HP-UX Floating-Point Guide

42 Chapter 2
Floating-Point Principles and the IEEE Standard for Binary Floating-Point Arithmetic
Floating-Point Formats
A denormalized value is represented by a zero exponent field and a
nonzero fraction (if the fraction were also zero, the floating-point value
would be zero). You can compute the value of a denormalized number by
interpreting the fraction as an integer and then multiplying this integer
by 2
−149
for single-precision numbers, by 2
−1074
for double-precision
numbers, and by 2
−16494
for quad-precision numbers. The maximum
fraction is always 2
k
− 1, where k is the number of bits in the fraction.
(Alternatively, you can compute the value by regarding the implicit bit as
0 and the exponent as 1 minus the bias.)
The purpose of denormalized values is to allow the spaces between zero
and the smallest magnitude normalized values to be divided up, so that
as values become smaller they underflow with a gradually increasing
loss of accuracy.
In the range of representable values, normalized values flow smoothly
into denormalized values, but there is an increasing loss of accuracy as
denormalized values become smaller and smaller. Table 2-2 shows the
range of positive denormalized values. (The hexadecimal representation
of the equivalent negative values begins with the digit 8; for example, the
minimum negative denormalized value in single-precision is 8000 0001.)
When used as operands, denormalized values are treated exactly like
normalized values in most instances. When a denormalized value is the
result of an arithmetic operation, however, an underflow exception
condition may occur. See “Underflow Conditions” on page 57 for more
information about underflow exceptions. Also, you should be aware that
denormalized values can significantly degrade performance. This issue is
addressed in “Denormalized Operands” on page 182.