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Unformatted text preview: 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 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. 4.8.4 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 0111E2 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 011E2 101 (c) 1.0001 0000 1000 0011 1001 100E2 101 4-22 Vol. 1 DATA TYPES 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. Table 4-8. Rounding Modes and Encoding of Rounding Control (RC) Field
Rounding Mode Round to nearest (even) Round down (toward -) Round up (toward +) RC Field Setting 00B Description Rounded result is the closest to the infinitely precise result. If two values are equally close, the result is the even value (that is, the one with the least-significant bit of zero). Default Rounded result is closest to but no greater than the infinitely precise result. Rounded result is closest to but no less than the infinitely precise result. Rounded result is closest to but no greater in absolute value than the infinitely precise result. 01B 10B Round toward 11B zero (Truncate) The round up and round down modes are termed directed rounding and can be used to implement interval arithmetic. Interval arithmetic is used to determine upper and lower bounds for the true result of a multistep computation,...
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- Winter '11