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Sunday, April 3, 2011

Computer Architecture # 02 : Data Representation: RANGE AND PRECISION IN FLOATING POINT NUMBERS (11)

A floating point representation allows a large range of expressible numbers to be represented in a small number of digits by separating the digits used for precision from the digits used for range. The base 10 floating point number representing Avogadro’s number is shown below:
+6.023 × 1023
Here, the range is represented by a power of 10, 1023 in this case, and the precision is represented by the digits in the fixed point number, 6.023 in this case. In discussing floating point numbers, the fixed point part is often referred to as the mantissa,

or significand of the number. Thus a floating point number can be characterized by a triple of numbers: sign, exponent, and significand.

The range is determined primarily by the number of digits in the exponent (two digits are used here) and the base to which it is raised (base 10 is used here) and the precision is determined primarily by the number of digits in the significand (four digits are used here). Thus the entire number can be represented by a sign and 6 digits, two for the exponent and four for the significand. Figure 2-7 shows how the triple of sign, exponent, significand, might be formatted in a computer.

Figure 2-7     Representation of a base 10 floating point number.
Notice how the digits are packed together with the sign first, followed by the exponent, followed by the significand. This ordering will turn out to be helpful in comparing two floating point numbers. The reader should be aware that the decimal point does not need to be stored with the number as long as the decimal point is always in the same position in the significand. (This will be discussed in Section 2.3.2.)

If we need a greater range, and if we are willing to sacrifice precision, then we can use just three digits in the fraction and have three digits left for the exponent without increasing the number of digits used in the representation. An alternative method of increasing the range is to increase the base, which has the effect of increasing the precision of the smallest numbers but decreasing the precision of the largest numbers. The range/precision trade-off is a major advantage of using a floating point representation, but the reduced precision can cause problems, sometimes leading to disaster, an example of which is described in Section 2.4.

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