DRAFT IEEE Standard for Binary Floating-Point Arithmetic - Sonic.net

DRAFT IEEE Standard for Floating-Point Arithmetic – 2003 August 12 10:20
The smallest positive normal number is b emin and the largest is
b emax (b − b 1-p ). The nonzero values with magnitude less than
b emin are called subnormal because their magnitudes lie between
zero and the smallest normal magnitude. One reason to
distinguish between normal and subnormal binary numbers is that
formats may encode these two kinds of numbers differently.
Every finite representable number is an integral multiple of the
smallest subnormal magnitude b emin b 1−p .
For any variable that has the value zero, the sign bit s provides an
extra bit of information. Although all formats have distinct
representations for +0 and –0, the signs are significant in some
circumstances, such as division by zero, but not in others (see
section 6.3). In this standard, 0 and ∞ are written without a sign
when the sign is not important.
[In this section, it may be appropriate to note and explain the differences between
the real numbers (or real numbers extended with signed infinities) and a format's
approximation to real numbers with respect to zero. In other words, while the
real number system only has a single zero, floating-point numbers have both
positive and negative zero for pragmatic reasons to make certain computations
easier to formulate (branch cuts, etc.) and to allow more regular rules (e.g.
defining sign of 1/0). -JDD]
Basic Format Parameters Defining Representable Values
Binary format b = 2 Decimal format b = 10
Parameter binary32 binary64 binary128 decimal32 decimal64 decimal128
p digits 24 53 113 7 16 34
emax +127 +1023 +16383 +96 +384 +6144
emin –126 –1022 –16382 –95 –383 –6143
Table 1a: Representable Value Parameters
[Note: These choices for emin, emax , and p meet the constraints of section 3.1
of IEEE 854:
• (emax - emin)/p shall exceed 4 and should exceed 10
• b p -1
>= 10 5 .
Copyright © 2003 by the Institute of Electrical and Electronics Engineers, Inc. This document is an unapproved
draft of a proposed IEEE-SA Standard - USE AT YOUR OWN RISK. See statement on page 1.
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