William Stallings �Computer Organization �and Architecture�7^th Edition

Chapter 9

Computer Arithmetic

Arithmetic & Logic Unit

• Does the calculations
• Everything else in the computer is there to service this unit
• Handles integers
• May handle floating point (real) numbers
• May be separate FPU (maths co-processor)
• May be on chip separate FPU (486DX +)

ALU Inputs and Outputs

Integer Representation

• Only have 0 & 1 to represent everything
• Positive numbers stored in binary
• e.g. 41=00101001
• No minus sign
• No period
• Sign-Magnitude
• Two’s compliment

Sign-Magnitude

• Left most bit is sign bit
• 0 means positive
• 1 means negative
• +18 = 00010010
• -18 = 10010010
• Problems
• Need to consider both sign and magnitude in arithmetic
• Two representations of zero (+0 and -0)

Two’s Compliment

• +3 = 00000011
• +2 = 00000010
• +1 = 00000001
• +0 = 00000000
• -1 = 11111111
• -2 = 11111110
• -3 = 11111101

Benefits

• One representation of zero
• Arithmetic works easily (see later)
• Negating is fairly easy
• 3 = 00000011
• Boolean complement gives 11111100
• Add 1 to LSB 11111101

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Geometric Depiction of Twos Complement Integers

Negation Special Case 1

• 0 = 00000000
• Bitwise not 11111111
• Add 1 to LSB +1
• Result 1 00000000
• Overflow is ignored, so:
• - 0 = 0 √

Negation Special Case 2

• -128 = 10000000
• bitwise not 01111111
• Add 1 to LSB +1
• Result 10000000
• So:
• -(-128) = -128 X
• Monitor MSB (sign bit)
• It should change during negation

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Range of Numbers

• 8 bit 2s compliment
• +127 = 01111111 = 2⁷ -1
• -128 = 10000000 = -2⁷
• 16 bit 2s compliment
• +32767 = 011111111 11111111 = 2¹⁵ - 1
• -32768 = 100000000 00000000 = -2¹⁵

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Conversion Between Lengths

• Positive number pack with leading zeros
• +18 = 00010010
• +18 = 00000000 00010010
• Negative numbers pack with leading ones
• -18 = 10010010
• -18 = 11111111 10010010
• i.e. pack with MSB (sign bit)

Addition and Subtraction

• Normal binary addition
• Monitor sign bit for overflow

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• Take twos compliment of substahend and add to minuend
• i.e. a - b = a + (-b)

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• So we only need addition and complement circuits

Hardware for Addition and Subtraction

Multiplication

• Complex
• Work out partial product for each digit
• Take care with place value (column)
• Add partial products

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Multiplication Example

• 1011 Multiplicand (11 dec)
• x 1101 Multiplier (13 dec)
• 1011 Partial products
• 0000 Note: if multiplier bit is 1 copy
• 1011 multiplicand (place value)
• 1011 otherwise zero
• 10001111 Product (143 dec)
• Note: need double length result

Unsigned Binary Multiplication

Execution of Example

Flowchart for Unsigned Binary Multiplication

Multiplying Negative Numbers

• This does not work!
• Solution 1
• Convert to positive if required
• Multiply as above
• If signs were different, negate answer
• Solution 2
• Booth’s algorithm

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Booth’s Algorithm

Example of Booth’s Algorithm

Division

• More complex than multiplication
• Negative numbers are really bad!
• Based on long division

Division of Unsigned Binary Integers

001111

1011

00001101

10010011

1011

001110

1011

1011

100

Quotient

Dividend

Remainder

Partial

Remainders

Divisor

Flowchart for Unsigned Binary Division

Real Numbers

• Numbers with fractions
• Could be done in pure binary
• 1001.1010 = 2⁴ + 2⁰ +2^-1 + 2^-3 =9.625
• Where is the binary point?
• Fixed?
• Very limited
• Moving?
• How do you show where it is?

Floating Point

• +/- .significand x 2^exponent
• Misnomer
• Point is actually fixed between sign bit and body of mantissa
• Exponent indicates place value (point position)

Floating Point Examples

Signs for Floating Point

• Mantissa is stored in 2s compliment
• Exponent is in excess or biased notation
• e.g. Excess (bias) 128 means
• 8 bit exponent field
• Pure value range 0-255
• Subtract 128 to get correct value
• Range -128 to +127

Normalization

• FP numbers are usually normalized
• i.e. exponent is adjusted so that leading bit (MSB) of mantissa is 1
• Since it is always 1 there is no need to store it
• (c.f. Scientific notation where numbers are normalized to give a single digit before the decimal point
• e.g. 3.123 x 10³)

FP Ranges

• For a 32 bit number
• 8 bit exponent
• +/- 2²⁵⁶ ≈ 1.5 x 10⁷⁷
• Accuracy
• The effect of changing lsb of mantissa
• 23 bit mantissa 2^-23 ≈ 1.2 x 10^-7
• About 6 decimal places

Expressible Numbers

Density of Floating Point Numbers

IEEE 754

• Standard for floating point storage
• 32 and 64 bit standards
• 8 and 11 bit exponent respectively
• Extended formats (both mantissa and exponent) for intermediate results

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IEEE 754 Formats

FP Arithmetic +/-

• Check for zeros
• Align significands (adjusting exponents)
• Add or subtract significands
• Normalize result

FP Addition & Subtraction Flowchart

FP Arithmetic x/÷

• Check for zero
• Add/subtract exponents
• Multiply/divide significands (watch sign)
• Normalize
• Round
• All intermediate results should be in double length storage

Floating Point Multiplication

Floating Point Division

Required Reading

• Stallings Chapter 9
• IEEE 754 on IEEE Web site