Floating Point/Normalization

Normalization
You are probably already familiar with most of these concepts in terms of scientific or exponential notation for floating point numbers. For example, the number 123456.06 could be expressed in exponential notation as 1.23456e+05, a shorthand notation indicating that the mantissa 1.23456 is multiplied by the base 10 raised to power 5.

More formally, the internal representation of a floating point number can be characterized in terms of the following parameters: The sign is either -1 or 1. The base or radix for exponentiation, an integer greater than 1. This is a constant for a particular representation. The exponent to which the base is raised. The upper and lower bounds of the exponent value are constants for a particular representation.

Sometimes, in the actual bits representing the floating point number, the exponent is biased by adding a constant to it, to make it always be represented as an unsigned quantity. This is only important if you have some reason to pick apart the bit fields making up the floating point number by hand, which is something for which the GNU library provides no support. So this is ignored in the discussion that follows. The mantissa or significand is an unsigned integer which is a part of each floating point number. The precision of the mantissa. If the base of the representation is b, then the precision is the number of base-b digits in the mantissa. This is a constant for a particular representation.

Many floating point representations have an implicit hidden bit in the mantissa. This is a bit which is present virtually in the mantissa, but not stored in memory because its value is always 1 in a normalized number. The precision figure (see previous article) includes any hidden bits.

Again, the GNU library provides no facilities for dealing with such low-level aspects of the representation. Floating point number is used to enhance the range of representation

The mantissa of a floating point number represents an implicit fraction whose denominator is the base raised to the power of the precision. Since the largest representable mantissa is one less than this denominator(base raised to the power of the precision), the value of the fraction is always strictly less than 1. The mathematical value of a floating point number is then the product of this fraction, the sign, and the base raised to the exponent.

We say that the floating point number is normalized if the fraction is at least 1/b, where b is the base. In other words, the mantissa would be too large to fit if it were multiplied by the base. Non-normalized numbers are sometimes called denormal; they contain less precision than the representation normally can hold.

If the number is not normalized, then you can subtract 1 from the exponent while multiplying the mantissa by the base, and get another floating point number with the same value. Normalization consists of doing this repeatedly until the number is normalized. Two distinct normalized floating point numbers cannot be equal in value.

(There is an exception to this rule: if the mantissa is zero, it is considered normalized. Another exception happens on certain machines where the exponent is as small as the representation can hold. Then it is impossible to subtract 1 from the exponent, so a number may be normalized even if its fraction is less than 1/b.)