 -6 1/5120 0000 010 -6 2/512…0 0000 110 -6 6/5120 0000 111 -6 7/5120 0001 000 -6 8/5120 0001 001 -6 9/512…0 0110 110 -1 28/320 0110 111 -1 30/320
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