Intel 80287, 80286 manual Computation Fundamentals, Number System

OVERVIEW OF NUMERIC PROCESSING

The instruction address saved in the 80287 will point to any prefixes that preceded the instruction. This is different from the 8087, for which the instruction address pointed only to the ESC instruction opcode.

COMPUTATION FUNDAMENTALS

This section covers 80287 programming concepts that are common to all applications. It describes the 80287's internal number system and the various types of numbers that can be employed in NPX programs. The most commonly used options for rounding, precision, and infinity (selected by fields in the control word) are described, with exhaustive coverage of less frequently used facilities deferred to later sections. Exception conditions that may arise during execution of NPX instructions are also described along with the options that are available for responding to these exceptions.

Number System

The system of real numbers that people use for pencil and paper calculations is conceptually infinite and continuous. There is no upper or lower limit to the magnitude of the numbers one can employ in a calculation, or to the precision (number of significant digits) that the numbers can represent. When considering any real number, there is always an infinity of numbers both larger and smaller. There is also an infinity of numbers between (i.e., with more significant digits than) any two real numbers. For example, between 2.5 and 2.6 are 2.51,2.5897,2.500001, etc.

While ideally it would be desirable for a computer to be able to operate on the entire real number system, in practice this is not possible. Computers, no matter how large, ultimately have fixed-size registers and memories that limit the system of numbers that can be accommodated. These limitations determine both the range and the precision of numbers. The result is a set of numbers that is finite and discrete, rather than infinite and continuous. This sequence is a subset of the r~al numbers that is designed to form a useful approximation of the real number system.

Figure 1-8 superimposes the basic 80287 real number system on a real number line (decimal numbers are shown for clarity, although the 80287 actually represents numbers in binary). The dots indicate the subset of real numbers the 80287 can represent as data and final results of calculations. The 80287's range is approximately ±4.19XlO·307 to ±1.67X10 308. Applications that are required to deal with data and final results outside this range are rare. For reference, the range of the IBM 370 is about ±0.54X10· 78 to ±0.72X1076.

The finite spacing in figure 1-8 illustrates that the NPX can represent a great many, but not all, of the real numbers in its range. There is always a gap between two adjacent 80287 numbers, and it is possible for the result of a calculation to fall in this space. When this occurs, the NPX rounds the true result to a number that it can represent. Thus, a real number that requires more digits than the 80287 can accommodate (e.g., a 20-digit number) is represented with some loss of accuracy. Notice also that the 80287's representable numbers are not distributed evenly along the real number line. In fact, an equal llulliber of representable !!umb~r~ p.xists between successive powers of 2 (i.e., as many representable numbers exist between 2 and 4 as between 65,536 and 131,072). Therefore, the gaps between repre- sentable numbers are larger as the numbers increase in magnitude. All integers in the range ± 2 64 (approximately ± 1018), however, are exactly representable.

In its internal operations, the 80287 actually employs a number system that is a substantial superset of

that shown in figure 1-8. The internal format (called temporary real) extends the 80287's range to about ± 3.4X 10-4932 to ± 1.2X 104932, and its precision to about 19 (equivalent decimal) digits. This

format is designed to provide extra range and precision for constants and intermediate results, and is not normally intended for data or final results.

1-14