سوال ۲۳ کنکور ارشد علوم کامپیوتر ۱۴۰۱

PASSAGE 2

It is important to distinguish between the syntax, the internal format of a floating-point datum, and the semantics, the associated interpretation or meaning. The term significand refers to the syntax that is interpreted as the mantissa; the exrad is interpreted as the exponent. Together, the mantissa and exponent define the value. First we will consider the significand. In all cases that have been commercially available, the sign is explicitly given as a bit toward the left of the floating-point datum. In most cases this is the same bit employed for the sign bit of an integer datum. Two situations are employed in the placement of the assumed radix point of the significand: to the extreme right and to the extreme left. If the radix point is on the right, it is apparent that the significand is an integral number. Presumably the significand would be shifted right (with an appropriate adjustment of the value of the exrad) until the rightmost base element (1 bit for binary, 3 for octal, 4 for both decimal and hexadecimal) was nonzero. Although exact conversions of external base 10 values to internal base 2 values, and vice versa, exist for integral values, the placement of the radix point on the right does not result in exact conversion accuracy, even for those values which can be entirely contained in the space provided for the significand, because of the effects of multiplying the also converted exponent as a power of the base. For larger values, an approximation is necessary that cannot be less accurate than if the radix point were assumed to be on the left. If the radix point is on the left, the significand is necessarily a fraction, and exact conversions from external base 10 values to internal base 2 values, and vice versa, are not possible.

Thus all significand values are approximations regardless of magnitude, although values beyond the capacity of the significand are approximated less accurately than those within. The significand is shifted left or right (toward the radix point) with appropriate adjustment of the exrad value, until the leftmost or rightmost element (in terms of the number base of the exrad) is nonzero in a process known as normalization. Whereas the maximum magnitude of a floating-point quantity is controlled by the length and coding scheme of the exrad, the precision or accuracy of a quantity is controlled by the effective length of the significand.

23- Normalization of a nonzero floating-point number is achieved by ----------.