Number Representation Fixed and Floating Point

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Number RepresentationFixed and Floating Point

• No Method Capable of Representing ALL Real
  Numbers Using Finite Register Lengths
• Must Use Approximations to Represent Values
• Concentrate on Two Forms
• Fixed Point
• Floating Point
• Others are
• Rational Number Systems uses ratios of integers
• Logarithmic Number Systems uses signs and
  logarithms of values

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Fixed Versus Floating Point

• Fixed Point Values Represent Values where Any Two
  Differ by 1 unit in the last place (ulp)
• Equal Spacing Between Numbers
• Floating Point Values Use Two Multi-Bit Words
• Mantissa
• Exponent
• Both Forms Must be Capable of Representing Signed
  Quantities
• Fixed Point Values CAN be Used to Represent
  Fractional Quantities

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Floating Point Characteristics

• Total Number of Representations Total Bit
  Strings
• For n-bit Register we have 2n
• Range of Value is Larger than Fixed Point
• Precision of Value is Smaller
• Distance Between Two Consecutive Values Increases

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Floating Point
s
e
m
s Sign Bit (signed magnitude) e Exponent (in
2s Complement Form) m Mantissa (significand or
fraction) mMAX1 - ulp 0,1)
hidden bit
float BIAS 127 (32 bits-23 for m and 8 for
e) double BIAS1023 (64 bits-52 for m and 11
for e) Sign of Exponent is Complement of its
MSb Thus, adding/subtracting bias is just
complementation of MSb
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Floating Point Example
double 00000000 bfe80000 Big Endian MSW has
Higher Address
s
m
e
1 011 1111 1110 1000 0000 0000 0000 0000
0000 0000 0000 0000 0000 0000 0000 0000 0000
s 1 e 1022 m 0.5 Value (-1)1?1.5
?2(1022-1023) Value -(1.5)(0.5) -0.75
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Floating Point Normalization

• Redundant /representations are Possible!

• Hidden Bit Helps
• Out of All Possible Representations, Choose One
  With Fewest Leading Zeros in Significand
• This is Normalization
• After Performing Arithmetic, Renormalization May
  
• Need to be Accomplished

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Floating Point Special Numbers
Value v when exponent e and fraction f are
special values (IEEE standard) Note NaN Not a
Number
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IEEE/ANSI 754/854 Standard
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Denormalized Numbers

• Allows for Gradual Degradation for Underflow

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Denormals
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Operations Internal Precision
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Floating Point Addition/Subtraction
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Floating Point Multiplication/Division
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Conversions and Roundings
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Exceptions
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Rounding Schemes
Signed Magnitude
Twos Complement
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Round to Nearest (Signed Magnitude)
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Rounding Comments
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Round to Nearest Even/Odd
Round to Nearest Even
Round to Nearest Odd (R)
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Jamming/von Neumann Rounding
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ROM Rounding
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Rounding
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Rounding Examples
Round Towards
Downward Directed Rounding
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Floating Point Operations
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Adders/Subtractors
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Operand Packing/Unpacking
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Other Key Parts of FP Add/Sub Unit
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Pre-Shifting
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Four-stage Combinational Shifter
Pre-shifts Operand by 0 to 15 Bits
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Leading Zeros/Ones Counting vs. Prediction
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Leading Zeros Prediction
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Guard Digits

• What is the smallest number of extra digits
  needed for rounding? post-normalization?
• Multiplication Double Length Result
• Add/Sub w/ differing exp. Can have Double
  Length Result
• FP Unit Provides One Length Result

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Significand Ranges

• Assume Significand M?(0,1-ulp
• Then Normalized M ranges as

• Multiplication prodM1?M2

• For postnormalization need at most one shift left
  to get

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Significand Ranges (cont)

• Division quotM1?M2

• Need at most one shift right to get

• Conclusion
• 1 Extra Digit Needed for Postnormalization
• 1 Extra Digit Needed for Round-to-Nearest
• 2 Extra Digits Needed
• G - guard
• R - round

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Sticky Bit in std754

• Round-to-Nearest-Even Requires 1 Extra Bit
• The sticky bit, S
• Turns out to be Logical-OR of Other Additional
  Bits

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Floating Point Multiplier
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Floating Point Divider