Intel ITANIUM ARCHITECTURE - SOFTWARE DEVELOPERS MANUAL VOLUME 1 REV 2.3 Manual page 1780

Figure 4-10.
When real numbers become very close to zero, the normalized-number format can no
longer be used to represent the numbers. This is because the range of the exponent is
not large enough to compensate for shifting the binary point to the right to eliminate
leading zeros.
When the biased exponent is zero, smaller numbers can only be represented by making
the integer bit (and perhaps other leading bits) of the significand zero. The numbers in
this range are called denormalized (or tiny) numbers. The use of leading zeros with
denormalized numbers allows smaller numbers to be represented. However, this
denormalization causes a loss of precision (the number of significant bits in the fraction
is reduced by the leading zeros).
When performing normalized floating-point computations, a processor normally
operates on normalized numbers and produces normalized numbers as results.
Denormalized numbers represent an underflow condition.
A denormalized number is computed through a technique called gradual underflow.
Table 4-2
the single-real format is being used, so the minimum exponent (unbiased) is 126
The true result in this example requires an exponent of 129
normalized number. Since 129
is denormalized by inserting leading zeros until the minimum exponent of 126
reached.
Table 4-2.
True Result
Denormalize
Denormalize
4:478
Real Numbers and NaNs
NaN
-Denormalized Finite

-
- Normalized Finite
Real Number and NaN Encodings For 32-bit Floating-point Format
E
S F
1 0 -0
0
-Denormalized
2
1 0
0.XXX
Finite
-Normalized
1 1...254
Any Value
Finite

1 255 0
-
1 2
255 1.0XX -SNaN
X
1
X 255 1.1XX
-QNaN
Notes
1. Sign bit ignored
2. Fractions must be non-zero
gives an example of gradual underflow in the denormalization process. Here
Denormalization Process
Operation
+Denormalized Finite
-0 +0
+Denormalized
+Normalized
+SNaN
+QNaN X
is beyond the allowable exponent range, the result
10
a
Sign Exponent
 129
0
 128
0
 127
0
NaN

+
+Normalized Finite
E F
S
+0 0 0 0
2
0 0
0.XXX
Finite
0 1...254
Any Value
Finite
+
0 255 0
1 2
X 255 1.0XX
1
1.1XX
255
in order to have a
10
Significand
1.01011100000...00
0.10101110000...00
0.01010111000...00
Volume 4: IA-32 SSE Instruction Reference
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