University of Washington
Sec on 2: Integer & Floa ng Point Numbers
Representa on of integers: unsigned and signed
Unsigned and signed integers in C
Arithme c and shi ing
Sign extension
Background: frac onal binary numbers
IEEE floa ng point standard
Floa ng point opera ons and rounding
Floa ng point in C
Shi ing and Sign Extension
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Shi Opera ons for unsigned integers
Le shi : x << y
x 00000110
Shi bit vector x le by y posi ons
<< 3 00110000
00010000
00110000
Throw away extra bits on le
>> 2 00000001 1
00011000
00000001
Fill with 0s on right
Right shi : x >> y
Shi bit vector x right by y posi ons
Throw away extra bits on right
x 11110010
Fill with 0s on le
144
<< 3 10010000
00010000
10010000
60 s
>> 2 00111100
00101000
00111100
Shi ing and Sign Extension
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Shi Opera ons for signed integers
Le shi : x << y
x 01100010
Equivalent to mul plying by 2y
16 s
<< 3 00010000
00010000
00010000
(if resul ng value fits, no 1s are lost)
Logical >> 2 00011000 24 s
00011000
00011000
Right shi : x >> y
Logical shi (for unsigned values)
24 s
Arithme c >> 2 00011000
00011000
00011000
Fill with 0s on le
Arithme c shi (for signed values)
x 10100010
Replicate most significant bit on le
16 s
Maintains sign of x
<< 3 00010000
00010000
00010000
Equivalent to dividing by 2y
40 s
Logical >> 2 00101000
00101000
00101000
Correct rounding (towards 0) requires
24
some care with signed numbers
Arithme c >> 2 11101000
11101000
11101000
Undefined behavior when
y < 0 or y e" word_size
Shi ing and Sign Extension
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Using Shi s and Masks
Extract the 2nd most significant byte of an integer:
First shi , then mask: ( x >> 16 ) & 0xFF
x 01100001 01100010 01100011 01100100
x >> 16 00000000 00000000 01100001 01100010
00010000
00010000
00011000
00011000
00000000 00000000 00000000 11111111
( x >> 16) & 0xFF
00000000 00000000 00000000 01100010
Extract the sign bit of a signed integer:
( x >> 31 ) & 1 need the  & 1 to clear out all other bits except LSB
Condi onals as Boolean expressions (assuming x is 0 or 1)
if (x) a=y else a=z; which is the same as a = x ? y : z;
Can be re wri en (assuming arithme c right shi ) as:
a = ( (x << 31) >> 31) & y + ((!x) << 31 ) >> 31 ) & z;
Shi ing and Sign Extension
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Sign Extension
Task:
Given w bit signed integer x
Convert it to w+k bit integer with same value
Rule:
Make k copies of sign bit:
X 2 = xw 1 ,& , xw 1 , xw 1 , xw 2 ,& , x0
w
k copies of MSB
X
" " "
" " "
X 2
" " " " " "
w
k
Shi ing and Sign Extension
University of Washington
Sign Extension Example
Conver ng from smaller to larger integer data type
C automa cally performs sign extension
short int x = 12345;
int ix = (int) x;
short int y = -12345;
int iy = (int) y;
Decimal Hex Binary
12345
x 30 39 00110000 01101101
12345
ix 00 00 30 39 00000000 00000000 00110000 01101101
-12345
y CF C7 11001111 11000111
iy -12345
FF FF CF C7 11111111 11111111 11001111 11000111
Shi ing and Sign Extension