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

Vol. 1 4-15
DATA TYPES
4.8.1 Real Number System
As shown in Figure 4-10, the real-number system comprises the continuum of real
numbers from minus infinity (−∞) to plus infinity (+ ∞).
Because the size and number of registers that any computer can have is limited, only
a subset of the real-number continuum can be used in real-number (floating-point)
calculations. As shown at the bottom of Figure 4-10, the subset of real numbers that
the IA-32 architecture supports represents an approximation of the real number
system. The range and precision of this real-number subset is determined by the
IEEE Standard 754 floating-point formats.
4.8.2 Floating-Point Format
To increase the speed and efficiency of real-number computations, computers and
microprocessors typically represent real numbers in a binary floating-point format.
In this format, a real number has three parts: a sign, a significand, and an exponent
(see Figure 4-11).
The sign is a binary value that indicates whether the number is positive (0) or nega-
tive (1). The significand has two parts: a 1-bit binary integer (also referred to as
the J-bit) and a binary fraction. The integer-bit is often not represented, but instead
is an implied value. The exponent is a binary integer that represents the base-2
power by which the significand is multiplied.
Table 4-5 shows how the real number 178.125 (in ordinary decimal format) is stored
in IEEE Standard 754 floating-point format. The table lists a progression of real
number notations that leads to the single-precision, 32-bit floating-point format. In
this format, the significand is normalized (see Section 4.8.2.1, “Normalized
Numbers”) and the exponent is biased (see Section 4.8.2.2, “Biased Exponent”). For
the single-precision floating-point format, the biasing constant is +127.