Fixed-Point Numbers
Positional representation: k whole and l fractional digits
Value of a number: x = (xk–1xk–2 . . . x1x0 . x–1x–2 . . . x–l )r = Σ xi r i
For example:
2.375 = (10.011)two = (1×21) + (0×20) + (0×2−1) + (1×2−2) + (1×2−3)
Numbers in the range [0, rk – ulp] representable, where ulp = r –l
Fixed-point arithmetic same as integer arithmetic
Two’s complement properties (including sign change) hold here as well:
(01.011)2’s-compl = (–0×21) + (1×20) + (0×2–1) + (1×2–2) + (1×2–3) = +1.375
(11.011)2’s-compl = (–1×21) + (1×20) + (0×2–1) + (1×2–2) + (1×2–3) = –0.625
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Radix Conversion for Fixed-Point Numbers
Convert the whole and fractional parts separately.
To convert the fractional part from an old radix r to a new radix R:
• Perform arithmetic in the new radix R
Evaluate a polynomial in r –1: (.011)two = 0 × 2–1 + 1 × 2–2 + 1 × 2–3
Simpler: View the fractional part as integer, convert, divide by r l
(.011)two = (?)ten
Multiply by 8 to make the number an integer: (011)two = (3)ten
Thus, (.011)two = (3 / 8)ten = (.375)ten
• Perform arithmetic in the old radix r
Multiply the given fraction by R, use the whole part as the MSD
and the fractional part to repeat the process
(.72)ten = (?)two
0.72 × 2 = 1.44, so the answer begins with 0.1
0.44 × 2 = 0.88, so the answer begins with 0.10
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Floating-Point Numbers
Useful for applications where very large and very small
numbers are needed simultaneously
• Fixed-point representation is not very good for dealing
with very large and extremely small #s simultaneously
x = (0000 0000 . 0000 1001)two
Small number
y = (1001 0000 . 0000 0000)two
Large number
• Neither y2 nor y / x is representable in the format above
• Floating-point representation is like scientific notation:
−20 000 000 = −2 × 10 7
0.000 000 007 = 7 × 10–9
Significand
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Exponent base
Exponent
Also, 7E−9
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ANSI/IEEE Stand. Floating-Point Format
(IEEE754)
Revision (IEEE 754-2008) is available.
Short (32-bit) format
8 bits,
bias = 127,
–126 to 127
23 bits for fractional part
(plus hidden 1 in integer part)
Sign Exponent
11 bits,
bias = 1023,
–1022 to 1023
Short exponent range is –127 to 128
but the two extreme values
are reserved for special operands
(similarly for the long format)
Significand
52 bits for fractional part
(plus hidden 1 in integer part)
Long (64-bit) format
Two ANSI/IEEE standard floating-point formats.
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Short and Long IEEE 754 Formats: Features
Some features of ANSI/IEEE standard floating-point formats
Feature
Word width in bits
Significand in bits
Significand range
Exponent bits
Exponent bias
Zero (±0)
Denormal
Single/Short
32
23 + 1 hidden
[1, 2 – 2–23]
8
127
e + bias = 0, f = 0
e + bias = 0, f ≠ 0
represents ±0.f × 2–126
e + bias = 255, f = 0
e + bias = 255, f ≠ 0
e + bias ∈ [1, 254]
e ∈ [–126, 127]
represents 1.f × 2e
Double/Long
64
52 + 1 hidden
[1, 2 – 2–52]
11
1023
e + bias = 0, f = 0
e + bias = 0, f ≠ 0
represents ±0.f × 2–1022
e + bias = 2047, f = 0
e + bias = 2047, f ≠ 0
e + bias ∈ [1, 2046]
e ∈ [–1022, 1023]
represents 1.f × 2e
min
2–126 ≅ 1.2 × 10–38
2–1022 ≅ 2.2 × 10–308
max
≅ 2128 ≅ 3.4 × 1038
≅ 21024 ≅ 1.8 × 10308
Infinity (±∞)
Not-a-number (NaN)
Ordinary number
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Simple Adders
Inputs
Outputs
x
y
c
s
0
0
1
1
0
1
0
1
0
0
0
1
0
1
1
0
x
c
Inputs
y
HA
s
Outputs
x
y
cin
cout
s
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
0
0
1
0
1
1
1
0
1
1
0
1
0
0
1
x
cout
y
FA
s
cin
S = x ⊕ y ⊕ cin
Cout = ycin + xy + xcin
Binary half-adder (HA) and full-adder (FA).
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Ripple-Carry Adder: Slow But Simple
x31
y31
x1
c31
c32
FA
cout
. . .
y1
c2
x0
y0
c0
c1
FA
FA
s1
s0
Critical path
s31
cin
Ripple-carry binary adder with 32-bit inputs and output.
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Carry Propagation
0111
+ 0001
1000
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•Carry Look-Ahead Adder
•uses the concepts of generating and propagating
carries
•Ci+1=xiyi+ci(xi+yi), xiyi = gi, xi+yi=pi
•Carry Bypass (=skip) Adder
-Looks for cases in which carry out of a set of bits is
identical to carry in
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Design of Fast Adders
•Carry Select Adder
-Don’t know carry in: so do both
-Use MUX to select the right sum
•Carry Save Adder
-produce the sum and carry bits separately and add
them at the end
-10011 + 00110 = (21+4) = 25, 01100+10011+00110 =
(25+12) = 37
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Combined
Addition/
Subtraction I.
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Two’s-Complement Addition and Subtraction
k
/
x
c in
Adder
k
/
y
k
/
k
/
x±y
c out
y or
y′
Add′Sub
Binary adder used as 2’s-complement adder/subtractor.
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Combined Addition/ Subtraction II.
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•Combined Addition/Subtraction II.
- 4 FAs, 4 XORs, 1 extra control bit.
Note: Combined Addition/Subtraction I.
- 4 FAs, 8 ANDs, 4 ORs, 2 extra control bits
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Magnitude Comparator
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A3=B3 ?
X3A2’B2
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Binary Multiplier
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Shift-Add Multiplication
Multiplicand
Multiplier
x
y
Partial
products
bit-matrix
y0
y
1
y2
y3
Product
z
x
x
x
x
20
21
22
23
Multiplication of 4-bit numbers in dot notation.
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Binary and Decimal Multiplication
Position
7 6 5 4 3 2 1 0
=========================
4
x2
1 0 1 0
y
0 0 1 1
=========================
(0)
z
0 0 0 0
+y0x24
1 0 1 0
––––––––––––––––––––––––––
2z (1)
0 1 0 1 0
z (1)
0 1 0 1 0
+y1x24
1 0 1 0
––––––––––––––––––––––––––
2z (2)
0 1 1 1 1 0
z (2)
0 1 1 1 1 0
+y2x24
0 0 0 0
––––––––––––––––––––––––––
2z (3)
0 0 1 1 1 1 0
z (3)
0 0 1 1 1 1 0
4
+y3x2
0 0 0 0
––––––––––––––––––––––––––
(4)
2z
0 0 0 1 1 1 1 0
z (4)
0 0 0 1 1 1 1 0
=========================
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Example
Position
7 6 5 4 3 2 1 0
=========================
4
x10
3 5 2 8
y
4 0 6 7
=========================
(0)
z
0 0 0 0
+y0x104 2 4 6 9 6
––––––––––––––––––––––––––
10z (1)
2 4 6 9 6
z (1)
0 2 4 6 9 6
+y1x104 2 1 1 6 8
––––––––––––––––––––––––––
10z (2)
2 3 6 3 7 6
z (2)
2 3 6 3 7 6
+y2x104 0 0 0 0 0
––––––––––––––––––––––––––
10z (3)
0 2 3 6 3 7 6
z (3)
0 2 3 6 3 7 6
4
+y3x10 1 4 1 1 2
––––––––––––––––––––––––––
10z (4)
1 4 3 4 8 3 7 6
z (4)
1 4 3 4 8 3 7 6
=========================
Step-by-step multiplication examples for 4-digit unsigned numbers.
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Example
Two’s-Complement Multiplication
Position
7 6 5 4 3 2 1 0
=========================
4
x2
1 0 1 0
y
0 0 1 1
=========================
(0)
z
0 0 0 0 0
+y0x24
1 1 0 1 0
––––––––––––––––––––––––––
(1)
2z
1 1 0 1 0
z (1)
1 1 1 0 1 0
4
+y1x2
1 1 0 1 0
––––––––––––––––––––––––––
(2)
2z
1 0 1 1 1 0
z (2)
1 1 0 1 1 1 0
+y2x24
0 0 0 0 0
––––––––––––––––––––––––––
2z (3)
1 1 0 1 1 1 0
z (3)
1 1 1 0 1 1 1 0
+(–y3x24) 0 0 0 0 0
––––––––––––––––––––––––––
2z (4)
1 1 1 0 1 1 1 0
z (4)
1 1 1 0 1 1 1 0
=========================
Position
7 6 5 4 3 2 1 0
=========================
4
x2
1 0 1 0
y
1 0 1 1
=========================
(0)
z
0 0 0 0 0
+y0x24
1 1 0 1 0
––––––––––––––––––––––––––
(1)
2z
1 1 0 1 0
z (1)
1 1 1 0 1 0
4
+y1x2
1 1 0 1 0
––––––––––––––––––––––––––
(2)
2z
1 0 1 1 1 0
z (2)
1 1 0 1 1 1 0
+y2x24
0 0 0 0 0
––––––––––––––––––––––––––
2z (3)
1 1 0 1 1 1 0
z (3)
1 1 1 0 1 1 1 0
+(–y3x24) 0 0 1 1 0
––––––––––––––––––––––––––
2z (4)
0 0 0 1 1 1 1 0
z (4)
0 0 0 1 1 1 1 0
=========================
Step-by-step multiplication examples for 2’s-complement numbers.
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Hardware Multipliers
Shift
Multiplier y
Doublewidth partial product z (j)
Shift
Multiplicand x
0
Mux
1
yj
Enable
Select
c out
Adder
c in
Add’Sub
Hardware multiplier based on the shift-add algorithm.
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