Computer Architecture # 03 : Arithmetic: FIXED POINT ADDITION AND SUBTRACTION (2)

3.2 FIXED POINT ADDITION AND SUBTRACTION
The addition of binary numbers and the concept of overflow were briefly discussed in Chapter 2. Here, we cover addition and subtraction of both signed and unsigned fixed point numbers in detail. Since the two’s complement representation of integers is almost universal in today’s computers, we will focus primarily on two’s complement operations. We will briefly cover operations on 1’s complement and BCD numbers, which have a foundational significance for other areas of computing, such as networking (for 1’s complement addition) and hand-held calculators (for BCD arithmetic.)

Computer Architecture # 03 : Arithmetic: OVERVIEW (1)

3.1 OVERVIEW
In the previous chapter we explored a few ways that numbers can be represented in a digital computer, but we only briefly touched upon arithmetic operations that can be performed on those numbers. In this chapter we cover four basic arithmetic operations: addition, subtraction, multiplication, and division. We begin by describing how these four operations can be performed on fixed point numbers, and continue with a description of how these four operations can be performed on floating point numbers. 

Computer Architecture # 02 : Data Representation: CHARACTER CODE (17)

2.5 CHARACTER CODE
Unlike real numbers, which have an infinite range, there is only a finite number of characters. An entire character set can be represented with a small number of bits per character. Three of the most common character representations, ASCII, EBCDIC, and Unicode, are described here.


2.5.1 THE ASCII CHARACTER SET
The American Standard Code for Information Interchange (ASCII) is summarized in Figure 2-13, using hexadecimal indices.

Computer Architecture # 02 : Data Representation: CASE STUDY: PATRIOT MISSILE DEFENSE FAILURE CAUSED BY LOSS OF PRECISION (16)

2.4 Case Study: Patriot Missile Defense Failure Caused by Loss of Precision
During the 1991-1992 Operation Desert Storm conflict between Coalition forces and Iraq, the Coalition used a military base in Dhahran, Saudi Arabia that was protected by six U.S. Patriot Missile batteries. The Patriot system was originally designed to be mobile and to operate for only a few hours in order to avoid detection.
The Patriot system tracks and intercepts certain types of objects, such as cruise missiles or Scud ballistic missiles, one of which hit a U.S. Army barracks at Dhahran on February 5, 1991, killing 28 Americans. The Patriot system failed to track and intercept the incoming Scud due to a loss of precision in converting integers to a floating point number representation.

Computer Architecture # 02 : Data Representation: THE IEEE 754 FLOATING POINT STANDARD (15)

2.3.5 THE IEEE 754 FLOATING POINT STANDARD

There are many ways to represent floating point numbers, a few of which we have already explored. Each representation has its own characteristics in terms of range, precision, and the number of representable numbers. In an effort to improve software portability and ensure uniform accuracy of floating point calculations, the IEEE 754 floating point standard for binary numbers was developed (IEEE, 1985). 

There are a few entrenched product lines that predate the standard that do not use it, such as the IBM/370, the DEC VAX, and the Cray line, but virtually all new architectures generally provide some level of IEEE 754 support.

Computer Architecture # 02 : Data Representation : ERROR IN FLOATING POINT REPRESENTATIONS (14)

2.3.4 ERROR IN FLOATING POINT REPRESENTATIONS
The fact that finite precision introduces error means that we should consider how great the error is (by “error”, we mean the distance between two adjacent representable numbers), and whether it is acceptable for our application. As an example of a potential pitfall, consider representing one million in floating point, and then subtracting one million 1’s from it. We may still be left with a million if the error is greater than 1.

Computer Architecture # 02 : Data Representation: EPRESENTING FLOATING POINT NUMBERS IN THE COMPUTER—PRELIMINARIES (13)

2.3.3 REPRESENTING FLOATING POINT NUMBERS IN THE COMPUTER—PRELIMINARIES

Let us design a simple floating point format to illustrate the important factors in representing oating point numbers on the computer. Our format may at first seem to be unnecessarily complex. We will represent the significand in signed magnitude format, with a single bit for the sign bit, and three exadecimal digits for the magnitude. The exponent will be a 3-bit excess-4 number, with a radix of 16.

Computer Architecture # 02 : Data Representation: NORMALIZATION, AND THE HIDDEN BIT (12)

2.3.2  NORMALIZATION, AND THE HIDDEN BIT

A potential problem with representing floating point numbers is that the same number can be represented in different ways, which makes comparisons and arithmetic operations difficult. For example, consider the numerically equivalent forms shown below:

3584.1 × 100 = 3.5841 × 103 = .35841 × 104.

In order to avoid multiple representations for the same number, floating point numbers are maintained in normalized form. That is, the radix point is shifted to the left or to the right and the exponent is adjusted accordingly until the radix point is to the left of the leftmost nonzero digit. So the rightmost number above is the

Computer Architecture # 02 : Data Representation: RANGE AND PRECISION IN FLOATING POINT NUMBERS (11)

2.3.1  RANGE AND PRECISION IN FLOATING POINT NUMBERS

A floating point representation allows a large range of expressible numbers to be represented in a small number of digits by separating the digits used for precision from the digits used for range. The base 10 floating point number representing Avogadro’s number is shown below:

+6.023 × 1023

Here, the range is represented by a power of 10, 1023 in this case, and the precision is represented by the digits in the fixed point number, 6.023 in this case. In discussing floating point numbers, the fixed point part is often referred to as the mantissa,

Computer Architecture # 02 : Data Representation: Floating Point Numbers (10)

2.3 Floating Point Numbers

The fixed point number representation, which we explored in Section 2.2, has a fixed position for the radix point, and a fixed number of digits to the left and right of the radix point. A fixed point representation may need a great many digits in order to represent a practical range of numbers. For example, a computer that can represent a number as large as a trillion1 maintains at least 40 bits to the left of the radix point since 240 ≈ 1012. If the same computer needs to represent one trillionth, then 40 bits must also be maintained to the right of the radix point, which results in a total of 80 bits per number.

Computer Architecture # 02 : Data Representation: BINARY CODED DECIMAL (9)

2.2.7 BINARY CODED DECIMAL

Numbers can be represented in the base 10 number system while still using a binary encoding. Each base 10 digit occupies four bit positions, which is known as binary coded decimal (BCD). Each BCD digit can take on any of 10 values. There are 24 = 16 possible bit patterns for each base 10 digit, and as a result, six bit patterns are unused for each digit. In Figure 2-6, there are four decimal significant digits, so 104 = 10,000 bit patterns are valid, even though 216 = 65,536 bit patterns are possible with 16 bits.

Computer Architecture # 02 : Data Representation : SIGNED FIXED POINT NUMBERS (8)

2.2.6 SIGNED FIXED POINT NUMBERS
Up to this point we have considered only the representation of unsigned fixed point numbers. The situation is quite different in representing signed fixed point numbers. There are four different ways of representing signed numbers that are commonly used: sign-magnitude, one’s complement, two’s complement, and excess notation. We will cover each in turn, using integers for our examples.

Throughout the discussion, the reader may wish to refer to Table 2.1 which shows for a 3-bit number how the various representations appear.