Denormalized Numbers, Infinities

Technical Considerations

Denormalized Numbers

Instead of using only normalized numbers and allowing this small gap around 0, PowerPC processor-based Macintosh computers use denormalized numbers, in which the leading implicit bit, b0, of the significand is 0 and the minimum exponent is used.

The use of denormalized numbers makes the following statement true for all real numbers:

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

Another advantage of denormalized numbers is that error analysis involving small values is much easier without the gap around zero (Demmel 1984). The computer determines that a floating-point number is denormalized (and therefore that its implicit leading bit is interpreted as 0) when the biased exponent field is filled with 0Õs and the fraction field is nonzero.

Infinities

An Infinity is a special bit pattern that can arise in one of two ways:

■When an operation (such as 1/0) should produce a mathematical infinity, the result is an Infinity.

■When an operation attempts to produce a number with a magnitude too great for the numberÕs intended floating-point data type, the result might be a value with the largest possible magnitude or it might be an Infinity (depending on the current rounding direction).

These bit patterns (as well as NaNs, introduced next) are recognized in subsequent operations and produce predictable results. The Infinities, one positive and one negative, generally behave as suggested by the theory of limits. For example:

■Adding 1 to +° yields +°

■Dividing -1 by +0 yields -°

■Dividing 1 by -° yields -0

The computer determines that a floating-point number is an Infinity if its exponent field is filled with 1Õs and its fraction field is filled with 0Õs. So, for example, in single format, if the sign bit is 1, the exponent field is 255 (which is the maximum biased exponent for the single format), and the fraction field is 0, the floating-point number represented is -°.

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