Lecture 2-3: Data Representation
Binary Integer Representation – Multiplication
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Recap
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Computers use finite-precision numbers of fixed length (typically
32-bit)
Two's complement representation allows us to efficiently
represent binary integers
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the first bit acts as a sign bit
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only one representation for 0
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there is one more negative number than there are positive numbers
Two's complement arithmetic
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addition is straightforward
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subtraction is performed by adding the negation of a number
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Hardware for Addition and Subtraction
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2C Multiplication
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Straightforward “unsigned” multiplication will not work if either
(or if both) the multiplicand or the multiplier are negative
1011
x 1101
00001011
multiplicand
multiplier
partial products
11
x 13
33
00000000
11
00101100
143
-5
x
-3
-113
01011000
10001111
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Why doesn't it work?
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If the multiplicand is negative
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the sign bits in the partial products do not align
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this can be solved using sign extension (see next slide)
If the multiplier is negative
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the 1s do not correspond with the shifts
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e.g. multiplying by 1101 (-3) should not have shifts in the first, third
and fourth partial products
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Conversion between different word lengths
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E.g. in Java: store an int as a long
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Converting signed magnitude is easy
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rule: move the sign bit to the new left-most position and fill with
zeros
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1001 0010 becomes 1000 0000 0001 0010
Two's complement rule for sign extension
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move the sign bit to the new left-most position and fill with copies of
the sign bit: for positive numbers fill with zeros, for negative
numbers fill with ones
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1110 1110 becomes 1111 1111 1110 1110
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2C Multiplication using Sign Extension
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When multiplying two 2C numbers where the multiplicand is
negative and the multiplier is positive, we must sign-extend the
partial products
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When multiplying two n-bit numbers, the product occupies 2n bits
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This means we must sign-extend the partial products to ensure they
each occupy 2n bits
1001 (-7)
x 0011 (3)
11111001 (-7)
11110010 (-14)
(1)11101011 (-21)
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How to solve the 2C Multiplication Problem
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A naïve way
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convert both operands into positive numbers
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to negate a number, complement that number and add 1
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do unsigned positional multiplication (see following slides)
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if the sign of the operands differ, negate the result
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this approach is not very efficient
A more efficient approach
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Booth's algorithm
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Unsigned Multiplication (1)
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Four registers are used in the order C, A, Q
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C is a 1-bit register initialised to 0
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A is initially set to 0 (each bit in the A register is set to 0)
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Q initially holds the multiplier
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M holds the multiplicand
Multiplication of two n-bit integers results in a product of up to 2n
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this holds a potential carry bit resulting from addition
the product is stored in the A and Q registers
This algorithm will not work for multiplying negative numbers that are
represented using two's complement
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Unsigned Multiplication (2)
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Unsigned Multiplication (3)
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If Q0 is 1(where Q0 is the right-most or least significant bit of Q)
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add the multiplicand to the number in the A register (the result is
stored in the A register with the C bit used for overflow)
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then shift all of the bits in the C, A and Q registers to the right by one
bit (Register Shift Right or RSR – see next slide)
Else if Q0 is 0
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shift all of the bits in the C, A and Q registers to the right by one bit
(RSR – see next slide)
Repeat this process for each bit of the original multiplier
The resulting 2n-bit product is contained in the A and Q registers
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Register Shifts
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Registers can have various operations performed on them
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e.g. register shift (left or right – RSL, RSR)
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RSL – the contents are shifted to the left by one bit
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the left-most bit is discarded, the right-most bit is filled with 0
0100 1000 0100 1001 becomes 1001 0000 1001 0010
this multiplies the original number by 2
RSR – the contents are shifted to the right by one bit
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the right-most bit is discarded, the left-most bit is filled with 0
0100 1000 0100 1001 becomes 0010 0100 0010 0100
this divides the original number by 2
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Unsigned Multiplication – Example
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Booth's Algorithm (1)
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Booth's algorithm can be used to multiply two numbers
represented using two's complement (2C)
Booth's algorithm will work for both positive and negative
numbers represented using 2C
The algorithm uses a combination of addition, subtraction and
Arithmetic Right Shifts (see later slide)
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subtraction is performed by taking the two's complement of the
subtrahend (the number to be subtracted) and adding it to the
minuend
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Booth's Algorithm (2)
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Booth's algorithm uses the following 4 registers (the first 3
registers are ordered A, Q, Q-1)
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A – each bit in A is initialised to 0
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A is reused and eventually stores the most-significant bits of the result
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A is the same size as M
Q is initialised to store the multiplier
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Q -1 is initialised to 0
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Q is reused and eventually stores the least-significant bits of the result
this is placed to the right of Q
this is a 1-bit register which remembers the least-significant bit of Q
(Q0)
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M stores the multiplicand
Booth's Algorithm (3)
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If Q0 and Q-1 are the same (e.g. Q0 = 1, Q-1 = 1 or Q0 = 0, Q-1 = 0)
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shift all of the bits of the A, Q, and Q-1 registers to the right
(Arithmetic Shift Right, ASR – see next slide)
Else
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if Q0 is 0 and Q-1 is 1, add the multiplicand to the contents of the A
register
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else if Q0 is 1 and Q-1 is 0, subtract the multiplicand from the
contents of the A register
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shift all of the bits of the A, Q, and Q-1 registers to the right (ASR)
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Repeat for each bit of the multiplier
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The product is stored in the A and Q registers
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Arithmetic Shifts
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For two's complement, RSL and RSR are not applicable
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the sign of the number being shifted must be preserved
An arithmetic shift preserves the sign of the left-most bit
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ASL – the contents are shifted to the left by one bit
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1000 1111 becomes 1001 1110
● 1101 becomes 1010
ASR – the contents are shifted to the right by one bit
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1101 becomes 1110
0101 becomes 0010
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Booth's Algorithm (4)
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Booth's Algorithm Example
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Why does Booth's algorithm work?
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Booth's algorithm acts on blocks of 1s
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M x (2n + 2n-1 + 2n-2 + .. + 2n-k) = M x (2n+1 – 2n-k)
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subtraction is only performed when a block of 1s begins, and
addition is only performed when the block ends – it shifts every time
(ASR)
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the algorithm therefore performs fewer additions and subtractions
(i.e. other algorithms perform one addition for every 1 that occurs in
the multiplier)
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Booth's Algorithm
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Booth's algorithm uses solely arithmetic shifts, addition and
subtraction
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where subtraction is negation followed by addition
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this means that multiplication is performed by combination of
addition and arithmetic shifts
Booth's algorithm works on two's complement and unsigned
multiplication
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Checkpoint – Two's Complement
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Conversion to and from 2C
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Conversion between different bit lengths
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Negating numbers
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Adding numbers
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detecting overflow
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Subtracting numbers
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Multiplying numbers
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Booth's algorithm
Further reading
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Computer Organisation and Architecture, William Stallings
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