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CS429: Computer Organization and Architecture
Integers
Dr. Bill Young
Department of Computer Science
University of Texas at Austin
Last updated: June 10, 2019 at 14:
CS429 Slideset 3: 1
Integers
Topics of this Slideset
Numeric Encodings: Unsigned and two’s complement Programming Implications: C promotion rules Basic operations:
addition, negation, multiplication Consequences of overflow Using shifts to perform power-of-2 multiply/divide
CS429 Slideset 3: 2
Integers
C Puzzles
i n t
x =
f o o ( ) ;
i n t
y =
b a r ( ) ;
u n s i g n e d
ux
= x ;
u n s i g n e d
uy
= y ;
Assume a machine with 32-bit, two’s complement integers. For each of the following, either:
Argue that is true for all argument values; Give an example where it’s not true. x < 0
→
((x*2) < 0
ux >= 0(x & 7) == 7
→
(x<<30) < 0
ux > -1x > y
→
-x < -y
x * x >= 0x > 0 && y > 0
→
x + y > 0
x >= 0
→
-x <= 0
x <= 0
→
-x >= 0
CS429 Slideset 3: 3
Integers
Encoding Integers: Unsigned
For unsigned integers, we treat all values as non-negative and use positional notation
as with non-negative decimal numbers.
Assume we have a
w
length bit string X.
Unsigned:
B2U
w
X
∑
w
−
1
i =
X
i^
×
i
CS429 Slideset 3: 4
Integers
Unsigned Integers: 4-bit System
CS429 Slideset 3: 5
Integers
Encoding Integers: Two’s Complement
Two’s complement is a way of encoding integers, including somepositive and negative values. It’s exactly like unsigned except
the
high order bit is given negative weight
Two’s complement:
B2T
w
X
X
w
−
1
×
w
−
1
∑
w
−
2
i =
X
i^
×
i
Decimal
Hex
Binary
3B 6D
C4 93
Sign Bit: For 2’s complement, the most significant bit indicates the sign.
0 for nonnegative 1 for negative
CS429 Slideset 3: 6
Integers
Encoding Example
x =
1 5 2 1 3 :
00111011
01101101
y =
−
15213:
11000100
10010011
Weight
15213
-
1
1
1
1
1
2
0
0
1
2
4
1
4
0
0
8
1
8
0
0
16
0
0
1
16
32
1
32
0
0
64
1
64
0
0
128
0
0
1
128
256
1
256
0
0
512
1
512
0
0
1024
0
0
1
1024
2048
1
2048
0
0
4096
1
4096
0
0
8192
1
8192
0
0
16384
0
0
1
16384
0
0
1
Sum
15213
-
CS429 Slideset 3: 7
Integers
Signed Integers: 4-bit System
CS429 Slideset 3: 8
Integers
Signed vs Unsigned in C
Constants
By default, constants are considered to be signed integers. They are unsigned if they have “U” as a suffix:
0U
4294967259U
Casting
Explicit casting between signed and unsigned is the same asU2T and T2U:^ i n t
tx ,
t y ;
u n s i g n e d
ux ,
uy ;
t x
=
( i n t
)
ux ;
uy
=
( u n s i g n e d
)
t y ;
Implicit casting also occurs via assignments and procedurecalls.^ t x
=
ux ;
uy
=
t y ;
CS429 Slideset 3: 13
Integers
Casting Surprises
Expression Evaluation
If you mix unsigned and signed in a single expression, signedvalues implicitly cast to unsigned. This includes when you compare using
Const 1
Const 2
Rel.
Evaluation
0U
unsigned
signed
0U
unsigned
signed
2147483647U
unsigned
signed
(unsigned) 1
unsigned
2147483648U
unsigned
(int) 2147483648U
signed
CS429 Slideset 3: 14
Integers
Sign Extension
Task:
Given a w-bit signed integer x, convert it to a w+k-bit
integer with the
same value
Rule:
Make k copies of the sign bit :
x
′
x
w
−
1
x
w
−
1
,^
x
w
−
2
w
0
Why does this work?
CS429 Slideset 3: 15
Integers
Sign Extension Example
s h o r t
i n t
x =
1 5 2 1 3 ;
i n t
i x
=
( i n t
)
x ;
s h o r t
i n t
y =
−
15213;
i n t
i y
=
( i n t
)
y ;
Decimal
Hex
Binary
x
15213
3B 6D
00111011 01101101
ix
15213
00 00 3B 6D
00000000 00000000 00111011 01101101
y
C4 93
11000100 10010011
iy
FF FF C4 93
11111111 11111111 11000100 10010011
In converting from smaller to larger signed integer data types, Cautomatically performs sign extension.
CS429 Slideset 3: 16
Integers
Why Use Unsigned?
Don’t use just to ensure numbers are nonzero.
Some C compilers generate less efficient code for unsigned.^ u n s i g n e d
i ;
f o r
( i =1;
i^
<
c n t ;
i ++)
a [ i ]
+= a [ i
−
1]
It’s easy to make mistakes.^ f o r
( i
=
cnt
−
2;
i^
=
0 ;
i^
−−
)
a [ i ]
+= a [ i +1]
Do use when performing modular arithmetic.
multiprecision arithmetic other esoteric stuff
Do use when you need extra bits of range.
CS429 Slideset 3: 17
Integers
Negating Two’s Complement
To find the negative of a number in two’s complement form:complement the bit pattern and add 1:
x
x
Example:
= 0x9C =
10
complement:
= 0x62 = 98
10
add 1:
= 0x63 = 99
10
Try it with:
and
CS429 Slideset 3: 18
Integers
Complement and Increment Examples
Decimal
Hex
Binary
x
3B 6D
˜x
C4 92
˜x+
C4 93
FF FF
CS429 Slideset 3: 19
Integers
Unsigned Addition
Given two w-bit unsigned quantities u, v, the true sum may be aw+1-bit quantity. Discard the carry bit
and treat the result as an unsigned integer.
Thus, unsigned addition implements
modular addition
UAdd
w
u
,^
v
u
v
) mod 2
w
UAdd
w
u
,^
v
{
u
v
u
v
^2
w
u
v
w
u
v
w
CS429 Slideset 3: 20
Integers
Detecting 2’s Complement Overflow
Task: Determine if
s
= TAdd
w
u
v
u
v
Claim:
We have overflow iff either:
u
,^
v
0 but
s
0 (NegOver)
u
,^
v
0 but
s
0 (PosOver)
Can compute this as:
ovf = (u<0 == v<0) && (u<0 != s<0);
On the machine, this causes the
overflow flag
to be set.
Why don’t we have to worry about the case where one input ispositive and one negative?
CS429 Slideset 3: 25
Integers
Properties of TAdd
TAdd is Isomorphic to UAdd. This is clear since they have identical bit patterns.
Tadd
w
u
,^
v
) = U2T(UAdd
w
(T2U(
u
,^
T2U(
v
Two’s Complement under TAdd forms a group.
Closed, commutative, associative, 0 is additive identity. Every element has an additive inverse:Let TComp
w
u
) = U2T(UComp
w
(T2U(
u
)), then
TAdd
w
u
,^
UComp
w
u
TComp
w
u
{
u
u
= TMin
w
TMin
w
u
= TMin
w
CS429 Slideset 3: 26
Integers
Multiplication
Computing the exact product of two w-bit numbers x, y.
This
is the same for both signed and unsigned. Ranges:
Unsigned: 0
x
y
w
2
2
w
w
up to 2
w
bits.
Two’s comp. min: x
y
w
−
1
w
−
1
2
w
−
2
w
−
1
, requires up
to 2
w
1 bits.
Two’s comp. max:
x
y
w
−
1
2
2
w
−
2
, requires up to
w
(but only for TMin
2 w
Maintaining the exact result
Would need to keep expanding the word size with eachproduct computed. Can be done in software with “arbitrary precision” arithmeticpackages.
CS429 Slideset 3: 27
Integers
Unsigned Multiplication in C
Given two w-bit unsigned quantities u, v, the true sum may be a2w-bit quantity. We just discard the most significant w bits,
treat the result as
an unsigned number.Thus, unsigned multiplication implements
modular
multiplication
UMult
w
u
,^
v
u
×
v
) mod 2
w
CS429 Slideset 3: 28
Integers
Unsigned vs. Signed Multiplication
Unsigned Multiplication^ u n s i g n e d
ux
=
( u n s i g n e d
)
x ;
u n s i g n e d
uy
=
( u n s i g n e d
)
y ;
u n s i g n e d
up =
ux
∗
uy ;
Truncates product to w-bit number:
up
= UMult
w
ux
uy
Modular arithmetic:
up
ux
·^
uy
) mod 2
w
Two’s Complement Multiplication^ i n t
x ,
y ;
i n t
p = x
∗
y ;
Compute exact product of two w-bit numbers x, y. Truncate result to w-bit number:
p
= TMult
w
x
y
CS429 Slideset 3: 29
Integers
Unsigned vs. Signed Multiplication
Unsigned Multiplication^ u n s i g n e d
ux
=
( u n s i g n e d
)
x ;
u n s i g n e d
uy
=
( u n s i g n e d
)
y ;
u n s i g n e d
up =
ux
∗
uy ;
Two’s Complement Multiplication^ i n t
x ,
y ;
i n t
p = x
∗
y ;
Relation
Signed multiplication gives same bit-level result as unsigned. up == (unsigned) p
CS429 Slideset 3: 30
Integers
Multiply with Shift
A left shift by
k
, is equivalent to multiplying by 2
k
. This is true for
both signed and unsigned values.
u << 1
u
×
u << 2
u
×
u << 3
u
×
u << 4
u
×
u << 5
u
×
u << 6
u
×
Compilers often use shifting for multiplication, since shift and addis much faster than multiply (on most machines).
u << 5 - u << 3
u * 24
CS429 Slideset 3: 31
Integers
Aside: Floor and Ceiling Functions
Two useful functions on real numbers are the
floor
and
ceiling
functions. Definition:
The floor function
r
, is the greatest integer less than
or equal to
r
.^14
.^14
Definition:
The ceiling function
r
, is the smallest integer greater
than or equal to
r
.^14
.^14
CS429 Slideset 3: 32
Integers
Properties of Two’s Complement Arithmetic
Isomorphic Algebras
Unsigned multiplication and addition: truncate to w bits Two’s complement multiplication and addition: truncate to wbits
Both form rings isomorphic to ring of integers mod
w
Comparison to Integer Arithmetic
Both are rings Integers obey ordering properties, e.g.
u
u
v
v
u
^0
v
u
v
These properties are not obeyed by two’s complementarithmetic.
TMax + 1
TMin
(for 16-bit words)
CS429 Slideset 3: 37
Integers
C Puzzle Answers
Assume a machine with 32-bit word size, two’s complementintegers.^ i n t
x =
f o o ( ) ;
i n t
y =
b a r ( ) ;
u n s i g n e d
ux
= x ;
u n s i g n e d
uy
= y ;
x < 0
→
((x*2) < 0
False:
TMin
ux >= 0
True:
0 = UMin
(x & 7) == 7
→
(x<<30) < 0
True:
x 1
= 1
ux > -
False:
0
x > y
→
-x < -y
False:
−
1, TMin
x * x >= 0
False:
30426
x > 0 && y > 0
→
x + y > 0
False:
TMax, TMax
x >= 0
→
-x <= 0
True:
-TMax
<
^0
x <= 0
→
-x >= 0
False:
TMin
CS429 Slideset 3: 38
Integers
