This preview shows page 1. Sign up to view the full content.
Unformatted text preview: of real numbers and special values can be encoded in the IEEE Standard 754 floatingpoint 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 412 shows how the encodings for these numbers and nonnumbers fit into the real number continuum. The encodings shown here are for the IEEE singleprecision floatingpoint 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 singleprecision floatingpoint 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 32Bit FloatingPoint 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 singleprecision floatingpoint format. 2. Fraction must be nonzero. 3. Sign bit ignored. Figure 412. Real Numbers and NaNs Vol. 1 417 DATA TYPES An IA32 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 nonnumber classes. 4.8.3.1 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. 4.8.3.2 Normalized and Denormalized Finite Numbers Nonzero, finite numbers are divided into two classes: normalized and denormalized. The normalized finite numbers comprise all the nonzero finite values that can be encoded in a normalized real number format between zero and . In the singleprecision floatingpoint format shown in Figure 412, 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 floatingpoint numbers become very close to zero, the normalizednumber 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...
View
Full
Document
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
 Watlins

Click to edit the document details