Clarification of definitions in IEEE 754

Question

I'm having trouble understanding what certain terms mean within IEEE Std 754-2008 (the standard for floating-point arithmetic). A few of their definitions seem somewhat circular, and so I'm just after some clarification.

In section 2.1, some basic definitions are listed. The ones I'm interested in are:

2.1.26 floating-point representation: An unencoded member of a floating-point format, representing a finite number, a signed infinity, a quiet NaN, or a signaling Nan. A representation of a finite number has three components: a sign, an exponent, and a significand; its numerical value is the signed product of its significand and its radix raised to the power of its exponent.

2.1.27 format: A set of representations of numerical values and symbols, perhaps accompanied by an encoding.

Then in section 3.1, we have the following:

Formats are characterized by their radix, precision, and exponent range, and each format can represent a unique set of floating-point data (see 3.3).

I'm confused as to what is meant by "a floating-point format". Is it just the set of floating-point representations with a given precision, and given bound on the exponent of the base?

That is, am I right in thinking that a floating point format is the set containing the two signed infinities, the two NaNs, and all numbers of the form (-1)^sign x b^exponent x significand, where sign is 0 or 1, exponent is some integer between two given constants, emin, and emax, and significand is a number greater than 0 and strictly less than 10, with up to p siginificant digits (where p is fixed)? Furthermore, am I right in thinking that a floating-point representation is just "some member" of the above set?

Answer 1

Your knowledge seems pretty correct about the terminology for the 5 commonly used formats. Each of these floating-point formats follow the same scheme and differ only in exponent size, exponent offset, and radix size. A number has a floating-point representation where the representation is however many bits that follow the format to describe the number.

However, as the Enron scandal, Madoff scandal, 07/08 crisis, etc... taught us, people in the financial industry make up numbers, as they've done here. The IEEE 754-2008 standard includes 3 decimal based floating point standards to give financial software easy to use decimal based rounding and exact representation of 0.1 . All 3 sizes use a scheme that has sign, exponent, and significant information, but has different ranges and different representations in bits.