Floating-Point Representation

In the IEEE 754-2008 standard (referred to as IEEE 754 henceforth), a floating-point representation is an unencoded member of a floating-point format which represents either a finite number, a signed infinity, or some kind of NaN. An element of the subset of floating-point representations consisting of finite numbers and signed infinities is called a floating-point number.

A floating-point representation of a finite real number has three components: A sign, an exponent, and a significand. The numerical value of a representation of a finite floating-point number is the signed product of its significand and its radix raised to the power of its exponent; in particular, note that the floating-point representation of a given value may not be unique, particularly when the radix is 10 (IEEE Computer Society 2008).

When implemented at the machine-level, representations of floating-point data are encoded as -element -ary sequences of digits - here, is a parameter determined by the precision of the representation - which are partitioned into three distinct fields. Due to the fact that values may have more than one representation, it may happen that encodings fail to be in one-to-one correspondence with the values they represent. Even so, IEEE 754 requires that all encoding procedures adhere to a strict set of guidelines in terms of how each representation is encoded, thereby guaranteeing that both the representation and the value of each floating-point datum be inferable from the fields of the encoding itself. What's more, the partitions of any encoding are formatted so that "auxiliary information" such as the payload of a NaN can be deduced by decoding such an encoded representation (IEEE Computer Society 2008, pp. 9-12).

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