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Unformatted text preview: of real numbers and special values can be encoded in the IEEE Standard 754 floating-point format. These numbers and values are generally divided into the following classes: Signed zeros Denormalized finite numbers Normalized finite numbers Signed infinities NaNs Indefinite numbers (The term NaN stands for "Not a Number.") Figure 4-12 shows how the encodings for these numbers and non-numbers fit into the real number continuum. The encodings shown here are for the IEEE single-precision floating-point format. The term "S" indicates the sign bit, "E" the biased exponent, and "Sig" the significand. The exponent values are given in decimal. The integer bit is shown for the significands, even though the integer bit is implied in single-precision floating-point format. NaN
- Denormalized Finite + Denormalized Finite - - Normalized Finite - 0+ 0
+ Normalized Finite + NaN S 1 1 E 0 0 Real Number and NaN Encodings For 32-Bit Floating-Point Format E Sig1 Sig1 S 0.000... 0.000... -0 0 +0 0 0.XXX...2 1.XXX... 1.000... 1.0XX...2 1.1XX... - Denormalized Finite - Normalized Finite - +Denormalized Finite 0 0 0.XXX...2 1 1...254 1 255 +Normalized 0 1...254 1.XXX... Finite + 0 255 1.000... 1.0XX...2 1.1XX... X3 255 X3 255
NOTES: SNaN QNaN SNaN X3 255 QNaN X3 255 1. Integer bit of fraction implied for single-precision floating-point format. 2. Fraction must be non-zero. 3. Sign bit ignored. Figure 4-12. Real Numbers and NaNs Vol. 1 4-17 DATA TYPES An IA-32 processor can operate on and/or return any of these values, depending on the type of computation being performed. The following sections describe these number and non-number classes. 126.96.36.199 Signed Zeros Zero can be represented as a +0 or a -0 depending on the sign bit. Both encodings are equal in value. The sign of a zero result depends on the operation being performed and the rounding mode being used. Signed zeros have been provided to aid in implementing interval arithmetic. The sign of a zero may indicate the direction from which underflow occurred, or it may indicate the sign of an that has been reciprocated. 188.8.131.52 Normalized and Denormalized Finite Numbers Non-zero, finite numbers are divided into two classes: normalized and denormalized. The normalized finite numbers comprise all the non-zero finite values that can be encoded in a normalized real number format between zero and . In the single-precision floating-point format shown in Figure 4-12, this group of numbers includes all the numbers with biased exponents ranging from 1 to 25410 (unbiased, the exponent range is from -12610 to +12710). When floating-point 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 per...
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This note was uploaded on 10/01/2013 for the course CPE 103 taught by Professor Watlins during the Winter '11 term at Mississippi State.
- Winter '11