Question

The biggest mysteries of the IEEE 754 Floating-Point Representation are “hidden bit” and “Bias.

Can someone explain to me why the "hidden bits" and "bias" are considered to be mysteries for the IEEE 754 floating point representation

Answer 1

FLOATING POINT REPRESENTATIONS:
It is assumed that the student is familiar with the discussion in Appendix B of the text by A. Tanenbaum,
Structured Computer Organization, 4th Ed., Prentice-Hall, 1999. Here we are concerned in particular with
the discussion of floating-point numbers and normalization in pp. 643-651.In the notation used here
floating point representations for real numbers have the form:
+-M x R^+-E

Here M is a signed mantissa, R is the radix of normalization, and E is a signed exponent. It is the signed
values M and E that are typically packed into computer \words" of varying length to encode single, double,
and extended precision (short, long, and extended) floating point numbers. The number of bits used in the
representation of M determines the precision of the representation. The number of bits used to represent
the signed value E determines the range of the representation. Only a finite number of values among those
in the representable range can be encoded in the n bits available, and this number is determined by the
precision.

Positive and negative floating point numbers are usually represented in one of two ways depending on the
computer's choice of arithmetic circuitry and representation for fixed point numbers. Machines that use one's
complement representations for fixed point numbers and have one's complement adders in their arithmetic units typically pack the positive floating point number into one (or more) word(s) and then complement all bits in this (these) word(s) to represent the negative floating point number.
Examples of these one's complement representations include the UNIVAC 1100 series machines and the CDC 6600, CYBER 70 series
and successor machines.

IEEE Floating Point Standard 754:

The IEEE Floating Point Standard uses four (or eight) 8-bit bytes to represent floating point numbers in machines whose fixed-point data use a two's complement representation for negative integers and that have
two's complement arithmetic circuitry.
A single precision floating point datum is packed into four consecutive 8-bit bytes viewed as a single 32-bit representation.
A double precision floating point datum is packed into eight consecutive 8-bit bytes viewed as a single 64-bit representation, where the bits in the second 32-bits (or second set of four bytes) is continuation of the low order bits in the mantissa (called the significant).
The mantissa (or significand) is a binary normalized mixed number (with integer and fraction parts) of the form 1:xxxx2 .
A single precision datum has an 8-bit characteristic that represents a signed exponent with bias (excess) (128 1)10 = 12710, and a double precision datum has an 11-bit characteristic that represents the signed exponent excess (1024 1)10 = 102310. Negative floating-point numbers are represented in a sign-magnitude format by packing the positive representation into the single (or double) precision format and then setting the SM bit only to one. Of the 32-bits used to represent single precision numbers only 23 remain in which to represent the binary normalized mixed number. Because every number is assumed normalized in the form 1:xxxx2 2^+-E, there is no need to represent the leading one and so only the x's in the fraction are put into the 23 remaining bits in the word, resulting in 24 bits of precision.
We imagine that the leading one lies under and is hidden by an opaque characteristic field; it is therefore, called a "hiddin bit"
When unpacking a floating-point number for output conversion or for arithmetic manipulation, the hidden bit must be inserted at its rightful place in order to become visible again.
A 32-bit (or 64-bit) floating-point word comprising four (or eight) bytes with all zeros is the representation for floating-point +0 by definition.
In addition to defining formats for normalized floating point representations and zero, the standard also specifies valid formats for denormalized numbers, infinity, and quantities that are not a number
(e.g., see A.Tanenbaum, Structured Computer Organization, 3rd Ed., Prentice-Hall, 1990, pp.565{572).

Precision of A floating-point representation:
In the IEEE-754 single precision floating-point representation, the mantissa is 23 bits long. This means that any two numbers in this representation cannot be closer than

1/2^23 = 1.1920928955078125*10^-7.

In double precision, this difference reduces to

1/2^52 = 2.220446049250313080847263336181640625*10^-16.