Vol. 1 4-17
DATA TYPES
the element. Then, if an application program attempts to access an element that it has not initialized, it can use the
NaN placed there by the compiler. If the invalid operation exception is unmasked, an interrupt will occur, and the
exception handler will be invoked. The exception handler can determine which element has been accessed, since
the operand address field of the exception pointer will point to the NaN, and the NaN will contain the index number
of the array element.
Quiet NaNs are often used to speed up debugging. In its early testing phase, a program often contains multiple
errors. An exception handler can be written to save diagnostic information in memory whenever it is invoked. After
storing the diagnostic data, it can supply a quiet NaN as the result of the erroneous instruction, and that NaN can
point to its associated diagnostic area in memory. The program will then continue, creating a different NaN for each
error. When the program ends, the NaN results can be used to access the diagnostic data saved at the time the
errors occurred. Many errors can thus be diagnosed and corrected in one test run.
In embedded applications that use computed results in further computations, an undetected QNaN can invalidate
all subsequent results. Such applications should therefore periodically check for QNaNs and provide a recovery
mechanism to be used if a QNaN result is detected.
4.8.3.7
QNaN Floating-Point Indefinite
For the floating-point data type encodings (single-precision, double-precision, and double-extended-precision),
one unique encoding (a QNaN) is reserved for representing the special value QNaN floating-point indefinite. The
x87 FPU and the SSE/SSE2/SSE3/SSE4.1/AVX extensions return these indefinite values as responses to some
masked floating-point exceptions. Table 4-3 shows the encoding used for the QNaN floating-point indefinite.
4.8.3.8
Half-Precision Floating-Point Operation
Half-precision floating-point values are not used by the processor directly for arithmetic operations. Two instruc-
tions, VCVTPH2PS, VCVTPS2PH, provide conversion only between half-precision and single-precision floating-point
values.
The SIMD floating-point exception behavior of VCVTPH2PS and VCVTPS2PH are described in Section 14.4.1.
4.8.4 Rounding
When performing floating-point operations, the processor produces an infinitely precise floating-point result in the
destination format (single-precision, double-precision, or double extended-precision floating-point) whenever
possible. However, because only a subset of the numbers in the real number continuum can be represented in IEEE
Standard 754 floating-point formats, it is often the case that an infinitely precise result cannot be encoded exactly
in the format of the destination operand.
For example, the following value (a) has a 24-bit fraction. The least-significant bit of this fraction (the underlined
bit) cannot be encoded exactly in the single-precision format (which has only a 23-bit fraction):
(a) 1.0001 0000 1000 0011 1001 0111E
2
101
To round this result (a), the processor first selects two representable fractions b and c that most closely bracket a
in value (b < a < c).
(b) 1.0001 0000 1000 0011 1001 011E
2
101
(c) 1.0001 0000 1000 0011 1001 100E
2
101
The processor then sets the result to b or to c according to the selected rounding mode. Rounding introduces an
error in a result that is less than one unit in the last place (the least significant bit position of the floating-point
value) to which the result is rounded.
The IEEE Standard 754 defines four rounding modes (see Table 4-8): round to nearest, round up, round down, and
round toward zero. The default rounding mode (for the Intel 64 and IA-32 architectures) is round to nearest. This
mode provides the most accurate and statistically unbiased estimate of the true result and is suitable for most
applications.