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

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

Images courtesy of Addison Wesley Longman, Inc. Copyright © 2001 Unsigned Notations Unsigned non-negativeUnsigned non-negative Unsigned two’s-complementUnsigned two’s-complement

Images courtesy of Addison Wesley Longman, Inc. Copyright © 2001 Unsigned Notations Unsigned non-negativeUnsigned non-negative Unsigned two’s-complementUnsigned two’s-complement

Images courtesy of Addison Wesley Longman, Inc. Copyright © 2001 Addition: X  X + Y

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Subtraction: X  X - Y

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Multiplication A non-optimal methodA non-optimal method z = 0 FOR i = 1 TO y DO { z = z + x }

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Images courtesy of Addison Wesley Longman, Inc. Copyright © 2001 Multiplication Calculating running totalsCalculating running totals

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Images courtesy of Addison Wesley Longman, Inc. Copyright © 2001 Shift-add Multiplication Algorithm C = 0, U = 0;

Images courtesy of Addison Wesley Longman, Inc. Copyright © 2001 Example: UV  X  Y (X = 1101, Y = 1001) C = 0, U = 0 0

Images courtesy of Addison Wesley Longman, Inc. Copyright © 2001 RTL Code C  0, U  0, i  n

Images courtesy of Addison Wesley Longman, Inc. Copyright © 2001 Example: UV  X  Y (X = 1101, Y = 1001) C  0, U  0, i  4 0

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Images courtesy of Addison Wesley Longman, Inc. Copyright © 2001 Optimizing the RTL Code UV  X  VUV  X  V Register Y not neededRegister Y not needed One operand is lostOne operand is lost

Images courtesy of Addison Wesley Longman, Inc. Copyright © 2001 Optimizing the RTL Code C  0, U  0

Images courtesy of Addison Wesley Longman, Inc. Copyright © 2001 Example C  0, U  0, i  4 0

Images courtesy of Addison Wesley Longman, Inc. Copyright © 2001 Booth’s Algorithm Multiplying unsigned 2’s-complement numbersMultiplying unsigned 2’s-complement numbers

Images courtesy of Addison Wesley Longman, Inc. Copyright © 2001 Example UV  X  Y, X = -3 (1101), Y = -5 (1011)UV  X  Y, X = -3 (1101), Y = -5 (1011)

Images courtesy of Addison Wesley Longman, Inc. Copyright © 2001 Example UV  X  Y, X = -3 (1101), Y = -5 (1011)UV  X  Y, X = -3 (1101), Y = -5 (1011)

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

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Optimized RTL Code

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Images courtesy of Addison Wesley Longman, Inc. Copyright © 2001 Division A non-optimal methodA non-optimal method

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Images courtesy of Addison Wesley Longman, Inc. Copyright © 2001 Division Shifting results to align remaindersShifting results to align remainders

Images courtesy of Addison Wesley Longman, Inc. Copyright © 2001 Shift-subtract Division Algorithm

Images courtesy of Addison Wesley Longman, Inc. Copyright © 2001 Example: UV  X (UV = , X = 1101)

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Hardware Implementation

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RTL Code

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Hardware Implementation

Images courtesy of Addison Wesley Longman, Inc. Copyright © 2001 Signed Notations Signed-magnitudeSigned-magnitude

Images courtesy of Addison Wesley Longman, Inc. Copyright © 2001 Signed Notations Signed-magnitudeSigned-magnitude Signed-2’s complementSigned-2’s complement

Images courtesy of Addison Wesley Longman, Inc. Copyright © 2001 Signed Notations Signed-magnitudeSigned-magnitude Signed-2’s complementSigned-2’s complement Value Signed-magnitude Signed-2’s complement

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

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

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

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

Examples

Examples

Hardware Implementation

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Example

Binary Coded Decimal (BCD) Every 4 bits = 1 decimal digitEvery 4 bits = 1 decimal digit 1 bit sign1 bit sign Example: +27 = Example: +27 =

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Images courtesy of Addison Wesley Longman, Inc. Copyright © 2001 Nine’s Complement Equivalent of 1’s complement in binaryEquivalent of 1’s complement in binary

Images courtesy of Addison Wesley Longman, Inc. Copyright © 2001 Nine’s Complement Equivalent of 1’s complement in binaryEquivalent of 1’s complement in binary

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

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

Examples

Examples

Hardware Implementation

Images courtesy of Addison Wesley Longman, Inc. Copyright © 2001 Multiplication dshr: decimal shift rightdshr: decimal shift right

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Arithmetic Pipelines Increase throughputIncrease throughput

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Example

Example

Steady State Speedup

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Wallace Trees Carry Save AdderCarry Save Adder

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Images courtesy of Addison Wesley Longman, Inc. Copyright © 2001 Wallace Trees Partial productsPartial products

Images courtesy of Addison Wesley Longman, Inc. Copyright © 2001 Wallace Trees Partial productsPartial products

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Example

4  4 Wallace Tree

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Images courtesy of Addison Wesley Longman, Inc. Copyright © 2001 Floating Point Numbers Sign, Significand, ExponentSign, Significand, Exponent Normalized:Normalized: =  =  10 4 NaNNaN BiasingBiasing

Images courtesy of Addison Wesley Longman, Inc. Copyright © 2001 Floating Point Numbers PrecisionPrecision GapGap RangeRange Round, Guard, Sticky bitsRound, Guard, Sticky bits

Images courtesy of Addison Wesley Longman, Inc. Copyright © 2001 Rounding Round toward nearestRound toward nearest Round toward 0Round toward 0 Round toward + Round toward +  Round toward - Round toward - 

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Addition and Subtraction

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

Images courtesy of Addison Wesley Longman, Inc. Copyright © 2001 Example: (.1101  2 3 ) + (.1110  2 2 )

Images courtesy of Addison Wesley Longman, Inc. Copyright © 2001 Example: (.1101  2 3 ) - (.1110  2 2 )

Images courtesy of Addison Wesley Longman, Inc. Copyright © 2001 Multiplication Check for special valuesCheck for special values Add exponentsAdd exponents Multiply significandsMultiply significands Normalize resultNormalize result

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

Images courtesy of Addison Wesley Longman, Inc. Copyright © 2001 Example: (.1101  2 3 )  (.1110  2 2 )

Images courtesy of Addison Wesley Longman, Inc. Copyright © 2001 IEEE 754 Floating Point Standard 1  significand < 21  significand < 2 Single of double precisionSingle of double precision

Images courtesy of Addison Wesley Longman, Inc. Copyright © 2001 IEEE 754 Floating Point Standard IEEE 754 Floating Point Standard

Images courtesy of Addison Wesley Longman, Inc. Copyright © 2001 IEEE 754 Floating Point Standard

Images courtesy of Addison Wesley Longman, Inc. Copyright © 2001 Denormalized Values Used to express smaller numbers than possible with normalized notationUsed to express smaller numbers than possible with normalized notation Smallest normalized value: (single precision)Smallest normalized value: (single precision) Smallest denormalized value: (single precision)Smallest denormalized value: (single precision)

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