Arithmetic Systems

1
Arithmetic Systems

• Kwang Hee Ko
• School of Mechatronics
• Gwangju Institute of Science and Technology

2
Introduction

• Representation of numbers and associated
  operations in digital computers.
• Various arithmetic systems are used.
• Integer Arithmetic
• Floating Point Arithmetic
• Rational Arithmetic
• Interval Arithmetic

3
Floating Point Arithmetic

• Current CAD/Graphics systems operate in floating
  point arithmetic (FPA).
• Basic arithmetic operations (especially division
  operation in FPA) lead to significant numerical
  errors.
• CAD systems frequently fail as a result of the
  limited precision that is inherent to the
  internal representation of floating point
  numbers.
• Any sequence of operations on a digital computer
  is essentially equivalent to a finite sequence of
  manipulations on a discrete grid of points.

4
Floating Point Arithmetic

• Representation of nonnegative real numbers
• The standard way to represent a nonnegative real
  number in decimal form
• An integer part, a fractional part, and a decimal
  point between them.
• 37.458, 0.003947
• Normalized scientific notation
• Shifting the decimal point and supplying
  appropriate powers of 10.
• 37.458 0.37458 x 102

5
Floating Point Arithmetic

• Normalized Floating-point Representation
• In Decimal System
• x 0.d1d2d3x10n
• d1,d2 are the decimal digits
• In Binary System
• x 0.b1b2b3x2n
• b1,b2 are the binary numbers, 0,1.
• In a digital computer, numbers are represented
  using binary system.

6
Floating Point Arithmetic

• Consider a floating point number x
  0.b1b2b3bpx2n
• b1,b2bp are the mantissa, b1? 0, p is the number
  of significant digits, and n is an integer
  exponent.
• Example p2 and -2 E 3.

7
Floating Point Arithmetic

• The resulting set of FP numbers is a finite
  subset of the rational numbers.
• They are distributed non-uniformly on the real
  number axis.
• The most real numbers cannot be represented
  exactly in a computer!!!!!!
• The result of a floating point calculation must
  often be rounded in order to fit back into its
  finite representation.
• This is a characteristic feature of floating
  point computation.
• Overflow/underflow occur when a number is outside
  the range that the computer can represent.

8
Floating Point Arithmetic

• Basic questions on floating point arithmetic
• What is the result of an operation when the
  infinitely precise result is not representable in
  the computer system?
• Are elementary operations like multiplication and
  addition commutative?
• What happens when we multiply two very large
  numbers?
• What if we divide a number by zero?
• What if we attempt to compute the square root of
  a negative number?

9
Floating Point Arithmetic

• The IEEE Standard 754 ensures that operations
  yield the mathematically expected results with
  the expected properties.
• It also ensures that exceptional cases yield
  specified results.

10
IEEE Standard 754

• It specifies
• Two basic floating point formats single and
  double
• Single
• Overall 32 bits, Significand 24 bits
• Double
• Overall 64 bits, Significand 53 bits
• Two classes of extended floating point formats
  single extended and double extended.

11
IEEE Standard 754

• It specifies
• Accuracy requirements on floating point
  operations
• Add, subtract, multiply, divide, square root,
  remainder, round number, compare, etc.
• If no exact result can be delivered, the
  operation must be specified such that the
  operation must minimally modify the exact result
  according to the rules of prescribed rounding
  modes.
• Five types of IEEE floating point exceptions
• Invalid operation, division by zero, overflow,
  underflow, inexact.

12
IEEE Standard 754

• Note on INEXACT exception
• This is signaled whenever the ideal result of an
  arithmetic operation would not fit into its
  intended destination, so the result had to be
  altered by rounding it off to fit.

13
IEEE Standard 754

• It specifies
• Four rounding directions
• Toward the nearest representable value, toward
  negative infinity, toward positive infinity,
  toward 0.
• Rounding precision
• If a system delivers results in double extended
  format, the user should be able to specify that
  such results are to be rounded to the precision
  of either the single or double format.

14
IEEE Standard 754 Formats

• Storage Formats
• It is a data structure specifying the fields that
  comprises a floating point numeral, the layout of
  those fields, and their arithmetic
  interpretation.
• It specifies how a floating point format is
  stored in memory.
• IEEE precision
• single float (C, C), REAL or REAL4 (Fortran)
• double double (C, C), DOUBLE PRECISION or
  REAL 8 (Fortran)
• Double extended long double (C, C), REAL 16
  (Fortran)

15
IEEE Standard 754 Formats

• Single Format continuous one 32-bit word
• 23-bit fraction, f
• 8-bit biased exponent, e
• 1-bit sign, s

16
IEEE Standard 754 Formats

• Single Format continuous one 32-bit word

17
IEEE Standard 754 Formats

• Single Format continuous one 32-bit word

18
IEEE Standard 754 Format

• Double Format two successive 32-bit words
• 52-bit fraction, f
• 11-bit biased exponent, e
• 1-bit sign, s

19
IEEE Standard 754 Format

• Double Format two successive 32-bit words

20
IEEE Standard 754 Format

• Double Format two successive 32-bit words

21
IEEE Standard 754 Format

• Double-Extended Format (for SPARC) four
  successive 32-bit words
• 112-bit fraction, f
• 15-bit biased exponent, e
• 1-bit sign, s

22
IEEE Standard 754 Format

• Double-Extended Format (for SPARC) four
  successive 32-bit words

23
IEEE Standard 754 Format

• Double-Extended Format (for SPARC) four
  successive 32-bit words

24
IEEE Standard 754 Format

• Double-Extended Format (for x86) ten successive
  32-bit words
• 63-bit fraction, f
• 1-bit explicit leading significand bit, j
• 15-bit biased exponent, e
• 1-bit sign, s

25
IEEE Standard 754 Format

• Double-Extended Format (for x86) ten successive
  32-bit words

26
IEEE Standard 754 Format

• Double-Extended Format (for x86) ten successive
  32-bit words

27
IEEE Standard 754 Format

• What is the number of significant decimal digits
  of a in the IEEE formats? Or how many decimal
  digits are to be trusted as accurate when one
  represents a in IEEE formats?

28
IEEE Standard 754 Format

• Underflow
• It occurs, roughly speaking, when the result of
  an arithmetic operation is so small that it
  cannot be stored in its intended destination
  format without suffering a rounding error that is
  larger than usual.

29
Rational Arithmetic

• Numbers are represented in rational form.
• The arithmetic is done with rational numbers
  without approximation.
• It is often important to know the exact value,
  not an approximate one.
• If arithmetic is done on fractions instead of on
  approximations to fractions, many computations
  can be done entirely without any accumulated
  rounding errors.
• Exact value of 1/3 instead of 0.3333333

30
Rational Arithmetic

• Features of Rational Arithmetic
• It is robust.
• It is generally memory intensive and time
  consuming
• Due to the growth of the number of digits needed
  to represent rational numbers.

31
Rational Arithmetic

• Rational numbers can be represented as pairs of
  integers (u/u), where u and u are relatively
  prime to each other and u gt 0 .
• (0/1) 0
• Basic Operators
• (u/u) (v/v) if and only if u v, u v
• (u/u) x (v/v) (w/w), where w uv/d, w
  uv/d, d gcd(uv,uv). (gcd -gt greatest common
  divisor)
• Division can be performed similarly.

32
Rational Arithmetic

• Basic Operators
• (u/u) (v/v) ((uv uv)/uv). Then reduce
  this fraction by using d gcd(uv uv,uv)
• Example
• (7/66) (17/12) (67/44)
• Can rational arithmetic represent all numbers?

33
Interval Arithmetic

• An interval number is defined to be an ordered
  pair of real numbers a,b with a b.
• A set of numbers, a,b xa x b
• Set operators are used.
• a,a degenerate interval value
• Equivalent to real numbers
• It is also called the range arithmetic.
• An upper and a lower bounds on each exact number
  are maintained during the calculations.

34
Interval Arithmetic

• The details of interval arithmetic will be
  introduced in the next lecture!!!