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Images courtesy of Addison Wesley Longman, Inc. Copyright © 2001 Images courtesy of Addison Wesley Longman, Inc. Copyright © 2001
Chapter 8Chapter 8Computer ArithmeticComputer Arithmetic
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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
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Unsigned NotationsUnsigned Notations
• Unsigned non-negativeUnsigned non-negative
• Unsigned twoUnsigned two’’s-complements-complement
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Unsigned NotationsUnsigned Notations
• Unsigned non-negativeUnsigned non-negative
• Unsigned twoUnsigned two’’s-complements-complement
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Addition: X Addition: X X + Y X + Y
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OverflowOverflow
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Subtraction: X Subtraction: X X - Y X - Y
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OverflowOverflow
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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 }
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MultiplicationMultiplication
• A more typical methodA more typical method
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MultiplicationMultiplication
• Calculating running totalsCalculating running totals
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MultiplicationMultiplication
• Shifting partial results to align sumsShifting partial results to align sums
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Shift-add Multiplication Shift-add Multiplication AlgorithmAlgorithm
C = 0, U = 0;
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Example: UV Example: UV X X Y Y(X = 1101, Y = 1001)(X = 1101, Y = 1001)
C = 0, U = 0 0
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RTL CodeRTL Code
C 0, U 0, i n
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Example: UV Example: UV X X Y Y(X = 1101, Y = 1001)(X = 1101, Y = 1001)
C 0, U 0, i 4 0
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Hardware ImplementationHardware Implementation
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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
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Optimizing the RTL CodeOptimizing the RTL Code
C 0, U 0
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ExampleExample
C 0, U 0, i 4 0
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BoothBooth’’s Algorithms Algorithm• Multiplying unsigned 2Multiplying unsigned 2’’s-complement s-complement
numbersnumbers
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ExampleExample• UV UV X X Y, X = -3 (1101), Y = -5 Y, X = -3 (1101), Y = -5
(1011)(1011)
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ExampleExample• UV UV X X Y, X = -3 (1101), Y = -5 Y, X = -3 (1101), Y = -5
(1011)(1011)
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RTL CodeRTL Code
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ExampleExample
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Optimized RTL CodeOptimized RTL Code
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Hardware ImplementationHardware Implementation
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DivisionDivision
• A non-optimal methodA non-optimal method
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DivisionDivision• A more typical methodA more typical method
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DivisionDivision• Shifting results to align remaindersShifting results to align remainders
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Shift-subtract Division AlgorithmShift-subtract Division Algorithm
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Example: UV Example: UV X X(UV = 1001 0011, X = 1101)(UV = 1001 0011, X = 1101)
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RTL CodeRTL Code
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ExampleExample
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Hardware ImplementationHardware Implementation
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Restoring Division AlgorithmRestoring Division Algorithm
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Overflow ComparisonOverflow Comparison
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ExampleExample
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RTL CodeRTL Code
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ExampleExample
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Hardware ImplementationHardware Implementation
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Signed NotationsSigned Notations
• Signed-magnitudeSigned-magnitude
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Signed NotationsSigned Notations
• Signed-magnitudeSigned-magnitude
• Signed-2Signed-2’’s complements complement
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Signed NotationsSigned Notations
• Signed-magnitudeSigned-magnitude
• 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
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Addition and SubtractionAddition and Subtraction
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RTL CodeRTL Code
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RTL CodeRTL Code
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ExamplesExamples
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ExamplesExamples
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ExamplesExamples
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Hardware ImplementationHardware Implementation
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MultiplicationMultiplication
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ExampleExample
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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
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BCD AdderBCD Adder
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NineNine’’s Complements Complement• Equivalent of 1Equivalent of 1’’s complement in binarys complement in binary
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NineNine’’s Complements Complement• Equivalent of 1Equivalent of 1’’s complement in binarys complement in binary
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RTL CodeRTL Code
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ExamplesExamples
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ExamplesExamples
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ExamplesExamples
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Hardware ImplementationHardware Implementation
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MultiplicationMultiplication
• dshr: decimal shift rightdshr: decimal shift right
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MultiplicationMultiplication
• dshr: decimal shift rightdshr: decimal shift right
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ExampleExample
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Arithmetic PipelinesArithmetic Pipelines
• Increase throughputIncrease throughput
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Arithmetic PipelinesArithmetic Pipelines
• Increase throughputIncrease throughput
• Speedup:Speedup:
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ExampleExample
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ExampleExample
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ExampleExample
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Steady State SpeedupSteady State Speedup
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Steady State SpeedupSteady State Speedup
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Speedup with Latch DelaysSpeedup with Latch Delays
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Speedup with Latch DelaysSpeedup with Latch Delays
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Lookup TablesLookup Tables
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ExampleExample
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Wallace TreesWallace Trees
• Carry Save AdderCarry Save Adder
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Wallace TreesWallace Trees
• Carry Save AdderCarry Save Adder
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Wallace TreesWallace Trees• Partial productsPartial products
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Wallace TreesWallace Trees
• Partial productsPartial products
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ExampleExample
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ExampleExample
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4 4 4 Wallace Tree 4 Wallace Tree
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8 8 8 Wallace Tree 8 Wallace Tree
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Floating Point NumbersFloating Point Numbers
• Sign, Significand, ExponentSign, Significand, Exponent
• Normalized:Normalized: -1234.5678 = -.12345678 -1234.5678 = -.12345678 10 1044
• NaNNaN
• BiasingBiasing
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Floating Point NumbersFloating Point Numbers
• PrecisionPrecision
• GapGap
• RangeRange
• Round, Guard, Sticky bitsRound, Guard, Sticky bits
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RoundingRounding
• Round toward nearestRound toward nearest
• Round toward 0Round toward 0
• Round toward +Round toward +• Round toward -Round toward -
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ExampleExample
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Addition and SubtractionAddition and Subtraction
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Addition and SubtractionAddition and Subtraction
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RTL CodeRTL Code
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Example: (.1101 Example: (.1101 2 233) + (.1110 ) + (.1110 2222))
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Example: (.1101 Example: (.1101 2 233) - (.1110 ) - (.1110 2222))
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MultiplicationMultiplication
• Check for special valuesCheck for special values
• Add exponentsAdd exponents
• Multiply significandsMultiply significands
• Normalize resultNormalize result
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RTL CodeRTL Code
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Example: (.1101 Example: (.1101 2 233) ) (.1110 (.1110 2222))
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IEEE 754 Floating Point IEEE 754 Floating Point StandardStandard
• 11 significand < 2 significand < 2
• Single of double precisionSingle of double precision
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IEEE 754 Floating Point StandardIEEE 754 Floating Point Standard
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IEEE 754 Floating Point StandardIEEE 754 Floating Point Standard
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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)
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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