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CSE 575
Computer Arithmetic
Spring 2005
Mary Jane Irwin
(www.cse.psu.edu/~mji)
Computer Arithmetic
CSE575 Introduction.1
© MJIrwin, PSU, 2005
Administrivia

Mary Jane Irwin
» 348C IST Bldg
» Office hours: by appointment
» [email protected]; www.cse.psu.edu/~mji



Optional: Digital Arithmetic, Ercogovac &
Lang, Morgan Kaufmann, 2004
Prerequisite: CSE 477 and CSE 431
Lectures are scheduled from 2:30 to 4:30
(but will often be over by 4:00, especially if a
colloquium is scheduled)
Computer Arithmetic
CSE575 Introduction.2
© MJIrwin, PSU, 2005
Course Grading
Homeworks (2 in total)
 Course project
150 pts
350 pts
TOTAL
500 pts

Computer Arithmetic
CSE575 Introduction.3
© MJIrwin, PSU, 2005
Remaining Lecture Schedule
Mar 15
Introduction, number repr
Dr. Irwin
Mar 17
Local project design review
Theo T.
Mar 22
Global project review
Dr. Vijay
Mar 24
Global project review
Dr. Vijay
Mar 29
Addition
Dr. Irwin
Apr 1
Redundant repr & its uses
Dr. Irwin
Apr 5
Multiplication
Dr. Irwin
Apr 7
Local/Global project review
Dr. Vijay
Apr 12
Division
Dr. Irwin
Chp 5
Apr 14
Flt point repr & operation
Dr. Irwin
Chp 8
Apr 19
Function evaluation
Dr. Irwin
Chp 10, 11
Apr 21
Final global project review
Dr. Vijay
Apr 26
Other # systems
Dr. Irwin
Apr 28
Final global project review
Dr. Vijay
Computer Arithmetic
CSE575 Introduction.4
Chp 1
Chp 2
Chp 4
© MJIrwin, PSU, 2005
Computer Arithmetic
“ is a subfield of digital computer organization
… deals with the hardware realization of
arithmetic functions … a major thrust <of
which> is the design of hardware algorithms
and circuits to enhance the speed of
numeric operations.”
Parhami
“ encompasses the study of number
representation, algorithms for operations on
numbers, implementations of arithmetic
units in hardware, and their use …”
Ercegovac & Lang
Computer Arithmetic
CSE575 Introduction.5
© MJIrwin, PSU, 2005
What Is Computer Arithmetic?


Hardware – Design
of efficient circuits
for arithmetic
operations (+, -, x, /,
sqrt, log, sine, …)
Issues

» Algorithm
» Accuracy
» Speed/power/area/
reliability trade-offs
» Test/verification
Computer Arithmetic

Software –
Numerical methods
for solving systems
of equations (CSE
45x and 55x)
Issues
» Algorithm
» Error analysis
» Computational
complexity
» Test/verification
CSE575 Introduction.6
© MJIrwin, PSU, 2005
Applications

General purpose systems
» Fast primitive operations for processor
datapaths

Special purpose systems
» Signal and image processing
» Real-time 3D graphics
» HDTV, image compression
» Network processors (data compression,
encryption/decryption)
»…
Computer Arithmetic
CSE575 Introduction.7
© MJIrwin, PSU, 2005
Design Axes
Architectural innovations
» Systolic arrays, wave pipelining, MMX,
razor pipelines
Technological advancements
» 3D, multivalued gates, nanotechnology, …
Representation (encoding) systems
» Redundant, residue, …
Arithmetic algorithms
» SD arithmetic, serial arithmetic (lsb and
msb first), saturating arithmetic, MACs,
parallel multipliers, …
Computer Arithmetic
CSE575 Introduction.8
© MJIrwin, PSU, 2005
Representing Numbers
infinite
(real numbers)
finite
(machine representation)
Computer Arithmetic
CSE575 Introduction.9
© MJIrwin, PSU, 2005
Number Representations
“Understanding different ways of
representing numbers, including their
relative cost-performance benefits
and conversions between various
representations, is an important
prerequisite for designing efficient
arithmetic algorithms or circuits.”
Parhami
Computer Arithmetic
CSE575 Introduction.10
© MJIrwin, PSU, 2005
Machine Representation Characteristics
1492
Positional - value of each symbol
depends on its relative position (1492 
1942)
 Fixed radix, r - unit of each position is a
constant multiple of the radix (base);
usually a positive integer
 Consecutive symbol (or digit) set, d, with
at least r symbols, e.g.,
d  [0, 1, 2, …, 8, 9]

Computer Arithmetic
CSE575 Introduction.11
© MJIrwin, PSU, 2005
Machine Repr. Characteristics, con’t
1492
Scale factor, s - implied radix point
(integer, fraction, or mixed number)
 Precision, n = k + l - granularity of
representation (k: # integer digits, l: #
of fractional digits), ulp = r0 = 1 for
integers, ulp = r-n for fractions
 Range – number of distinct values from
largest to smallest; depends on n, s, & r

Computer Arithmetic
CSE575 Introduction.12
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Aside: Some Notation

X - (capital letters) full precision operand
consisting of multiple bits (base 2) or digits
(base 2 and higher)
X =  xiri = (xk-1 xk-2 … x1 x0. x-1 x-2 … x-l+1 x-l )r


xi - (small letters) single bit or digit values (i
negative then a fractional bit/digit, i positive
then an integer bit/digit)
Xi - (capital letters) full precision result from
the ith iteration of a computation
Computer Arithmetic
CSE575 Introduction.13
© MJIrwin, PSU, 2005
Binary Encoding Examples
Digit vector of length n = k + l
( xk-1 xk-2 … x1 x0. x-1 x-2 … x-l+1 x-l )r=  xiri

-16 -14 -12 -10 -8 -6 -4
-2
0
2
4
6
8 10 12 14 16
n=4
Unsigned integer
Sign magnitude
integer
2+2 floating point
Sx2E, S  (0,1,2,3)
E  (-2,-1,0,1)
Computer Arithmetic
CSE575 Introduction.14
© MJIrwin, PSU, 2005
Fixed-radix Positional # Systems
0110
Decimal Equilavent?

Binary: r=2, d[0,1]
6

Balanced ternary: r=3, d[-1,0,1]
12

Negabinary: r=-2, d[0,1]
2
Redundant (signed digit):
r=2, d[-1,0,1]
6

Fractional radix: r=1/2, d[0,1]
¾

Irrational radix: r=sqrt(2), d[0,1]

Complex-radix: r=2j, d[0,1,2,3]

Computer Arithmetic
CSE575 Introduction.16
2 + √2
© MJIrwin, PSU, 2005
Real Number Representations

Floating point (approximation)

Slash (rational) representations
» Keep numbers as num/denom pairs
– Exact arithmetic (e.g., 2/3)
– Division is multiplication

Logarithmic representation
» Keep numbers as log2 |A|
– Easy multiply and divide
– Large dynamic range, but low precision
Computer Arithmetic
CSE575 Introduction.17
© MJIrwin, PSU, 2005
Number Radix Conversion

Assume that you are familiar with
techniques for
» converting to/from radix 2, 4, 8, 10, 16, …
» doing arithmetic in the old radix
» doing arithmetic in the new radix
» using Horner’s rule shortcut to convert
between radices
Computer Arithmetic
CSE575 Introduction.18
© MJIrwin, PSU, 2005
Choosing a “Best” Radix

Conversion
» I/O is in decimal, so r=10 (BCD) would
require no conversion, but wastes storage
space and arithetic unit area and delay

Compatibility
» CMOS circuitry supports two states (on,
off) only, so r=2 (or some power of 2)

Reliability
» circuitry supports binary logic only (as
opposed to multivalued logic)
Computer Arithmetic
CSE575 Introduction.19
© MJIrwin, PSU, 2005
Choosing a “Best” Radix con’t

Efficiency
The # of different representations is
M=rn
Assuming the component cost for repr is proportional
to r (with binary logic it’s really prop. to log2r for r
a power of 2), then the cost of the repr is
$ = K1 n r
$ = K1 (ln M)/(ln r) r = K2 r/(ln r)
To minimize $, take derivative at zero wrt r
d$/dr = (K2 (ln r -1))/ (ln r2)  0
So, ln r = 1 (r = e = 2.71828…) is the most efficient
radix! Cost of r=2 and r=4 are about equal.
Computer Arithmetic
CSE575 Introduction.20
© MJIrwin, PSU, 2005
Binary Full Adder (FA) Review
2ci+1 + si = xi + yi + ci
addend augend
xi y i
ci+1
carry-out
FA
carry-in
ci
si sum
si = xi  yi  ci
y
ci
ci+1
s
0
0
0
0
0
0
0
1
0
1
0
1
0
0
1
0
1
1
1
0
1
0
0
0
1
1
0
1
1
0
1
1
0
1
0
1
1
1
1
1
(odd parity function)
ci+1 = xi&yi | xi&ci | yi&ci
Computer Arithmetic
x
(majority function)
CSE575 Introduction.21
© MJIrwin, PSU, 2005
Ripple Carry Adder (RCA)
cout=c4
x3 y 3
x2 y2
x1 y1
x0 y0
FA
FA
FA
FA
s3
s2
s1
s0
sign bits
c0=cin
mod(2k) adder
Can we use it to do both addition and subtraction?
How about negation?
How do we determine overflow, zero result, …?
How do we make it fast, power efficient, reliable, …?
Computer Arithmetic
CSE575 Introduction.22
© MJIrwin, PSU, 2005
Representing Signed Numbers
Signed magnitude
 Complement

» Two’s complement
» One’s complement


Biased (excess)
sign indicated by
most significant
digit
Choice depends on area, delay, power,
reliability of arithmetic hardware
(adder) implementation
Computer Arithmetic
CSE575 Introduction.23
© MJIrwin, PSU, 2005
Sign Magnitude (SM)

most significant digit is the sign indicator,
remaining digits are the magnitude
0000
1111
0001
1110
0010
-7 0 +1
-6
+2
1101
-5
+3 0011
1100 -4
SM +4 0100
1011 -3
+5 0101
-2
+6
1010
0110
-1 -0 +7
1001
0111
1000
increment
X = xk-1 xk-2 … x1 x0
k-2
X = (-1)xk-12ixi
i=0
Computer Arithmetic
CSE575 Introduction.24
© MJIrwin, PSU, 2005
Characteristics of Binary SM






k-bit symmetric range of [-2k-1+ulp, 2k-1-ulp]
two representations for zero
simple negation (complement sign bit)
addition of unlike sign numbers must be
handled differently than addition of like sign
numbers (SM not a modular system)
symmetric shift – left: shift in zeros; right:
shift zeros into magnitude - while maintaining
sign bit
extend - left (sign): pad with zeros; right: pad
with zeros – while maintaining sign bit
Computer Arithmetic
CSE575 Introduction.25
© MJIrwin, PSU, 2005
SM Adder/Subtractor
sign of X sign of Y
X
Y
Selective
compl X complement
Xcompl
add/sub
Control
cout
sign
compl S
sign of S
Computer Arithmetic
cin
Binary
adder
Selective
complement
S =X ± Y
CSE575 Introduction.26
© MJIrwin, PSU, 2005
Complement Representations

negation constant, M, so that negative
representation of Y is M-Y
0
M-1
M-2
M-3
-2
-3
M-N
Computer Arithmetic
-1 0 +1
unsigned
representations
1
2
+2
+3 3
signed
+4 4
values
increment
-N
+P
P
MN+P+1
CSE575 Introduction.27
© MJIrwin, PSU, 2005
Addition Rules
Desired
Operation
Computation to be
performed mod M
Correct result
with no overflow
Overflow
condition
(+X) + (+Y)
(+X) – (-Y)
X+Y
X+Y
X+Y>P
(+X) + (-Y)
(+X) – (+Y)
X + (M-Y)
X-Y if Y <= X
M-(Y-X) if Y > X
NA
(-X) + (+Y)
(-X) – (-Y)
(M-X) + Y
Y-X if X <=Y
M-(X-Y) if X > Y
NA
(-X) + (-Y)
(-X) – (+Y)
(M-X) + (M-Y)
M – (X + Y)
X+Y>N

with a selection negation unit available all
the operations are the same!
Computer Arithmetic
CSE575 Introduction.28
© MJIrwin, PSU, 2005
Complement Rep Options

Two auxiliary operations required
» negation (computing M-Y)
» computation of residues mod M

Select M so these two operations are simple

Radix complement
M= rk
» negation –
» mod M –

Diminished radix complement
M = rk – ulp
» negation –
» mod M Computer Arithmetic
CSE575 Introduction.29
© MJIrwin, PSU, 2005
Two’s Complement (2’sc)

M = 2k (for k=4, l=0, M = 24 =16)
0000
1111
0001
1110
0010
-1 0 +1
-2
+2
1101
-3
+3 0011
1100 -4
2s’c +4 0100
1011 -5
+5 0101
-6
+6
1010
0110
-7 -8 +7
1001
0111
1000
increment
a mod(2k) system
k-2
X = -2k-1xk-1 + 2ixi
i=0
Computer Arithmetic
CSE575 Introduction.30
© MJIrwin, PSU, 2005
Characteristics of 2’sc

asymmetric range of [-2k-1, 2k-1 - ulp]
» so negation can lead to overflow!


one representation for zero (all zeros)
negation – complement all the bits and add 1
2s’c(Y) = 2k – Y = ((2k – ulp) – Y) + ulp = Ycompl + ulp




mod - simply drop the high order carry-out
RCA with negation logic can handle both add
and subtract
asymmetric shift – left: shift in zeros; right:
shift in sign
extend - left (sign): replicate sign bit; right:
pad with zeros
Computer Arithmetic
CSE575 Introduction.31
© MJIrwin, PSU, 2005
2’sc Negation Example

RC(Y) = rk – Y
rK =
-Y =
1
-
0
yk-1
0
yk-2
...
...
0 0
y1 y0
0 r-1
r-1
...
r-1
-yk-1 -yk-2
...
-y1 -y0
r
complement
the digit
Computer Arithmetic
CSE575 Introduction.33
© MJIrwin, PSU, 2005
2s’c Adder/Subtractor
X
overflow: ck-1  ck
cout = ck
Y
Selective
complement
Y or Ycompl
Binary
adder
add/sub
0 for addition
1 for subtraction
cin
X±Y


mitigates disadvantage of negation
removes disadvantage of asymmetric range
» if operand to be negated is -2k-1, it is represented in
two parts: Ycompl (which is 2k-1 -1) and cin
Computer Arithmetic
CSE575 Introduction.34
© MJIrwin, PSU, 2005
Overflow

Looking at the just the sign bit results
no
overflow
here
ever!
overflow =
ck-1  ck
xk-1
yk-1
ck-1
ck
sk-1
0
0
0
0
0
0
0
0
1
0
1
+2k-1
negative? - overflow
0
1
0
0
1
-2k-1
negative? - okay
0
1
1
1
0
0
positive? - okay
1
0
0
0
1
-2k-1
negative? - okay
1
0
1
1
0
0
positive? - okay
1
1
0
1
0
-2k
positive? - overflow
1
1
1
1
1
-2k-1
negative? - okay
+2k
-2k-1
-2k-1 -2k-1 +2k-1
Computer Arithmetic
CSE575 Introduction.35
positive? - okay
bit position weights
© MJIrwin, PSU, 2005
One’s Complement (1’sc)

M = 2k – 1 (for k=4, l=0, M = 24 – 1 = 15)
0000
1111
0001
1110
0010
-0 0 +1
-1
+2
1101
-2
+3 0011
1100 -3
1s’c +4 0100
1011 -4
+5 0101
-5
+6
1010
0110
-6 -7 +7
1001
0111
1000
increment
a mod(2k-1) system
k-2
X = -(2k-1-1)xk-1 + 2ixi
i=0
Computer Arithmetic
CSE575 Introduction.36
© MJIrwin, PSU, 2005
Characteristics of 1’sc



symmetric range of [-2k-1 + ulp, 2k-1 - ulp]
two representations for zero
negation – complement all the bits
1s’c(Y) = 2k – 1 - Y = (2k – ulp) – Y = Ycompl

mod - end around carry
» reduces magnitude by 2k - ulp if cout = 1



RCA with negation logic and end around carry
can handle both add and subtract
symmetric shift – left: shift in sign; right:
shift in sign
extend - left (sign): replicate sign bit; right:
replicate sign bit
Computer Arithmetic
CSE575 Introduction.37
© MJIrwin, PSU, 2005
End Around Carry


When adding two 1’sc [mod(2k – ulp)] numbers
that results in a high order carry out, result
will be off by 2k - ulp
End-around carry
» Drop a high order carry-out of 1 and
simultaneously insert a carry-in of 1 into position l
» Since the dropped carry is worth 2k units and the
inserted carry is worth ulp, the combined effect
is to reduce the magnitude of the result by
2k – ulp

Overflow – ? for you to consider
Computer Arithmetic
CSE575 Introduction.38
© MJIrwin, PSU, 2005
1s’c Adder/Subtractor
X
overflow: ck-1  ck
cout = ck
Y
Selective
complement
Y or Ycompl
Binary
adder
cin
add/sub
0 for addition
1 for subtraction
end around carry
X±Y
Does the end around carry terminate?
Assuming so, how long do we have to wait?
Computer Arithmetic
CSE575 Introduction.39
© MJIrwin, PSU, 2005
Complement Rep Options Revisited

Two auxiliary operations required
» negation (computing M-Y)
» computation of residues mod M


Select M so these two operations are simple
Radix complement
M= rk
» negation – replace each digit by its radix complement
[(r-1) - xi] and add a unit in the ls digit
» mod M – ignore the high order carry out

Diminished radix complement
M = rk – ulp
» negation – replace each nonzero digit by its radix
complement
» mod M - do end around carry
Computer Arithmetic
CSE575 Introduction.40
© MJIrwin, PSU, 2005
Biased (Excess)

most significant digit is the sign indicator,
remaining digits are the magnitude
0000
1111
0001
1110
0010
+7 -8 -7
+6
-6
1101
0011
+5
-5
1100 +4
XC8 -4 0100
1011 +3
-3 0101
+2
-2
1010
0110
+1 0 -1
1001
0111
1000
Computer Arithmetic
CSE575 Introduction.41
increment
a mod(2k) system
© MJIrwin, PSU, 2005
Characteristics of Biased



similar to 2s’c with sign bit inverted
(0 -> negative; 1 -> positive)
bias is always in the sign position
addition requires subtraction of bias
X + Y + bias = (X + bias) + (Y + bias) - bias
» with a bias of 2k-1 just complement the msb

used to represent the exponent in floating
point notation
» simplifies comparison of floating point numbers
(with leading excess exponent, integer comparison
logic suffices)
Computer Arithmetic
CSE575 Introduction.42
© MJIrwin, PSU, 2005
Indirect Signed Arithmetic

When hardware uses
unsigned operands
» some multiplication and
division schemes
» floating point mantissa
units

Sign removal
Sign
logic
Unsigned
operation
Adjustment
Other examples of pre/post operand processing
» dealing with unacceptable or inconvenient operands
– infinity, NAN
– sin(2 + X) = sin( - X) = sin X
» more efficient division when divisor can be scaled
(along with the dividend) to be in a certain range
Computer Arithmetic
CSE575 Introduction.43
© MJIrwin, PSU, 2005
Key References
Biannual Proc. Of the Computer Arithmetic Symposia, 1975 – 2003.
Cavanagh, Digital Computer Arithmetic Design and Implementation,
McGraw Hill, 1984.
Ercegovac & Lang, Digital Arithmetic, Morgan Kaufmann, 2004.
Flynn & Oberman, Advanced Computer Arithmetic Design, Wiley, 2001.
Gosling, Design of Arithmetic Units for Digital Computers, Macmillan,
1980.
Koren, Computer Arithmetic Algorithms, Prentice-Hall, 1993.
Omondi, Computer Arithmetic Systems: Algorithms, Architectures,
and Implementation, Prentice-Hall, 1994.
Parhami, Computer Arithmetic, Oxford Press, 1999.
Scott, Computer Number Systems and Arithmetic, Prentice-Hall, 1985.
Spaniol, Computer Arithmetic Logic and Design, Wiley, 1981.
Swartzlander, Ed., Computer Arithmetic, Dowden, 1980.
Swartzlander, Ed., Computer Arithmetic II, IEEE Press, 1990.
Waser & Flynn, Introduction to Arithmetic for Digital Systems
Designers, Holt, 1982.
Computer Arithmetic
CSE575 Introduction.44
© MJIrwin, PSU, 2005