Perkuliahan 7

 Komputer Aritmatika

Komputer Aritmatika

• 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

Two’s Compliment

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

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 

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

Addition and Subtraction

• Normal binary addition
• Monitor sign bit for overflow
• 
  
• Take twos compliment of substahend and add to minuend
• i.e. a - b = a + (-b)
• 
  
• So we only need addition and complement circuits

Hardware for Addition and Subtraction

Multiplication Example

• 1011   Multiplicand (11 dec)
•  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

Booth’s Algorithm

Example of Booth’s Algorithm

Example

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)

Expressible 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

FP Arithmetic +/-

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

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 8
• IEEE 754  on IEEE Web site