4:478
Volume 4: IA-32 SSE Instruction Reference
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.
gives an example of gradual underflow in the denormalization process. Here
the single-real format is being used, so the minimum exponent (unbiased) is
126
10
.
The true result in this example requires an exponent of
129
10
in order to have a
normalized number. Since
129
10
is beyond the allowable exponent range, the result
is denormalized by inserting leading zeros until the minimum exponent of
126
10
is
reached.
Figure 4-10. Real Numbers and NaNs
Table 4-2.
Denormalization Process
Operation
Sign
Exponent
a
Significand
True Result
0
129
1.01011100000...00
Denormalize
0
128
0.10101110000...00
Denormalize
0
127
0.01010111000...00
1
0
0
S
E
F
-0
1
0
-Denormalized
Finite
NaN
1 1...254
Any Value
-Normalized
Finite
1
255
0
-
255
1.0XX
2
-SNaN
255
1.1XX
-QNaN
Notes
1. Sign bit ignored
2. Fractions must be non-zero
0
0
0
S
E
F
0
0
NaN
0 1...254
Any Value
0
255
0
X
1
255
1.0XX
2
255
1.1XX
+0
+Denormalized
Finite
+Normalized
Finite
+
+SNaN
+QNaN X
1
X
1
X
1
Real Number and NaN Encodings For 32-bit Floating-point Format
-Denormalized Finite
-
Normalized Finite
-0
-
+
+Denormalized Finite
+Normalized Finite
+0
0.XXX
2
0.XXX
2
Summary of Contents for ITANIUM ARCHITECTURE - SOFTWARE DEVELOPERS VOLUME 3 REV 2.3
Page 1: ......
Page 11: ...x Intel Itanium Architecture Software Developer s Manual Rev 2 3 ...
Page 13: ...1 2 Intel Itanium Architecture Software Developer s Manual Rev 2 3 ...
Page 33: ...1 22 Volume 1 Part 1 Introduction to the Intel Itanium Architecture ...
Page 57: ...1 46 Volume 1 Part 1 Execution Environment ...
Page 147: ...1 136 Intel Itanium Architecture Software Developer s Manual Rev 2 3 ...
Page 149: ...1 138 Volume 1 Part 2 About the Optimization Guide ...
Page 191: ...1 180 Volume 1 Part 2 Predication Control Flow and Instruction Stream ...
Page 230: ......
Page 248: ...236 Intel Itanium Architecture Software Developer s Manual Rev 2 3 ...
Page 250: ...2 2 Intel Itanium Architecture Software Developer s Manual Rev 2 3 ...
Page 264: ...2 16 Volume 2 Part 1 Intel Itanium System Environment ...
Page 380: ...2 132 Volume 2 Part 1 Interruptions ...
Page 398: ...2 150 Volume 2 Part 1 Register Stack Engine ...
Page 486: ...2 238 Volume 2 Part 1 IA 32 Interruption Vector Descriptions ...
Page 750: ...2 502 Intel Itanium Architecture Software Developer s Manual Rev 2 3 ...
Page 754: ...2 506 Volume 2 Part 2 About the System Programmer s Guide ...
Page 796: ...2 548 Volume 2 Part 2 Interruptions and Serialization ...
Page 808: ...2 560 Volume 2 Part 2 Context Management ...
Page 842: ...2 594 Volume 2 Part 2 Floating point System Software ...
Page 850: ...2 602 Volume 2 Part 2 IA 32 Application Support ...
Page 862: ...2 614 Volume 2 Part 2 External Interrupt Architecture ...
Page 870: ...2 622 Volume 2 Part 2 Performance Monitoring Support ...
Page 891: ......
Page 1099: ...3 200 Volume 3 Instruction Reference padd Interruptions Illegal Operation fault ...
Page 1295: ...3 396 Volume 3 Resource and Dependency Semantics ...
Page 1296: ......
Page 1302: ...402 Intel Itanium Architecture Software Developer s Manual Rev 2 3 ...
Page 1494: ...4 192 Volume 4 Base IA 32 Instruction Reference FWAIT Wait See entry for WAIT ...
Page 1647: ...Volume 4 Base IA 32 Instruction Reference 4 345 ROL ROR Rotate See entry for RCL RCR ROL ROR ...
Page 1884: ...4 582 Volume 4 IA 32 SSE Instruction Reference ...
Page 1885: ...Index Intel Itanium Architecture Software Developer s Manual Rev 2 3 Index ...
Page 1886: ...Index Intel Itanium Architecture Software Developer s Manual Rev 2 3 ...
Page 1898: ...INDEX Index 12 Index for Volumes 1 2 3 and 4 ...