User's Manual

inter
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.
1-19