Lecture 12-13 notes

1
Lecture 12-13 notes

• Reading Section 3.4, 3.5, 3.6
• Multiplication
• Unsigned multiplication
• Hardware implementation
• Division
• Floating point

2
Unisigned shift-add multiplier (version 1)

• 64-bit Multiplicand reg, 64-bit ALU, 64-bit
  Product reg, 32-bit multiplier reg

Shift Left
Multiplicand
64 bits
Multiplier
Shift Right
64-bit ALU
32 bits
Write
Product
Control
64 bits
Multiplier datapath control
3
MULTIPLY HARDWARE Version 2

• 32-bit Multiplicand reg, 32 -bit ALU, 64-bit
  Product reg, 32-bit Multiplier reg

Multiplicand
32 bits
Multiplier
Shift Right
32-bit ALU
32 bits
Shift Right
Product
Control
Write
64 bits
4
Whats going on?
0
0
0
0
B0
B1
B2
B3
P0
P1
P2
P3
P4
P5
P6
P7

• Multiplicand stays still and product moves right

5
Multiplier is Negative

• Convert to positive-gtmult-gtsign conversion
• Sign extended algorithm
• 1 0 0 1 1 (-13)
• x 0 1 0 1 1 (11)
• 1 1 1 1 1 1 0 0 1 1
• 1 1 1 1 1 0 0 1 1
• 0 0 0 0 0 0 0 0
• 1 1 1 0 0 1 1
• 0 0 0 0 0 0 0
• 1 1 0 1 1 1 0 0 0 1 (-143)

6
Fast Hardware

• Use multiple hardware ALUs
• Binary tree type structured
• Parallel binary addition

7
Long Divide Paper Pencil

• 1001 Quotient
• Divisor 1000 1001010 Dividend 1000
  10 101 1010
  1000 10 Remainder
• Dividend Quotient x Divisor Remainder

8
DIVIDE HARDWARE Version 1

• 64-bit Divisor reg, 64-bit ALU, 64-bit Remainder
  reg, 32-bit Quotient reg

Shift Right
Divisor
64 bits
Quotient
Shift Left
64-bit ALU
32 bits
Write
Remainder
Control
64 bits
9

• Initialization
• Set 32-bit Quotient reg to 0
• Place the divisor in the high half of the 64-bit
  divisor reg
• Remainder reg initialized with dividend

10
Binary representation of fraction

• (1001.1001)2 1 x 2 3 0 x 2 2 0 x 2 1 1 x 2
  0
• 1 x 2-1 0 x 2-2 0 x
  2-3 1 x 2-4
• (9.5625)10
• (0.625) 10 (0.5 0.125) 10
• 1 x 2-1 0 x 2-2 1 x 2-3
• (0.101) 2

2-1 2-2 2-3 2-4 2-5 2-6
0.5 0.25 0.125 0.0625 0.03125 0.0150625
11
Scientific Notation
exponent
decimal point
Sign, magnitude
23
-24
6.02 x 10 1.673 x 10
radix (base)
Mantissa
Sign, magnitude

• Issues
• Arithmetic (, -, , / )
• Representation, Normal form
• Range and Precision
• Rounding
• Exceptions (e.g., divide by zero, overflow,
  underflow)

12
IEEE 754 Floating-Point
1
8
23
single precision
S
E
F
fraction sign magnitude, normalized binary
significand w/ hidden integer bit 1.F
exponent excess 127 binary integer
actual exponent is e E - 127
0 lt E lt 255
S
E-127
N (-1) 2 (1.F)
0 0 00000000 0 . . . 0 -1.5 1
01111111 10 . . . 0
Magnitude of numbers that can be represented is
in the range
-126
127
23
)
2
(1.0)
(2 - 2
to
2
which is approximately
-38
38
to
3.40 x 10
1.8 x 10
(integer comparison valid on IEEE Fl.Pt. numbers
of same sign!)
13
Example -0.75 in float point

• -0.75-(0.50.25) -(0.11)2
• In scientific notation, the value is -0.112 x 20
• normalized scientific notation
• -1.12 x 2-1
• In single precision
• (-1) S x (1 fraction) x 2 (exponent-127)
• S 1
• fraction 10000000000000000000000
• Exponent 126 01111110

10000000000000000000000
01111110
1
14
Floating Point Addition Algorithm

• Add x0.5 and y 0.4375 in binary
• (2) x 1.000 x 2-1, y -1.110 x 2-2. right shift
  the smaller exponent (y) so that both have same
  exponent value
• y -0.111 x 2-1
• (3) Add the fraction parts
• 1.000 x 2-1 -0.111 x 2-1 0.001 x 2-1
• (4) left shift result to normalize
• 0.001 x 2-11.000 x 2-4
• (5) Round (not needed in this example)