Download Ch. 3

CS 265
Computer Architecture
Wei Lu, Ph.D., P.Eng.
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Part 1: Data Representation
Our goal:
• revisit and re-establish fundamental of mathematics for
the computer architecture course
Overview:
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what are bits & bytes?
bit and arithmetic value?
bit and hexadecimal?
character set in PCs?
how to do binary arithmetic
operation (+,-,*,/)?
• integer representation in
PCs: signed and unsigned
• binary and floating point
• floating point representation?
• order of byte in computers
big endian vs. little endian
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Conversion in Between
Float and Binary
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Floating point
• Floating point means a number with an integral part and a
fractional part
• Converting floating point in between decimal and binary
• Place values:
25 24 23 22 21 20 2-1
32 16 8
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Integral part
2-2
2-3
2-4
2-5
1 0.5 0.25 0.125 0.0625 0.03125
Fractional part
Example: 4.5 (decimal) = 100.1 (binary)
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Converting floating point
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Example:
Method:
6 = 110
.25 x 2 = 0.5
.5 x 2 = 1.0
– 110.01
• Example:
6 = 110
.2 x 2 = 0.4
.4 x 2 = 0.8
.8 x 2 = 1.6
.6 x 2 = 1.2
.2 x 2 = 0.4
6.25 = 110.01
(Integral part: convert in the usual way)
(Fractional part: successive multiplication by 2)
(Stop when fractional part is 0)
6.2 ≅ 110.001100110011…
(repeats)
– 110.0011 0011 0011
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One more example to convert floating point
• Example:
6.4 ≅ ?
110.0110 0110 0110…
How the floating binary number stored in computer?
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How Float Represented in Computer
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Floating point representation in computer
• Computers use a form of scientific notation for floatingpoint representation
• Numbers written in scientific notation have three basic
components:
Example:
– 1.101 x 2 010
Sign
Mantissa
use base 2 instead of base 10
Exponent
– Exponent specifies the order of magnitude in a base
– Mantissa specifies most significant part of value
– Sign specifies negative or positive
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Floating point representation in computer
Computer representation of a floating-point number
consists of three fixed-size fields:
This is the standard arrangement of these fields.
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Floating point representation in computer
(1) The one-bit sign field is the sign of the stored value.
(2) The size of the exponent field, determines the range of values
that can be represented.
(3) The size of the significand determines the precision of the
representation.
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Floating point representation in computer
• single-precision (32 bits) floating point representation
defined by IEEE standard 754
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1 sign bit
"biased" exponent (8 bits)
"normalized" mantissa (23 bits)
Floating point = (-1)s x (1.m) x 2(e-bias)
where,
bias = 127
Notice that the 1 in 1.m is always assumed.
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Floating point representation in computer
• double-precision (64 bits) floating point representation
defined by IEEE standard 754
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1 sign bit
"biased" exponent (11 bits)
"normalized" mantissa (52 bits)
Floating point = (-1)s x (1.m) x 2(e-bias)
where,
bias = 1023
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Floating point representation: example 1
• Given decimal real number 6.25, what is its single
precision floating number stored in PC?
Step 1. 6.25 (decimal) = 110.01 (binary)
Step 2. Move the radix point until a single 1 appears on the left,
and multiply by the corresponding power of 2 1.1001 x 22
… so the sign bit is 0 (positive)
… the “biased” exponent is 2 + 127 = 129 = 10000001 = e
… and the “normalized” mantissa is 1001 (drop the 1, rest zero-fill).
0 10000001 10010000000000000000000
0100 0000 1100 1000 0000 0000 0000 0000
40C80000 hexadecimal
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Floating point representation: example 2
• Given decimal real number 6.2, what is its single precision
floating number stored in PC?
Step 1. 6.2 (decimal) = 110.001100110011… (binary)
Step 2. Move the radix point until a single 1 appears on the left,
and multiply by the corresponding power of 2
= 1.10001100110011… x 22
… so the sign bit is 0 (positive)
… the “biased” exponent is 2 + 127 = 129= 10000001
… and the “normalized” mantissa is 10001100110011…
0 10000001 10001100110011001100110
0100 0000 1100 0110 0110 0110 0110 0110
40C66666 hexadecimal
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Floating point representation: example 3
• Double value in memory (in hex):
4024 0000 0000 0000
what’s its binary:
0100 0000 0010 0100 0000 0000 0000 0000 …
what’s its decimal?
s (bit 64) = 0 = positive number
e (bits 52 to 62) = 100 0000 0010 = 1024 + 2 = 1026
m (bits 0 to 51) = .0100 0000 0000 0000 0000
Value = (-1)0 x (1.01)b x 2 (1026-1023)
= 1x (20+2-2)x23=(1+1/4)x8=8+2=10
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Floating point representation: example 4
• What decimal floating-point number is represented by
0xC1870000?
1100 0001 1000 0111 0000 0000 0000 0000
1 10000011 00001110000000000000000
Sign
Biased
exponent
Normalized
mantissa
… so the sign is negative
… the “unbiased” exponent is 131 - 127 = 4
… and the “unnormalized” mantissa is 1.00001110000000000000000
(add the 1 left of the radix point).
Move the radix point 4 places to the right:
10000.111
• -10000.111 = -16.875
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Byte Order:
Big Endian vs. Little Endian
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Byte order
• Byte ordering, or endianness, is a major computer
architectural consideration
• Byte ordering describes how data stored in memory,
which is visible and important to programmer
• More details later in the course, here we introduce basic
concept
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Byte order
• If we have a two-byte integer, the integer may be stored
so that the least significant byte is followed by the most
significant byte or vice versa.
– In little endian machines, the least significant byte is
stored in lowest memory location. (DEC, IBM (Intel
Pentium))
– Big endian machines store the most significant byte in
lowest memory. (Motorola, Sun SPARC)
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Big and little endian: an example
• Given 32 bits (4 bytes) integer (or single-precision floating point
numbers) 0x 90AB12CD, the 4 bytes are: 90, AB, 12, CD, How can
we store these values into memory?
Big Endian
• In big endian, we store the most
significant byte in the smallest
address. Here's how it would look:
Address
Value
1000
90
1001
AB
1002
12
1003
CD
Little Endian
• In little endian, we store the least
significant byte in the smallest
address. Here's how it would look:
Address
Value
1000
CD
1001
12
1002
AB
1003
90
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Endianness: consequence for programmer
• For JAVA programmer, we have to know:
- Everything in Java binary format files is stored big-endian, i.e.
MSB (Most Significant Byte) first.
- JAVA is platform independent because and Java hides internal
endianness from us no matter what internal byte order used (i.e.
Mac is big-endian order and intel PC is little-endian)
• For C programmer, we have to know:
- Most files compiled by C use little-endian in windows PC
• For both C and JAVA programmer, we have to fix the byte
order problem when transferring data files between computers for
which the byte ordering differs
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Summary: conceptual issues
• Bits grouped into sets to represent
– Integers
– Characters
– Floating point values
• Signed integers can be represented as
– Sign magnitude
– One's complement
– Two's complement
• Bytes of integer can be numbered in
– Big-endian order and Little-endian order
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Summary: technical issues
• Conversion in between Binary, Hexadecimal, Decimal,
Single Float and Double
• Signed Integer represented by
– Sign magnitude
– One's complement
– Two's complement
• The consequence of byte order to our programmers
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Thank you for your attendance
Any questions?
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