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Part V
Real Arithmetic
Number Representation
Numbers and ArithmeticRepresenting Signed Numbers Redundant Number SystemsResidue Number Systems
Addition / Subtraction
Basic Addition and CountingCarry-Look ahead Adders Variations in Fast AddersMultioperand Addition
Multiplication
Basic Multiplication SchemesHigh-Radix Multipliers Tree and Array MultipliersVariations in Multipliers
Division
Basic Division SchemesHigh-Radix Dividers Variations in DividersDivision by Convergence
Real Arithmetic
Floating-Point ReperesentationsFloating-Point Operations Errors and Error ControlPrecise and Certifiable Arithmetic
Function Evaluation
Square-Rooting MethodsThe CORDIC Algorithms Variations in Function EvaluationArithmetic by Table Lookup
Implementation Topics
High-Throughput ArithmeticLow-Power Arithmetic Fault-Tolerant ArithmeticPast, Present, and Future
Parts Chapters I.
II.
III.
IV.
V.
VI.
VII.
1.2. 3.4. 5.6. 7.8.
10.9. 11.12.
25.26. 27.28.
21.22. 23.24.
17.18. 19.20.
13.14. Elementary Operations 15.16.
- Reconfigurable Arithmetic Appendix: Past, Present, and Future
V Real Arithmetic
Topics in This Part
Chapter 17 Floating-Point Representations
Chapter 18 Floating-Point Operations
Chapter 19 Errors and Error Control
Chapter 20 Precise and Certifiable Arithmetic
Review floating-point numbers, arithmetic, and errors:
- How to combine wide range with high precision
- Format and arithmetic ops; the IEEE standard
- Causes and consequence of computation errors
- When can we trust computation results?
Floating-Point Representations: Topics
Topics in This Chapter
17.1 Floating-Point Numbers
17.2 The IEEE Floating-Point Standard
17.3 Basic Floating-Point Algorithms
17.4 Conversions and Exceptions
17.5 Rounding Schemes
17.6 Logarithmic Number Systems
17.1 Floating-Point Numbers
No finite number system can represent all real numbers Various systems can be used for a subset of real numbers
Fixed-point w. f Rational p / q Floating-point s be Logarithmic log bx
Fixed-point numbers
x = (0000 0000. 0000 1001)two Small number y = (1001 0000. 0000 0000)two Large number
Low precision and/or range Difficult arithmetic Most common scheme Limiting case of floating-point
Floating-point numbers
x = s be^ or significand baseexponent
A floating-point number comes with two signs:
Number sign, usually appears as a separate bit Exponent sign, usually embedded in the biased exponent
Square of neither number representable
x = 1.001 2 –^5 y = 1.001 2 +
17.2 The IEEE Floating-Point Standard
Short (32-bit) format
Long (64-bit) format
Sign Exponent Significand
8 bits, bias = 127,
11 bits, bias = 1023,
52 bits for fractional part (plus hidden 1 in integer part)
23 bits for fractional part (plus hidden 1 in integer part)
Fig. 17.3 The IEEE standard floating-point number representation formats.
IEEE 754-2008 Standard (supersedes IEEE 754-1985) Also includes half- & quad-word binary, plus some decimal formats
Overview of IEEE 754-2008 Standard Formats
Feature Single / Short Double / Long –––––––––––––––––––––––––––––––––––––––––––––––––––––––– Word width (bits) 32 64 Significand bits 23 + 1 hidden 52 + 1 hidden Significand range [1, 2 – 2 –^23 ] [1, 2 – 2 –^52 ] Exponent bits 8 11 Exponent bias 127 1023 Zero (0) e + bias = 0, f = 0 e + bias = 0, f = 0 Denormal e + bias = 0, f 0 e + bias = 0, f 0 represents 0. f 2 –^126 represents 0. f 2 –^1022 Infinity () e + bias = 255, f = 0 e + bias = 2047, f = 0 Not-a-number (NaN) e + bias = 255, f 0 e + bias = 2047, f 0 Ordinary number e + bias [1, 254] e + bias [1, 2046] e [–126, 127] e [–1022, 1023] represents 1. f 2 e^ represents 1. f 2 e min 2 –^126 1.2 10 –^38 2 –^1022 2.2 10 –^308 max 2128 3.4 1038 21024 1.8 10308 ––––––––––––––––––––––––––––––––––––––––––––––––––––––––
Table 17.1 Some features of the IEEE 754-2008 standard floating-point number representation formats
Special Operands and Subnormals
Operations on special operands: Ordinary number (+) = 0 (+) Ordinary number = NaN + Ordinary number = NaN
Biased value 0 1 2... 253 254 255
Ordinary FLP numbers 0 , Subnormal , NaN ( 0. f 2 –^126 )
(1. f 2 e^ )
(1.00…01 – 1.00…00)2–^126 = 2–^149
(^0 ) Denormals – 2
......
min
...
Fig. 17.4 Subnormals in the IEEE single-precision format.
Subnormals
Extended Formats
Short (32-bit) format
Long (64-bit) format
Sign Exponent Significand
8 bits, bias = 127,
11 bits, bias = 1023,
- 1022 to 1023 52 bits for fractional part (plus hidden 1 in integer part)
23 bits for fractional part (plus hidden 1 in integer part)
11 bits 32 bits
15 bits 64 bits
Double extended [-16 382, 16 383]
Single extended [-1022, 1023]
Bias is unspecified, but exponent range must include:
Single extended
Double extended
17.3 Basic Floating-Point Algorithms
( s 1 b e^1 ) + ( s 2 b e^2 ) = ( s 1 b e^1 ) + ( s 2 / b e^1 – e^2 ) b e^1 = ( s 1 s 2 / b e^1 – e^2 ) b e^1 = s b e
Assume e 1 e 2; alignment shift ( preshift ) is needed if e 1 > e 2
Operands after alignment shift:
x = 2 1. 00101101
y = 2 0. 000111101101
Numbers to be added:
x = 2 1. 00101101
y = 2 1. 11101101
(^5)
5
Extra bits to be rounded off
Operand with smaller exponent to be preshifted
Result of addition:
s = 2 1. 010010111101
s = 2 1. 01001100 Rounded sum
5
1
5
5
Example:
Addition
Rounding, overflow, and underflow issues discussed later
Floating-Point Multiplication and Division
Because s 1 s 2 [1, 4), postshifting may be needed for normalization
( s 1 b e^1 ) ( s 2 b e^2 ) = ( s 1 s 2 ) b e 1+ e^2
Multiplication
Overflow or underflow can occur during multiplication or normalization
Because s 1 / s 2 (0.5, 2), postshifting may be needed for normalization
( s 1 b e^1 ) / ( s 2 b e^2 ) = ( s 1 / s 2 ) b e^1 - e^2
Division
Overflow or underflow can occur during division or normalization
17.4 Conversions and Exceptions
Conversions from fixed- to floating-point
Conversions between floating-point formats
Conversion from high to lower precision: Rounding
The IEEE 754-2008 standard includes five rounding modes:
Round to nearest, ties away from 0 (rtna) Round to nearest, ties to even (rtne) [default rounding mode] Round toward zero (inward) Round toward + (upward) Round toward – (downward)
Exceptions in Floating-Point Arithmetic
Divide by zero
Overflow
Underflow
Inexact result: Rounded value not the same as original
Invalid operation: examples include
Addition (+) + (–) Multiplication 0 Division 0 / 0 or / Square-rooting operand < 0
Produce NaN as their results
Truncation or Chopping
Fig. 17.5 Truncation or chopping of a signed-magnitude number (same as round toward 0).
chop( x )
x
4 3 2 1
Fig. 17.6 Truncation or chopping of a 2’s-complement number (same as downward-directed rounding).
chop( x ) = down( x )
x
4 3 2 1
Round to Nearest Number
Fig. 17.7 Rounding of a signed-magnitude value to the nearest number.
Rounding has a slight upward bias. Consider rounding ( xk – 1 xk – 2 ... x 1 x 0. x – 1 x – 2 )two to an integer ( yk – 1 yk – 2 ... y 1 y 0. )two The four possible cases, and their representation errors are: x – 1 x – 2 Round Error 00 down 0 01 down – 0. 10 up 0. 11 up 0. With equal prob., mean = 0.
For certain calculations, the probability of getting a midpoint value can be much higher than 2– l
rtn( x )
x
4
3
2
1
rtna( x )
