
``decimal`` --- Decimal fixed point and floating point arithmetic
*****************************************************************

New in version 2.4.

The ``decimal`` module provides support for decimal floating point
arithmetic.  It offers several advantages over the ``float`` datatype:

* Decimal "is based on a floating-point model which was designed with
  people in mind, and necessarily has a paramount guiding principle --
  computers must provide an arithmetic that works in the same way as
  the arithmetic that people learn at school." -- excerpt from the
  decimal arithmetic specification.

* Decimal numbers can be represented exactly.  In contrast, numbers
  like ``1.1`` do not have an exact representation in binary floating
  point. End users typically would not expect ``1.1`` to display as
  ``1.1000000000000001`` as it does with binary floating point.

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

* The decimal module incorporates a notion of significant places so
  that ``1.30 + 1.20`` is ``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, ``1.3 *
  1.2`` gives ``1.56`` while ``1.30 * 1.20`` gives ``1.5600``.

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

  >>> getcontext().prec = 6
  >>> Decimal(1) / Decimal(7)
  Decimal('0.142857')
  >>> getcontext().prec = 28
  >>> Decimal(1) / Decimal(7)
  Decimal('0.1428571428571428571428571429')

* 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. This includes an option
  to enforce exact arithmetic by using exceptions to block any inexact
  operations.

* The decimal module was designed to support "without prejudice, both
  exact unrounded decimal arithmetic (sometimes called fixed-point
  arithmetic) and rounded floating-point arithmetic."  -- excerpt from
  the decimal arithmetic specification.

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 zeros.  Decimals also include special values such as
``Infinity``, ``-Infinity``, and ``NaN``.  The standard also
differentiates ``-0`` from ``+0``.

The context for arithmetic is an environment specifying precision,
rounding rules, limits on exponents, flags indicating the results of
operations, and trap enablers which determine whether signals are
treated as exceptions.  Rounding options include ``ROUND_CEILING``,
``ROUND_DOWN``, ``ROUND_FLOOR``, ``ROUND_HALF_DOWN``,
``ROUND_HALF_EVEN``, ``ROUND_HALF_UP``, ``ROUND_UP``, and
``ROUND_05UP``.

Signals are groups of exceptional conditions arising during the course
of computation.  Depending on the needs of the application, signals
may be ignored, considered as informational, or treated as exceptions.
The signals in the decimal module are: ``Clamped``,
``InvalidOperation``, ``DivisionByZero``, ``Inexact``, ``Rounded``,
``Subnormal``, ``Overflow``, and ``Underflow``.

For each signal there is a flag and a trap enabler.  When a signal is
encountered, its flag is set to one, 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.

See also:

   * IBM's General Decimal Arithmetic Specification, The General
     Decimal Arithmetic Specification.

   * IEEE standard 854-1987, Unofficial IEEE 854 Text.


Quick-start Tutorial
====================

The usual start to using decimals is importing the module, viewing the
current context with ``getcontext()`` and, if necessary, setting new
values for precision, rounding, or enabled traps:

   >>> from decimal import *
   >>> getcontext()
   Context(prec=28, rounding=ROUND_HALF_EVEN, Emin=-999999999, Emax=999999999,
           capitals=1, flags=[], traps=[Overflow, DivisionByZero,
           InvalidOperation])

   >>> getcontext().prec = 7       # Set a new precision

Decimal instances can be constructed from integers, strings, or
tuples.  To create a Decimal from a ``float``, first convert it to a
string.  This serves as an explicit reminder of the details of the
conversion (including representation error).  Decimal numbers include
special values such as ``NaN`` which stands for "Not a number",
positive and negative ``Infinity``, and ``-0``.

>>> getcontext().prec = 28
>>> 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(2) ** Decimal('0.5')
Decimal('1.414213562373095048801688724')
>>> Decimal('NaN')
Decimal('NaN')
>>> Decimal('-Infinity')
Decimal('-Infinity')

The significance of a new Decimal is determined solely by the number
of digits input.  Context precision and rounding only come into play
during arithmetic operations.

   >>> getcontext().prec = 6
   >>> 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')

Decimals interact well with much of the rest of Python.  Here is a
small decimal floating point flying circus:

   >>> 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)     # round() first converts to binary floating point
   1.3
   >>> int(a)
   1
   >>> a * 5
   Decimal('6.70')
   >>> a * b
   Decimal('2.5058')
   >>> c % a
   Decimal('0.77')

And some mathematical functions are also available to Decimal:

>>> getcontext().prec = 28
>>> Decimal(2).sqrt()
Decimal('1.414213562373095048801688724')
>>> Decimal(1).exp()
Decimal('2.718281828459045235360287471')
>>> Decimal('10').ln()
Decimal('2.302585092994045684017991455')
>>> Decimal('10').log10()
Decimal('1')

The ``quantize()`` method rounds a number to a fixed exponent.  This
method is useful for monetary applications that often round results to
a fixed number of places:

>>> Decimal('7.325').quantize(Decimal('.01'), rounding=ROUND_DOWN)
Decimal('7.32')
>>> Decimal('7.325').quantize(Decimal('1.'), rounding=ROUND_UP)
Decimal('8')

As shown above, the ``getcontext()`` function accesses the current
context and allows the settings to be changed.  This approach meets
the needs of most applications.

For more advanced work, it may be useful to create alternate contexts
using the Context() constructor.  To make an alternate active, use the
``setcontext()`` function.

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

   >>> myothercontext = Context(prec=60, rounding=ROUND_HALF_DOWN)
   >>> setcontext(myothercontext)
   >>> Decimal(1) / Decimal(7)
   Decimal('0.142857142857142857142857142857142857142857142857142857142857')

   >>> ExtendedContext
   Context(prec=9, rounding=ROUND_HALF_EVEN, Emin=-999999999, Emax=999999999,
           capitals=1, flags=[], traps=[])
   >>> 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

Contexts also have signal flags for monitoring exceptional conditions
encountered during computations.  The flags remain set until
explicitly cleared, so it is best to clear the flags before each set
of monitored computations by using the ``clear_flags()`` method.

   >>> setcontext(ExtendedContext)
   >>> getcontext().clear_flags()
   >>> Decimal(355) / Decimal(113)
   Decimal('3.14159292')
   >>> getcontext()
   Context(prec=9, rounding=ROUND_HALF_EVEN, Emin=-999999999, Emax=999999999,
           capitals=1, flags=[Rounded, Inexact], traps=[])

The *flags* entry shows that the rational approximation to ``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 ``traps`` field
of a context:

   >>> setcontext(ExtendedContext)
   >>> Decimal(1) / Decimal(0)
   Decimal('Infinity')
   >>> getcontext().traps[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

Most programs adjust the current context only once, at the beginning
of the program.  And, in many applications, data is converted to
``Decimal`` with a single cast inside a loop.  With context set and
decimals created, the bulk of the program manipulates the data no
differently than with other Python numeric types.


Decimal objects
===============

class class decimal.Decimal([value[, context]])

   Construct a new ``Decimal`` object based from *value*.

   *value* can be an integer, string, tuple, or another ``Decimal``
   object. If no *value* is given, returns ``Decimal('0')``.  If
   *value* is a string, it should conform to the decimal numeric
   string syntax after leading and trailing whitespace characters are
   removed:

      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

   If *value* is a unicode string then other Unicode decimal digits
   are also permitted where ``digit`` appears above.  These include
   decimal digits from various other alphabets (for example, Arabic-
   Indic and Devanāgarī digits) along with the fullwidth digits
   ``u'\uff10'`` through ``u'\uff19'``.

   If *value* is a ``tuple``, it should have three components, a sign
   (``0`` for positive or ``1`` for negative), a ``tuple`` of digits,
   and an integer exponent. For example, ``Decimal((0, (1, 4, 1, 4),
   -3))`` returns ``Decimal('1.414')``.

   The *context* precision does not affect how many digits are stored.
   That is determined exclusively by the number of digits in *value*.
   For example, ``Decimal('3.00000')`` records all five zeros even if
   the context precision is only three.

   The purpose of the *context* argument is determining what to do if
   *value* is a malformed string.  If the context traps
   ``InvalidOperation``, an exception is raised; otherwise, the
   constructor returns a new Decimal with the value of ``NaN``.

   Once constructed, ``Decimal`` objects are immutable.

   Changed in version 2.6: leading and trailing whitespace characters
   are permitted when creating a Decimal instance from a string.

   Decimal floating point objects share many properties with the other
   built-in numeric types such as ``float`` and ``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 ``float`` or ``long``).

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

   adjusted()

      Return the adjusted exponent after shifting out the
      coefficient's rightmost digits until only the lead digit
      remains: ``Decimal('321e+5').adjusted()`` returns seven.  Used
      for determining the position of the most significant digit with
      respect to the decimal point.

   as_tuple()

      Return a *named tuple* representation of the number:
      ``DecimalTuple(sign, digits, exponent)``.

      Changed in version 2.6: Use a named tuple.

   canonical()

      Return the canonical encoding of the argument.  Currently, the
      encoding of a ``Decimal`` instance is always canonical, so this
      operation returns its argument unchanged.

      New in version 2.6.

   compare(other[, context])

      Compare the values of two Decimal instances.  This operation
      behaves in the same way as the usual comparison method
      ``__cmp__()``, except that ``compare()`` returns a Decimal
      instance rather than an integer, and if either operand is a NaN
      then the result is a NaN:

         a or b is a NaN ==> Decimal('NaN')
         a < b           ==> Decimal('-1')
         a == b          ==> Decimal('0')
         a > b           ==> Decimal('1')

   compare_signal(other[, context])

      This operation is identical to the ``compare()`` method, except
      that all NaNs signal.  That is, if neither operand is a
      signaling NaN then any quiet NaN operand is treated as though it
      were a signaling NaN.

      New in version 2.6.

   compare_total(other)

      Compare two operands using their abstract representation rather
      than their numerical value.  Similar to the ``compare()``
      method, but the result gives a total ordering on ``Decimal``
      instances.  Two ``Decimal`` instances with the same numeric
      value but different representations compare unequal in this
      ordering:

      >>> Decimal('12.0').compare_total(Decimal('12'))
      Decimal('-1')

      Quiet and signaling NaNs are also included in the total
      ordering.  The result of this function is ``Decimal('0')`` if
      both operands have the same representation, ``Decimal('-1')`` if
      the first operand is lower in the total order than the second,
      and ``Decimal('1')`` if the first operand is higher in the total
      order than the second operand.  See the specification for
      details of the total order.

      New in version 2.6.

   compare_total_mag(other)

      Compare two operands using their abstract representation rather
      than their value as in ``compare_total()``, but ignoring the
      sign of each operand. ``x.compare_total_mag(y)`` is equivalent
      to ``x.copy_abs().compare_total(y.copy_abs())``.

      New in version 2.6.

   conjugate()

      Just returns self, this method is only to comply with the
      Decimal Specification.

      New in version 2.6.

   copy_abs()

      Return the absolute value of the argument.  This operation is
      unaffected by the context and is quiet: no flags are changed and
      no rounding is performed.

      New in version 2.6.

   copy_negate()

      Return the negation of the argument.  This operation is
      unaffected by the context and is quiet: no flags are changed and
      no rounding is performed.

      New in version 2.6.

   copy_sign(other)

      Return a copy of the first operand with the sign set to be the
      same as the sign of the second operand.  For example:

      >>> Decimal('2.3').copy_sign(Decimal('-1.5'))
      Decimal('-2.3')

      This operation is unaffected by the context and is quiet: no
      flags are changed and no rounding is performed.

      New in version 2.6.

   exp([context])

      Return the value of the (natural) exponential function ``e**x``
      at the given number.  The result is correctly rounded using the
      ``ROUND_HALF_EVEN`` rounding mode.

      >>> Decimal(1).exp()
      Decimal('2.718281828459045235360287471')
      >>> Decimal(321).exp()
      Decimal('2.561702493119680037517373933E+139')

      New in version 2.6.

   fma(other, third[, context])

      Fused multiply-add.  Return self*other+third with no rounding of
      the intermediate product self*other.

      >>> Decimal(2).fma(3, 5)
      Decimal('11')

      New in version 2.6.

   is_canonical()

      Return ``True`` if the argument is canonical and ``False``
      otherwise.  Currently, a ``Decimal`` instance is always
      canonical, so this operation always returns ``True``.

      New in version 2.6.

   is_finite()

      Return ``True`` if the argument is a finite number, and
      ``False`` if the argument is an infinity or a NaN.

      New in version 2.6.

   is_infinite()

      Return ``True`` if the argument is either positive or negative
      infinity and ``False`` otherwise.

      New in version 2.6.

   is_nan()

      Return ``True`` if the argument is a (quiet or signaling) NaN
      and ``False`` otherwise.

      New in version 2.6.

   is_normal()

      Return ``True`` if the argument is a *normal* finite non-zero
      number with an adjusted exponent greater than or equal to
      *Emin*. Return ``False`` if the argument is zero, subnormal,
      infinite or a NaN.  Note, the term *normal* is used here in a
      different sense with the ``normalize()`` method which is used to
      create canonical values.

      New in version 2.6.

   is_qnan()

      Return ``True`` if the argument is a quiet NaN, and ``False``
      otherwise.

      New in version 2.6.

   is_signed()

      Return ``True`` if the argument has a negative sign and
      ``False`` otherwise.  Note that zeros and NaNs can both carry
      signs.

      New in version 2.6.

   is_snan()

      Return ``True`` if the argument is a signaling NaN and ``False``
      otherwise.

      New in version 2.6.

   is_subnormal()

      Return ``True`` if the argument is subnormal, and ``False``
      otherwise. A number is subnormal is if it is nonzero, finite,
      and has an adjusted exponent less than *Emin*.

      New in version 2.6.

   is_zero()

      Return ``True`` if the argument is a (positive or negative) zero
      and ``False`` otherwise.

      New in version 2.6.

   ln([context])

      Return the natural (base e) logarithm of the operand.  The
      result is correctly rounded using the ``ROUND_HALF_EVEN``
      rounding mode.

      New in version 2.6.

   log10([context])

      Return the base ten logarithm of the operand.  The result is
      correctly rounded using the ``ROUND_HALF_EVEN`` rounding mode.

      New in version 2.6.

   logb([context])

      For a nonzero number, return the adjusted exponent of its
      operand as a ``Decimal`` instance.  If the operand is a zero
      then ``Decimal('-Infinity')`` is returned and the
      ``DivisionByZero`` flag is raised.  If the operand is an
      infinity then ``Decimal('Infinity')`` is returned.

      New in version 2.6.

   logical_and(other[, context])

      ``logical_and()`` is a logical operation which takes two
      *logical operands* (see *Logical operands*).  The result is the
      digit-wise ``and`` of the two operands.

      New in version 2.6.

   logical_invert([context])

      ``logical_invert()`` is a logical operation.  The result is the
      digit-wise inversion of the operand.

      New in version 2.6.

   logical_or(other[, context])

      ``logical_or()`` is a logical operation which takes two *logical
      operands* (see *Logical operands*).  The result is the digit-
      wise ``or`` of the two operands.

      New in version 2.6.

   logical_xor(other[, context])

      ``logical_xor()`` is a logical operation which takes two
      *logical operands* (see *Logical operands*).  The result is the
      digit-wise exclusive or of the two operands.

      New in version 2.6.

   max(other[, context])

      Like ``max(self, other)`` except that the context rounding rule
      is applied before returning and that ``NaN`` values are either
      signaled or ignored (depending on the context and whether they
      are signaling or quiet).

   max_mag(other[, context])

      Similar to the ``max()`` method, but the comparison is done
      using the absolute values of the operands.

      New in version 2.6.

   min(other[, context])

      Like ``min(self, other)`` except that the context rounding rule
      is applied before returning and that ``NaN`` values are either
      signaled or ignored (depending on the context and whether they
      are signaling or quiet).

   min_mag(other[, context])

      Similar to the ``min()`` method, but the comparison is done
      using the absolute values of the operands.

      New in version 2.6.

   next_minus([context])

      Return the largest number representable in the given context (or
      in the current thread's context if no context is given) that is
      smaller than the given operand.

      New in version 2.6.

   next_plus([context])

      Return the smallest number representable in the given context
      (or in the current thread's context if no context is given) that
      is larger than the given operand.

      New in version 2.6.

   next_toward(other[, context])

      If the two operands are unequal, return the number closest to
      the first operand in the direction of the second operand.  If
      both operands are numerically equal, return a copy of the first
      operand with the sign set to be the same as the sign of the
      second operand.

      New in version 2.6.

   normalize([context])

      Normalize the number by stripping the rightmost trailing zeros
      and converting any result equal to ``Decimal('0')`` to
      ``Decimal('0e0')``. Used for producing canonical values for
      members of an equivalence class. For example,
      ``Decimal('32.100')`` and ``Decimal('0.321000e+2')`` both
      normalize to the equivalent value ``Decimal('32.1')``.

   number_class([context])

      Return a string describing the *class* of the operand.  The
      returned value is one of the following ten strings.

      * ``"-Infinity"``, indicating that the operand is negative
        infinity.

      * ``"-Normal"``, indicating that the operand is a negative
        normal number.

      * ``"-Subnormal"``, indicating that the operand is negative and
        subnormal.

      * ``"-Zero"``, indicating that the operand is a negative zero.

      * ``"+Zero"``, indicating that the operand is a positive zero.

      * ``"+Subnormal"``, indicating that the operand is positive and
        subnormal.

      * ``"+Normal"``, indicating that the operand is a positive
        normal number.

      * ``"+Infinity"``, indicating that the operand is positive
        infinity.

      * ``"NaN"``, indicating that the operand is a quiet NaN (Not a
        Number).

      * ``"sNaN"``, indicating that the operand is a signaling NaN.

      New in version 2.6.

   quantize(exp[, rounding[, context[, watchexp]]])

      Return a value equal to the first operand after rounding and
      having the exponent of the second operand.

      >>> Decimal('1.41421356').quantize(Decimal('1.000'))
      Decimal('1.414')

      Unlike other operations, if the length of the coefficient after
      the quantize operation would be greater than precision, then an
      ``InvalidOperation`` is signaled. This guarantees that, unless
      there is an error condition, the quantized exponent 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.

      If the exponent of the second operand is larger than that of the
      first then rounding may be necessary.  In this case, the
      rounding mode is determined by the ``rounding`` argument if
      given, else by the given ``context`` argument; if neither
      argument is given the rounding mode of the current thread's
      context is used.

      If *watchexp* is set (default), then an error is returned
      whenever the resulting exponent is greater than ``Emax`` or less
      than ``Etiny``.

   radix()

      Return ``Decimal(10)``, the radix (base) in which the
      ``Decimal`` class does all its arithmetic.  Included for
      compatibility with the specification.

      New in version 2.6.

   remainder_near(other[, context])

      Compute the modulo as either a positive or negative value
      depending on which is closest to zero.  For instance,
      ``Decimal(10).remainder_near(6)`` returns ``Decimal('-2')``
      which is closer to zero than ``Decimal('4')``.

      If both are equally close, the one chosen will have the same
      sign as *self*.

   rotate(other[, context])

      Return the result of rotating the digits of the first operand by
      an amount specified by the second operand.  The second operand
      must be an integer in the range -precision through precision.
      The absolute value of the second operand gives the number of
      places to rotate.  If the second operand is positive then
      rotation is to the left; otherwise rotation is to the right. The
      coefficient of the first operand is padded on the left with
      zeros to length precision if necessary.  The sign and exponent
      of the first operand are unchanged.

      New in version 2.6.

   same_quantum(other[, context])

      Test whether self and other have the same exponent or whether
      both are ``NaN``.

   scaleb(other[, context])

      Return the first operand with exponent adjusted by the second.
      Equivalently, return the first operand multiplied by
      ``10**other``.  The second operand must be an integer.

      New in version 2.6.

   shift(other[, context])

      Return the result of shifting the digits of the first operand by
      an amount specified by the second operand.  The second operand
      must be an integer in the range -precision through precision.
      The absolute value of the second operand gives the number of
      places to shift.  If the second operand is positive then the
      shift is to the left; otherwise the shift is to the right.
      Digits shifted into the coefficient are zeros.  The sign and
      exponent of the first operand are unchanged.

      New in version 2.6.

   sqrt([context])

      Return the square root of the argument to full precision.

   to_eng_string([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 ``Decimal('123E+1')`` to
      ``Decimal('1.23E+3')``

   to_integral([rounding[, context]])

      Identical to the ``to_integral_value()`` method.  The
      ``to_integral`` name has been kept for compatibility with older
      versions.

   to_integral_exact([rounding[, context]])

      Round to the nearest integer, signaling ``Inexact`` or
      ``Rounded`` as appropriate if rounding occurs.  The rounding
      mode is determined by the ``rounding`` parameter if given, else
      by the given ``context``.  If neither parameter is given then
      the rounding mode of the current context is used.

      New in version 2.6.

   to_integral_value([rounding[, context]])

      Round to the nearest integer without signaling ``Inexact`` or
      ``Rounded``.  If given, applies *rounding*; otherwise, uses the
      rounding method in either the supplied *context* or the current
      context.

      Changed in version 2.6: renamed from ``to_integral`` to
      ``to_integral_value``.  The old name remains valid for
      compatibility.


Logical operands
----------------

The ``logical_and()``, ``logical_invert()``, ``logical_or()``, and
``logical_xor()`` methods expect their arguments to be *logical
operands*.  A *logical operand* is a ``Decimal`` instance whose
exponent and sign are both zero, and whose digits are all either ``0``
or ``1``.


Context objects
===============

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

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

decimal.getcontext()

   Return the current context for the active thread.

decimal.setcontext(c)

   Set the current context for the active thread to *c*.

Beginning with Python 2.5, you can also use the ``with`` statement and
the ``localcontext()`` function to temporarily change the active
context.

decimal.localcontext([c])

   Return a context manager that will set the current context for the
   active thread to a copy of *c* on entry to the with-statement and
   restore the previous context when exiting the with-statement. If no
   context is specified, a copy of the current context is used.

   New in version 2.5.

   For example, the following code sets the current decimal precision
   to 42 places, performs a calculation, and then automatically
   restores the previous context:

      from decimal import localcontext

      with localcontext() as ctx:
          ctx.prec = 42   # Perform a high precision calculation
          s = calculate_something()
      s = +s  # Round the final result back to the default precision

New contexts can also be created using the ``Context`` constructor
described below. In addition, the module provides three pre-made
contexts:

class class decimal.BasicContext

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

   Because many of the traps are enabled, this context is useful for
   debugging.

class class decimal.ExtendedContext

   This is a standard context defined by the General Decimal
   Arithmetic Specification.  Precision is set to nine.  Rounding is
   set to ``ROUND_HALF_EVEN``.  All flags are cleared.  No traps are
   enabled (so that exceptions are not raised during computations).

   Because the traps are disabled, this context is useful for
   applications that prefer to have result value of ``NaN`` or
   ``Infinity`` instead of raising exceptions.  This allows an
   application to complete a run in the presence of conditions that
   would otherwise halt the program.

class class decimal.DefaultContext

   This context is used by the ``Context`` constructor as a prototype
   for new contexts.  Changing a field (such a precision) has the
   effect of changing the default for new contexts created by the
   ``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 as
   described below.

   The default values are precision=28, rounding=ROUND_HALF_EVEN, and
   enabled traps for Overflow, InvalidOperation, and DivisionByZero.

In addition to the three supplied contexts, new contexts can be
created with the ``Context`` constructor.

class class decimal.Context(prec=None, rounding=None, traps=None, flags=None, Emin=None, Emax=None, capitals=1)

   Creates a new context.  If a field is not specified or is ``None``,
   the default values are copied from the ``DefaultContext``.  If the
   *flags* field is not specified or is ``None``, all flags are
   cleared.

   The *prec* field is a positive integer that sets the precision for
   arithmetic operations in the context.

   The *rounding* option is one of:

   * ``ROUND_CEILING`` (towards ``Infinity``),

   * ``ROUND_DOWN`` (towards zero),

   * ``ROUND_FLOOR`` (towards ``-Infinity``),

   * ``ROUND_HALF_DOWN`` (to nearest with ties going towards zero),

   * ``ROUND_HALF_EVEN`` (to nearest with ties going to nearest even
     integer),

   * ``ROUND_HALF_UP`` (to nearest with ties going away from zero), or

   * ``ROUND_UP`` (away from zero).

   * ``ROUND_05UP`` (away from zero if last digit after rounding
     towards zero would have been 0 or 5; otherwise towards zero)

   The *traps* and *flags* fields list any signals to be set.
   Generally, new contexts should only set traps and leave the flags
   clear.

   The *Emin* and *Emax* fields are integers specifying the outer
   limits allowable for exponents.

   The *capitals* field is either ``0`` or ``1`` (the default). If set
   to ``1``, exponents are printed with a capital ``E``; otherwise, a
   lowercase ``e`` is used: ``Decimal('6.02e+23')``.

   Changed in version 2.6: The ``ROUND_05UP`` rounding mode was added.

   The ``Context`` class defines several general purpose methods as
   well as a large number of methods for doing arithmetic directly in
   a given context. In addition, for each of the ``Decimal`` methods
   described above (with the exception of the ``adjusted()`` and
   ``as_tuple()`` methods) there is a corresponding ``Context``
   method.  For example, ``C.exp(x)`` is equivalent to
   ``x.exp(context=C)``.

   clear_flags()

      Resets all of the flags to ``0``.

   copy()

      Return a duplicate of the context.

   copy_decimal(num)

      Return a copy of the Decimal instance num.

   create_decimal(num)

      Creates a new Decimal instance from *num* but using *self* as
      context. Unlike the ``Decimal`` constructor, the 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.  Another benefit is
      that rounding immediately eliminates unintended effects from
      digits beyond the current precision. In the following example,
      using unrounded inputs means that adding zero to a sum can
      change the result:

         >>> getcontext().prec = 3
         >>> Decimal('3.4445') + Decimal('1.0023')
         Decimal('4.45')
         >>> Decimal('3.4445') + Decimal(0) + Decimal('1.0023')
         Decimal('4.44')

      This method implements the to-number operation of the IBM
      specification. If the argument is a string, no leading or
      trailing whitespace is permitted.

   Etiny()

      Returns a value equal to ``Emin - prec + 1`` which is the
      minimum exponent value for subnormal results.  When underflow
      occurs, the exponent is set to ``Etiny``.

   Etop()

      Returns a value equal to ``Emax - prec + 1``.

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

   abs(x)

      Returns the absolute value of *x*.

   add(x, y)

      Return the sum of *x* and *y*.

   canonical(x)

      Returns the same Decimal object *x*.

   compare(x, y)

      Compares *x* and *y* numerically.

   compare_signal(x, y)

      Compares the values of the two operands numerically.

   compare_total(x, y)

      Compares two operands using their abstract representation.

   compare_total_mag(x, y)

      Compares two operands using their abstract representation,
      ignoring sign.

   copy_abs(x)

      Returns a copy of *x* with the sign set to 0.

   copy_negate(x)

      Returns a copy of *x* with the sign inverted.

   copy_sign(x, y)

      Copies the sign from *y* to *x*.

   divide(x, y)

      Return *x* divided by *y*.

   divide_int(x, y)

      Return *x* divided by *y*, truncated to an integer.

   divmod(x, y)

      Divides two numbers and returns the integer part of the result.

   exp(x)

      Returns *e ** x*.

   fma(x, y, z)

      Returns *x* multiplied by *y*, plus *z*.

   is_canonical(x)

      Returns True if *x* is canonical; otherwise returns False.

   is_finite(x)

      Returns True if *x* is finite; otherwise returns False.

   is_infinite(x)

      Returns True if *x* is infinite; otherwise returns False.

   is_nan(x)

      Returns True if *x* is a qNaN or sNaN; otherwise returns False.

   is_normal(x)

      Returns True if *x* is a normal number; otherwise returns False.

   is_qnan(x)

      Returns True if *x* is a quiet NaN; otherwise returns False.

   is_signed(x)

      Returns True if *x* is negative; otherwise returns False.

   is_snan(x)

      Returns True if *x* is a signaling NaN; otherwise returns False.

   is_subnormal(x)

      Returns True if *x* is subnormal; otherwise returns False.

   is_zero(x)

      Returns True if *x* is a zero; otherwise returns False.

   ln(x)

      Returns the natural (base e) logarithm of *x*.

   log10(x)

      Returns the base 10 logarithm of *x*.

   logb(x)

      Returns the exponent of the magnitude of the operand's MSD.

   logical_and(x, y)

      Applies the logical operation *and* between each operand's
      digits.

   logical_invert(x)

      Invert all the digits in *x*.

   logical_or(x, y)

      Applies the logical operation *or* between each operand's
      digits.

   logical_xor(x, y)

      Applies the logical operation *xor* between each operand's
      digits.

   max(x, y)

      Compares two values numerically and returns the maximum.

   max_mag(x, y)

      Compares the values numerically with their sign ignored.

   min(x, y)

      Compares two values numerically and returns the minimum.

   min_mag(x, y)

      Compares the values numerically with their sign ignored.

   minus(x)

      Minus corresponds to the unary prefix minus operator in Python.

   multiply(x, y)

      Return the product of *x* and *y*.

   next_minus(x)

      Returns the largest representable number smaller than *x*.

   next_plus(x)

      Returns the smallest representable number larger than *x*.

   next_toward(x, y)

      Returns the number closest to *x*, in direction towards *y*.

   normalize(x)

      Reduces *x* to its simplest form.

   number_class(x)

      Returns an indication of the class of *x*.

   plus(x)

      Plus corresponds to the unary prefix plus operator in Python.
      This operation applies the context precision and rounding, so it
      is *not* an identity operation.

   power(x, y[, modulo])

      Return ``x`` to the power of ``y``, reduced modulo ``modulo`` if
      given.

      With two arguments, compute ``x**y``.  If ``x`` is negative then
      ``y`` must be integral.  The result will be inexact unless ``y``
      is integral and the result is finite and can be expressed
      exactly in 'precision' digits. The result should always be
      correctly rounded, using the rounding mode of the current
      thread's context.

      With three arguments, compute ``(x**y) % modulo``.  For the
      three argument form, the following restrictions on the arguments
      hold:

         * all three arguments must be integral

         * ``y`` must be nonnegative

         * at least one of ``x`` or ``y`` must be nonzero

         * ``modulo`` must be nonzero and have at most 'precision'
           digits

      The value resulting from ``Context.power(x, y, modulo)`` is
      equal to the value that would be obtained by computing ``(x**y)
      % modulo`` with unbounded precision, but is computed more
      efficiently.  The exponent of the result is zero, regardless of
      the exponents of ``x``, ``y`` and ``modulo``.  The result is
      always exact.

      Changed in version 2.6: ``y`` may now be nonintegral in
      ``x**y``. Stricter requirements for the three-argument version.

   quantize(x, y)

      Returns a value equal to *x* (rounded), having the exponent of
      *y*.

   radix()

      Just returns 10, as this is Decimal, :)

   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.

   remainder_near(x, y)

      Returns ``x - y * n``, where *n* is the integer nearest the
      exact value of ``x / y`` (if the result is 0 then its sign will
      be the sign of *x*).

   rotate(x, y)

      Returns a rotated copy of *x*, *y* times.

   same_quantum(x, y)

      Returns True if the two operands have the same exponent.

   scaleb(x, y)

      Returns the first operand after adding the second value its exp.

   shift(x, y)

      Returns a shifted copy of *x*, *y* times.

   sqrt(x)

      Square root of a non-negative number to context precision.

   subtract(x, y)

      Return the difference between *x* and *y*.

   to_eng_string(x)

      Converts a number to a string, using scientific notation.

   to_integral_exact(x)

      Rounds to an integer.

   to_sci_string(x)

      Converts a number to a string using scientific notation.


Signals
=======

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

The context flag is set 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
``DivisionByZero`` trap is set, then a ``DivisionByZero`` exception is
raised upon encountering the condition.

class class decimal.Clamped

   Altered an exponent to fit representation constraints.

   Typically, clamping occurs when an exponent falls outside the
   context's ``Emin`` and ``Emax`` limits.  If possible, the exponent
   is reduced to fit by adding zeros to the coefficient.

class class decimal.DecimalException

   Base class for other signals and a subclass of ``ArithmeticError``.

class class decimal.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, returns
   ``Infinity`` or ``-Infinity`` with the sign determined by the
   inputs to the calculation.

class class decimal.Inexact

   Indicates that rounding occurred and the result is not exact.

   Signals when 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.

class class decimal.InvalidOperation

   An invalid operation was performed.

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

      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

class class decimal.Overflow

   Numerical overflow.

   Indicates the exponent is larger than ``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 ``Infinity``.  In either case, ``Inexact`` and
   ``Rounded`` are also signaled.

class class decimal.Rounded

   Rounding occurred though possibly no information was lost.

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

class class decimal.Subnormal

   Exponent was lower than ``Emin`` prior to rounding.

   Occurs when an operation result is subnormal (the exponent is too
   small). If not trapped, returns the result unchanged.

class class decimal.Underflow

   Numerical underflow with result rounded to zero.

   Occurs when a subnormal result is pushed to zero by rounding.
   ``Inexact`` and ``Subnormal`` are also signaled.

The following table summarizes the hierarchy of signals:

   exceptions.ArithmeticError(exceptions.StandardError)
       DecimalException
           Clamped
           DivisionByZero(DecimalException, exceptions.ZeroDivisionError)
           Inexact
               Overflow(Inexact, Rounded)
               Underflow(Inexact, Rounded, Subnormal)
           InvalidOperation
           Rounded
           Subnormal


Floating Point Notes
====================


Mitigating round-off error with increased precision
---------------------------------------------------

The use of decimal floating point eliminates decimal representation
error (making it possible to represent ``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
breakdown of the associative and distributive properties of addition:

   # Examples from Seminumerical Algorithms, Section 4.2.2.
   >>> from decimal import Decimal, getcontext
   >>> getcontext().prec = 8

   >>> u, v, w = Decimal(11111113), Decimal(-11111111), Decimal('7.51111111')
   >>> (u + v) + w
   Decimal('9.5111111')
   >>> u + (v + w)
   Decimal('10')

   >>> u, v, w = Decimal(20000), Decimal(-6), Decimal('6.0000003')
   >>> (u*v) + (u*w)
   Decimal('0.01')
   >>> u * (v+w)
   Decimal('0.0060000')

The ``decimal`` module makes it possible to restore the identities by
expanding the precision sufficiently to avoid loss of significance:

   >>> getcontext().prec = 20
   >>> u, v, w = Decimal(11111113), Decimal(-11111111), Decimal('7.51111111')
   >>> (u + v) + w
   Decimal('9.51111111')
   >>> u + (v + w)
   Decimal('9.51111111')
   >>>
   >>> u, v, w = Decimal(20000), Decimal(-6), Decimal('6.0000003')
   >>> (u*v) + (u*w)
   Decimal('0.0060000')
   >>> u * (v+w)
   Decimal('0.0060000')


Special values
--------------

The number system for the ``decimal`` module provides special values
including ``NaN``, ``sNaN``, ``-Infinity``, ``Infinity``, and two
zeros, ``+0`` and ``-0``.

Infinities can be constructed directly with:  ``Decimal('Infinity')``.
Also, they can arise from dividing by zero when the ``DivisionByZero``
signal is not trapped.  Likewise, when the ``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 ``NaN``, or if the
``InvalidOperation`` signal is trapped, raise an exception.  For
example, ``0/0`` returns ``NaN`` which means "not a number".  This
variety of ``NaN`` is quiet and, once created, will flow through other
computations always resulting in another ``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 ``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 behavior of Python's comparison operators can be a little
surprising where a ``NaN`` is involved.  A test for equality where one
of the operands is a quiet or signaling ``NaN`` always returns
``False`` (even when doing ``Decimal('NaN')==Decimal('NaN')``), while
a test for inequality always returns ``True``.  An attempt to compare
two Decimals using any of the ``<``, ``<=``, ``>`` or ``>=`` operators
will raise the ``InvalidOperation`` signal if either operand is a
``NaN``, and return ``False`` if this signal is not trapped.  Note
that the General Decimal Arithmetic specification does not specify the
behavior of direct comparisons; these rules for comparisons involving
a ``NaN`` were taken from the IEEE 854 standard (see Table 3 in
section 5.7).  To ensure strict standards-compliance, use the
``compare()`` and ``compare-signal()`` methods instead.

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, both
positive and negative zeros 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:

>>> 1 / Decimal('Infinity')
Decimal('0E-1000000026')


Working with threads
====================

The ``getcontext()`` function accesses a different ``Context`` object
for each thread.  Having separate thread contexts means that threads
may make changes (such as ``getcontext.prec=10``) without interfering
with other threads.

Likewise, the ``setcontext()`` function automatically assigns its
target to the current thread.

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

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

   # Set applicationwide defaults for all threads about to be launched
   DefaultContext.prec = 12
   DefaultContext.rounding = ROUND_DOWN
   DefaultContext.traps = ExtendedContext.traps.copy()
   DefaultContext.traps[InvalidOperation] = 1
   setcontext(DefaultContext)

   # Afterwards, the threads can be started
   t1.start()
   t2.start()
   t3.start()
    . . .


Recipes
=======

Here are a few recipes that serve as utility functions and that
demonstrate ways to work with the ``Decimal`` class:

   def moneyfmt(value, places=2, curr='', sep=',', dp='.',
                pos='', neg='-', trailneg=''):
       """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, space, or blank)
       dp:      decimal point indicator (comma or period)
                only specify as blank when places is zero
       pos:     optional sign for positive numbers: '+', space or blank
       neg:     optional sign for negative numbers: '-', '(', space or blank
       trailneg:optional trailing minus indicator:  '-', ')', space or blank

       >>> d = Decimal('-1234567.8901')
       >>> moneyfmt(d, curr='$')
       '-$1,234,567.89'
       >>> moneyfmt(d, places=0, sep='.', dp='', neg='', trailneg='-')
       '1.234.568-'
       >>> moneyfmt(d, curr='$', neg='(', trailneg=')')
       '($1,234,567.89)'
       >>> moneyfmt(Decimal(123456789), sep=' ')
       '123 456 789.00'
       >>> moneyfmt(Decimal('-0.02'), neg='<', trailneg='>')
       '<0.02>'

       """
       q = Decimal(10) ** -places      # 2 places --> '0.01'
       sign, digits, exp = value.quantize(q).as_tuple()
       result = []
       digits = map(str, digits)
       build, next = result.append, digits.pop
       if sign:
           build(trailneg)
       for i in range(places):
           build(next() if digits else '0')
       build(dp)
       if not digits:
           build('0')
       i = 0
       while digits:
           build(next())
           i += 1
           if i == 3 and digits:
               i = 0
               build(sep)
       build(curr)
       build(neg if sign else pos)
       return ''.join(reversed(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
       lasts, t, s, n, na, d, da = 0, three, 3, 1, 0, 0, 24
       while s != lasts:
           lasts = s
           n, na = n+na, na+8
           d, da = d+da, da+32
           t = (t * n) / d
           s += t
       getcontext().prec -= 2
       return +s               # unary plus applies 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
       i, lasts, s, fact, num = 0, 0, 1, 1, 1
       while s != lasts:
           lasts = s
           i += 1
           fact *= i
           num *= x
           s += num / fact
       getcontext().prec -= 2
       return +s

   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
       i, lasts, s, fact, num, sign = 0, 0, 1, 1, 1, 1
       while s != lasts:
           lasts = s
           i += 2
           fact *= i * (i-1)
           num *= x * x
           sign *= -1
           s += num / fact * sign
       getcontext().prec -= 2
       return +s

   def sin(x):
       """Return the sine 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
       i, lasts, s, fact, num, sign = 1, 0, x, 1, x, 1
       while s != lasts:
           lasts = s
           i += 2
           fact *= i * (i-1)
           num *= x * x
           sign *= -1
           s += num / fact * sign
       getcontext().prec -= 2
       return +s


Decimal FAQ
===========

Q. It is cumbersome to type ``decimal.Decimal('1234.5')``.  Is there a
way to minimize typing when using the interactive interpreter?

A. Some users abbreviate the constructor to just a single letter:

>>> D = decimal.Decimal
>>> D('1.23') + D('3.45')
Decimal('4.68')

Q. In a fixed-point application with two decimal places, some inputs
have many places and need to be rounded.  Others are not supposed to
have excess digits and need to be validated.  What methods should be
used?

A. The ``quantize()`` method rounds to a fixed number of decimal
places. If the ``Inexact`` trap is set, it is also useful for
validation:

   >>> TWOPLACES = Decimal(10) ** -2       # same as Decimal('0.01')

   >>> # Round to two places
   >>> Decimal('3.214').quantize(TWOPLACES)
   Decimal('3.21')

   >>> # Validate that a number does not exceed two places
   >>> Decimal('3.21').quantize(TWOPLACES, context=Context(traps=[Inexact]))
   Decimal('3.21')

   >>> Decimal('3.214').quantize(TWOPLACES, context=Context(traps=[Inexact]))
   Traceback (most recent call last):
      ...
   Inexact: None

Q. Once I have valid two place inputs, how do I maintain that
invariant throughout an application?

A. Some operations like addition, subtraction, and multiplication by
an integer will automatically preserve fixed point.  Others
operations, like division and non-integer multiplication, will change
the number of decimal places and need to be followed-up with a
``quantize()`` step:

>>> a = Decimal('102.72')           # Initial fixed-point values
>>> b = Decimal('3.17')
>>> a + b                           # Addition preserves fixed-point
Decimal('105.89')
>>> a - b
Decimal('99.55')
>>> a * 42                          # So does integer multiplication
Decimal('4314.24')
>>> (a * b).quantize(TWOPLACES)     # Must quantize non-integer multiplication
Decimal('325.62')
>>> (b / a).quantize(TWOPLACES)     # And quantize division
Decimal('0.03')

In developing fixed-point applications, it is convenient to define
functions to handle the ``quantize()`` step:

   >>> def mul(x, y, fp=TWOPLACES):
   ...     return (x * y).quantize(fp)
   >>> def div(x, y, fp=TWOPLACES):
   ...     return (x / y).quantize(fp)

   >>> mul(a, b)                       # Automatically preserve fixed-point
   Decimal('325.62')
   >>> div(b, a)
   Decimal('0.03')

Q. There are many ways to express the same value.  The numbers
``200``, ``200.000``, ``2E2``, and ``02E+4`` all have the same value
at various precisions. Is there a way to transform them to a single
recognizable canonical value?

A. The ``normalize()`` method maps all equivalent values to a single
representative:

>>> values = map(Decimal, '200 200.000 2E2 .02E+4'.split())
>>> [v.normalize() for v in values]
[Decimal('2E+2'), Decimal('2E+2'), Decimal('2E+2'), Decimal('2E+2')]

Q. Some decimal values always print with exponential notation.  Is
there a way to get a non-exponential representation?

A. For some values, exponential notation is the only way to express
the number of significant places in the coefficient.  For example,
expressing ``5.0E+3`` as ``5000`` keeps the value constant but cannot
show the original's two-place significance.

If an application does not care about tracking significance, it is
easy to remove the exponent and trailing zeroes, losing significance,
but keeping the value unchanged:

   >>> def remove_exponent(d):
   ...     return d.quantize(Decimal(1)) if d == d.to_integral() else d.normalize()

   >>> remove_exponent(Decimal('5E+3'))
   Decimal('5000')

Q. Is there a way to convert a regular float to a ``Decimal``?

A. Yes, all binary floating point numbers can be exactly expressed as
a Decimal.  An exact conversion may take more precision than intuition
would suggest, so we trap ``Inexact`` to signal a need for more
precision:

   def float_to_decimal(f):
       "Convert a floating point number to a Decimal with no loss of information"
       n, d = f.as_integer_ratio()
       numerator, denominator = Decimal(n), Decimal(d)
       ctx = Context(prec=60)
       result = ctx.divide(numerator, denominator)
       while ctx.flags[Inexact]:
           ctx.flags[Inexact] = False
           ctx.prec *= 2
           result = ctx.divide(numerator, denominator)
       return result

   >>> float_to_decimal(math.pi)
   Decimal('3.141592653589793115997963468544185161590576171875')

Q. Why isn't the ``float_to_decimal()`` routine included in the
module?

A. There is some question about whether it is advisable to mix binary
and decimal floating point.  Also, its use requires some care to avoid
the representation issues associated with binary floating point:

>>> float_to_decimal(1.1)
Decimal('1.100000000000000088817841970012523233890533447265625')

Q. Within a complex calculation, how can I make sure that I haven't
gotten a spurious result because of insufficient precision or rounding
anomalies.

A. The decimal module makes it easy to test results.  A best practice
is to re-run calculations using greater precision and with various
rounding modes. Widely differing results indicate insufficient
precision, rounding mode issues, ill-conditioned inputs, or a
numerically unstable algorithm.

Q. I noticed that context precision is applied to the results of
operations but not to the inputs.  Is there anything to watch out for
when mixing values of different precisions?

A. Yes.  The principle is that all values are considered to be exact
and so is the arithmetic on those values.  Only the results are
rounded.  The advantage for inputs is that "what you type is what you
get".  A disadvantage is that the results can look odd if you forget
that the inputs haven't been rounded:

   >>> getcontext().prec = 3
   >>> Decimal('3.104') + Decimal('2.104')
   Decimal('5.21')
   >>> Decimal('3.104') + Decimal('0.000') + Decimal('2.104')
   Decimal('5.20')

The solution is either to increase precision or to force rounding of
inputs using the unary plus operation:

   >>> getcontext().prec = 3
   >>> +Decimal('1.23456789')      # unary plus triggers rounding
   Decimal('1.23')

Alternatively, inputs can be rounded upon creation using the
``Context.create_decimal()`` method:

>>> Context(prec=5, rounding=ROUND_DOWN).create_decimal('1.2345678')
Decimal('1.2345')
