where
± is the sign stored in the sign bit (1 is negative, 0 is
positive).
significand has the form b
0
.b
1
b
2
b
3
. . . b
precision-1
where
b
1
b
2
b
3
. . . b
precision-1
are the bits in the fraction field
and b
0
is an implicit bit whose value is interpreted
as described in the sections ÒNormalized NumbersÓ
and ÒDenormalized NumbersÓ below. The significand
is sometimes called the mantissa.
exponent is the value of the exponent field.
bias is the bias of the exponent. The bias is a
predefined value (127 for single format, 1023 for
double and double-double formats) that is added to
the exponent when it is stored in the exponent
field. When the floating-point number is evaluated,
the bias is subtracted to return the correct
exponent. The minimum biased exponent field (all
0Õs) and maximum biased exponent field (all 1Õs)
are assigned special floating-point values.
In a numeric data format, each valid representation belongs to
exactly one of these classes, which are described in the sections
that follow:
■ normalized numbers
■ denormalized numbers
■ Infinities
■ NaNs
■ zeroes
Normalized Numbers
The numeric data formats represent most floating-point numbers
as normalized numbers, meaning that the implicit leading bit,
b
0
, of the significand is 1. Normalization maximizes the resolution
of the data type and ensures that representations are unique.
Using only normalized representations creates a gap around the
value 0. If a computer supports only the normalized numbers, it
must round all tiny values to 0. For example, suppose such a
computer must perform the operation
x-y, where x and y are very
close to, but not equal to, each other. If the difference between
x
and y is smaller than the smallest normalized number, the
computer must deliver 0 as the result. Thus, for such flush-to-
zero systems, the following statement is not true for all real
numbers:
x-y=0 if and only if x=y
Technical Considerations
iMalc Manual
Technical Considerations
37
