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
64 bits
Write

4. What’s going on? Multiplicand stay’s still and product moves right A0A0 A1 A2 A3 B0 A0 A1 A2 A3 B1 A0 A1 A2 A3 B2 A0 A1 A2 A3 B3 P7 P6 P5 P4 P3 P2 P1 P0
Multiplicand stay’s still and product moves right

5. Multiplier is Negative
Convert to positive->mult->sign conversion
Sign extended algorithm:
(-13)
x (11)
(-143)

6. Fast Hardware Use multiple hardware ALUs Parallel binary addition
Binary tree type structured
Parallel binary addition

7. Long Divide: Paper & Pencil
1001 Quotient
Divisor Dividend – – 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
( )2 = 1 x x x x 2 0
x x x x 2-4
= (9.5625)10
(0.625) 10 = ( ) 10
= 1 x x 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

11. Scientific Notation Issues: Arithmetic (+, -, *, / )
exponent
decimal point
Sign, magnitude 23
-24
6.02 x 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 exponent:
excess 127
binary integer
fraction:
sign + magnitude, normalized
binary significand w/ hidden
integer bit: 1.F
actual exponent is
e = E - 127
0 < E < 255 S
E-127
N = (-1) (1.F)
0 = =
Magnitude of numbers that can be represented is in the range:
-126
127 23 2
(1.0) to 2
(2 - 2 )
which is approximately:
-38 38
1.8 x 10 to
3.40 x 10
(integer comparison valid on IEEE Fl.Pt. numbers of same sign!)

13. Example: -0.75 in float point
-0.75=-( )= -(0.11)2
In scientific notation, the value is 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 =
Exponent = 126 = 1

14. Floating Point Addition Algorithm
Add x=0.5 and y=– in binary
(2) x= x 2-1, y= x 2-2. right shift the smaller exponent (y) so that both have same exponent value
y= x 2-1
(3) Add the fraction parts:
1.000 x x 2-1 = x 2-1
(4) left shift result to normalize
0.001 x 2-1=1.000 x 2-4
(5) Round (not needed in this example)