Floating Point Numbers

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1 Floating Point Numbers Summer 8 Fractional numbers Fractional numbers fixed point Floating point numbers the IEEE 7 floating point standard Floating point operations Rounding modes CMPE Summer 8 Slides by ADB
2 Positional representation of fractional numbers In base Multiplier Number Position Decimal point CMPE Summer 8 Slides by ADB Positional representation of fractional numbers In base Multiplier Number Position Binary point CMPE Summer 8 Slides by ADB
3 Fractional numbers fixed point Fixedpoint representation How much information is necessary to store? How do you choose a format for the bits? Fixedpoint operations Addition Align binary points, and add straight down Multiplication??? CMPE Summer 8 Slides by ADB Decimal to binary conversion Convert A =. to base CMPE Summer 8 Slides by ADB
4 Fixedpoint number density CMPE Summer 8 Slides by ADB 7 Scientific notation In base Example:. 8 In base Example:. (= 8. ) The general form r = Sign significand base Exponent CMPE Summer 8 Slides by ADB 8
5 Singleprecision IEEE 7 floatingpoint numbers e x p o n e n t Sign significand Onebit sign Eightbit exponent bit significand That s the fractional part CMPE Summer 8 Slides by ADB 9 Singleprecision IEEE 7 floatingpoint numbers Normalized numbers: only one nonzero bit to the left of the binary point Adjust the exponent as needed r = ( ) S F E Implicit leading in the significand (the hidden bit ) r = ( ) S ( + F) E Bias notation to represent the exponent With the bias B = 7 r = ( ) S ( + F) EB CMPE Summer 8 Slides by ADB
6 How to convert a base number into IEEE 7 singleprecision floating point Convert the number to binary The big part And the fractional part Normalize Isolate the hidden one Remove the significand s hidden one Add bias to the exponent Represent the number CMPE Summer 8 Slides by ADB Example: Sign Convert to binary Normalize Remove hidden one Add bias to exponent The end significand CMPE Summer 8 Slides by ADB
7 Doubleprecision IEEE 7 floatingpoint numbers e x p o n e n t Sign significand significand, continued Onebit sign Elevenbit exponent bit significand That s the fractional part CMPE Summer 8 Slides by ADB Summary of IEEE 7 formats Precision Parameter Single Single Ext. Double Double Ext. Bits for F Bits for E 8 E max E min 8 Total bits 79 For every precision, there are reserved exponents, used for special quantities: E min (i.e., E=) is used for zero and denorms E max + (i.e., E= or E=7, with bias) is used for NaN and infinity CMPE Summer 8 Slides by ADB 7
8 Special quantities: Infinity This special quantity avoids halt on overflow Much safer than returning the largest possible number Representation: E = E max + E = in single precision with bias E = 7 in double precision with bias F = Sign (+ or ) Sign significand CMPE Summer 8 Slides by ADB Special quantities: Infinity Examples of operations that return ±Inf Operation / / Inf Sqrt (+Inf) / Inf Result CMPE Summer 8 Slides by ADB 8
9 Special quantities: NaN (Not a Number) This special quantity avoids halt on invalid operations Representation: E = E max + E = in single precision with bias E = 7 in double precision with bias F Sign significand CMPE Summer 8 Slides by ADB 7 Special quantities: NaN (Not a Number) Examples of operations that return NaN Operation Result +/ / /+ / / Inf log(+) log( ) CMPE Summer 8 Slides by ADB 8 9
10 Special quantities: Zero Representation: E = E min (i.e., E = ) F = Sign: + or Sign significand CMPE Summer 8 Slides by ADB 9 Special quantities: Zero Examples of operations that involve ± Operation + / Sqrt NaN produced by CMPE Summer 8 Slides by ADB
11 Floating point numbers range What is the largest number we can represent in IEEE 7 singleprecision floating point? What is the smallest number? Sign significand CMPE Summer 8 Slides by ADB Floating point numbers range What is the largest number we can represent in IEEE 7 doubleprecision floating point? What is the smallest number? CMPE Summer 8 Slides by ADB
12 Floating point numbers: density Fact : Floats are not reals E.g., / Fact : Floats are not decimals E.g.,. (base ) =. (base ) Fact : Not even all the integers in the range are represented E.g.,,, (base ) = (base ) CMPE Summer 8 Slides by ADB Floating point numbers: density Close to : high density Far from : low density CMPE Summer 8 Slides by ADB
13 Special quantities: Denormals These are numbers smaller than ^(E min ) Fill the gap between ^(E min ) and (gradual underflow) Representation: E = E min (i.e., E = ) F The number represented is.f  f  f  ^(E min ) Sign significand CMPE Summer 8 Slides by ADB Special quantities: Denormals From to ^(E min ) CMPE Summer 8 Slides by ADB
14 Summary of IEEE 7 numbers Single precision Exponent Significand Double precision Exponent Significand Object anything anything 7 7 CMPE Summer 8 Slides by ADB 7 Floating point operations: addition Addition: C = A + B Step : Align the significands Are the exponents different? Shift the smaller number to the right and adjust the exponent Step : Add the significands Don t forget the sign Step : Normalize the sum Check for overflow or underflow CMPE Summer 8 Slides by ADB 8
15 Floating point operations: addition A =. (base ) B =. (base ) Step : Align the significands Step : Add the significands Step : Normalize the sum CMPE Summer 8 Slides by ADB 9 Floating point operations: multiplication C = A B Step : Add the exponents Step : Multiply the significands Without the sign Step : Normalize the product Step : Set the sign of the product Both positive? Result is positive Both negative? Result is positive Different signs? Result is negative CMPE Summer 8 Slides by ADB
16 Floating point operations: multiplication A =. (base ) B =. (base ) Step : Add the exponents Step : Multiply the significands Step : Normalize the product Step : Set the sign of the product CMPE Summer 8 Slides by ADB Rounding What is rounding, and when do we need to round? Numbers that we can represent, and numbers that we can not represent: What is fair? CMPE Summer 8 Slides by ADB
17 Rounding modes IEEE 7 defines four rounding modes Towards +Inf Towards Inf Towards To nearest CMPE Summer 8 Slides by ADB Rounding modes We need some extra bits In IEEE7, the internal representation uses at least more bits in the significand Guard Round Sticky CMPE Summer 8 Slides by ADB 7
18 Rounding modes CMPE Summer 8 Slides by ADB Recommended exercises Ex. A: Represent the binary number A =. in IEEE 7 singleprecision and doubleprecision. Ex. B: Give the IEEE 7 singleprecision representation of the number (thirtyfour million and one). Give its rounded representation using all four rounding modes, and for each one, give the decimal value of the number actually represented. CMPE Summer 8 Slides by ADB 8
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