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Unformatted text preview: t binary integer (also referred to as the J-bit) and a binary fraction. The integer-bit is often not represented, but instead is an implied value. The exponent is a binary integer that represents the base-2 power by which the significand is multiplied. Table 4-5 shows how the real number 178.125 (in ordinary decimal format) is stored in IEEE Standard 754 floating-point format. The table lists a progression of real number notations that leads to the single-precision, 32-bit floating-point format. In this format, the significand is normalized (see Section 126.96.36.199, "Normalized Numbers") and the exponent is biased (see Section 188.8.131.52, "Biased Exponent"). For the single-precision floating-point format, the biasing constant is +127. 4-14 Vol. 1 DATA TYPES -100 Binary Real Number System 10 -1 0 -10 1 100 Subset of binary real numbers that can be represented with IEEE single-precision (32-bit) floating-point format 10 -1 0 100 -100 -10 1 +10 10.0000000000000000000000 Precision 1.11111111111111111111111 24 Binary Digits Numbers within this range cannot be represented. Figure 4-10. Binary Real Number System Sign Exponent Significand Fraction Integer or J-Bit Figure 4-11. Binary Floating-Point Format Vol. 1 4-15 DATA TYPES Table 4-5. Real and Floating-Point Number Notation
Notation Ordinary Decimal Scientific Decimal Scientific Binary Scientific Binary (Biased Exponent) IEEE Single-Precision Format 178.125 1.78125E10 2 1.0110010001E2111 1.0110010001E210000110 Sign 0 Biased Exponent 10000110 Normalized Significand 0110010001000000000000 0 1. (Implied) Value 184.108.40.206 Normalized Numbers In most cases, floating-point numbers are encoded in normalized form. This means that except for zero, the significand is always made up of an integer of 1 and the following fraction: 1.fff...ff For values less than 1, leading zeros are eliminated. (For each leading zero eliminated, the exponent is decremented by one.) Representing numbers in normalized form maximizes the number of significant digits that can be accommodated in a significand of a given width. To summarize, a normalized real number consists of a normalized significand that represents a real number between 1 and 2 and an exponent that specifies the number's binary point. 220.127.116.11 Biased Exponent In the IA-32 architecture, the exponents of floating-point numbers are encoded in a biased form. This means that a constant is added to the actual exponent so that the biased exponent is always a positive number. The value of the biasing constant depends on the number of bits available for representing exponents in the floatingpoint format being used. The biasing constant is chosen so that the smallest normalized number can be reciprocated without overflow. See Section 4.2.2, "Floating-Point Data Types," for a list of the biasing constants that the IA-32 architecture uses for the various sizes of floating-point data-types. 4-16 Vol. 1 DATA TYPES 4.8.3 Real Number and Non-number Encodings A variety...
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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