Chapter 6. Arithmetic
1
Outline







A basic operation in all digital computers is
the addition or subtraction of two numbers.
ALU – AND, OR, NOT, XOR
Unsigned/signed numbers
Addition/subtraction
Multiplication
Division
Floating number operation
2
Adders
3
Addition of Unsigned Numbers
– Half Adder
x
0
0
1
1
+ y
+ 0
+ 1
+ 0
+ 1
c s
0 0
0 1
0 1
1 0
Carry
Sum
(a) The four possible cases
Carry
Sum
x
y
c
0
0
0
0
0
1
0
1
1
0
0
1
1
1
1
0
s
(b) Truth table
x
s
y
x
y
s
HA
c
c
4
(c) Circuit
(d) Graphical symbol
Addition and Subtraction of
Signed Numbers
xi
yi
Carry-in ci
Sumsi
Carry-outci +1
0
0
0
0
1
1
1
1
0
0
1
1
0
0
1
1
0
1
0
1
0
1
0
1
0
1
1
0
1
0
0
1
0
0
0
1
0
1
1
1
si = xi yi ci + xi yi ci + xi yi ci + xi yi ci = x i  yi  ci
ci +1 = yi ci + xi ci + xi yi
Example:
X
7
+Y = +6
Z
13
0
= + 00
1
1
1
1
1
1
1
1
0
0
1
0
1
0
Carry-out
ci+1
xi
yi
si
Carry-in
ci
Legend for stagei
5
Figure 6.1. Logic specification for a stage of binary addition.
Addition and Subtraction of
Signed Numbers

A full adder (FA)
yi
c
i
xi
yi
xi
c
si
i
ci
xi
ci + 1
yi
Full adder
(FA)
c
i +1
x
i
yi
ci
s
i
(a) Logic for a single stage
6
Addition and Subtraction of
Signed Numbers


n-bit ripple-carry adder
Overflow?
xn - 1
cn
yn - 1
x1
y1
cn - 1
x0
y0
c1
FA
FA
FA
sn - 1
Most significant bit
(MSB) position
s1
s0
c0
Least significant bit
(LSB) position
(b) An n-bit ripple-carry adder
7
Addition and Subtraction of
Signed Numbers

kn-bit ripple-carry adder
xkn - 1 ykn - 1
x2n - 1 y2n - 1 xn y n
n-bit
adder
c kn
s
kn - 1
cn
n-bit
adder
s(
k - 1) n
s
2n - 1
xn - 1 y n - 1 x 0 y 0
s
n
n-bit
adder
s
n- 1
c
0
s
0
(c) Cascade of k n-bit adders
Figure 6.2. Logic for addition of binary vectors.
8
Addition and Subtraction of
Signed Numbers
yn - 1
y1
y0
Add/Sub
control
xn - 1
x1
x0
n-bit adder
cn
c
0
sn -
1
s1
s0
Figure 6.3. Binary addition-subtraction logic netw
ork.

Addition/subtraction logic unit
9
Make Addition Faster
10
Ripple-Carry Adder (RCA)





Straight-forward design
Simple circuit structure
Easy to understand
Most power efficient
Slowest (too long critical path)
11
Adders

We can view addition in terms of generate,
G[i], and propagate, P[i].
12
Carry-lookahead Logic
Carry Generate Gi = Ai Bi
must generate carry when A = B = 1
Carry Propagate Pi = Ai xor Bi
carry-in will equal carry-out here
Sum and Carry can be reexpressed in terms of generate/propagate/Ci:
Si = Ai xor Bi xor Ci = Pi xor Ci
Ci+1 = Ai Bi + Ai Ci + Bi Ci
= Ai Bi + Ci (Ai + Bi)
= Ai Bi + Ci (Ai xor Bi)
= Gi + Ci Pi
13
Carry-lookahead Logic
Reexpress the carry logic as follows:
C1 = G0 + P0 C0
C2 = G1 + P1 C1 = G1 + P1 G0 + P1 P0 C0
C3 = G2 + P2 C2 = G2 + P2 G1 + P2 P1 G0 + P2 P1 P0 C0
C4 = G3 + P3 C3 = G3 + P3 G2 + P3 P2 G1 + P3 P2 P1 G0 + P3 P2 P1 P0 C0
Each of the carry equations can be implemented in a two-level logic
network
Variables are the adder inputs and carry in to stage 0!
14
Carry-lookahead
Implementation
Ai
Pi @ 1 gate dela y
Bi
Si @ 2 gate dela ys
Ci
Gi @ 1 g ate d elay
C0
P0
C1
G0
C0
P0
P1
P2
C2
G1
P2
G2
G1
Increasingly complex logic
C0
P0
P1
P2
P3
G0
P1
P2
C0
P0
P1
G0
P1
Adder with Propagate and
Generate Outputs
C3
G0
P1
P2
P3
G1
P2
P3
G2
P3
C4
G3
15
Carry-lookahead Logic
Cascaded Carry Lookahead
C0
A0
Carry lookahead
logic generates
individual carries
S 0 @2
B0
C 1 @3
A1
S 1 @4
B1
sums computed
much faster
C 2 @3
A2
S 2 @4
B2
C 3 @3
A3
S 3 @4
B3
C 4 @3
16
x15-12
y15-12
x11-8
y11-8
c12
c16
4-bit adder
s15-12
G3
I
P3
I
x7-4
y7-4
c8
4-bit adder
G2
I
I
P2
x3-0
y3-0
c4
4-bit adder
s11-8
4-bit adder
s7-4
I
G1
I
P1
.
c0
s3-0
G0I
P0I
Carry -lookahead logic
G0
II
P0
II
Figure 6.5. 16-bit carry-lookahead adder built from 4-bit adders (see b).
Figure 6.4
Carry-lookahead Logic
17
Carry-lookahead Logic
4
4
4
A [15- 12] B [15- 12]
4-bit Adder
G
P
C16
4
4
4
A [1 1-8]
B [1 1-8]
4-bit Adder
G
P
C 12
@8
4
S [15- 12]
4
@8
P3
@3
G3
@5
C3
A [3-0]
B [3-0]
4-bit Adder
P
G
C4
@7
4
P2
@3
G2
@5
C2
@5
@0
@4
@2 @3
P1
G1
@4
@2 @3
C1
P0
G0
C16
C4
C0
S [3-0]
S [7-4]
@2
4
4
A [7-4]
B [7-4]
4-bit Adder
G
P
C8
S [1 1-8]
@2
4
C0
Lookahead Car r y Unit
P 3-0
@3
G 3-0
C0
@0
@5
4 bit adders with internal carry lookahead
second level carry lookahead unit, extends lookahead to 16 bits
Group Propagate P = P3 P2 P1 P0
Group Generate G = G3 + G2P3 + G1P3P2 + G0P3P2P1
18
Unsigned
Multiplication
19
Manual Multiplication
Algorithm
1
1
0
1
(13) Multiplicand M
1
0
1
1
(11) Multiplier Q
1
1
0
1
1
1
0
1
0
0
0
0
1
1
0
1
0
0
0
1
´
1
1
1
1
(143) Product P
(a) Manual multiplication algorithm
20
Array Multiplication
Multiplicand
Partial product 0
(PP0)
m3 0
m2 0
m1 0
m0
q0
0
PP1
q1
0
PP2
p1
q2
0
PP3
q3
0
p7
p6
p5
p4
p0
p2
PP4 =p7 , p6, ...p0 = Product
p3
Bit of incoming partial product (PP
i)
mj
qi
Typical cell
Carry-out
FA
Carry-in
Bit of outgoing partial product [PP(
i +1)]
(b) Array implementation
21
X3
Y3
X2
Y2
X1
Y1
X0
Y0
X3Y0 X2Y0 X1Y0 X0Y0
X 3Y 1 X 2Y 1 X 1Y 1 X 0Y 1
X3Y2 X2Y2 X1Y2 X0Y2
X3Y3 X2Y3 X1Y3 X0Y3
P7
P6
P5
P4
P3
P2
P1
P0
22
Another Version of 4×4 Array
Multiplier
23
Array Multiplication



What is the critical path (worst case signal
propagation delay path)?
Assuming that there are two gate delays from
the inputs to the outputs of a full adder block,
the path has a total of 6(n-1)-1 gate delays,
including the initial AND gate delay in all cells,
for the nn array.
Any advantages/disadvantages?
24
Sequential Circuit Binary
Multiplier
Register A (initially 0)
M
1 1 0 1
Shift right
an - 1
C
a0
qn - 1
q0
Multiplier Q
Add/Noadd
control
n-bit
adder
Control
sequencer
MUX
0
0
mn - 1
m0
Multiplicand M
Initial configuration
0
C
0 0 0 0
A
1 0 1 1
Q
0
0
1 1 0 1
0 1 1 0
1 0 1 1
1 1 0 1
Add
Shift
First cycle
1
0
0 0 1 1
1 0 0 1
1 1 0 1
1 1 1 0
Add
Shift
Second cycle
0
0
1 0 0 1
0 1 0 0
1 1 1 0
1 1 1 1
No add
Shift
Third cycle
1
0
0 0 0 1
1 0 0 0
1 1 1 1
1 1 1 1
Add
Shift
Fourth cycle
Product
(b) Multiplication example
(a) Register configuration
25
Signed Multiplication
26
Signed Multiplication

Considering 2’s-complement signed operands, what will happen to (13)(+11) if following the same method of unsigned multiplication?
Sign extension is
shown in blue
1
0
0
1
1
0
1
0
1
1
1
1
1
1
1
1
1
0
0
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
1
1
0
1
1
1
0
0
0
1
( - 13)
( + 11)
( - 143)
27
Figure 6.8. Sign extension of negative multiplicand.
Signed Multiplication



For a negative multiplier, a straightforward
solution is to form the 2’s-complement of both
the multiplier and the multiplicand and
proceed as in the case of a positive multiplier.
This is possible because complementation of
both operands does not change the value or
the sign of the product.
A technique that works equally well for both
negative and positive multipliers – Booth
algorithm.
28
Booth Algorithm

Consider in a multiplication, the multiplier is
positive 0011110, how many appropriately
shifted versions of the multiplicand are added
in a standard procedure?
0
0
0
1
0
1
0
0
1
0
1
0
0
1 0 1 1 0
0 +1 +1 + 1+1
1
0
0
1
0
1
1
0
0
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
1
0
1
0
0
1
1
0
1
0
1
0
1
0
0
1
0
1
0
0
0
1
1
0
29
Booth Algorithm

Since 0011110 = 0100000 – 0000010, if we
use the expression to the right, what will
happen?
0 1
0 +1
0
0
1
0
1 0
0 -1
1
0
0
1
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
1
0
0
1
0
0
0
0
0
0
1
0
0
0
1
0
0
1
0
0
0
1
0
0
0
0
0
0
0
0
0
1
0
0
0
1
0
0
0
0
0
0
0
0
0
0
1
0
0
1
0
0
0
0
1
0
1
0
1
0
0
0
1
1
0
2's complement of
the multiplicand
30
Booth Algorithm

In general, in the Booth scheme, -1 times the shifted multiplicand
is selected when moving from 0 to 1, and +1 times the shifted
multiplicand is selected when moving from 1 to 0, as the
multiplier is scanned from right to left.
0
0
1
0
0 +1 -1 +1
1
1
0 - 1
0
0
1
1
1
0
1
0
0 +1
0
0 - 1 +1 - 1 + 1
1
1
0
0
0 - 1
0
0
Figure 6.10. Booth recoding of a multiplier.
31
Booth Algorithm
0 1 1 0 1
´ 1 1 0 1 0
( + 13)
(- 6)
0 1 1 0 1
0 - 1 +1 - 1 0
0
1
0
1
0
0
1
0
1
0
0
1
0
1
0
0
1
0
0
0
0
1
1
0
0
0
0
1
1
0
0 0 0 0
0 1 1
0 1
1
1 1 1 0 1 1 0 0 1 0
( - 78)
Figure 6.11. Booth multiplication with a negative multiplier.
32
Booth Algorithm
Multiplier
Version of multiplicand
selected by biti
Bit i
Bit i -1
0
0
0 ×M
0
1
+ 1 ×M
1
0
 1 ×M
1
1
0 ×M
Figure 6.12. Booth multiplier recoding table.
33
Booth Algorithm


Best case – a long string of 1’s (skipping over 1s)
Worst case – 0’s and 1’s are alternating
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
Worst-case
multiplier
+1 - 1 +1 - 1 +1 - 1 +1 - 1 +1 - 1 +1 - 1 +1 - 1 +1 - 1
1
1
0
0
0
1
0
1
1
0
1
1
1
1
0
0
Ordinary
multiplier
0 -1 0
0 +1 - 1 +1
0 - 1 +1
0
0
0 -1
0
0
0
0
1
0
0
0
1
1
0
0
1
1
1
1
0
1
Good
multiplier
34
0
0
0 +1
0
0
0
0 -1
0
0
0 +1
0
0 -1
Fast Multiplication
35
Bit-Pair Recoding of
Multipliers

Bit-pair recoding halves the maximum number of
summands (versions of the multiplicand).
Sign extension
Implied 0 to right of LSB
1
1
0
1
0
1
0
0
1 +1 1
0
0
1
2
0
(a) Example of bit-pair recoding derived from Booth recoding
36
Bit-Pair Recoding of
Multipliers
Multiplier bit-pair
Multiplier bit on the right
Multiplicand
selected at position i
i +1
i
i 1
0
0
0
0
×M
0
0
1
+1
×M
0
1
0
+1
×M
0
1
1
+2
×M
1
0
0
2
×M
1
0
1
1
×M
1
1
0
1
×M
1
1
1
0
×M
(b) Table of multiplicand selection decisions
37
Bit-Pair Recoding of
Multipliers
0 1 1 0 1
0 - 1 +1 - 1 0
0 1 1 0 1 ( + 13)
´ 1 1 0 1 0 (- 6 )
0
1
0
1
0
0
1
0
1
0
0
1
0
1
0
0
1
0
0
0
0
1
1
0
0
0
0
1
1
0
0 0 0 0
0 1 1
0 1
1
1 1 1 0 1 1 0 0 1 0 ( - 78)
0 1 1 0 1
0
-1
-2
1 1 1 1 1 0 0 1 1 0
1 1 1 1 0 0 1 1
0 0 0 0 0 0
1 1 1 0 1 1 0 0 1 0
38
Figure 6.15. Multiplication requiring only n/2 summands.
Carry-Save Addition of
Summands
1
1
0
1
(13) Multiplicand M
1
0
1
1
(11) Multiplier Q
1
1
0
1
1
1
0
1
0
0
0
0
1
1
0
1
0
0
0
1
´
1
1
1
1
(143) Product P
(a) Manual multiplication algorithm
39
Carry-Save Addition of
Summands
Multiplicand
Partial product 0
(PP0)
m3 0
m2 0
m1 0
m0
q0
0
PP1
q1
0
PP2
p1
q2
0
PP3
q3
0
p7
p6
p5
p4
p0
p2
PP4 =p7 , p6, ...p0 = Product
p3
Bit of incoming partial product (PP
i)
mj
qi
Typical cell
Carry-out
FA
Carry-in
Bit of outgoing partial product [PP(
i +1)]
(b) Array implementation
40
Carry-Save Addition of
Summands
0
m3q0
m3q1
FA
m3q2
FA
m3q3
FA
p7
p6
FA
m2q3
FA
p5
FA
m1q0
m1q1
m0q0
m0q1
FA
m1q2
FA
0
m0q2
FA
m1q3
p4
m2q0
m2q1
FA
m2q2
FA
0
m0q3
FA
p3
0
p2
p1
p0
(a) Ripple-carry array (Figure 6.6 structure)

CSA speeds up the addition process.
41
Carry-Save Addition of
Summands
0
m3q0
m2q2
FA
m2q3
m3q3
FA
FA
p7
p6
FA
p5
m2q0
m2q1
m3q1
m3q2
m1q3
FA
FA
p4
m1q0
m1q1
m1q2
FA
m0q3
FA
FA
0
0
FA
p3
m0q0
m0q1
m0q2
FA
FA
0
p2
p1
p0
(b) Carry -sav e array
Figure 6.16. Ripple-carry and carry-save arrays for the
multiplication operationx M
Q = P for 4-bit operands.
Figure 6.16. Ripple-carry and carry-save arrays for the multiplication operation M  Q = P for 4-bit operands.
42
Carry-Save Addition of
Summands






The delay through the carry-save array is somewhat
less than delay through the ripple-carry array. This is
because the S and C vector outputs from each row
are produced in parallel in one full-adder delay.
Consider the addition of many summands, we can:
Group the summands in threes and perform carry-save addition on
each of these groups in parallel to generate a set of S and C vectors
in one full-adder delay
Group all of the S and C vectors into threes, and perform carry-save
addition on them, generating a further set of S and C vectors in one
more full-adder delay
Continue with this process until there are only two vectors remaining
They can be added in a RCA or CLA to produce the desired product
43
Carry-Save Addition of
Summands
1
0
1
1
0
1
(45)
M
1
1
1
1
1
1
(63)
Q
1
0
1
1
0
1
A
1
0
1
1
0
1
1
0
1
1
0
1
1
0
1
1
0
1
1
0
1
1
0
1
1
0
1
1
0
1
0
1
1
0
0
0
X
1
B
C
D
E
F
1
0
0
1
1
(2,835)
Product
Figure 6.17. A multiplication example used to illustrate carry-save addition as shown in Figure 6.18.
44
0
1
0
1
1
0
1
M
x 1
1
1
1
1
1
Q
1
0
1
1
0
1
1
0
1
1
0
1
1
0
1
1
0
1
1
1
0
0
0
0
1
0
0
1
1
1
1
0
0
1
0
1
1
0
1
1
0
1
1
0
1
1
0
1
1
0
1
1
1
0
0
0
0
1
0
1
1
1
1
0
0
1
1
0
0
0
0
1
0
0
1
1
1
1
0
0
S
2
C
2
1
1
1
1
0
1
0
1
0
0
0
1
0
0
0
0
1
0
1
1
0
0
0
0
0
1
1
1
1
0
0
0
1
0
1
1
1
0
1
0
0
1
+ 0
1
0
1
0
1
0
0
0
0
0
0
1
S1
1
C
1
S2
0
0
1
1
0
0
C
F
0
1
1
E
0
1
S
1
D
1
0
B
C
1
1
A
S
1
3
C3
C2
0
0
1
S4
1
C
4
1
Product
Figure 6.18. The multiplication example from Figure 6.17 performed using
carry-save addition.
45
+
Product
Figure 6.19. Schematic representation of the carry-save
addition operations in Figure 6.18.
Carry-Save Addition of
Summands
Figure 6.19. Schematic representation of the carry-save addition operations in Figure 6.18.
46
Carry-Save Addition of
Summands





When the number of summands is large, the
time saved is proportionally much greater.
Some omitted issues:
Sign-extension
Computation width of the final CLA/RCA
Bit-pair recoding
47
Integer Division
48
Manual Division
13
21
274
26
14
13
1
10101
1101 100010010
1101
10000
1101
1110
1101
1
Figure 6.20. Longhand division examples.
49
Longhand Division Steps



Position the divisor appropriately with respect to the
dividend and performs a subtraction.
If the remainder is zero or positive, a quotient bit of
1 is determined, the remainder is extended by
another bit of the dividend, the divisor is
repositioned, and another subtraction is performed.
If the remainder is negative, a quotient bit of 0 is
determined, the dividend is restored by adding back
the divisor, and the divisor is repositioned for
another subtraction.
50
Circuit Arrangement
Shift left
an
an - 1
a0
q
q
n- 1
A
0
Dividend Q
Quotient
setting
Add/Subtract
n +1-bit
adder
Control
sequencer
0
mn - 1
m0
Divisor M
Figure 6.21. Circuit arrangement for binary division.
51
Restoring Division




Shift A and Q left one binary position
Subtract M from A, and place the answer
back in A
If the sign of A is 1, set q0 to 0 and add M
back to A (restore A); otherwise, set q0 to 1
Repeat these steps n times
52
11
Examples
10
1000
11
1
10
Initially
0
0
Shift
0
Subtract 1
Set q0
1
Restore
0
Shift
0
Subtract 1
0
0
0
1
1
0
0
0
1
1
0
1
0
0
1
1
0 0 0
0 0 1
1 1 0
0
1
1
1
0
1
1
0
1
1 1 1 1
1
0 0 0 1
Shift
0 0 1 0
Subtract 1 1 1 0
Set q0
0 0 0 0
1
1
0
0
1
Shift
0 0 0 1
Subtract 1 1 1 0
Set q0
1 1 1 1
Restore
1
0 0 0 1
0
1
1
1
0
0 0 0 1
0 0 1
Remainder
Quotient
Set q0
Restore
1 0 0 0
0 0 0
First cycle
0 0 0 0
0 0 0
Second cycle
0 0 0 0
0 0 0
Third cycle
1
Fourth cycle
0 0 1 0
53
Figure 6.22. A restoring-division example.
Nonrestoring Division






Avoid the need for restoring A after an
unsuccessful subtraction.
Any idea?
Step 1: (Repeat n times)
If the sign of A is 0, shift A and Q left one bit position and
subtract M from A; otherwise, shift A and Q left and add
M to A.
Now, if the sign of A is 0, set q0 to 1; otherwise, set q0 to
0.
Step2: If the sign of A is 1, add M to A
54
Examples
Initially
0
0
Shift
0
Subtract 1
Set q0
1
0
0
0
1
0
0
0
1
0
1
0
0
0
1
1
1
1 0 0 0
1 1 1 0
0 0 0 0
0 0 0
Shift
Add
Set q0
1 1 1 0 0
0 0 0 1 1
0 0 0
1 1 1 1 1
0 0 0 0
Shift
Add
Set q0
1 1 1 1 0
0 0 0 1 1
0 0 0
0 0 0 0 1
0 0 0 1
Shift
0 0 0 1 0
Subtract 1 1 1 0 1
Set q0
1 1 1 1 1
First cycle
Second cycle
Third cycle
0 0 1
Fourth cycle
0 0 1 0
Quotient
Add
1 1 1 1 1
0 0 0 1 1
0 0 0 1 0
Restore remainder
Remainder
55
Figure 6.23. A nonrestoring-division example.
Floating-Point Numbers
and Operations
56
Floating-Point Numbers


So far we have dealt with fixed-point numbers (what
is it?), and have considered them as integers.
Floating-point numbers: the binary point is just to the
right of the sign bit.
B  b 0 .b 1 b  2  b  ( n 1 )
F ( B )   b0  2

0
 b 1  2
1
 b 2  2
2
 ( n 1 )
Where the range of F is:
1  F  1 2

   b  ( n 1 )  2
 ( n 1 )
The position of the binary point is variable and is
automatically adjusted as computation proceeds.
57
Floating-Point Numbers





What are needed to represent a floating-point
decimal number?
Sign
Mantissa (the significant digits)
Exponent to an implied base (scale factor)
“Normalized” – the decimal point is placed to
the right of the first (nonzero) significant digit.
58
IEEE Standard for FloatingPoint Numbers



Think about this number (all digits are decimal):
±X1.X2X3X4X5X6X7×10±Y1Y2
It is possible to approximate this mantissa precision
and scale factor range in a binary representation
that occupies 32 bits: 24-bit mantissa (1 sign bit for
signed number), 8-bit exponent.
Instead of the signed exponent, E, the value actually
stored in the exponent field is an unsigned integer
E’=E+127, so called excess-127 format
59
IEEE Standard
32 bits
S
Sign of
number :
0 signifies+
1 signifies -
E
M
8-bit signed
exponent in
excess-127
representation
23-bit
mantissa fraction
E - 127
Value represented= ±1.M ´ 2
(a) Single precision
0 00 10 1 00 0 0 0 10 1 0 . . .
0
 0 ´ 2- 87
Value represented
= 1.001010
(101000)2=4010, 40-127=-87
(b) Example of a single-precision number
64 bits
S
E
M
Sign
11-bit excess-1023
exponent
52-bit
mantissa fraction
Value represented
= ±1.M ´ 2
E - 1023
(c) Double precision
60
Figure 6.24. IEEE standard floating-point formats.
IEEE Standard



For excess-127 format, 0 ≤ E’ ≤ 255.
However, 0 and 255 are used to represent
special value. So actually 1 ≤ E’ ≤ 254. That
means -126 ≤ E ≤ 127.
Single precision uses 32-bit. The value range
is from 2-126 to 2+127.
Double precision used 64-bit. The value
range is from 2-1022 to 2+1023.
61
Two Aspects

If a number is not normalized, it can always be put in normalized
form by shifting the fraction and adjusting the exponent.
excess-127 exponent
0 1 0 0 0 1 0 0 0 0 0 1 0 1 1 0 ...
(There is no implicit 1 to the left of the binary point.)
(100001000)2=13610, 136-127=-9
Value represented= + 0.0010110 ´ 29
(a) Unnormalized value
0 1 0 0 0 0 1 0 1 0 1 1 0 ...
6+127=133. 13310, = (100000101)2
6
Value represented= + 1.0110 ´ 2
(b) Normalized version
62
Figure 6.25. Floating-point normalization in IEEE single-precision format.
Two Aspects


As computations proceed, a number that
does not fall in the representable range of
normal numbers might be generated.
It requires an exponent less than -126
(underflow) or greater than +127 (overflow).
Both are exceptions that need to be
considered.
63
Special Values





The end value 0 and 255 are used to represent
special values.
When E’=0 and M=0, the value exact 0 is
represented. (±0)
When E’=255 and M=0, the value ∞ is represented.
(± ∞)
When E’=0 and M≠0, denormal numbers are
represented. The value is ±0.M2-126.
When E’=255 and M≠0, Not a Number (NaN).
64
Exceptions


A processor must set exception flags if any of
the following occur in performing operations:
underflow, overflow, divide by zero, inexact,
invalid.
When exception occurs, the results are set to
special values.
65
Arithmetic Operations on
Floating-Point Numbers

Add/Subtract rule


Choose the number with the smaller exponent and shift its mantissa right a
number of steps equal to the difference in exponents.
Set the exponent of the result equal to the larger exponent.
Perform addition/subtraction on the mantissas and determine the sign of the
result.
Normalize the resulting value, if necessary.

Multiply rule


Add the exponents and subtract 127.
Multiply the mantissas and determine the sign of the result.
Normalize the resulting value, if necessary.

Divide rule

Subtract the exponents and add 127.
Divide the mantissas and determine the sign of the result.
Normalize the resulting value, if necessary.





66
Guard Bits and Truncation


During the intermediate steps, it is important
to retain extra bits, often called guard bits, to
yield the maximum accuracy in the final
results.
Removing the guard bits in generating a final
result requires truncation of the extended
mantissa – how?
67
Guard Bits and Truncation







Chopping – biased, 0 to 1 at LSB. 0.b-1b-2b-3000 -- 0.b-1b-2b-31110.b-1b-2b-3
Von Neumann Rounding (any of the bits to be removed are 1, the
LSB of the retained bits is set to 1) – unbiased, -1 to +1 at LSB.
All 6-bit fractions
with b-4is
b-5better
b6 not equal
000
are truncated
to 0.b-1b-21
Why unbiased
rounding
for to
the
cases
that many
operands are involved?
Rounding (A 1 is added to the LSB position of the bits to be
retained if there is a 1 in the MSB position of the bits being
removed) – unbiased, -½ to +½ at LSB.
Round to the nearest number or nearest even number in case of a tie
(0.b-1b-20000 - 0.b-1b-20, 0.b-1b-21100 - 0.b-1b-21+0.001)
Best accuracy
Most difficult to implement
68
Implementing Floating-Point
Operations




Hardware/software
In most general-purpose processors, floatingpoint operations are available at the machineinstruction level, implemented in hardware.
In high-performance processors, a significant
portion of the chip area is assigned to
floating-point operations.
Addition/subtraction circuitry
69
EA 
EB 
MA
32-bit operands
MB
A : SA, EA , M A
B : SB, EB , M B
8-bit
subtractor
M of number
with smaller E
SWAP
M of number
with larger E
SHIFTER
sign
SA SB
n bits
to right
n = EA  - EB 
Add /
Subtract
Combinational
Add/Sub
CONTROL
network
Mantissa
adder/subtractor
Sign
EA 
EB 
MagnitudeM
Leading zeros
detector
MUX
X
E
Normalize and
round
8-bit
subtractor
E - X
R : SR
ER
MR
32-bit
result
R = A+B
70
Figure 6.26. Floating-point addition-subtraction unit.
Requirements for Homework6


5.6. (a): 3 credits
5.6. (b):
 Draw a figure to show how program words are mapped on the
cache blocks: 4
 sequence of reads from the main memory blocks into cache
blocks:4
 total time for reading the blocks from the main memory into the
cache:4
 Executing the program out of the cache:



Outer loop excluding Inner loop:4
Inner loop:4
End section of program:4
Total execution time:3
Due time: class on Oct. 18


71
Hints for Homework6




Assume that consecutive addresses refer to consecutive words. The
cycle time is for one word
Assume this problem does not use load-through, which means when
a read miss occurs, the block of words that contains the requested
word is copied from the main MEM into the cache, after the entire
block is loaded into the cache, the particular word requested is
forwarded to the processor
Total time for reading the blocks from the main memory into the
cache: the number of readsx128x10
Executing the program out of the cache

MEM word size for instructionsxloopNumx1




Outer loop excluding Inner loop: (outer loop word size-inner loop word
size)x10x1
Inner loop: inner loop word sizex20x10x1
MEM word size from MEM 23 to 1200 is 1200-22
MEM word size from MEM 1200 to 1500(end) is 1500-1200
72
Homework 7
1.
2.
3.
4.
5.
6.
1.
2.
Addition and Subtraction of Signed Numbers 5-9, Oct. 20 (Barret, Felix,
Washington)
Carry-lookahead Addition 11-18, Oct. 20 (Kyle White, Jose Jo)
Unsigned Multiplication 20-25, Oct. 20 (Tannet Garrett, Garth Gergerich,
Gabriel Graderson)
Signed Multiplication 26-28 (Shen)
Booth Alg. 29-34, Oct. 25 (Ashraf Hajiyer)
Fast Multiplication
Bit-Pair Recoding of Multipliers 36-38, Oct. 25(Alex, Suzanne, Scott)
Carry-Save Addition of Summands 39-47, Oct. 25 (Jason, Jordan,
Chris)
Integer Division
7.
1.
2.
Restoring Division 49-52, Oct. . 27 (Kyle, Brandan, Alex Shipman)
Nonrestoring Division 53-55, Oct. 27 (Zach, Eric, Chase)
Each presentation is limited to 15 minutes including 2 minutes for questions
73
Exercise for Oct.23



Read “Booth’s algorithm and Bit-Pair
Recoding” in the textbook(6.4.1 & 6.5.1)
Calculate 2’s complement multiplication
(+4)×(-7) using Booth’s algorithm and BitPair Recoding. (Booth’s algorithm and BitPair Recoding will be introduced on Oct.25)
You don’t need to hand in this exercise
74