1. Topic 3d Representation of Real Numbers
Introduction to Computer Systems Engineering
(CPEG 323)
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2. Recap What can be represented in n bits? But, what about?
Unsigned 0 to 2n - 1
2’s Complement -2n-1 to 2n-1 - 1
BCD 0 to 10n/4 - 1
But, what about?
very large numbers 9,349,398,989,787,762,244,859,087,678
very small number
rationals /3
Irrationals …., …..,
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3. Conceptual Overview: Finite-precision numbers
Negative underflow
Positive underflow
Expressible negative numbers
zero
Expressible positive numbers
Negative overflow
Positive overflow
……………….
Is there any “underflow” region in integer representation?
Between two adjacent integer numbers, there is no other integer.
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4. Representing Real Numbers
How to represent fractional parts to the right of the ``decimal'' point?
A number like 0.12 (i.e., (1/2)10) not represented well by integers 0 or 1!
Two ways to represent real numbers better:
Fixed point
Floating point
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5. Fixed-Point Data Format4 2 1
1/2
1/4 ……
1/8 S 1 1 1 . 1
Sign
Integer
Fractional
When representing a real value, you have two choices: the fixed-point format or the floating-point format.
Any way, to avoid nonlinear effects introduced by overflow, underflow, saturation, and wrapping during computation, control of each variable’s range has to be done by appropriate scaling of operands.
In the floating-point case, the exponent of a variable is part of the run-time representation and is computed automatically by the floating-point ALU. But in the fixed-point case, the exponent of each variable is implicit and determined by the programmer off-line using the predicted dynamic range of the variable. Therefore, the scaling of fixed-point operands has to be done explicitly by the programmer. This is generally a tedious and error-prone process.
In fact, you may regard the fixed-point as a form of integer with an additional hypothetical binary point. In our research, the number of bits to the left of the binary-point excluding the sign bit is called Integer Word-Length or IWL.
hypothetical binary point
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6. Fixed Point Pros Cons Add two reals just by adding the integers
Much less logic than floating point
Faster
Often used in digital signal processing
Cons
The range of numbers is narrow
number=
It is much more economical to represent as 4*1026
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7. Recall Scientific Notation
decimal point
exponent
-23
6.02 x 10
Significand
radix (base)
Issues:
Arithmetic (+, -, *, / )
Representation, Normal form
Range and Precision(Determined by?)
Rounding and errors
Exceptions (e.g., divide by zero, overflow, underflow)
Properties
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8. Scientific Notation: Normalized
12.35 x 10^-9 ?
1.235 x 10^-8 ?
scientific notation: has a single digit to the left of the decimal point
Normalized scientific notation: such a single digit must be non-zero.
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9. Floating Point Representation
Numerical Form: (–1)s M 2E
Sign bit s determines whether number is negative or positive
Significand M normally a fractional value in range [1.0,2.0).
Exponent E weights value by power of two
Encoding
MSB is sign bit: S=0/1
exp field encodes E
frac field encodes M s
exp
frac
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10. IEEE 754 standard
Three formats: single/double/extended precision (32,64,80 bits).
Single precision: s
exp
frac
1 bit bits bits
Double precision: see page 192 in P&H book
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11. IEEE 754 Standard Floating Point Representation
the representation:
(–1)s (1+ Fraction) 2E-bias
where E is the exponent part in the notation
Sign bit s determines whether number is negative or positive
Fraction is normally a fractional value in range between 0 and 1
A leading 1 added to the fraction is “implicit”
Exponent using a “biased” notation”
For single precision – the bias is 127
That is, when you convert the notation to the number it represents, the exponential part used should be
reading E out from the IEEE 754 notation, and calculated as E-127.
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12. Advantage of using the biased notation for exponents
Under the single-precision IEEE 754 standard: bias = 127
If the real exponent is +1, what is the biased exponent ? Answer: = 128! (Note = +1!)
How about if the real exponent is -1 ?
Answer: = 126! (Note = -1!)
How about if the real exponent is -127 ?
E -127 = -127, then E = ?
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13. Observations
Using biased notation, the exponential part in the single-precision IEEE 754 notation itself never get nagative.
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14. Normalized Encoding Example
Value: Float F = ;
= X 213
Significand
M =
frac =
(23 bits! With leading 1 hiding!)
Exponent
E = 1310, Bias =
Exp = =
Floating Point Representation:
Binary:
exp
frac
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15. Denormalized Values Condition Significand value M = 0.xxx…x Cases
 exp = 000…0
Significand value M = 0.xxx…x
xxx…x: bits of frac
Cases
frac = 000…0
Represents value 0
Note that have distinct values +0 and –0
frac  000…0
Numbers very close to 0.0
Lose precision as get smaller
“Gradual underflow”
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16. Special Values (infinity) Condition Not-a-Number (NaN) exp = 111…1
Operation that overflows
Both positive and negative
E.g., 1.0/0.0 = 1.0/0.0 = +, 1.0/0.0 = 
Not-a-Number (NaN)
Represents case when no numeric value can be determined
E.g., sqrt(–1), 
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17. IEEE 754: Summary Normalized: ± 0<exp<max Any bit pattern ±
Denormalized:
Any nonzero bit pattern
zero:
Infinite:
11…1
NaN:
11…1
Any nonzero bit pattern
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18. IEEE754: Summary (Cont.) + 0 +0  -Normalized -Denorm +Denorm
NaN
NaN 0 +0
Negative overflow
Positive underflow
Negative underflow
Positive overflow
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19. FP Addition Operands Exact Result: (–1)s M 2E (–1)s1 M1 2E1
Assume E1 > E2
Exact Result: (–1)s M 2E
Exponent E: E1
Sign s, significand M: Result of signed align & add
E1–E2
(–1)s1 M1
(–1)s2 M2 +
(–1)s M
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20. Decimal Number Conversion
Convert Binary to Decimal (base 2 to base 10)
x x x x x. d d d d d d …2
… …
= 1*23+1*22+0*21+1*20+0*2-1+1*2-2+1*2-3 =
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21. Decimal Number Conversion (Cont.)
Convert Decimal Integer to binary Integer
divide the decimal value by 2 and then write down the remainder from bottom to top (Divide 2 and Get the Remainders)
3710 = ?2
Quotient reminder
37÷2 = … 1
18÷2 = … 0
9÷2 = … 1
4÷2 = … 0
2÷2= … 0
1÷2 = … 1
Can it always be converted into an accurate binary number?
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YES!
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22. Decimal Number Conversion (Cont.)
Convert Decimal Fraction to Binary Fraction
multiply the decimal value by 2 and then write down the integer number from top to bottom (Multiply 2 and Get the Integers) .
= ?2
.375 * 2 =
.75 * 2 =
.5 * 2 =
Where to put the binary point?
Prior to the First number!
0.0112
Can it always be converted into an accurate binary number?
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NO!
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23. Decimal Number Conversion (Cont.)
Convert Decimal Number to Binary Binary
Put Together: Integer Part . Fraction Part =
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24. Summary Computer arithmetic is constrained by limited precision
Bit patterns have no inherent meaning but standards do exist
two’s complement
IEEE 754 floating point
OPcode determines “meaning” of the bit patterns
Performance and accuracy are important so there are many complexities in real machines (i.e., algorithms and implementation).
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