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Multiplication
CPSC 252 Computer Organization
Ellen Walker, Hiram College
Multiplication
• Multiplication result needs twice as many bits– E.g. 7*7 = 49 (7 is 3 bits, 49 is 6 bits)
• Multiplication is more complex than addition/subtraction– Develop algorithm, implement using
simpler hardware
Multiplication Algorithms
• Repeated addition– Easy to implement in software– Can only multiply positive numbers directly– Time depends on magnitude of operands
• Shift-and add– Efficient use of hardware– Time depends only on width of operands– No special case for negative numbers
Shift and Add Algorithm
• For each bit in multiplier– If bit = 1, add multiplicand to result– Shift multiplicand left by 1
• This is the “usual” multiplication algorithm you learned many years ago (but simpler in binary)
Shift and Add example
10101 21
X 101 5
---------
10101
000000 (included for completeness)
+ 1010100
------------------
1101001 105 (1+8+32+64)
Hardware to Multiply
• 64-bit register for multiplicand– 32 bits for original value– Option to shift left 32 times
• 64-bit ALU– Add shifted multiplicand to product
• 32-bit register for multiplier– Shift right after each step (low bits fall off)
• Control hardware
Multiplication: Implementation
DatapathControl
MultiplicandShift left
64 bits
64-bit ALU
ProductWrite
64 bits
Control test
MultiplierShift right
32 bits
32nd repetition?
1a. Add multiplicand to product and
place the result in Product register
Multiplier0 = 01. Test
Multiplier0
Start
Multiplier0 = 1
2. Shift the Multiplicand register left 1 bit
3. Shift the Multiplier register right 1 bit
No: < 32 repetitions
Yes: 32 repetitions
Done
How Long Does it Take?
• 32 shift/add cycles
• Each cycle has 3 steps (add, shift left, shift right)
• Without parallelism, that’s 96 cycles per multiply
Improvements
• Save hardware by using 1 64-bit register for the product and multiplier– Left part is product (so far)– Right part is unused multiplier– Add a bit to the product and lose a bit from the
multiplier each cycle– Shift product right & add instead of shift
multiplicand left & add!
• Shift and add in the same cycle (in parallel)• This revision requires only 32 cycles
Final Version
Multiplicand
32 bits
32-bit ALU
ProductWrite
64 bits
Controltest
Shift right
32nd repetition?
Product0 = 01. Test
Product0
Start
Product0 = 1
3. Shift the Product register right 1 bit
No: < 32 repetitions
Yes: 32 repetitions
Done
What goes here?
•Multiplier starts in right half of product
Parallel Multiplication
• Instead of 32 cycles, use 32 adders
• Each adds 0 or multiplicand
• Arrange in tree so results cascade properly
• Result pulls each bit from the appropriate adder
Parallel Multiplication (cont)
010101 x 000101 (6 bits) 000000 000101 (initial product)
+010101 Rightmost bit is 1, add multiplicand
010101 000101
0010101 00010 Shift product right
00010101 0001 Shift product right
+010101 Rightmost bit is 1, add multiplicand
01101001 0001
001101001 000 Shift product right
0001101001 00 Shift product right
00001101001 0 Shift product right
000001101001 Shift product right, answer is 105
Negative numbers
• Convert to positive, multiply, then negate if initial signs were different
• Or, consider 3 cases:– Both negative: convert to positive and
multiply– One negative: make multiplier positive,
multiplicand negative– Negative multiplicand: when high bit is 1,
shift in 1’s (shift right arithmetic)
MIPS multiplication
• 2 32-bit registers: Hi and Lomflo $s0 move from lo (to
$s0)
mfhi $s0 move from hi (to $s0)
• Multiplication Operationsmult $s0, $s1 {Hi,Lo}= $s0 x $s1
multu $s0, $s1 unsigned version
