Motorola CPU32 Bedienungsanleitung PDF herunterladen (Seite 29)

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Introduction

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M68000 FAMILY PROGRAMMER’S REFERENCE MANUAL

MOTOROLA

1.6.1 Normalized Numbers

Normalized numbers encompass all numbers with exponents laying between the maximum

and minimum values. Normalized numbers can be positive or negative. For normalized

numbers in single and double precision the implied integer bit is one. In extended precision,

the mantissa’s MSB, the explicit integer bit, can only be a one (see Figure 1-13); and the

exponent can be zero.

1.6.2 Denormalized Numbers

Denormalized numbers represent real values near the underﬂow threshold. The detection

of the underﬂow for a given data format and operation occurs when the result’s exponent is

less than or equal to the minimum exponent value. Denormalized numbers can be positive

or negative. For denormalized numbers in single and double precision the implied integer

bit is a zero. In extended precision, the mantissa’s MSB, the explicit integer bit, can only be

a zero (see Figure 1-14).

Traditionally, the detection of underﬂow causes ﬂoating-point number systems to perform a

"ﬂush-to-zero". This leaves a large gap in the number line between the smallest magnitude

normalized number and zero. The IEEE 754 standard implements gradual underﬂows: the

result mantissa is shifted right (denormalized) while the result exponent is incremented until

reaching the minimum value. If all the mantissa bits of the result are shifted off to the right

during this denormalization, the result becomes zero. Usually a gradual underﬂow limits the

potential underﬂow damage to no more than a round-off error. This underﬂow and

denormalization description ignores the effects of rounding and the user-selectable

rounding modes. Thus, the large gap in the number line created by "ﬂush-to-zero" number

systems is ﬁlled with representable (denormalized) numbers in the IEEE "gradual

underﬂow" ﬂoating-point number system.

Since the extended-precision data format has an explicit integer bit, a number can be

formatted with a nonzero exponent, less than the maximum value, and a zero integer bit.

The IEEE 754 standard does not deﬁne a zero integer bit. Such a number is an

unnormalized number. Hardware does not directly support denormalized and unnormalized

numbers, but implicitly supports them by trapping them as unimplemented data types,

allowing efﬁcient conversion in software.

Figure 1-13. Normalized Number Format

Figure 1-14. Denormalized Number Format

MIN < EXPONENT < MAX

MANTISSA = ANY BIT PATTERN

SIGN OF MANTISSA, 0 OR 1

EXPONENT = 0

MANTISSA = ANY NONZERO BIT PATTERN

SIGN OF MANTISSA, 0 OR 1

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