Intel 80286, 80287 manual Precision Control, ·6. Rounding Modes

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OVERVIEW OF NUMERIC PROCESSING

in arithmetic and store operations when the format of the destination cannot exactly represent the infinitely precise true result. For example, a real number may be rounded if it is stored in a shorter real format, or in an integer format. Or, the infinitely precise true result may be rounded when it is returned to a register.

The NPX has four rounding modes, selectable by the RC field in the control word (see figure 1-5). Given a true result b that cannot be represented by the target data type, the 80287 determines the two representable numbers a and c that most closely bracket b in value (a < b < c). The processor then rounds (changes) b to a or to c according to the mode selected by the RC field as shown in table 1-6. Round introduces an error in a result that is less than one unit in the last place to which the result is rounded. "Round to nearest" is the default mode and is suitable for most applications; it provides the most accurate and statistically unbiased estimate of the true result. The chop mode is provided for integer arithmetic applications.

"Round up" and "round down" are termed directed rounding and can be used to implement interval arithmetic. Interval arithmetic generates a certifiable result independent of the occurrence of rounding and other errors. The upper and lower bounds of an interval may be computed by executing an algorithm twice, rounding up in one pass and down in the other.

Precision Control

The 80287 allows results to be calculated with either 64, 53, or 24 bits of precision in the significand as selected by the precision control (PC) field of the control word. The default setting, and the one that is best suited for most applications, is the full 64 bits of significance provided by the temporary- real format. The other settings are required by the proposed IEEE standard, and are provided to obtain compatibility with the specifications of certain existing programming languages. Specifying less preci- sion nullifies the advantages of the temporary real format's extended fraction length, and does not increase execution speed. When reduced precision is specified, the rounding of the fractional value clears the unused bits on the right to zeros.

Infinity Control

The 80287's system of real numbers may be closed by either of two models of infinity. These two means of closing the number system, projective and affine closure, are illustrated schematically in figure 1-10. The setting of the IC field in the control word selects one model or the other. The default

Table 1·6. Rounding Modes

RC Field	Rounding Mode	Rounding Action
00	Round to nearest	Closer to b of a or c; if equally close,
		select even number (the one whose
		least significant bit is zero).
01	Round down (toward -00)	a
10	Round up (toward +00)	c
11	Chop (toward 0)	Smaller in magnitude of a or c

NOTE: a < b < c; a and c are representable, b is not.

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