Intel 64 and IA-32 Architectures Software Developers Manual Volume 1, Basic Architecture

Vol. 1 4-17

DATA TYPES

4.8.2.1 Normalized Numbers

In most cases, floating-point numbers are encoded in normalized form. This means

that except for zero, the significand is always made up of an integer of 1 and the

following fraction:

1.fff...ff

For values less than 1, leading zeros are eliminated. (For each leading zero elimi-

nated, the exponent is decremented by one.)

Representing numbers in normalized form maximizes the number of significant digits

that can be accommodated in a significand of a given width. To summarize, a normal-

ized real number consists of a normalized significand that represents a real number

between 1 and 2 and an exponent that specifies the number’s binary point.

4.8.2.2 Biased Exponent

In the IA-32 architecture, the exponents of floating-point numbers are encoded in a

biased form. This means that a constant is added to the actual exponent so that the

biased exponent is always a positive number. The value of the biasing constant

depends on the number of bits available for representing exponents in the floating-

point format being used. The biasing constant is chosen so that the smallest normal-

ized number can be reciprocated without overflow.

See Section 4.2.2, “Floating-Point Data Types,” for a list of the biasing constants that

the IA-32 architecture uses for the various sizes of floating-point data-types.

Table 4-5. Real and Floating-Point Number Notation

Notation Value

Ordinary Decimal 178.125

Scientific Decimal 1

.78125E

Scientific Binary 1

.0110010001E

111

Scientific Binary

(Biased Exponent)

1.0110010001E

10000110

IEEE Single-Precision Format Sign Biased Exponent Normalized Significand

0 10000110 0110010001000000000000

1. (Implied)