path: root/Doc/lib/libdecimal.tex

\section{\module{decimal} ---
         Decimal floating point arithmetic}

\declaremodule{standard}{decimal}
\modulesynopsis{Implementation of the General Decimal Arithmetic 
Specification.}

\moduleauthor{Eric Price}{eprice at tjhsst.edu}
\moduleauthor{Facundo Batista}{facundo at taniquetil.com.ar}
\moduleauthor{Raymond Hettinger}{python at rcn.com}
\moduleauthor{Aahz}{aahz at pobox.com}
\moduleauthor{Tim Peters}{tim.one at comcast.net}

\sectionauthor{Raymond D. Hettinger}{python at rcn.com}

\versionadded{2.4}

The \module{decimal} module provides support for decimal floating point
arithmetic.  It offers several advantages over the \class{float()} datatype:

\begin{itemize}

\item Decimal numbers can be represented exactly.  In contrast, numbers like
\constant{1.1} do not have an exact representations in binary floating point.
End users typically wound not expect \constant{1.1} to display as
\constant{1.1000000000000001} as it does with binary floating point.

\item The exactness carries over into arithmetic.  In decimal floating point,
\samp{0.1 + 0.1 + 0.1 - 0.3} is exactly equal to zero.  In binary floating
point, result is \constant{5.5511151231257827e-017}.  While near to zero, the
differences prevent reliable equality testing and differences can accumulate.
For this reason, decimal would be preferred in accounting applications which
have strict equality invariants.

\item The decimal module incorporates notion of significant places so that
\samp{1.30 + 1.20} is \constant{2.50}.  The trailing zero is kept to indicate
significance.  This is the customary presentation for monetary applications. For
multiplication, the ``schoolbook'' approach uses all the figures in the
multiplicands.  For instance, \samp{1.3 * 1.2} gives \constant{1.56} while
\samp{1.30 * 1.20} gives \constant{1.5600}.

\item Unlike hardware based binary floating point, the decimal module has a user
settable precision (defaulting to 28 places) which can be as large as needed for
a given problem:

\begin{verbatim}
>>> getcontext().prec = 6
>>> Decimal(1) / Decimal(7)
Decimal("0.142857")
>>> getcontext().prec = 28
>>> Decimal(1) / Decimal(7)
Decimal("0.1428571428571428571428571429")
\end{verbatim}

\item Both binary and decimal floating point are implemented in terms of published
standards.  While the built-in float type exposes only a modest portion of its
capabilities, the decimal module exposes all required parts of the standard.
When needed, the programmer has full control over rounding and signal handling.

\end{itemize}


The module design is centered around three concepts:  the decimal number, the
context for arithmetic, and signals.

A decimal number is immutable.  It has a sign, coefficient digits, and an
exponent.  To preserve significance, the coefficient digits do not truncate
trailing zeroes.  Decimals also include special values such as
\constant{Infinity} (the result of \samp{1 / 0}), \constant{-Infinity},
(the result of \samp{-1 / 0}), and \constant{NaN} (the result of
\samp{0 / 0}).  The standard also differentiates \constant{-0} from
\constant{+0}.
                                                   
The context for arithmetic is an environment specifying precision, rounding
rules, limits on exponents, flags that indicate the results of operations,
and trap enablers which determine whether signals are to be treated as
exceptions.  Rounding options include \constant{ROUND_CEILING},
\constant{ROUND_DOWN}, \constant{ROUND_FLOOR}, \constant{ROUND_HALF_DOWN},
\constant{ROUND_HALF_EVEN}, \constant{ROUND_HALF_UP}, and \constant{ROUND_UP}.

Signals are types of information that arise during the course of a
computation.  Depending on the needs of the application, some signals may be
ignored, considered as informational, or treated as exceptions. The signals in
the decimal module are: \constant{Clamped}, \constant{InvalidOperation},
\constant{ConversionSyntax}, \constant{DivisionByZero},
\constant{DivisionImpossible}, \constant{DivisionUndefined},
\constant{Inexact}, \constant{InvalidContext}, \constant{Rounded},
\constant{Subnormal}, \constant{Overflow}, and \constant{Underflow}.

For each signal there is a flag and a trap enabler.  When a signal is
encountered, its flag incremented from zero and, then, if the trap enabler
is set to one, an exception is raised.  Flags are sticky, so the user
needs to reset them before monitoring a calculation.


\begin{seealso}
  \seetext{IBM's General Decimal Arithmetic Specification,
           \citetitle[http://www2.hursley.ibm.com/decimal/decarith.html]
           {The General Decimal Arithmetic Specification}.}

  \seetext{IEEE standard 854-1987,
           \citetitle[http://www.cs.berkeley.edu/\textasciitilde ejr/projects/754/private/drafts/854-1987/dir.html]
           {Unofficial IEEE 854 Text}.} 
\end{seealso}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Quick-start Tutorial \label{decimal-tutorial}}

The normal start to using decimals is to import the module, and then use
\function{getcontext()} to view the context and, if necessary, set the context
precision, rounding, or trap enablers:

\begin{verbatim}
>>> from decimal import *
>>> getcontext()
Context(prec=28, rounding=ROUND_HALF_EVEN, Emin=-999999999, Emax=999999999,
        setflags=[], settraps=[])

>>> getcontext().prec = 7
\end{verbatim}

Decimal instances can be constructed from integers, strings or tuples.  To
create a Decimal from a \class{float}, first convert it to a string.  This
serves as an explicit reminder of the details of the conversion (including
representation error).  Malformed strings signal \constant{ConversionSyntax}
and return a special kind of Decimal called a \constant{NaN} which stands for
``Not a number''. Positive and negative \constant{Infinity} is yet another
special kind of Decimal.        

\begin{verbatim}
>>> Decimal(10)
Decimal("10")
>>> Decimal("3.14")
Decimal("3.14")
>>> Decimal((0, (3, 1, 4), -2))
Decimal("3.14")
>>> Decimal(str(2.0 ** 0.5))
Decimal("1.41421356237")
>>> Decimal("NaN")
Decimal("NaN")
>>> Decimal("-Infinity")
Decimal("-Infinity")
\end{verbatim}

Creating decimals is unaffected by context precision.  Their level of
significance is completely determined by the number of digits input.  It is
the arithmetic operations that are governed by context.

\begin{verbatim}
>>> getcontext().prec = 6
>>> Decimal('3.0000')
Decimal("3.0000")
>>> Decimal('3.0')
Decimal("3.0")
>>> Decimal('3.1415926535')
Decimal("3.1415926535")
>>> Decimal('3.1415926535') + Decimal('2.7182818285')
Decimal("5.85987")
>>> getcontext().rounding = ROUND_UP
>>> Decimal('3.1415926535') + Decimal('2.7182818285')
Decimal("5.85988")
\end{verbatim}

Decimals interact well with much of the rest of python.  Here is a small
decimal floating point flying circus:
    
\begin{verbatim}    
>>> data = map(Decimal, '1.34 1.87 3.45 2.35 1.00 0.03 9.25'.split())
>>> max(data)
Decimal("9.25")
>>> min(data)
Decimal("0.03")
>>> sorted(data)
[Decimal("0.03"), Decimal("1.00"), Decimal("1.34"), Decimal("1.87"),
 Decimal("2.35"), Decimal("3.45"), Decimal("9.25")]
>>> sum(data)
Decimal("19.29")
>>> a,b,c = data[:3]
>>> str(a)
'1.34'
>>> float(a)
1.3400000000000001
>>> round(a, 1)
1.3
>>> int(a)
1
>>> a * 5
Decimal("6.70")
>>> a * b
Decimal("2.5058")
>>> c % a
Decimal("0.77")
\end{verbatim}

The \function{getcontext()} function accesses the current context.  This one
context is sufficient for many applications; however, for more advanced work,
multiple contexts can be created using the Context() constructor.  To make a
new context active, use the \function{setcontext()} function.

In accordance with the standard, the \module{Decimal} module provides two
ready to use standard contexts, \constant{BasicContext} and
\constant{ExtendedContext}. The former is especially useful for debugging
because many of the traps are enabled:

\begin{verbatim}
>>> myothercontext = Context(prec=60, rounding=ROUND_HALF_DOWN)
>>> myothercontext
Context(prec=60, rounding=ROUND_HALF_DOWN, Emin=-999999999, Emax=999999999,
        setflags=[], settraps=[])
>>> ExtendedContext
Context(prec=9, rounding=ROUND_HALF_EVEN, Emin=-999999999, Emax=999999999,
        setflags=[], settraps=[])
>>> setcontext(myothercontext)
>>> Decimal(1) / Decimal(7)
Decimal("0.142857142857142857142857142857142857142857142857142857142857")
>>> setcontext(ExtendedContext)
>>> Decimal(1) / Decimal(7)
Decimal("0.142857143")
>>> Decimal(42) / Decimal(0)
Decimal("Infinity")
>>> setcontext(BasicContext)
>>> Decimal(42) / Decimal(0)
Traceback (most recent call last):
  File "<pyshell#143>", line 1, in -toplevel-
    Decimal(42) / Decimal(0)
DivisionByZero: x / 0
\end{verbatim}

Besides using contexts to control precision, rounding, and trapping signals,
they can be used to monitor flags which give information collected during
computation.  The flags remain set until explicitly cleared, so it is best to
clear the flags before each set of monitored computations by using the
\method{clear_flags()} method.

\begin{verbatim}
>>> setcontext(ExtendedContext)
>>> Decimal(355) / Decimal(113)
Decimal("3.14159292")
>>> getcontext()
Context(prec=9, rounding=ROUND_HALF_EVEN, Emin=-999999999, Emax=999999999,
        setflags=['Inexact', 'Rounded'], settraps=[])
\end{verbatim}

The \var{setflags} entry shows that the rational approximation to
\constant{Pi} was rounded (digits beyond the context precision were thrown
away) and that the result is inexact (some of the discarded digits were
non-zero).

Individual traps are set using the dictionary in the \member{trap_enablers}
field of a context:

\begin{verbatim}
>>> Decimal(1) / Decimal(0)
Decimal("Infinity")
>>> getcontext().trap_enablers[DivisionByZero] = 1
>>> Decimal(1) / Decimal(0)

Traceback (most recent call last):
  File "<pyshell#112>", line 1, in -toplevel-
    Decimal(1) / Decimal(0)
DivisionByZero: x / 0
\end{verbatim}

To turn all the traps on or off all at once, use a loop.  Also, the
\method{dict.update()} method is useful for changing a handfull of values.

\begin{verbatim}
>>> getcontext.clear_flags()
>>> for sig in getcontext().trap_enablers:
...     getcontext().trap_enablers[sig] = 1

>>> getcontext().trap_enablers.update({Rounded:0, Inexact:0, Subnormal:0})
>>> getcontext()
Context(prec=9, rounding=ROUND_HALF_EVEN, Emin=-999999999, Emax=999999999,
        setflags=[], settraps=['Underflow', 'DecimalException', 'Clamped',
        'InvalidContext', 'InvalidOperation', 'ConversionSyntax',
        'DivisionByZero', 'DivisionImpossible', 'DivisionUndefined',
        'Overflow'])
\end{verbatim}

Applications typically set the context once at the beginning of a program
and no further changes are needed.  For many applications, the data resides
in a resource external to the program and is converted to \class{Decimal} with
a single cast inside a loop.  Afterwards, decimals are as easily manipulated
as other Python numeric types.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Decimal objects \label{decimal-decimal}}

\begin{classdesc}{Decimal}{\optional{value \optional{, context}}}
  Constructs a new \class{Decimal} object based from \var{value}.

  \var{value} can be an integer, string, tuple, or another \class{Decimal}
  object. If no \var{value} is given, returns \code{Decimal("0")}.  If
  \var{value} is a string, it should conform to the decimal numeric string
  syntax:
    
  \begin{verbatim}
    sign           ::=  '+' | '-'
    digit          ::=  '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
    indicator      ::=  'e' | 'E'
    digits         ::=  digit [digit]...
    decimal-part   ::=  digits '.' [digits] | ['.'] digits
    exponent-part  ::=  indicator [sign] digits
    infinity       ::=  'Infinity' | 'Inf'
    nan            ::=  'NaN' [digits] | 'sNaN' [digits]
    numeric-value  ::=  decimal-part [exponent-part] | infinity
    numeric-string ::=  [sign] numeric-value | [sign] nan  
  \end{verbatim}

  If \var{value} is a \class{tuple}, it should have three components,
  a sign (\constant{0} for positive or \constant{1} for negative),
  a \class{tuple} of digits, and an exponent represented as an integer.
  For example, \samp{Decimal((0, (1, 4, 1, 4), -3))} returns
  \code{Decimal("1.414")}.

  The supplied \var{context} or, if not specified, the current context
  governs only the handling of malformed strings not conforming to the
  numeric string syntax.  If the context traps \constant{ConversionSyntax},
  an exception is raised; otherwise, the constructor returns a new Decimal
  with the value of \constant{NaN}.

  The context serves no other purpose.  The number of significant digits
  recorded is determined solely by the \var{value} and the \var{context}
  precision is not a factor.  For example, \samp{Decimal("3.0000")} records
  all four zeroes even if the context precision is only three.

  Once constructed, \class{Decimal} objects are immutable.
\end{classdesc}

Decimal floating point objects share many properties with the other builtin
numeric types such as \class{float} and \class{int}.  All of the usual
math operations and special methods apply.  Likewise, decimal objects can
be copied, pickled, printed, used as dictionary keys, used as set elements,
compared, sorted, and coerced to another type (such as \class{float}
or \class{long}).

In addition to the standard numeric properties, decimal floating point objects
have a number of more specialized methods:

\begin{methoddesc}{adjusted}{}
  Return the adjusted exponent after shifting out the coefficient's rightmost
  digits until only the lead digit remains: \code{Decimal("321e+5").adjusted()}
  returns seven.  Used for determining the place value of the most significant
  digit.
\end{methoddesc}

\begin{methoddesc}{as_tuple}{}
  Returns a tuple representation of the number:
  \samp{(sign, digittuple, exponent)}.
\end{methoddesc}

\begin{methoddesc}{compare}{other\optional{, context}}
  Compares like \method{__cmp__()} but returns a decimal instance:
  \begin{verbatim}
        a or b is a NaN ==> Decimal("NaN")
        a < b           ==> Decimal("-1")
        a == b          ==> Decimal("0")
        a > b           ==> Decimal("1")
  \end{verbatim}
\end{methoddesc}

\begin{methoddesc}{max}{other\optional{, context}}
  Like \samp{max(self, other)} but returns \constant{NaN} if either is a
  \constant{NaN}.  Applies the context rounding rule before returning.
\end{methoddesc}

\begin{methoddesc}{min}{other\optional{, context}}
  Like \samp{min(self, other)} but returns \constant{NaN} if either is a
  \constant{NaN}.  Applies the context rounding rule before returning.
\end{methoddesc}

\begin{methoddesc}{normalize}{\optional{context}}
  Normalize the number by stripping the rightmost trailing zeroes and
  converting any result equal to \constant{Decimal("0")} to
  \constant{Decimal("0e0")}. Used for producing canonical values for members
  of an equivalence class. For example, \code{Decimal("32.100")} and
  \code{Decimal("0.321000e+2")} both normalize to the equivalent value
  \code{Decimal("32.1")},
\end{methoddesc}                                              

\begin{methoddesc}{quantize}
  {\optional{exp \optional{, rounding\optional{, context\optional{, watchexp}}}}}
  Quantize makes the exponent the same as \var{exp}.  Searches for a
  rounding method in \var{rounding}, then in \var{context}, and then
  in the current context.

  If \var{watchexp} is set (default), then an error is returned whenever
  the resulting exponent is greater than \member{Emax} or less than
  \member{Etiny}.
\end{methoddesc} 

\begin{methoddesc}{remainder_near}{other\optional{, context}}
  Computed the modulo as either a positive or negative value depending
  on which is closest to zero.  For instance,
  \samp{Decimal(10).remainder_near(6)} returns \code{Decimal("-2")}
  which is closer to zero than \code{Decimal("4")}.

  If both are equally close, the one chosen will have the same sign
  as \var{self}.
\end{methoddesc}  

\begin{methoddesc}{same_quantum}{other\optional{, context}}
  Test whether self and other have the same exponent or whether both
  are \constant{NaN}.
\end{methoddesc}

\begin{methoddesc}{sqrt}{\optional{context}}
  Return the square root to full precision.
\end{methoddesc}                    
 
\begin{methoddesc}{to_eng_string}{\optional{context}}
  Convert to an engineering-type string.

  Engineering notation has an exponent which is a multiple of 3, so there
  are up to 3 digits left of the decimal place.  For example, converts
  \code{Decimal('123E+1')} to \code{Decimal("1.23E+3")}
\end{methoddesc}  

\begin{methoddesc}{to_integral}{\optional{rounding\optional{, context}}}                   
  Rounds to the nearest integer without signaling \constant{Inexact}
  or \constant{Rounded}.  If given, applies \var{rounding}; otherwise,
  uses the rounding method in either the supplied \var{context} or the
  current context.
\end{methoddesc} 

    
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%            
\subsection{Context objects \label{decimal-decimal}}

Contexts are environments for arithmetic operations.  They govern the precision,
rules for rounding, determine which signals are treated as exceptions, and set limits
on the range for exponents.

Each thread has its own current context which is accessed or changed using
the \function{getcontext()} and \function{setcontext()} functions:

\begin{funcdesc}{getcontext}{}
  Return the current context for the active thread.                                          
\end{funcdesc}            

\begin{funcdesc}{setcontext}{c}
  Set the current context for the active thread to \var{c}.                                          
\end{funcdesc}  

New contexts can formed using the \class{Context} constructor described below.
In addition, the module provides three pre-made contexts:                                          


\begin{classdesc*}{BasicContext}
  This is a standard context defined by the General Decimal Arithmetic
  Specification.  Precision is set to nine.  Rounding is set to
  \constant{ROUND_HALF_UP}.  All flags are cleared.  All traps are enabled
  (treated as exceptions) except \constant{Inexact}, \constant{Rounded}, and
  \constant{Subnormal}.

  Because many of the traps are enabled, this context is useful for debugging.
\end{classdesc*}

\begin{classdesc*}{ExtendedContext}
  This is a standard context defined by the General Decimal Arithmetic
  Specification.  Precision is set to nine.  Rounding is set to
  \constant{ROUND_HALF_EVEN}.  All flags are cleared.  No traps are enabled
  (so that exceptions are not raised during computations).

  Because the trapped are disabled, this context is useful for applications
  that prefer to have result value of \constant{NaN} or \constant{Infinity}
  instead of raising exceptions.  This allows an application to complete a
  run in the presense of conditions that would otherwise halt the program.
\end{classdesc*}

\begin{classdesc*}{DefaultContext}
  This class is used by the \class{Context} constructor as a prototype for
  new contexts.  Changing a field (such a precision) has the effect of
  changing the default for new contexts creating by the \class{Context}
  constructor.

  This context is most useful in multi-threaded environments.  Changing one of
  the fields before threads are started has the effect of setting system-wide
  defaults.  Changing the fields after threads have started is not recommended
  as it would require thread synchronization to prevent race conditions.

  In single threaded environments, it is preferable to not use this context
  at all.  Instead, simply create contexts explicitly.  This is especially
  important because the default values context may change between releases
  (with initial release having precision=28, rounding=ROUND_HALF_EVEN,
  cleared flags, and no traps enabled).
\end{classdesc*}


In addition to the three supplied contexts, new contexts can be created
with the \class{Context} constructor.

\begin{classdesc}{Context}{prec=None, rounding=None, trap_enablers=None,
        flags=None, Emin=None, Emax=None, capitals=1}
  Creates a new context.  If a field is not specified or is \constant{None},
  the default values are copied from the \constant{DefaultContext}.  If the
  \var{flags} field is not specified or is \constant{None}, all flags are
  cleared.

  The \var{prec} field is a positive integer that sets the precision for
  arithmetic operations in the context.

  The \var{rounding} option is one of:
      \constant{ROUND_CEILING} (towards \constant{Infinity}),
      \constant{ROUND_DOWN} (towards zero),
      \constant{ROUND_FLOOR} (towards \constant{-Infinity}),
      \constant{ROUND_HALF_DOWN} (towards zero),
      \constant{ROUND_HALF_EVEN},
      \constant{ROUND_HALF_UP} (away from zero), or
      \constant{ROUND_UP} (away from zero).

  The \var{trap_enablers} and \var{flags} fields are mappings from signals
  to either \constant{0} or \constant{1}.

  The \var{Emin} and \var{Emax} fields are integers specifying the outer
  limits allowable for exponents.

  The \var{capitals} field is either \constant{0} or \constant{1} (the
  default). If set to \constant{1}, exponents are printed with a capital
  \constant{E}; otherwise, lowercase is used:  \constant{Decimal('6.02e+23')}.
\end{classdesc}

The \class{Context} class defines several general methods as well as a
large number of methods for doing arithmetic directly from the context.

\begin{methoddesc}{clear_flags}{}
  Sets all of the flags to \constant{0}.
\end{methoddesc}  

\begin{methoddesc}{copy}{}
  Returns a duplicate of the context.
\end{methoddesc}  

\begin{methoddesc}{create_decimal}{num}
  Creates a new Decimal instance but using \var{self} as context.
  Unlike the \class{Decimal} constructor, context precision,
  rounding method, flags, and traps are applied to the conversion.

  This is useful because constants are often given to a greater
  precision than is needed by the application.
\end{methoddesc} 

\begin{methoddesc}{Etiny}{}
  Returns a value equal to \samp{Emin - prec + 1} which is the minimum
  exponent value for subnormal results.  When underflow occurs, the
  exponont is set to \constant{Etiny}.
\end{methoddesc} 

\begin{methoddesc}{Etop}{}
  Returns a value equal to \samp{Emax - prec + 1}.
\end{methoddesc} 


The usual approach to working with decimals is to create \class{Decimal}
instances and then apply arithmetic operations which take place within the
current context for the active thread.  An alternate approach is to use
context methods for calculating within s specific context.  The methods are
similar to those for the \class{Decimal} class and are only briefly recounted
here.

\begin{methoddesc}{abs}{x}
  Returns the absolute value of \var{x}.
\end{methoddesc}

\begin{methoddesc}{add}{x, y}
  Return the sum of \var{x} and \var{y}.
\end{methoddesc}
   
\begin{methoddesc}{compare}{x, y}
  Compares values numerically.
  
  Like \method{__cmp__()} but returns a decimal instance:
  \begin{verbatim}
        a or b is a NaN ==> Decimal("NaN")
        a < b           ==> Decimal("-1")
        a == b          ==> Decimal("0")
        a > b           ==> Decimal("1")
  \end{verbatim}                                          
\end{methoddesc}

\begin{methoddesc}{divide}{x, y}
  Return \var{x} divided by \var{y}.
\end{methoddesc}   
  
\begin{methoddesc}{divmod}{x, y}
  Divides two numbers and returns the integer part of the result.
\end{methoddesc} 

\begin{methoddesc}{max}{x, y}
  Compare two values numerically and returns the maximum.

  If they are numerically equal then the left-hand operand is chosen as the
  result.
\end{methoddesc} 
 
\begin{methoddesc}{min}{x, y}
  Compare two values numerically and returns the minimum.

  If they are numerically equal then the left-hand operand is chosen as the
  result.
\end{methoddesc}

\begin{methoddesc}{minus}{x}
  Minus corresponds to the unary prefix minus operator in Python.
\end{methoddesc}

\begin{methoddesc}{multiply}{x, y}
  Return the product of \var{x} and \var{y}.
\end{methoddesc}

\begin{methoddesc}{normalize}{x}
  Normalize reduces an operand to its simplest form.

  Essentially a plus operation with all trailing zeros removed from the
  result.
\end{methoddesc}
  
\begin{methoddesc}{plus}{x}
  Minus corresponds to the unary prefix plus operator in Python.
\end{methoddesc}

\begin{methoddesc}{power}{x, y\optional{, modulo}}
  Return \samp{x ** y} to the \var{modulo} if given.

  The right-hand operand must be a whole number whose integer part (after any
  exponent has been applied) has no more than 9 digits and whose fractional
  part (if any) is all zeros before any rounding. The operand may be positive,
  negative, or zero; if negative, the absolute value of the power is used, and
  the left-hand operand is inverted (divided into 1) before use.

  If the increased precision needed for the intermediate calculations exceeds
  the capabilities of the implementation then an \constant{InvalidOperation}
  condition is signaled.

  If, when raising to a negative power, an underflow occurs during the
  division into 1, the operation is not halted at that point but continues. 
\end{methoddesc}

\begin{methoddesc}{quantize}{x, y}
  Returns a value equal to \var{x} after rounding and having the
  exponent of v\var{y}.

  Unlike other operations, if the length of the coefficient after the quantize
  operation would be greater than precision then an
  \constant{InvalidOperation} is signaled. This guarantees that, unless there
  is an error condition, the exponent of the result of a quantize is always
  equal to that of the right-hand operand.

  Also unlike other operations, quantize never signals Underflow, even
  if the result is subnormal and inexact.  
\end{methoddesc} 

\begin{methoddesc}{remainder}{x, y}
  Returns the remainder from integer division.

  The sign of the result, if non-zero, is the same as that of the original
  dividend. 
\end{methoddesc}
 
\begin{methoddesc}{remainder_near}{x, y}
  Computed the modulo as either a positive or negative value depending
  on which is closest to zero.  For instance,
  \samp{Decimal(10).remainder_near(6)} returns \code{Decimal("-2")}
  which is closer to zero than \code{Decimal("4")}.

  If both are equally close, the one chosen will have the same sign
  as \var{self}.
\end{methoddesc}

\begin{methoddesc}{same_quantum}{x, y}
  Test whether \var{x} and \var{y} have the same exponent or whether both are
  \constant{NaN}.
\end{methoddesc}

\begin{methoddesc}{sqrt}{}
  Return the square root to full precision.
\end{methoddesc}                    

\begin{methoddesc}{substract}{x, y}
  Return the difference between \var{x} and \var{y}.
\end{methoddesc}
 
\begin{methoddesc}{to_eng_string}{}
  Convert to engineering-type string.

  Engineering notation has an exponent which is a multiple of 3, so there
  are up to 3 digits left of the decimal place.  For example, converts
  \code{Decimal('123E+1')} to \code{Decimal("1.23E+3")}
\end{methoddesc}  

\begin{methoddesc}{to_integral}{x}                  
  Rounds to the nearest integer without signaling \constant{Inexact}
  or \constant{Rounded}.                                        
\end{methoddesc} 

\begin{methoddesc}{to_sci_string}{}
  Converts a number to a string using scientific notation.
\end{methoddesc} 



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%            
\subsection{Signals \label{decimal-signals}}

Signals represent conditions that arise during computation.
Each corresponds to one context flag and one context trap enabler.

The context flag is incremented whenever the condition is encountered.
After the computation, flags may be checked for informational
purposes (for instance, to determine whether a computation was exact).
After checking the flags, be sure to clear all flags before starting
the next computation.

If the context's trap enabler is set for the signal, then the condition
causes a Python exception to be raised.  For example, if the
\class{DivisionByZero} trap is set, the a \exception{DivisionByZero}
exception is raised upon encountering the condition.


\begin{classdesc*}{Clamped}
    Altered an exponent to fit representation constraints.

    Typically, clamping occurs when an exponent falls outside the context's
    \member{Emin} and \member{Emax} limits.  If possible, the exponent is
    reduced to fit by adding zeroes to the coefficient.
\end{classdesc*}

\begin{classdesc*}{ConversionSyntax}
    Trying to convert a malformed string such as:  \code{Decimal('jump')}.

    Decimal converts only strings conforming to the numeric string
    syntax.  If this signal is not trapped, returns \constant{NaN}.
\end{classdesc*}

\begin{classdesc*}{DecimalException}
    Base class for other signals.
\end{classdesc*}

\begin{classdesc*}{DivisionByZero}
    Signals the division of a non-infinite number by zero.

    Can occur with division, modulo division, or when raising a number to
    a negative power.  If this signal is not trapped, return
    \constant{Infinity} or \constant{-Infinity} with sign determined by
    the inputs to the calculation.
\end{classdesc*}

\begin{classdesc*}{DivisionImpossible}
    Error performing a division operation.  Caused when an intermediate result
    has more digits that the allowed by the current precision.  If not trapped,
    returns \constant{NaN}.
\end{classdesc*}


\begin{classdesc*}{DivisionUndefined}
    This is a subclass of \class{DivisionByZero}.

    It occurs only in the context of division operations.
\end{classdesc*}

\begin{classdesc*}{Inexact}
    Indicates that rounding occurred and the result is not exact.

    Signals whenever non-zero digits were discarded during rounding.
    The rounded result is returned.  The signal flag or trap is used
    to detect when results are inexact.
\end{classdesc*}


\begin{classdesc*}{InvalidContext}
    This is a subclass of \class{InvalidOperation}.

    Indicates an error within the Context object such as an unknown
    rounding operation.  If not trapped, returns \constant{NaN}.
\end{classdesc*}

\begin{classdesc*}{InvalidOperation}
    An invalid operation was performed.

    Indicates that an operation was requested that does not make sense.
    If not trapped, returns \constant{NaN}.  Possible causes include:

    \begin{verbatim}
        Infinity - Infinity
        0 * Infinity
        Infinity / Infinity
        x % 0
        Infinity % x
        x._rescale( non-integer )
        sqrt(-x) and x > 0
        0 ** 0
        x ** (non-integer)
        x ** Infinity      
    \end{verbatim}    
\end{classdesc*}

\begin{classdesc*}{Overflow}
    Numerical overflow.

    Indicates the exponent is larger than \member{Emax} after rounding has
    occurred.  If not trapped, the result depends on the rounding mode, either
    pulling inward to the largest representable finite number or rounding
    outward to \constant{Infinity}.  In either case, \class{Inexact} and
    \class{Rounded} are also signaled.   
\end{classdesc*}


\begin{classdesc*}{Rounded}
    Rounding occurred though possibly no information was lost.

    Signaled whenever rounding discards digits; even if those digits are
    zero (such as rounding \constant{5.00} to \constant{5.0}).   If not
    trapped, returns the result unchanged.  This signal is used to detect
    loss of significant digits.
\end{classdesc*}

\begin{classdesc*}{Subnormal}
    Exponent was lower than \member{Emin} prior to rounding.
          
    Occurs when an operation result is subnormal (the exponent is too small).
    If not trapped, returns the result unchanged.
\end{classdesc*}

\begin{classdesc*}{Underflow}
    Numerical underflow with result rounded to zero.

    Occurs when a subnormal result is pushed to zero by rounding.
    \class{Inexact} and \class{Subnormal} are also signaled.
\end{classdesc*}

The following table summarizes the hierarchy of signals:

\begin{verbatim}    
    exceptions.ArithmeticError(exceptions.StandardError)
        DecimalException
            Clamped
            DivisionByZero(DecimalException, exceptions.ZeroDivisionError)
            Inexact
                Overflow(Inexact, Rounded)
                Underflow(Inexact, Rounded, Subnormal)
            InvalidOperation
                ConversionSyntax
                DivisionImpossible
                DivisionUndefined(InvalidOperation, exceptions.ZeroDivisionError)
                InvalidContext
            Rounded
            Subnormal
\end{verbatim}            



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Working with threads \label{decimal-threads}}

The \function{getcontext()} function accesses a different \class{Context}
object for each thread.  Having separate thread contexts means that threads
may make changes (such as \code{getcontext.prec=10}) without interfering with
other threads and without needing mutexes.

Likewise, the \function{setcontext()} function automatically assigns its target
to the current thread.

If \function{setcontext()} has not been called before \function{getcontext()},
then \function{getcontext()} will automatically create a new context for use
in the current thread.

The new context is copied from a prototype context called \var{DefaultContext}.
To control the defaults so that each thread will use the same values
throughout the application, directly modify the \var{DefaultContext} object.
This should be done \emph{before} any threads are started so that there won't
be a race condition with threads calling \function{getcontext()}. For example:

\begin{verbatim}
# Set applicationwide defaults for all threads about to be launched
DefaultContext.prec=12
DefaultContext.rounding=ROUND_DOWN
DefaultContext.trap_enablers=dict.fromkeys(Signals, 0)
setcontext(DefaultContext)

# Now start all of the threads
t1.start()
t2.start()
t3.start()
 . . .
\end{verbatim}
 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Recipes \label{decimal-recipes}}

Here are some functions demonstrating ways to work with the
\class{Decimal} class:

\begin{verbatim}
from decimal import Decimal, getcontext
getcontext().prec = 28

def moneyfmt(value, places=2, curr='$', sep=',', dp='.', pos='', neg='-'):
    """Convert Decimal to a money formatted string.

    places:  required number of places after the decimal point
    curr:    optional currency symbol before the sign (may be blank)
    sep:     optional grouping separator (comma, period, or blank)
    dp:      decimal point indicator (comma or period)
             only set to blank if places is zero
    pos:     optional sign for positive numbers ("+" or blank)
    neg:     optional sign for negative numbers ("-" or blank)
             leave blank to separately add brackets or a trailing minus

    >>> d = Decimal('-1234567.8901')
    >>> moneyfmt(d)
    '-$1,234,567.89'
    >>> moneyfmt(d, places=0, curr='', sep='.', dp='')
    '-1.234.568'
    >>> '($%s)' % moneyfmt(d, curr='', neg='')
    '($1,234,567.89)'
    """
    q = Decimal((0, (1,), -places))    # 2 places --> '0.01'
    sign, digits, exp = value.quantize(q).as_tuple()
    result = []
    digits = map(str, digits)
    build, next = result.append, digits.pop    
    for i in range(places):
        build(next())
    build(dp)
    try:
        while 1:
            for i in range(3):
                build(next())
            if digits:
                build(sep)
    except IndexError:
        pass
    build(curr)
    if sign:
        build(neg)
    else:
        build(pos)
    result.reverse()
    return ''.join(result)

def pi():
    """Compute Pi to the current precision.

    >>> print pi()
    3.141592653589793238462643383
    """
    getcontext().prec += 2  # extra digits for intermediate steps
    three = Decimal(3)      # substitute "three=3.0" for regular floats
    lastc, t, c, n, na, d, da = 0, three, 3, 1, 0, 0, 24
    while c != lastc:
        lastc = c
        n, na = n+na, na+8
        d, da = d+da, da+32
        t = (t * n) / d
        c += t
    getcontext().prec -= 2
    return c + 0            # Adding zero causes rounding to the new precision

def exp(x):
    """Return e raised to the power of x.  Result type matches input type.

    >>> print exp(Decimal(1))
    2.718281828459045235360287471
    >>> print exp(Decimal(2))
    7.389056098930650227230427461
    >>> print exp(2.0)
    7.38905609893
    >>> print exp(2+0j)
    (7.38905609893+0j)
    """
    getcontext().prec += 2  # extra digits for intermediate steps
    i, laste, e, fact, num = 0, 0, 1, 1, 1
    while e != laste:
        laste = e    
        i += 1
        fact *= i
        num *= x     
        e += num / fact   
    getcontext().prec -= 2        
    return e + 0

def cos(x):
    """Return the cosine of x as measured in radians.

    >>> print cos(Decimal('0.5'))
    0.8775825618903727161162815826
    >>> print cos(0.5)
    0.87758256189
    >>> print cos(0.5+0j)
    (0.87758256189+0j)
    """
    getcontext().prec += 2  # extra digits for intermediate steps
    i, laste, e, fact, num, sign = 0, 0, 1, 1, 1, 1
    while e != laste:
        laste = e    
        i += 2
        fact *= i * (i-1)
        num *= x * x
        sign *= -1
        e += num / fact * sign 
    getcontext().prec -= 2        
    return e + 0

def sin(x):
    """Return the cosine of x as measured in radians.

    >>> print sin(Decimal('0.5'))
    0.4794255386042030002732879352
    >>> print sin(0.5)
    0.479425538604
    >>> print sin(0.5+0j)
    (0.479425538604+0j)
    """
    getcontext().prec += 2  # extra digits for intermediate steps
    i, laste, e, fact, num, sign = 1, 0, x, 1, x, 1
    while e != laste:
        laste = e    
        i += 2
        fact *= i * (i-1)
        num *= x * x
        sign *= -1
        e += num / fact * sign 
    getcontext().prec -= 2        
    return e + 0

\end{verbatim}