Apple 1.1.2 manual Normalized Numbers

Technical Considerations

where

±is the sign stored in the sign bit (1 is negative, 0 is positive).

significand	has the form b0.b1b2b3. . . bprecision-1where
	b1b2b3. . . bprecision-1are the bits in the fraction field
	and b0 is an implicit bit whose value is interpreted
	as described in the sections ÒNormalized NumbersÓ
	and ÒDenormalized NumbersÓ below. The significand
	is sometimes called the mantissa.
exponent	is the value of the exponent field.
bias	is the bias of the exponent. The bias is a
	predefined value (127 for single format, 1023 for
	double and double-double formats) that is added to
	the exponent when it is stored in the exponent
	field. When the floating-point number is evaluated,
	the bias is subtracted to return the correct
	exponent. The minimum biased exponent field (all
	0Õs) and maximum biased exponent field (all 1Õs)
	are assigned special floating-point values.

In a numeric data format, each valid representation belongs to exactly one of these classes, which are described in the sections that follow:

■normalized numbers

■denormalized numbers

■Infinities

■NaNs

■zeroes

Normalized Numbers

The numeric data formats represent most floating-point numbers as normalized numbers, meaning that the implicit leading bit, b0, of the significand is 1. Normalization maximizes the resolution of the data type and ensures that representations are unique.

Using only normalized representations creates a gap around the value 0. If a computer supports only the normalized numbers, it must round all tiny values to 0. For example, suppose such a computer must perform the operation x-y, where x and y are very close to, but not equal to, each other. If the difference between x and y is smaller than the smallest normalized number, the computer must deliver 0 as the result. Thus, for such flush-to- zero systems, the following statement is not true for all real numbers:

x-y=0if and only if x=y

Technical Considerations

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