\subsection{Floating Point Notes \label{decimal-notes}}

The use of decimal floating point eliminates decimal representation error

(making it possible to represent \constant{0.1} exactly); however, some

operations can still incur round-off error when non-zero digits exceed the

fixed precision.

The effects of round-off error can be amplified by the addition or subtraction

of nearly offsetting quantities resulting in loss of significance. Knuth

provides two instructive examples where rounded floating point arithmetic with

insufficient precision causes the break down of the associative and

distributive properties of addition:

\begin{verbatim}

\end{verbatim}

The \module{decimal} module makes it possible to restore the identities

by expanding the precision sufficiently to avoid loss of significance:

\begin{verbatim}

\end{verbatim}

The number system for the \module{decimal} module provides special

values including \constant{NaN}, \constant{sNaN}, \constant{-Infinity},

\constant{Infinity}, and two zeroes, \constant{+0} and \constant{-0}.

Infinities can constructed directly with: \code{Decimal('Infinity')}. Also,

they can arise from dividing by zero when the \exception{DivisionByZero}

signal is not trapped. Likewise, when the \exception{Overflow} signal is not

trapped, infinity can result from rounding beyond the limits of the largest

representable number.

The infinities are signed (affine) and can be used in arithmetic operations

where they get treated as very large, indeterminate numbers. For instance,

adding a constant to infinity gives another infinite result.

Some operations are indeterminate and return \constant{NaN} or when the

\exception{InvalidOperation} signal is trapped, raise an exception. For

example, \code{0/0} returns \constant{NaN} which means ``not a number''. This

variety of \constant{NaN} is quiet and, once created, will flow through other

computations always resulting in another \constant{NaN}. This behavior can be

useful for a series of computations that occasionally have missing inputs ---

it allows the calculation to proceed while flagging specific results as

invalid.

A variant is \constant{sNaN} which signals rather than remaining quiet

after every operation. This is a useful return value when an invalid

result needs to interrupt a calculation for special handling.

The signed zeros can result from calculations that underflow.

They keep the sign that would have resulted if the calculation had

been carried out to greater precision. Since their magnitude is

zero, the positive and negative zero are treated as equal and their

sign is informational.

In addition to the two signed zeros which are distinct, yet equal,

there are various representations of zero with differing precisions,

yet equivalent in value. This takes a bit of getting used to. For

an eye accustomed to normalized floating point representations, it

is not immediately obvious that the following calculation returns

a value equal to zero:

\begin{verbatim}

\end{verbatim}