2
:mod:`decimal` --- Decimal fixed point and floating point arithmetic
3
====================================================================
6
:synopsis: Implementation of the General Decimal Arithmetic Specification.
9
.. moduleauthor:: Eric Price <eprice at tjhsst.edu>
10
.. moduleauthor:: Facundo Batista <facundo at taniquetil.com.ar>
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.. moduleauthor:: Raymond Hettinger <python at rcn.com>
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.. moduleauthor:: Aahz <aahz at pobox.com>
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.. moduleauthor:: Tim Peters <tim.one at comcast.net>
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.. sectionauthor:: Raymond D. Hettinger <python at rcn.com>
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.. import modules for testing inline doctests with the Sphinx doctest builder
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# make sure each group gets a fresh context
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The :mod:`decimal` module provides support for decimal floating point
30
arithmetic. It offers several advantages over the :class:`float` datatype:
32
* Decimal "is based on a floating-point model which was designed with people
33
in mind, and necessarily has a paramount guiding principle -- computers must
34
provide an arithmetic that works in the same way as the arithmetic that
35
people learn at school." -- excerpt from the decimal arithmetic specification.
37
* Decimal numbers can be represented exactly. In contrast, numbers like
38
:const:`1.1` and :const:`2.2` do not have exact representations in binary
39
floating point. End users typically would not expect ``1.1 + 2.2`` to display
40
as :const:`3.3000000000000003` as it does with binary floating point.
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* The exactness carries over into arithmetic. In decimal floating point, ``0.1
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+ 0.1 + 0.1 - 0.3`` is exactly equal to zero. In binary floating point, the result
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is :const:`5.5511151231257827e-017`. While near to zero, the differences
45
prevent reliable equality testing and differences can accumulate. For this
46
reason, decimal is preferred in accounting applications which have strict
49
* The decimal module incorporates a notion of significant places so that ``1.30
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+ 1.20`` is :const:`2.50`. The trailing zero is kept to indicate significance.
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This is the customary presentation for monetary applications. For
52
multiplication, the "schoolbook" approach uses all the figures in the
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multiplicands. For instance, ``1.3 * 1.2`` gives :const:`1.56` while ``1.30 *
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1.20`` gives :const:`1.5600`.
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* Unlike hardware based binary floating point, the decimal module has a user
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alterable precision (defaulting to 28 places) which can be as large as needed for
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>>> from decimal import *
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>>> getcontext().prec = 6
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>>> Decimal(1) / Decimal(7)
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>>> getcontext().prec = 28
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>>> Decimal(1) / Decimal(7)
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Decimal('0.1428571428571428571428571429')
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* Both binary and decimal floating point are implemented in terms of published
69
standards. While the built-in float type exposes only a modest portion of its
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capabilities, the decimal module exposes all required parts of the standard.
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When needed, the programmer has full control over rounding and signal handling.
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This includes an option to enforce exact arithmetic by using exceptions
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to block any inexact operations.
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* The decimal module was designed to support "without prejudice, both exact
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unrounded decimal arithmetic (sometimes called fixed-point arithmetic)
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and rounded floating-point arithmetic." -- excerpt from the decimal
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arithmetic specification.
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The module design is centered around three concepts: the decimal number, the
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context for arithmetic, and signals.
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A decimal number is immutable. It has a sign, coefficient digits, and an
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exponent. To preserve significance, the coefficient digits do not truncate
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trailing zeros. Decimals also include special values such as
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:const:`Infinity`, :const:`-Infinity`, and :const:`NaN`. The standard also
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differentiates :const:`-0` from :const:`+0`.
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The context for arithmetic is an environment specifying precision, rounding
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rules, limits on exponents, flags indicating the results of operations, and trap
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enablers which determine whether signals are treated as exceptions. Rounding
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options include :const:`ROUND_CEILING`, :const:`ROUND_DOWN`,
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:const:`ROUND_FLOOR`, :const:`ROUND_HALF_DOWN`, :const:`ROUND_HALF_EVEN`,
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:const:`ROUND_HALF_UP`, :const:`ROUND_UP`, and :const:`ROUND_05UP`.
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Signals are groups of exceptional conditions arising during the course of
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computation. Depending on the needs of the application, signals may be ignored,
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considered as informational, or treated as exceptions. The signals in the
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decimal module are: :const:`Clamped`, :const:`InvalidOperation`,
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:const:`DivisionByZero`, :const:`Inexact`, :const:`Rounded`, :const:`Subnormal`,
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:const:`Overflow`, and :const:`Underflow`.
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For each signal there is a flag and a trap enabler. When a signal is
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encountered, its flag is set to one, then, if the trap enabler is
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set to one, an exception is raised. Flags are sticky, so the user needs to
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reset them before monitoring a calculation.
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* IBM's General Decimal Arithmetic Specification, `The General Decimal Arithmetic
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Specification <http://speleotrove.com/decimal/>`_.
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.. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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.. _decimal-tutorial:
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The usual start to using decimals is importing the module, viewing the current
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context with :func:`getcontext` and, if necessary, setting new values for
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precision, rounding, or enabled traps::
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>>> from decimal import *
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Context(prec=28, rounding=ROUND_HALF_EVEN, Emin=-999999999, Emax=999999999,
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capitals=1, flags=[], traps=[Overflow, DivisionByZero,
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>>> getcontext().prec = 7 # Set a new precision
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Decimal instances can be constructed from integers, strings, floats, or tuples.
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Construction from an integer or a float performs an exact conversion of the
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value of that integer or float. Decimal numbers include special values such as
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:const:`NaN` which stands for "Not a number", positive and negative
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:const:`Infinity`, and :const:`-0`.
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>>> getcontext().prec = 28
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Decimal('3.140000000000000124344978758017532527446746826171875')
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>>> Decimal((0, (3, 1, 4), -2))
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>>> Decimal(str(2.0 ** 0.5))
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Decimal('1.41421356237')
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>>> Decimal(2) ** Decimal('0.5')
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Decimal('1.414213562373095048801688724')
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>>> Decimal('-Infinity')
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The significance of a new Decimal is determined solely by the number of digits
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input. Context precision and rounding only come into play during arithmetic
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.. doctest:: newcontext
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>>> getcontext().prec = 6
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>>> Decimal('3.1415926535')
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Decimal('3.1415926535')
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>>> Decimal('3.1415926535') + Decimal('2.7182818285')
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>>> getcontext().rounding = ROUND_UP
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>>> Decimal('3.1415926535') + Decimal('2.7182818285')
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Decimals interact well with much of the rest of Python. Here is a small decimal
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floating point flying circus:
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:options: +NORMALIZE_WHITESPACE
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>>> data = map(Decimal, '1.34 1.87 3.45 2.35 1.00 0.03 9.25'.split())
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[Decimal('0.03'), Decimal('1.00'), Decimal('1.34'), Decimal('1.87'),
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Decimal('2.35'), Decimal('3.45'), Decimal('9.25')]
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>>> round(a, 1) # round() first converts to binary floating point
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And some mathematical functions are also available to Decimal:
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>>> getcontext().prec = 28
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>>> Decimal(2).sqrt()
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Decimal('1.414213562373095048801688724')
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Decimal('2.718281828459045235360287471')
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>>> Decimal('10').ln()
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Decimal('2.302585092994045684017991455')
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>>> Decimal('10').log10()
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The :meth:`quantize` method rounds a number to a fixed exponent. This method is
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useful for monetary applications that often round results to a fixed number of
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>>> Decimal('7.325').quantize(Decimal('.01'), rounding=ROUND_DOWN)
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>>> Decimal('7.325').quantize(Decimal('1.'), rounding=ROUND_UP)
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As shown above, the :func:`getcontext` function accesses the current context and
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allows the settings to be changed. This approach meets the needs of most
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For more advanced work, it may be useful to create alternate contexts using the
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Context() constructor. To make an alternate active, use the :func:`setcontext`
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In accordance with the standard, the :mod:`decimal` module provides two ready to
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use standard contexts, :const:`BasicContext` and :const:`ExtendedContext`. The
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former is especially useful for debugging because many of the traps are
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.. doctest:: newcontext
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:options: +NORMALIZE_WHITESPACE
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>>> myothercontext = Context(prec=60, rounding=ROUND_HALF_DOWN)
245
>>> setcontext(myothercontext)
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>>> Decimal(1) / Decimal(7)
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Decimal('0.142857142857142857142857142857142857142857142857142857142857')
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Context(prec=9, rounding=ROUND_HALF_EVEN, Emin=-999999999, Emax=999999999,
251
capitals=1, flags=[], traps=[])
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>>> setcontext(ExtendedContext)
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>>> Decimal(1) / Decimal(7)
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Decimal('0.142857143')
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>>> Decimal(42) / Decimal(0)
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>>> setcontext(BasicContext)
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>>> Decimal(42) / Decimal(0)
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Traceback (most recent call last):
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File "<pyshell#143>", line 1, in -toplevel-
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Decimal(42) / Decimal(0)
263
DivisionByZero: x / 0
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Contexts also have signal flags for monitoring exceptional conditions
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encountered during computations. The flags remain set until explicitly cleared,
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so it is best to clear the flags before each set of monitored computations by
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using the :meth:`clear_flags` method. ::
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>>> setcontext(ExtendedContext)
271
>>> getcontext().clear_flags()
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>>> Decimal(355) / Decimal(113)
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Decimal('3.14159292')
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Context(prec=9, rounding=ROUND_HALF_EVEN, Emin=-999999999, Emax=999999999,
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capitals=1, flags=[Rounded, Inexact], traps=[])
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The *flags* entry shows that the rational approximation to :const:`Pi` was
279
rounded (digits beyond the context precision were thrown away) and that the
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result is inexact (some of the discarded digits were non-zero).
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Individual traps are set using the dictionary in the :attr:`traps` field of a
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.. doctest:: newcontext
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>>> setcontext(ExtendedContext)
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>>> Decimal(1) / Decimal(0)
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>>> getcontext().traps[DivisionByZero] = 1
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>>> Decimal(1) / Decimal(0)
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Traceback (most recent call last):
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File "<pyshell#112>", line 1, in -toplevel-
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Decimal(1) / Decimal(0)
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DivisionByZero: x / 0
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Most programs adjust the current context only once, at the beginning of the
298
program. And, in many applications, data is converted to :class:`Decimal` with
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a single cast inside a loop. With context set and decimals created, the bulk of
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the program manipulates the data no differently than with other Python numeric
303
.. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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.. class:: Decimal([value [, context]])
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Construct a new :class:`Decimal` object based from *value*.
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*value* can be an integer, string, tuple, :class:`float`, or another :class:`Decimal`
317
object. If no *value* is given, returns ``Decimal('0')``. If *value* is a
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string, it should conform to the decimal numeric string syntax after leading
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and trailing whitespace characters are removed::
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digit ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
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indicator ::= 'e' | 'E'
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digits ::= digit [digit]...
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decimal-part ::= digits '.' [digits] | ['.'] digits
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exponent-part ::= indicator [sign] digits
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infinity ::= 'Infinity' | 'Inf'
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nan ::= 'NaN' [digits] | 'sNaN' [digits]
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numeric-value ::= decimal-part [exponent-part] | infinity
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numeric-string ::= [sign] numeric-value | [sign] nan
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If *value* is a unicode string then other Unicode decimal digits
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are also permitted where ``digit`` appears above. These include
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decimal digits from various other alphabets (for example,
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Arabic-Indic and Devanāgarī digits) along with the fullwidth digits
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``u'\uff10'`` through ``u'\uff19'``.
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If *value* is a :class:`tuple`, it should have three components, a sign
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(:const:`0` for positive or :const:`1` for negative), a :class:`tuple` of
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digits, and an integer exponent. For example, ``Decimal((0, (1, 4, 1, 4), -3))``
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returns ``Decimal('1.414')``.
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If *value* is a :class:`float`, the binary floating point value is losslessly
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converted to its exact decimal equivalent. This conversion can often require
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53 or more digits of precision. For example, ``Decimal(float('1.1'))``
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``Decimal('1.100000000000000088817841970012523233890533447265625')``.
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The *context* precision does not affect how many digits are stored. That is
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determined exclusively by the number of digits in *value*. For example,
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``Decimal('3.00000')`` records all five zeros even if the context precision is
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The purpose of the *context* argument is determining what to do if *value* is a
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malformed string. If the context traps :const:`InvalidOperation`, an exception
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is raised; otherwise, the constructor returns a new Decimal with the value of
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Once constructed, :class:`Decimal` objects are immutable.
361
.. versionchanged:: 2.6
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leading and trailing whitespace characters are permitted when
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creating a Decimal instance from a string.
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.. versionchanged:: 2.7
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The argument to the constructor is now permitted to be a :class:`float` instance.
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Decimal floating point objects share many properties with the other built-in
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numeric types such as :class:`float` and :class:`int`. All of the usual math
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operations and special methods apply. Likewise, decimal objects can be
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copied, pickled, printed, used as dictionary keys, used as set elements,
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compared, sorted, and coerced to another type (such as :class:`float` or
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There are some small differences between arithmetic on Decimal objects and
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arithmetic on integers and floats. When the remainder operator ``%`` is
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applied to Decimal objects, the sign of the result is the sign of the
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*dividend* rather than the sign of the divisor::
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>>> Decimal(-7) % Decimal(4)
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The integer division operator ``//`` behaves analogously, returning the
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integer part of the true quotient (truncating towards zero) rather than its
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floor, so as to preserve the usual identity ``x == (x // y) * y + x % y``::
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>>> Decimal(-7) // Decimal(4)
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The ``%`` and ``//`` operators implement the ``remainder`` and
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``divide-integer`` operations (respectively) as described in the
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Decimal objects cannot generally be combined with floats in
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arithmetic operations: an attempt to add a :class:`Decimal` to a
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:class:`float`, for example, will raise a :exc:`TypeError`.
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There's one exception to this rule: it's possible to use Python's
402
comparison operators to compare a :class:`float` instance ``x``
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with a :class:`Decimal` instance ``y``. Without this exception,
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comparisons between :class:`Decimal` and :class:`float` instances
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would follow the general rules for comparing objects of different
406
types described in the :ref:`expressions` section of the reference
407
manual, leading to confusing results.
409
.. versionchanged:: 2.7
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A comparison between a :class:`float` instance ``x`` and a
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:class:`Decimal` instance ``y`` now returns a result based on
412
the values of ``x`` and ``y``. In earlier versions ``x < y``
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returned the same (arbitrary) result for any :class:`Decimal`
414
instance ``x`` and any :class:`float` instance ``y``.
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In addition to the standard numeric properties, decimal floating point
417
objects also have a number of specialized methods:
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.. method:: adjusted()
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Return the adjusted exponent after shifting out the coefficient's
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rightmost digits until only the lead digit remains:
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``Decimal('321e+5').adjusted()`` returns seven. Used for determining the
425
position of the most significant digit with respect to the decimal point.
428
.. method:: as_tuple()
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Return a :term:`named tuple` representation of the number:
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``DecimalTuple(sign, digits, exponent)``.
433
.. versionchanged:: 2.6
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.. method:: canonical()
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Return the canonical encoding of the argument. Currently, the encoding of
440
a :class:`Decimal` instance is always canonical, so this operation returns
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its argument unchanged.
443
.. versionadded:: 2.6
445
.. method:: compare(other[, context])
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Compare the values of two Decimal instances. This operation behaves in
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the same way as the usual comparison method :meth:`__cmp__`, except that
449
:meth:`compare` returns a Decimal instance rather than an integer, and if
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either operand is a NaN then the result is a NaN::
452
a or b is a NaN ==> Decimal('NaN')
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a < b ==> Decimal('-1')
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a == b ==> Decimal('0')
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a > b ==> Decimal('1')
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.. method:: compare_signal(other[, context])
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This operation is identical to the :meth:`compare` method, except that all
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NaNs signal. That is, if neither operand is a signaling NaN then any
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quiet NaN operand is treated as though it were a signaling NaN.
463
.. versionadded:: 2.6
465
.. method:: compare_total(other)
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Compare two operands using their abstract representation rather than their
468
numerical value. Similar to the :meth:`compare` method, but the result
469
gives a total ordering on :class:`Decimal` instances. Two
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:class:`Decimal` instances with the same numeric value but different
471
representations compare unequal in this ordering:
473
>>> Decimal('12.0').compare_total(Decimal('12'))
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Quiet and signaling NaNs are also included in the total ordering. The
477
result of this function is ``Decimal('0')`` if both operands have the same
478
representation, ``Decimal('-1')`` if the first operand is lower in the
479
total order than the second, and ``Decimal('1')`` if the first operand is
480
higher in the total order than the second operand. See the specification
481
for details of the total order.
483
.. versionadded:: 2.6
485
.. method:: compare_total_mag(other)
487
Compare two operands using their abstract representation rather than their
488
value as in :meth:`compare_total`, but ignoring the sign of each operand.
489
``x.compare_total_mag(y)`` is equivalent to
490
``x.copy_abs().compare_total(y.copy_abs())``.
492
.. versionadded:: 2.6
494
.. method:: conjugate()
496
Just returns self, this method is only to comply with the Decimal
499
.. versionadded:: 2.6
501
.. method:: copy_abs()
503
Return the absolute value of the argument. This operation is unaffected
504
by the context and is quiet: no flags are changed and no rounding is
507
.. versionadded:: 2.6
509
.. method:: copy_negate()
511
Return the negation of the argument. This operation is unaffected by the
512
context and is quiet: no flags are changed and no rounding is performed.
514
.. versionadded:: 2.6
516
.. method:: copy_sign(other)
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Return a copy of the first operand with the sign set to be the same as the
519
sign of the second operand. For example:
521
>>> Decimal('2.3').copy_sign(Decimal('-1.5'))
524
This operation is unaffected by the context and is quiet: no flags are
525
changed and no rounding is performed.
527
.. versionadded:: 2.6
529
.. method:: exp([context])
531
Return the value of the (natural) exponential function ``e**x`` at the
532
given number. The result is correctly rounded using the
533
:const:`ROUND_HALF_EVEN` rounding mode.
536
Decimal('2.718281828459045235360287471')
537
>>> Decimal(321).exp()
538
Decimal('2.561702493119680037517373933E+139')
540
.. versionadded:: 2.6
542
.. method:: from_float(f)
544
Classmethod that converts a float to a decimal number, exactly.
546
Note `Decimal.from_float(0.1)` is not the same as `Decimal('0.1')`.
547
Since 0.1 is not exactly representable in binary floating point, the
548
value is stored as the nearest representable value which is
549
`0x1.999999999999ap-4`. That equivalent value in decimal is
550
`0.1000000000000000055511151231257827021181583404541015625`.
552
.. note:: From Python 2.7 onwards, a :class:`Decimal` instance
553
can also be constructed directly from a :class:`float`.
557
>>> Decimal.from_float(0.1)
558
Decimal('0.1000000000000000055511151231257827021181583404541015625')
559
>>> Decimal.from_float(float('nan'))
561
>>> Decimal.from_float(float('inf'))
563
>>> Decimal.from_float(float('-inf'))
566
.. versionadded:: 2.7
568
.. method:: fma(other, third[, context])
570
Fused multiply-add. Return self*other+third with no rounding of the
571
intermediate product self*other.
573
>>> Decimal(2).fma(3, 5)
576
.. versionadded:: 2.6
578
.. method:: is_canonical()
580
Return :const:`True` if the argument is canonical and :const:`False`
581
otherwise. Currently, a :class:`Decimal` instance is always canonical, so
582
this operation always returns :const:`True`.
584
.. versionadded:: 2.6
586
.. method:: is_finite()
588
Return :const:`True` if the argument is a finite number, and
589
:const:`False` if the argument is an infinity or a NaN.
591
.. versionadded:: 2.6
593
.. method:: is_infinite()
595
Return :const:`True` if the argument is either positive or negative
596
infinity and :const:`False` otherwise.
598
.. versionadded:: 2.6
602
Return :const:`True` if the argument is a (quiet or signaling) NaN and
603
:const:`False` otherwise.
605
.. versionadded:: 2.6
607
.. method:: is_normal()
609
Return :const:`True` if the argument is a *normal* finite non-zero
610
number with an adjusted exponent greater than or equal to *Emin*.
611
Return :const:`False` if the argument is zero, subnormal, infinite or a
612
NaN. Note, the term *normal* is used here in a different sense with
613
the :meth:`normalize` method which is used to create canonical values.
615
.. versionadded:: 2.6
617
.. method:: is_qnan()
619
Return :const:`True` if the argument is a quiet NaN, and
620
:const:`False` otherwise.
622
.. versionadded:: 2.6
624
.. method:: is_signed()
626
Return :const:`True` if the argument has a negative sign and
627
:const:`False` otherwise. Note that zeros and NaNs can both carry signs.
629
.. versionadded:: 2.6
631
.. method:: is_snan()
633
Return :const:`True` if the argument is a signaling NaN and :const:`False`
636
.. versionadded:: 2.6
638
.. method:: is_subnormal()
640
Return :const:`True` if the argument is subnormal, and :const:`False`
641
otherwise. A number is subnormal is if it is nonzero, finite, and has an
642
adjusted exponent less than *Emin*.
644
.. versionadded:: 2.6
646
.. method:: is_zero()
648
Return :const:`True` if the argument is a (positive or negative) zero and
649
:const:`False` otherwise.
651
.. versionadded:: 2.6
653
.. method:: ln([context])
655
Return the natural (base e) logarithm of the operand. The result is
656
correctly rounded using the :const:`ROUND_HALF_EVEN` rounding mode.
658
.. versionadded:: 2.6
660
.. method:: log10([context])
662
Return the base ten logarithm of the operand. The result is correctly
663
rounded using the :const:`ROUND_HALF_EVEN` rounding mode.
665
.. versionadded:: 2.6
667
.. method:: logb([context])
669
For a nonzero number, return the adjusted exponent of its operand as a
670
:class:`Decimal` instance. If the operand is a zero then
671
``Decimal('-Infinity')`` is returned and the :const:`DivisionByZero` flag
672
is raised. If the operand is an infinity then ``Decimal('Infinity')`` is
675
.. versionadded:: 2.6
677
.. method:: logical_and(other[, context])
679
:meth:`logical_and` is a logical operation which takes two *logical
680
operands* (see :ref:`logical_operands_label`). The result is the
681
digit-wise ``and`` of the two operands.
683
.. versionadded:: 2.6
685
.. method:: logical_invert([context])
687
:meth:`logical_invert` is a logical operation. The
688
result is the digit-wise inversion of the operand.
690
.. versionadded:: 2.6
692
.. method:: logical_or(other[, context])
694
:meth:`logical_or` is a logical operation which takes two *logical
695
operands* (see :ref:`logical_operands_label`). The result is the
696
digit-wise ``or`` of the two operands.
698
.. versionadded:: 2.6
700
.. method:: logical_xor(other[, context])
702
:meth:`logical_xor` is a logical operation which takes two *logical
703
operands* (see :ref:`logical_operands_label`). The result is the
704
digit-wise exclusive or of the two operands.
706
.. versionadded:: 2.6
708
.. method:: max(other[, context])
710
Like ``max(self, other)`` except that the context rounding rule is applied
711
before returning and that :const:`NaN` values are either signaled or
712
ignored (depending on the context and whether they are signaling or
715
.. method:: max_mag(other[, context])
717
Similar to the :meth:`.max` method, but the comparison is done using the
718
absolute values of the operands.
720
.. versionadded:: 2.6
722
.. method:: min(other[, context])
724
Like ``min(self, other)`` except that the context rounding rule is applied
725
before returning and that :const:`NaN` values are either signaled or
726
ignored (depending on the context and whether they are signaling or
729
.. method:: min_mag(other[, context])
731
Similar to the :meth:`.min` method, but the comparison is done using the
732
absolute values of the operands.
734
.. versionadded:: 2.6
736
.. method:: next_minus([context])
738
Return the largest number representable in the given context (or in the
739
current thread's context if no context is given) that is smaller than the
742
.. versionadded:: 2.6
744
.. method:: next_plus([context])
746
Return the smallest number representable in the given context (or in the
747
current thread's context if no context is given) that is larger than the
750
.. versionadded:: 2.6
752
.. method:: next_toward(other[, context])
754
If the two operands are unequal, return the number closest to the first
755
operand in the direction of the second operand. If both operands are
756
numerically equal, return a copy of the first operand with the sign set to
757
be the same as the sign of the second operand.
759
.. versionadded:: 2.6
761
.. method:: normalize([context])
763
Normalize the number by stripping the rightmost trailing zeros and
764
converting any result equal to :const:`Decimal('0')` to
765
:const:`Decimal('0e0')`. Used for producing canonical values for attributes
766
of an equivalence class. For example, ``Decimal('32.100')`` and
767
``Decimal('0.321000e+2')`` both normalize to the equivalent value
770
.. method:: number_class([context])
772
Return a string describing the *class* of the operand. The returned value
773
is one of the following ten strings.
775
* ``"-Infinity"``, indicating that the operand is negative infinity.
776
* ``"-Normal"``, indicating that the operand is a negative normal number.
777
* ``"-Subnormal"``, indicating that the operand is negative and subnormal.
778
* ``"-Zero"``, indicating that the operand is a negative zero.
779
* ``"+Zero"``, indicating that the operand is a positive zero.
780
* ``"+Subnormal"``, indicating that the operand is positive and subnormal.
781
* ``"+Normal"``, indicating that the operand is a positive normal number.
782
* ``"+Infinity"``, indicating that the operand is positive infinity.
783
* ``"NaN"``, indicating that the operand is a quiet NaN (Not a Number).
784
* ``"sNaN"``, indicating that the operand is a signaling NaN.
786
.. versionadded:: 2.6
788
.. method:: quantize(exp[, rounding[, context[, watchexp]]])
790
Return a value equal to the first operand after rounding and having the
791
exponent of the second operand.
793
>>> Decimal('1.41421356').quantize(Decimal('1.000'))
796
Unlike other operations, if the length of the coefficient after the
797
quantize operation would be greater than precision, then an
798
:const:`InvalidOperation` is signaled. This guarantees that, unless there
799
is an error condition, the quantized exponent is always equal to that of
800
the right-hand operand.
802
Also unlike other operations, quantize never signals Underflow, even if
803
the result is subnormal and inexact.
805
If the exponent of the second operand is larger than that of the first
806
then rounding may be necessary. In this case, the rounding mode is
807
determined by the ``rounding`` argument if given, else by the given
808
``context`` argument; if neither argument is given the rounding mode of
809
the current thread's context is used.
811
If *watchexp* is set (default), then an error is returned whenever the
812
resulting exponent is greater than :attr:`Emax` or less than
817
Return ``Decimal(10)``, the radix (base) in which the :class:`Decimal`
818
class does all its arithmetic. Included for compatibility with the
821
.. versionadded:: 2.6
823
.. method:: remainder_near(other[, context])
825
Return the remainder from dividing *self* by *other*. This differs from
826
``self % other`` in that the sign of the remainder is chosen so as to
827
minimize its absolute value. More precisely, the return value is
828
``self - n * other`` where ``n`` is the integer nearest to the exact
829
value of ``self / other``, and if two integers are equally near then the
832
If the result is zero then its sign will be the sign of *self*.
834
>>> Decimal(18).remainder_near(Decimal(10))
836
>>> Decimal(25).remainder_near(Decimal(10))
838
>>> Decimal(35).remainder_near(Decimal(10))
841
.. method:: rotate(other[, context])
843
Return the result of rotating the digits of the first operand by an amount
844
specified by the second operand. The second operand must be an integer in
845
the range -precision through precision. The absolute value of the second
846
operand gives the number of places to rotate. If the second operand is
847
positive then rotation is to the left; otherwise rotation is to the right.
848
The coefficient of the first operand is padded on the left with zeros to
849
length precision if necessary. The sign and exponent of the first operand
852
.. versionadded:: 2.6
854
.. method:: same_quantum(other[, context])
856
Test whether self and other have the same exponent or whether both are
859
.. method:: scaleb(other[, context])
861
Return the first operand with exponent adjusted by the second.
862
Equivalently, return the first operand multiplied by ``10**other``. The
863
second operand must be an integer.
865
.. versionadded:: 2.6
867
.. method:: shift(other[, context])
869
Return the result of shifting the digits of the first operand by an amount
870
specified by the second operand. The second operand must be an integer in
871
the range -precision through precision. The absolute value of the second
872
operand gives the number of places to shift. If the second operand is
873
positive then the shift is to the left; otherwise the shift is to the
874
right. Digits shifted into the coefficient are zeros. The sign and
875
exponent of the first operand are unchanged.
877
.. versionadded:: 2.6
879
.. method:: sqrt([context])
881
Return the square root of the argument to full precision.
884
.. method:: to_eng_string([context])
886
Convert to a string, using engineering notation if an exponent is needed.
888
Engineering notation has an exponent which is a multiple of 3. This
889
can leave up to 3 digits to the left of the decimal place and may
890
require the addition of either one or two trailing zeros.
892
For example, this converts ``Decimal('123E+1')`` to ``Decimal('1.23E+3')``.
894
.. method:: to_integral([rounding[, context]])
896
Identical to the :meth:`to_integral_value` method. The ``to_integral``
897
name has been kept for compatibility with older versions.
899
.. method:: to_integral_exact([rounding[, context]])
901
Round to the nearest integer, signaling :const:`Inexact` or
902
:const:`Rounded` as appropriate if rounding occurs. The rounding mode is
903
determined by the ``rounding`` parameter if given, else by the given
904
``context``. If neither parameter is given then the rounding mode of the
905
current context is used.
907
.. versionadded:: 2.6
909
.. method:: to_integral_value([rounding[, context]])
911
Round to the nearest integer without signaling :const:`Inexact` or
912
:const:`Rounded`. If given, applies *rounding*; otherwise, uses the
913
rounding method in either the supplied *context* or the current context.
915
.. versionchanged:: 2.6
916
renamed from ``to_integral`` to ``to_integral_value``. The old name
917
remains valid for compatibility.
919
.. _logical_operands_label:
924
The :meth:`logical_and`, :meth:`logical_invert`, :meth:`logical_or`,
925
and :meth:`logical_xor` methods expect their arguments to be *logical
926
operands*. A *logical operand* is a :class:`Decimal` instance whose
927
exponent and sign are both zero, and whose digits are all either
928
:const:`0` or :const:`1`.
930
.. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
938
Contexts are environments for arithmetic operations. They govern precision, set
939
rules for rounding, determine which signals are treated as exceptions, and limit
940
the range for exponents.
942
Each thread has its own current context which is accessed or changed using the
943
:func:`getcontext` and :func:`setcontext` functions:
946
.. function:: getcontext()
948
Return the current context for the active thread.
951
.. function:: setcontext(c)
953
Set the current context for the active thread to *c*.
955
Beginning with Python 2.5, you can also use the :keyword:`with` statement and
956
the :func:`localcontext` function to temporarily change the active context.
959
.. function:: localcontext([c])
961
Return a context manager that will set the current context for the active thread
962
to a copy of *c* on entry to the with-statement and restore the previous context
963
when exiting the with-statement. If no context is specified, a copy of the
964
current context is used.
966
.. versionadded:: 2.5
968
For example, the following code sets the current decimal precision to 42 places,
969
performs a calculation, and then automatically restores the previous context::
971
from decimal import localcontext
973
with localcontext() as ctx:
974
ctx.prec = 42 # Perform a high precision calculation
975
s = calculate_something()
976
s = +s # Round the final result back to the default precision
978
with localcontext(BasicContext): # temporarily use the BasicContext
979
print Decimal(1) / Decimal(7)
980
print Decimal(355) / Decimal(113)
982
New contexts can also be created using the :class:`Context` constructor
983
described below. In addition, the module provides three pre-made contexts:
986
.. class:: BasicContext
988
This is a standard context defined by the General Decimal Arithmetic
989
Specification. Precision is set to nine. Rounding is set to
990
:const:`ROUND_HALF_UP`. All flags are cleared. All traps are enabled (treated
991
as exceptions) except :const:`Inexact`, :const:`Rounded`, and
994
Because many of the traps are enabled, this context is useful for debugging.
997
.. class:: ExtendedContext
999
This is a standard context defined by the General Decimal Arithmetic
1000
Specification. Precision is set to nine. Rounding is set to
1001
:const:`ROUND_HALF_EVEN`. All flags are cleared. No traps are enabled (so that
1002
exceptions are not raised during computations).
1004
Because the traps are disabled, this context is useful for applications that
1005
prefer to have result value of :const:`NaN` or :const:`Infinity` instead of
1006
raising exceptions. This allows an application to complete a run in the
1007
presence of conditions that would otherwise halt the program.
1010
.. class:: DefaultContext
1012
This context is used by the :class:`Context` constructor as a prototype for new
1013
contexts. Changing a field (such a precision) has the effect of changing the
1014
default for new contexts created by the :class:`Context` constructor.
1016
This context is most useful in multi-threaded environments. Changing one of the
1017
fields before threads are started has the effect of setting system-wide
1018
defaults. Changing the fields after threads have started is not recommended as
1019
it would require thread synchronization to prevent race conditions.
1021
In single threaded environments, it is preferable to not use this context at
1022
all. Instead, simply create contexts explicitly as described below.
1024
The default values are precision=28, rounding=ROUND_HALF_EVEN, and enabled traps
1025
for Overflow, InvalidOperation, and DivisionByZero.
1027
In addition to the three supplied contexts, new contexts can be created with the
1028
:class:`Context` constructor.
1031
.. class:: Context(prec=None, rounding=None, traps=None, flags=None, Emin=None, Emax=None, capitals=1)
1033
Creates a new context. If a field is not specified or is :const:`None`, the
1034
default values are copied from the :const:`DefaultContext`. If the *flags*
1035
field is not specified or is :const:`None`, all flags are cleared.
1037
The *prec* field is a positive integer that sets the precision for arithmetic
1038
operations in the context.
1040
The *rounding* option is one of:
1042
* :const:`ROUND_CEILING` (towards :const:`Infinity`),
1043
* :const:`ROUND_DOWN` (towards zero),
1044
* :const:`ROUND_FLOOR` (towards :const:`-Infinity`),
1045
* :const:`ROUND_HALF_DOWN` (to nearest with ties going towards zero),
1046
* :const:`ROUND_HALF_EVEN` (to nearest with ties going to nearest even integer),
1047
* :const:`ROUND_HALF_UP` (to nearest with ties going away from zero), or
1048
* :const:`ROUND_UP` (away from zero).
1049
* :const:`ROUND_05UP` (away from zero if last digit after rounding towards zero
1050
would have been 0 or 5; otherwise towards zero)
1052
The *traps* and *flags* fields list any signals to be set. Generally, new
1053
contexts should only set traps and leave the flags clear.
1055
The *Emin* and *Emax* fields are integers specifying the outer limits allowable
1058
The *capitals* field is either :const:`0` or :const:`1` (the default). If set to
1059
:const:`1`, exponents are printed with a capital :const:`E`; otherwise, a
1060
lowercase :const:`e` is used: :const:`Decimal('6.02e+23')`.
1062
.. versionchanged:: 2.6
1063
The :const:`ROUND_05UP` rounding mode was added.
1065
The :class:`Context` class defines several general purpose methods as well as
1066
a large number of methods for doing arithmetic directly in a given context.
1067
In addition, for each of the :class:`Decimal` methods described above (with
1068
the exception of the :meth:`adjusted` and :meth:`as_tuple` methods) there is
1069
a corresponding :class:`Context` method. For example, for a :class:`Context`
1070
instance ``C`` and :class:`Decimal` instance ``x``, ``C.exp(x)`` is
1071
equivalent to ``x.exp(context=C)``. Each :class:`Context` method accepts a
1072
Python integer (an instance of :class:`int` or :class:`long`) anywhere that a
1073
Decimal instance is accepted.
1076
.. method:: clear_flags()
1078
Resets all of the flags to :const:`0`.
1082
Return a duplicate of the context.
1084
.. method:: copy_decimal(num)
1086
Return a copy of the Decimal instance num.
1088
.. method:: create_decimal(num)
1090
Creates a new Decimal instance from *num* but using *self* as
1091
context. Unlike the :class:`Decimal` constructor, the context precision,
1092
rounding method, flags, and traps are applied to the conversion.
1094
This is useful because constants are often given to a greater precision
1095
than is needed by the application. Another benefit is that rounding
1096
immediately eliminates unintended effects from digits beyond the current
1097
precision. In the following example, using unrounded inputs means that
1098
adding zero to a sum can change the result:
1100
.. doctest:: newcontext
1102
>>> getcontext().prec = 3
1103
>>> Decimal('3.4445') + Decimal('1.0023')
1105
>>> Decimal('3.4445') + Decimal(0) + Decimal('1.0023')
1108
This method implements the to-number operation of the IBM specification.
1109
If the argument is a string, no leading or trailing whitespace is
1112
.. method:: create_decimal_from_float(f)
1114
Creates a new Decimal instance from a float *f* but rounding using *self*
1115
as the context. Unlike the :meth:`Decimal.from_float` class method,
1116
the context precision, rounding method, flags, and traps are applied to
1121
>>> context = Context(prec=5, rounding=ROUND_DOWN)
1122
>>> context.create_decimal_from_float(math.pi)
1124
>>> context = Context(prec=5, traps=[Inexact])
1125
>>> context.create_decimal_from_float(math.pi)
1126
Traceback (most recent call last):
1130
.. versionadded:: 2.7
1134
Returns a value equal to ``Emin - prec + 1`` which is the minimum exponent
1135
value for subnormal results. When underflow occurs, the exponent is set
1141
Returns a value equal to ``Emax - prec + 1``.
1143
The usual approach to working with decimals is to create :class:`Decimal`
1144
instances and then apply arithmetic operations which take place within the
1145
current context for the active thread. An alternative approach is to use
1146
context methods for calculating within a specific context. The methods are
1147
similar to those for the :class:`Decimal` class and are only briefly
1153
Returns the absolute value of *x*.
1156
.. method:: add(x, y)
1158
Return the sum of *x* and *y*.
1161
.. method:: canonical(x)
1163
Returns the same Decimal object *x*.
1166
.. method:: compare(x, y)
1168
Compares *x* and *y* numerically.
1171
.. method:: compare_signal(x, y)
1173
Compares the values of the two operands numerically.
1176
.. method:: compare_total(x, y)
1178
Compares two operands using their abstract representation.
1181
.. method:: compare_total_mag(x, y)
1183
Compares two operands using their abstract representation, ignoring sign.
1186
.. method:: copy_abs(x)
1188
Returns a copy of *x* with the sign set to 0.
1191
.. method:: copy_negate(x)
1193
Returns a copy of *x* with the sign inverted.
1196
.. method:: copy_sign(x, y)
1198
Copies the sign from *y* to *x*.
1201
.. method:: divide(x, y)
1203
Return *x* divided by *y*.
1206
.. method:: divide_int(x, y)
1208
Return *x* divided by *y*, truncated to an integer.
1211
.. method:: divmod(x, y)
1213
Divides two numbers and returns the integer part of the result.
1221
.. method:: fma(x, y, z)
1223
Returns *x* multiplied by *y*, plus *z*.
1226
.. method:: is_canonical(x)
1228
Returns ``True`` if *x* is canonical; otherwise returns ``False``.
1231
.. method:: is_finite(x)
1233
Returns ``True`` if *x* is finite; otherwise returns ``False``.
1236
.. method:: is_infinite(x)
1238
Returns ``True`` if *x* is infinite; otherwise returns ``False``.
1241
.. method:: is_nan(x)
1243
Returns ``True`` if *x* is a qNaN or sNaN; otherwise returns ``False``.
1246
.. method:: is_normal(x)
1248
Returns ``True`` if *x* is a normal number; otherwise returns ``False``.
1251
.. method:: is_qnan(x)
1253
Returns ``True`` if *x* is a quiet NaN; otherwise returns ``False``.
1256
.. method:: is_signed(x)
1258
Returns ``True`` if *x* is negative; otherwise returns ``False``.
1261
.. method:: is_snan(x)
1263
Returns ``True`` if *x* is a signaling NaN; otherwise returns ``False``.
1266
.. method:: is_subnormal(x)
1268
Returns ``True`` if *x* is subnormal; otherwise returns ``False``.
1271
.. method:: is_zero(x)
1273
Returns ``True`` if *x* is a zero; otherwise returns ``False``.
1278
Returns the natural (base e) logarithm of *x*.
1281
.. method:: log10(x)
1283
Returns the base 10 logarithm of *x*.
1288
Returns the exponent of the magnitude of the operand's MSD.
1291
.. method:: logical_and(x, y)
1293
Applies the logical operation *and* between each operand's digits.
1296
.. method:: logical_invert(x)
1298
Invert all the digits in *x*.
1301
.. method:: logical_or(x, y)
1303
Applies the logical operation *or* between each operand's digits.
1306
.. method:: logical_xor(x, y)
1308
Applies the logical operation *xor* between each operand's digits.
1311
.. method:: max(x, y)
1313
Compares two values numerically and returns the maximum.
1316
.. method:: max_mag(x, y)
1318
Compares the values numerically with their sign ignored.
1321
.. method:: min(x, y)
1323
Compares two values numerically and returns the minimum.
1326
.. method:: min_mag(x, y)
1328
Compares the values numerically with their sign ignored.
1331
.. method:: minus(x)
1333
Minus corresponds to the unary prefix minus operator in Python.
1336
.. method:: multiply(x, y)
1338
Return the product of *x* and *y*.
1341
.. method:: next_minus(x)
1343
Returns the largest representable number smaller than *x*.
1346
.. method:: next_plus(x)
1348
Returns the smallest representable number larger than *x*.
1351
.. method:: next_toward(x, y)
1353
Returns the number closest to *x*, in direction towards *y*.
1356
.. method:: normalize(x)
1358
Reduces *x* to its simplest form.
1361
.. method:: number_class(x)
1363
Returns an indication of the class of *x*.
1368
Plus corresponds to the unary prefix plus operator in Python. This
1369
operation applies the context precision and rounding, so it is *not* an
1373
.. method:: power(x, y[, modulo])
1375
Return ``x`` to the power of ``y``, reduced modulo ``modulo`` if given.
1377
With two arguments, compute ``x**y``. If ``x`` is negative then ``y``
1378
must be integral. The result will be inexact unless ``y`` is integral and
1379
the result is finite and can be expressed exactly in 'precision' digits.
1380
The result should always be correctly rounded, using the rounding mode of
1381
the current thread's context.
1383
With three arguments, compute ``(x**y) % modulo``. For the three argument
1384
form, the following restrictions on the arguments hold:
1386
- all three arguments must be integral
1387
- ``y`` must be nonnegative
1388
- at least one of ``x`` or ``y`` must be nonzero
1389
- ``modulo`` must be nonzero and have at most 'precision' digits
1391
The value resulting from ``Context.power(x, y, modulo)`` is
1392
equal to the value that would be obtained by computing ``(x**y)
1393
% modulo`` with unbounded precision, but is computed more
1394
efficiently. The exponent of the result is zero, regardless of
1395
the exponents of ``x``, ``y`` and ``modulo``. The result is
1398
.. versionchanged:: 2.6
1399
``y`` may now be nonintegral in ``x**y``.
1400
Stricter requirements for the three-argument version.
1403
.. method:: quantize(x, y)
1405
Returns a value equal to *x* (rounded), having the exponent of *y*.
1410
Just returns 10, as this is Decimal, :)
1413
.. method:: remainder(x, y)
1415
Returns the remainder from integer division.
1417
The sign of the result, if non-zero, is the same as that of the original
1420
.. method:: remainder_near(x, y)
1422
Returns ``x - y * n``, where *n* is the integer nearest the exact value
1423
of ``x / y`` (if the result is 0 then its sign will be the sign of *x*).
1426
.. method:: rotate(x, y)
1428
Returns a rotated copy of *x*, *y* times.
1431
.. method:: same_quantum(x, y)
1433
Returns ``True`` if the two operands have the same exponent.
1436
.. method:: scaleb (x, y)
1438
Returns the first operand after adding the second value its exp.
1441
.. method:: shift(x, y)
1443
Returns a shifted copy of *x*, *y* times.
1448
Square root of a non-negative number to context precision.
1451
.. method:: subtract(x, y)
1453
Return the difference between *x* and *y*.
1456
.. method:: to_eng_string(x)
1458
Convert to a string, using engineering notation if an exponent is needed.
1460
Engineering notation has an exponent which is a multiple of 3. This
1461
can leave up to 3 digits to the left of the decimal place and may
1462
require the addition of either one or two trailing zeros.
1465
.. method:: to_integral_exact(x)
1467
Rounds to an integer.
1470
.. method:: to_sci_string(x)
1472
Converts a number to a string using scientific notation.
1474
.. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
1477
.. _decimal-signals:
1482
Signals represent conditions that arise during computation. Each corresponds to
1483
one context flag and one context trap enabler.
1485
The context flag is set whenever the condition is encountered. After the
1486
computation, flags may be checked for informational purposes (for instance, to
1487
determine whether a computation was exact). After checking the flags, be sure to
1488
clear all flags before starting the next computation.
1490
If the context's trap enabler is set for the signal, then the condition causes a
1491
Python exception to be raised. For example, if the :class:`DivisionByZero` trap
1492
is set, then a :exc:`DivisionByZero` exception is raised upon encountering the
1498
Altered an exponent to fit representation constraints.
1500
Typically, clamping occurs when an exponent falls outside the context's
1501
:attr:`Emin` and :attr:`Emax` limits. If possible, the exponent is reduced to
1502
fit by adding zeros to the coefficient.
1505
.. class:: DecimalException
1507
Base class for other signals and a subclass of :exc:`ArithmeticError`.
1510
.. class:: DivisionByZero
1512
Signals the division of a non-infinite number by zero.
1514
Can occur with division, modulo division, or when raising a number to a negative
1515
power. If this signal is not trapped, returns :const:`Infinity` or
1516
:const:`-Infinity` with the sign determined by the inputs to the calculation.
1521
Indicates that rounding occurred and the result is not exact.
1523
Signals when non-zero digits were discarded during rounding. The rounded result
1524
is returned. The signal flag or trap is used to detect when results are
1528
.. class:: InvalidOperation
1530
An invalid operation was performed.
1532
Indicates that an operation was requested that does not make sense. If not
1533
trapped, returns :const:`NaN`. Possible causes include::
1540
x._rescale( non-integer )
1551
Indicates the exponent is larger than :attr:`Emax` after rounding has
1552
occurred. If not trapped, the result depends on the rounding mode, either
1553
pulling inward to the largest representable finite number or rounding outward
1554
to :const:`Infinity`. In either case, :class:`Inexact` and :class:`Rounded`
1560
Rounding occurred though possibly no information was lost.
1562
Signaled whenever rounding discards digits; even if those digits are zero
1563
(such as rounding :const:`5.00` to :const:`5.0`). If not trapped, returns
1564
the result unchanged. This signal is used to detect loss of significant
1568
.. class:: Subnormal
1570
Exponent was lower than :attr:`Emin` prior to rounding.
1572
Occurs when an operation result is subnormal (the exponent is too small). If
1573
not trapped, returns the result unchanged.
1576
.. class:: Underflow
1578
Numerical underflow with result rounded to zero.
1580
Occurs when a subnormal result is pushed to zero by rounding. :class:`Inexact`
1581
and :class:`Subnormal` are also signaled.
1583
The following table summarizes the hierarchy of signals::
1585
exceptions.ArithmeticError(exceptions.StandardError)
1588
DivisionByZero(DecimalException, exceptions.ZeroDivisionError)
1590
Overflow(Inexact, Rounded)
1591
Underflow(Inexact, Rounded, Subnormal)
1596
.. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
1601
Floating Point Notes
1602
--------------------
1605
Mitigating round-off error with increased precision
1606
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
1608
The use of decimal floating point eliminates decimal representation error
1609
(making it possible to represent :const:`0.1` exactly); however, some operations
1610
can still incur round-off error when non-zero digits exceed the fixed precision.
1612
The effects of round-off error can be amplified by the addition or subtraction
1613
of nearly offsetting quantities resulting in loss of significance. Knuth
1614
provides two instructive examples where rounded floating point arithmetic with
1615
insufficient precision causes the breakdown of the associative and distributive
1616
properties of addition:
1618
.. doctest:: newcontext
1620
# Examples from Seminumerical Algorithms, Section 4.2.2.
1621
>>> from decimal import Decimal, getcontext
1622
>>> getcontext().prec = 8
1624
>>> u, v, w = Decimal(11111113), Decimal(-11111111), Decimal('7.51111111')
1626
Decimal('9.5111111')
1630
>>> u, v, w = Decimal(20000), Decimal(-6), Decimal('6.0000003')
1634
Decimal('0.0060000')
1636
The :mod:`decimal` module makes it possible to restore the identities by
1637
expanding the precision sufficiently to avoid loss of significance:
1639
.. doctest:: newcontext
1641
>>> getcontext().prec = 20
1642
>>> u, v, w = Decimal(11111113), Decimal(-11111111), Decimal('7.51111111')
1644
Decimal('9.51111111')
1646
Decimal('9.51111111')
1648
>>> u, v, w = Decimal(20000), Decimal(-6), Decimal('6.0000003')
1650
Decimal('0.0060000')
1652
Decimal('0.0060000')
1658
The number system for the :mod:`decimal` module provides special values
1659
including :const:`NaN`, :const:`sNaN`, :const:`-Infinity`, :const:`Infinity`,
1660
and two zeros, :const:`+0` and :const:`-0`.
1662
Infinities can be constructed directly with: ``Decimal('Infinity')``. Also,
1663
they can arise from dividing by zero when the :exc:`DivisionByZero` signal is
1664
not trapped. Likewise, when the :exc:`Overflow` signal is not trapped, infinity
1665
can result from rounding beyond the limits of the largest representable number.
1667
The infinities are signed (affine) and can be used in arithmetic operations
1668
where they get treated as very large, indeterminate numbers. For instance,
1669
adding a constant to infinity gives another infinite result.
1671
Some operations are indeterminate and return :const:`NaN`, or if the
1672
:exc:`InvalidOperation` signal is trapped, raise an exception. For example,
1673
``0/0`` returns :const:`NaN` which means "not a number". This variety of
1674
:const:`NaN` is quiet and, once created, will flow through other computations
1675
always resulting in another :const:`NaN`. This behavior can be useful for a
1676
series of computations that occasionally have missing inputs --- it allows the
1677
calculation to proceed while flagging specific results as invalid.
1679
A variant is :const:`sNaN` which signals rather than remaining quiet after every
1680
operation. This is a useful return value when an invalid result needs to
1681
interrupt a calculation for special handling.
1683
The behavior of Python's comparison operators can be a little surprising where a
1684
:const:`NaN` is involved. A test for equality where one of the operands is a
1685
quiet or signaling :const:`NaN` always returns :const:`False` (even when doing
1686
``Decimal('NaN')==Decimal('NaN')``), while a test for inequality always returns
1687
:const:`True`. An attempt to compare two Decimals using any of the ``<``,
1688
``<=``, ``>`` or ``>=`` operators will raise the :exc:`InvalidOperation` signal
1689
if either operand is a :const:`NaN`, and return :const:`False` if this signal is
1690
not trapped. Note that the General Decimal Arithmetic specification does not
1691
specify the behavior of direct comparisons; these rules for comparisons
1692
involving a :const:`NaN` were taken from the IEEE 854 standard (see Table 3 in
1693
section 5.7). To ensure strict standards-compliance, use the :meth:`compare`
1694
and :meth:`compare-signal` methods instead.
1696
The signed zeros can result from calculations that underflow. They keep the sign
1697
that would have resulted if the calculation had been carried out to greater
1698
precision. Since their magnitude is zero, both positive and negative zeros are
1699
treated as equal and their sign is informational.
1701
In addition to the two signed zeros which are distinct yet equal, there are
1702
various representations of zero with differing precisions yet equivalent in
1703
value. This takes a bit of getting used to. For an eye accustomed to
1704
normalized floating point representations, it is not immediately obvious that
1705
the following calculation returns a value equal to zero:
1707
>>> 1 / Decimal('Infinity')
1708
Decimal('0E-1000000026')
1710
.. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
1713
.. _decimal-threads:
1715
Working with threads
1716
--------------------
1718
The :func:`getcontext` function accesses a different :class:`Context` object for
1719
each thread. Having separate thread contexts means that threads may make
1720
changes (such as ``getcontext.prec=10``) without interfering with other threads.
1722
Likewise, the :func:`setcontext` function automatically assigns its target to
1725
If :func:`setcontext` has not been called before :func:`getcontext`, then
1726
:func:`getcontext` will automatically create a new context for use in the
1729
The new context is copied from a prototype context called *DefaultContext*. To
1730
control the defaults so that each thread will use the same values throughout the
1731
application, directly modify the *DefaultContext* object. This should be done
1732
*before* any threads are started so that there won't be a race condition between
1733
threads calling :func:`getcontext`. For example::
1735
# Set applicationwide defaults for all threads about to be launched
1736
DefaultContext.prec = 12
1737
DefaultContext.rounding = ROUND_DOWN
1738
DefaultContext.traps = ExtendedContext.traps.copy()
1739
DefaultContext.traps[InvalidOperation] = 1
1740
setcontext(DefaultContext)
1742
# Afterwards, the threads can be started
1748
.. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
1751
.. _decimal-recipes:
1756
Here are a few recipes that serve as utility functions and that demonstrate ways
1757
to work with the :class:`Decimal` class::
1759
def moneyfmt(value, places=2, curr='', sep=',', dp='.',
1760
pos='', neg='-', trailneg=''):
1761
"""Convert Decimal to a money formatted string.
1763
places: required number of places after the decimal point
1764
curr: optional currency symbol before the sign (may be blank)
1765
sep: optional grouping separator (comma, period, space, or blank)
1766
dp: decimal point indicator (comma or period)
1767
only specify as blank when places is zero
1768
pos: optional sign for positive numbers: '+', space or blank
1769
neg: optional sign for negative numbers: '-', '(', space or blank
1770
trailneg:optional trailing minus indicator: '-', ')', space or blank
1772
>>> d = Decimal('-1234567.8901')
1773
>>> moneyfmt(d, curr='$')
1775
>>> moneyfmt(d, places=0, sep='.', dp='', neg='', trailneg='-')
1777
>>> moneyfmt(d, curr='$', neg='(', trailneg=')')
1779
>>> moneyfmt(Decimal(123456789), sep=' ')
1781
>>> moneyfmt(Decimal('-0.02'), neg='<', trailneg='>')
1785
q = Decimal(10) ** -places # 2 places --> '0.01'
1786
sign, digits, exp = value.quantize(q).as_tuple()
1788
digits = map(str, digits)
1789
build, next = result.append, digits.pop
1792
for i in range(places):
1793
build(next() if digits else '0')
1801
if i == 3 and digits:
1805
build(neg if sign else pos)
1806
return ''.join(reversed(result))
1809
"""Compute Pi to the current precision.
1812
3.141592653589793238462643383
1815
getcontext().prec += 2 # extra digits for intermediate steps
1816
three = Decimal(3) # substitute "three=3.0" for regular floats
1817
lasts, t, s, n, na, d, da = 0, three, 3, 1, 0, 0, 24
1824
getcontext().prec -= 2
1825
return +s # unary plus applies the new precision
1828
"""Return e raised to the power of x. Result type matches input type.
1830
>>> print exp(Decimal(1))
1831
2.718281828459045235360287471
1832
>>> print exp(Decimal(2))
1833
7.389056098930650227230427461
1840
getcontext().prec += 2
1841
i, lasts, s, fact, num = 0, 0, 1, 1, 1
1848
getcontext().prec -= 2
1852
"""Return the cosine of x as measured in radians.
1854
>>> print cos(Decimal('0.5'))
1855
0.8775825618903727161162815826
1858
>>> print cos(0.5+0j)
1862
getcontext().prec += 2
1863
i, lasts, s, fact, num, sign = 0, 0, 1, 1, 1, 1
1870
s += num / fact * sign
1871
getcontext().prec -= 2
1875
"""Return the sine of x as measured in radians.
1877
>>> print sin(Decimal('0.5'))
1878
0.4794255386042030002732879352
1881
>>> print sin(0.5+0j)
1885
getcontext().prec += 2
1886
i, lasts, s, fact, num, sign = 1, 0, x, 1, x, 1
1893
s += num / fact * sign
1894
getcontext().prec -= 2
1898
.. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
1906
Q. It is cumbersome to type ``decimal.Decimal('1234.5')``. Is there a way to
1907
minimize typing when using the interactive interpreter?
1909
A. Some users abbreviate the constructor to just a single letter:
1911
>>> D = decimal.Decimal
1912
>>> D('1.23') + D('3.45')
1915
Q. In a fixed-point application with two decimal places, some inputs have many
1916
places and need to be rounded. Others are not supposed to have excess digits
1917
and need to be validated. What methods should be used?
1919
A. The :meth:`quantize` method rounds to a fixed number of decimal places. If
1920
the :const:`Inexact` trap is set, it is also useful for validation:
1922
>>> TWOPLACES = Decimal(10) ** -2 # same as Decimal('0.01')
1924
>>> # Round to two places
1925
>>> Decimal('3.214').quantize(TWOPLACES)
1928
>>> # Validate that a number does not exceed two places
1929
>>> Decimal('3.21').quantize(TWOPLACES, context=Context(traps=[Inexact]))
1932
>>> Decimal('3.214').quantize(TWOPLACES, context=Context(traps=[Inexact]))
1933
Traceback (most recent call last):
1937
Q. Once I have valid two place inputs, how do I maintain that invariant
1938
throughout an application?
1940
A. Some operations like addition, subtraction, and multiplication by an integer
1941
will automatically preserve fixed point. Others operations, like division and
1942
non-integer multiplication, will change the number of decimal places and need to
1943
be followed-up with a :meth:`quantize` step:
1945
>>> a = Decimal('102.72') # Initial fixed-point values
1946
>>> b = Decimal('3.17')
1947
>>> a + b # Addition preserves fixed-point
1951
>>> a * 42 # So does integer multiplication
1953
>>> (a * b).quantize(TWOPLACES) # Must quantize non-integer multiplication
1955
>>> (b / a).quantize(TWOPLACES) # And quantize division
1958
In developing fixed-point applications, it is convenient to define functions
1959
to handle the :meth:`quantize` step:
1961
>>> def mul(x, y, fp=TWOPLACES):
1962
... return (x * y).quantize(fp)
1963
>>> def div(x, y, fp=TWOPLACES):
1964
... return (x / y).quantize(fp)
1966
>>> mul(a, b) # Automatically preserve fixed-point
1971
Q. There are many ways to express the same value. The numbers :const:`200`,
1972
:const:`200.000`, :const:`2E2`, and :const:`.02E+4` all have the same value at
1973
various precisions. Is there a way to transform them to a single recognizable
1976
A. The :meth:`normalize` method maps all equivalent values to a single
1979
>>> values = map(Decimal, '200 200.000 2E2 .02E+4'.split())
1980
>>> [v.normalize() for v in values]
1981
[Decimal('2E+2'), Decimal('2E+2'), Decimal('2E+2'), Decimal('2E+2')]
1983
Q. Some decimal values always print with exponential notation. Is there a way
1984
to get a non-exponential representation?
1986
A. For some values, exponential notation is the only way to express the number
1987
of significant places in the coefficient. For example, expressing
1988
:const:`5.0E+3` as :const:`5000` keeps the value constant but cannot show the
1989
original's two-place significance.
1991
If an application does not care about tracking significance, it is easy to
1992
remove the exponent and trailing zeros, losing significance, but keeping the
1995
def remove_exponent(d):
1996
'''Remove exponent and trailing zeros.
1998
>>> remove_exponent(Decimal('5E+3'))
2002
return d.quantize(Decimal(1)) if d == d.to_integral() else d.normalize()
2004
Q. Is there a way to convert a regular float to a :class:`Decimal`?
2006
A. Yes, any binary floating point number can be exactly expressed as a
2007
Decimal though an exact conversion may take more precision than intuition would
2012
>>> Decimal(math.pi)
2013
Decimal('3.141592653589793115997963468544185161590576171875')
2015
Q. Within a complex calculation, how can I make sure that I haven't gotten a
2016
spurious result because of insufficient precision or rounding anomalies.
2018
A. The decimal module makes it easy to test results. A best practice is to
2019
re-run calculations using greater precision and with various rounding modes.
2020
Widely differing results indicate insufficient precision, rounding mode issues,
2021
ill-conditioned inputs, or a numerically unstable algorithm.
2023
Q. I noticed that context precision is applied to the results of operations but
2024
not to the inputs. Is there anything to watch out for when mixing values of
2025
different precisions?
2027
A. Yes. The principle is that all values are considered to be exact and so is
2028
the arithmetic on those values. Only the results are rounded. The advantage
2029
for inputs is that "what you type is what you get". A disadvantage is that the
2030
results can look odd if you forget that the inputs haven't been rounded:
2032
.. doctest:: newcontext
2034
>>> getcontext().prec = 3
2035
>>> Decimal('3.104') + Decimal('2.104')
2037
>>> Decimal('3.104') + Decimal('0.000') + Decimal('2.104')
2040
The solution is either to increase precision or to force rounding of inputs
2041
using the unary plus operation:
2043
.. doctest:: newcontext
2045
>>> getcontext().prec = 3
2046
>>> +Decimal('1.23456789') # unary plus triggers rounding
2049
Alternatively, inputs can be rounded upon creation using the
2050
:meth:`Context.create_decimal` method:
2052
>>> Context(prec=5, rounding=ROUND_DOWN).create_decimal('1.2345678')
added
removed
python-2.7.12-docs-html/_sources/library/decimal.txt
