User Guide

254 x87 Floating-Point Programming
AMD64 Technology 24592—Rev. 3.15—November 2009
In common engineering and scientific usage, floating-point numbers—also called real numbers—are
represented in base (radix) 10. A non-zero number consists of a sign, a normalized significand, and a
signed exponent, as in:
+2.71828 e0
Both large and small numbers are representable in this notation, subject to the limits of data-type
precision. For example, a million in base-10 notation appears as +1.00000 e6 and -0.0000383 is
represented as -3.83000 e-5. A non-zero number can always be written in normalized form—that is,
with a leading non-zero digit immediately before the decimal point. Thus, a normalized significand in
base-10 notation is a number in the range [1,10). The signed exponent specifies the number of
positions that the decimal point is shifted.
Unlike the common engineering and scientific usage described above, x87 floating-point numbers are
represented in base (radix) 2. Like its base-10 counterpart, a normalized base-2 significand is written
with its leading non-zero digit immediately to the left of the radix point. In base-2 arithmetic, a non-
zero digit is always a one, so the range of a binary significand is [1,2):
+1.fraction ±exponent
The leading non-zero digit is called the integer bit, and in the x87 double-extended-precision floating-
point format this integer bit is explicit, as shown in Figure 6-8. In t he x87 single-precision and the
double-precision floating-point formats, the integer bit is simply omitted (and called the hidden
integer bit), because its implied value is always 1 in a normalized significand (0 in a denormalized
significand), and the omission allows an extra bit of precision.
The following sections describe the supported number representations.
Normalized Numbers. Normalized floating-point numbers are the most frequent operands for x87
instructions. These are finite, non-zero, positive or negative numbers in which the integer bit is 1, the
biased exponent is non-zero and non-maximum, and the fraction is any representable value. Thus, the
significand is within the range of [1, 2). Whenever possible, the processor represents a floating-point
result as a normalized number.
Denormalized (Tiny) Numbers. Denormalized numbers (also called tiny numbers) are smaller than
the smallest representable normalized numbers. They arise through an underflow condition, when the
exponent of a result lies below the representable minimum exponent. These are finite, non-zero,
positive or negative numbers in which the integer bit is 0, the biased exponent is 0, and the fraction is
non-zero.
The processor generates a denormalized-operand exception (DE) when an instruction uses a
denormalized source operand. The processor may generate an underflow exception (UE) when an
instruction produces a rounded, non-zero result that is too small to be represented as a normalized
floating-point number in the destination format, and thus is represented as a denormalized number. If a
result, after rounding, is too small to be represented as the minimum denormalized number, it is
represented as zero. (See “Exceptions” on page 278 for specific details.)