Data formats for single and double precision numbers differ slightly from the data formats
for extended precision numbers in the representation of the mantissa. A normalized man-
tissa, for all three precisions, is always in the range [1.0... 2.0). The extended precision
data format explicitly represents the entire mantissa, including the explicit integer part bit.
However, for single and double precision data formats, only the fractional portion of the
mantissa is explicitly represented and the integer part, always one, is implied.
The IEEE standard has created the term "significand" to bridge this difference and to avoid
the historical implications of the term mantissa. The IEEE standard defines a significand
as the component of a binary floating-point number that consists of an explicit or implicit
leading bit to the left of the implied binary point. This'manual uses the terms mantissa
and significand, defined as follows, interchangeably.
Single Precision Mantissa = Single Precision Significand
= 1.<23-Bit Fraction Field>
Double Precision Mantissa = Double Precision Significand
= 1.<52-Bit Fraction Field>
Extended Precision Mantissa = Extended Precision Significand
= 1.Fraction
= <64-Bit Mantissa Field>
NOTE
Throughout this manual, ranges are specified using traditional set notation with
the format "bound... bound" specifyingthe boundaries of the range. The type
of brackets enclosing the range defines whether the endpoint is inclusive or ex-
clusive. A square bracket indicates inclusive, and a parenthesis indicates exclu-
sive. For example, the range specification "[1.0...2.0]" defines the range of numbers
greater than or equal to 1.0 and less than or equal to 2.0. The range specification
"(0.0... + inf]" defines the range of numbers greater than 0.0 and less than or equal
to positive infinity.
Each of the three floating-point data formats can represent five unique floating-point data
types:
Normalized Numbers
Denormalized Numbers
Zeros
Infinities
Not-A-Numbers (NANs)
The normalized data type never uses the maximum or minimum exponent value for a
given format (except for the extended precision format, see following note). These exponent
values in each precision are reserved for representing the special data types: zeros, infin-
ities, denormalized numbers, and NANs. Details of each type of number for each format
are shown in 3.6 DATA FORMAT DETAILS.
NOTE
There is a subtle difference between the definition of an extended precision num-
ber with an exponent equal to zero and a single or double precision number with
an exponent equal to zero. If the exponent of a single or double precision number
is zero, the number is defined to be denormalized, and the implied integer bit is
also a zero. However, an extended precision number with an exponent of zero
may have an explicit integer bit equal to one, which results in a normalized number
(even though the exponent is equal to the minimum value).
MC68881/MC68882 USER'S MANUAL
FREESCALE
3-3
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