Images courtesy of Addison Wesley Longman, Inc. Copyright © 2001 Chapter 8 Computer Arithmetic.

Images courtesy of Addison Wesley Longman, Inc. Copyright © 2001 Images courtesy of Addison Wesley Longman, Inc. Copyright © 2001

Chapter 8Chapter 8Computer ArithmeticComputer Arithmetic


Chapter OutlineChapter Outline

• Unsigned notationsUnsigned notations

• Signed notationsSigned notations

• Binary Coded DecimalBinary Coded Decimal

• Specialized arithmetic hardwareSpecialized arithmetic hardware

• Floating point numbersFloating point numbers

• IEEE 754 floating point standardIEEE 754 floating point standard


Unsigned NotationsUnsigned Notations

• Unsigned non-negativeUnsigned non-negative

• Unsigned twoUnsigned two’’s-complements-complement


Addition: X Addition: X X + Y X + Y


OverflowOverflow


Subtraction: X Subtraction: X X - Y X - Y


OverflowOverflow


MultiplicationMultiplication

• A non-optimal methodA non-optimal method

z = 0 z = 0

FOR i = 1 TO y DO { z = z + x }FOR i = 1 TO y DO { z = z + x }



• A more typical methodA more typical method



• Calculating running totalsCalculating running totals



• Shifting partial results to align sumsShifting partial results to align sums


Shift-add Multiplication Shift-add Multiplication AlgorithmAlgorithm

C = 0, U = 0;


Example: UV Example: UV X X Y Y(X = 1101, Y = 1001)(X = 1101, Y = 1001)

C = 0, U = 0 0


RTL CodeRTL Code

C 0, U 0, i n


Example: UV Example: UV X X Y Y(X = 1101, Y = 1001)(X = 1101, Y = 1001)

C 0, U 0, i 4 0


Hardware ImplementationHardware Implementation


Optimizing the RTL CodeOptimizing the RTL Code

• UV UV X X V V

• Register Y not neededRegister Y not needed

• One operand is lostOne operand is lost


Optimizing the RTL CodeOptimizing the RTL Code

C 0, U 0


ExampleExample

C 0, U 0, i 4 0


BoothBooth’’s Algorithms Algorithm• Multiplying unsigned 2Multiplying unsigned 2’’s-complement s-complement

numbersnumbers


ExampleExample• UV UV X X Y, X = -3 (1101), Y = -5 Y, X = -3 (1101), Y = -5

(1011)(1011)


RTL CodeRTL Code


ExampleExample


Optimized RTL CodeOptimized RTL Code


DivisionDivision

• A non-optimal methodA non-optimal method


DivisionDivision• A more typical methodA more typical method


DivisionDivision• Shifting results to align remaindersShifting results to align remainders


Shift-subtract Division AlgorithmShift-subtract Division Algorithm


Example: UV Example: UV X X(UV = 1001 0011, X = 1101)(UV = 1001 0011, X = 1101)


RTL CodeRTL Code


ExampleExample


Restoring Division AlgorithmRestoring Division Algorithm


Overflow ComparisonOverflow Comparison


ExampleExample


RTL CodeRTL Code


ExampleExample


Signed NotationsSigned Notations

• Signed-magnitudeSigned-magnitude




• Signed-2Signed-2’’s complements complement




• Signed-2Signed-2’’s complements complementValue Signed-magnitude Signed-2Value Signed-magnitude Signed-2’’s complements complement

+3 0 0011 0 0011+3 0 0011 0 0011

-3 1 0011 1 1101-3 1 0011 1 1101


Addition and SubtractionAddition and Subtraction


RTL CodeRTL Code


ExamplesExamples


ExampleExample


Binary Coded Decimal (BCD)Binary Coded Decimal (BCD)

• Every 4 bits = 1 decimal digitEvery 4 bits = 1 decimal digit

• 1 bit sign1 bit sign

• Example: +27 = 0 0010 0111Example: +27 = 0 0010 0111


BCD AdderBCD Adder


NineNine’’s Complements Complement• Equivalent of 1Equivalent of 1’’s complement in binarys complement in binary


RTL CodeRTL Code


ExamplesExamples



• dshr: decimal shift rightdshr: decimal shift right


ExampleExample


Arithmetic PipelinesArithmetic Pipelines

• Increase throughputIncrease throughput


Arithmetic PipelinesArithmetic Pipelines

• Increase throughputIncrease throughput

• Speedup:Speedup:


ExampleExample


Steady State SpeedupSteady State Speedup


Speedup with Latch DelaysSpeedup with Latch Delays


Lookup TablesLookup Tables


ExampleExample


Wallace TreesWallace Trees

• Carry Save AdderCarry Save Adder


Wallace TreesWallace Trees• Partial productsPartial products


Wallace TreesWallace Trees

• Partial productsPartial products


ExampleExample


4 4 4 Wallace Tree 4 Wallace Tree


8 8 8 Wallace Tree 8 Wallace Tree


Floating Point NumbersFloating Point Numbers

• Sign, Significand, ExponentSign, Significand, Exponent

• Normalized:Normalized: -1234.5678 = -.12345678 -1234.5678 = -.12345678 10 1044

• NaNNaN

• BiasingBiasing


Floating Point NumbersFloating Point Numbers

• PrecisionPrecision

• GapGap

• RangeRange

• Round, Guard, Sticky bitsRound, Guard, Sticky bits


RoundingRounding

• Round toward nearestRound toward nearest

• Round toward 0Round toward 0

• Round toward +Round toward +• Round toward -Round toward -


ExampleExample


Addition and SubtractionAddition and Subtraction


RTL CodeRTL Code


Example: (.1101 Example: (.1101 2 233) + (.1110 ) + (.1110 2222))


Example: (.1101 Example: (.1101 2 233) - (.1110 ) - (.1110 2222))



• Check for special valuesCheck for special values

• Add exponentsAdd exponents

• Multiply significandsMultiply significands

• Normalize resultNormalize result


RTL CodeRTL Code


Example: (.1101 Example: (.1101 2 233) ) (.1110 (.1110 2222))


IEEE 754 Floating Point IEEE 754 Floating Point StandardStandard

• 11 significand < 2 significand < 2

• Single of double precisionSingle of double precision


IEEE 754 Floating Point StandardIEEE 754 Floating Point Standard


Denormalized ValuesDenormalized Values

• Used to express smaller numbers than Used to express smaller numbers than possible with normalized notationpossible with normalized notation

• Smallest normalized value: 2Smallest normalized value: 2-126-126 (single (single precision)precision)

• Smallest denormalized value: 2Smallest denormalized value: 2-149-149 (single precision)(single precision)


SummarySummary

• Unsigned notationsUnsigned notations

• Signed notationsSigned notations

• Binary Coded DecimalBinary Coded Decimal

• Specialized arithmetic hardwareSpecialized arithmetic hardware

• Floating point numbersFloating point numbers

• IEEE 754 floating point standardIEEE 754 floating point standard

Images courtesy of Addison Wesley Longman, Inc. Copyright © 2001 Chapter 8 Computer Arithmetic.

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Transcript of Images courtesy of Addison Wesley Longman, Inc. Copyright © 2001 Chapter 8 Computer Arithmetic.