2213 lines
76 KiB
ReStructuredText
2213 lines
76 KiB
ReStructuredText
:mod:`decimal` --- Decimal fixed point and floating point arithmetic
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====================================================================
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.. module:: decimal
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:synopsis: Implementation of the General Decimal Arithmetic Specification.
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.. moduleauthor:: Eric Price <eprice at tjhsst.edu>
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.. 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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.. moduleauthor:: Stefan Krah <skrah at bytereef.org>
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.. sectionauthor:: Raymond D. Hettinger <python at rcn.com>
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**Source code:** :source:`Lib/decimal.py`
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.. import modules for testing inline doctests with the Sphinx doctest builder
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.. testsetup:: *
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import decimal
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import math
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from decimal import *
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# make sure each group gets a fresh context
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setcontext(Context())
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.. testcleanup:: *
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# make sure other tests (outside this file) get a fresh context
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setcontext(Context())
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--------------
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The :mod:`decimal` module provides support for fast correctly-rounded
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decimal floating point arithmetic. It offers several advantages over the
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:class:`float` datatype:
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* Decimal "is based on a floating-point model which was designed with people
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in mind, and necessarily has a paramount guiding principle -- computers must
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provide an arithmetic that works in the same way as the arithmetic that
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people learn at school." -- excerpt from the decimal arithmetic specification.
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* Decimal numbers can be represented exactly. In contrast, numbers like
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:const:`1.1` and :const:`2.2` do not have exact representations in binary
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floating point. End users typically would not expect ``1.1 + 2.2`` to display
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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
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prevent reliable equality testing and differences can accumulate. For this
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reason, decimal is preferred in accounting applications which have strict
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equality invariants.
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* 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
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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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a given problem:
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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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Decimal('0.142857')
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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
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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`, :const:`Underflow` and :const:`FloatOperation`.
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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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.. seealso::
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* IBM's General Decimal Arithmetic Specification, `The General Decimal Arithmetic
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Specification <http://speleotrove.com/decimal/decarith.html>`_.
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.. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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.. _decimal-tutorial:
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Quick-start Tutorial
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--------------------
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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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>>> getcontext()
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Context(prec=28, rounding=ROUND_HALF_EVEN, Emin=-999999, Emax=999999,
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capitals=1, clamp=0, flags=[], traps=[Overflow, DivisionByZero,
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InvalidOperation])
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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(10)
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Decimal('10')
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>>> Decimal('3.14')
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Decimal('3.14')
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>>> Decimal(3.14)
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Decimal('3.140000000000000124344978758017532527446746826171875')
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>>> Decimal((0, (3, 1, 4), -2))
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Decimal('3.14')
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>>> Decimal(str(2.0 ** 0.5))
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Decimal('1.4142135623730951')
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>>> Decimal(2) ** Decimal('0.5')
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Decimal('1.414213562373095048801688724')
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>>> Decimal('NaN')
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Decimal('NaN')
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>>> Decimal('-Infinity')
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Decimal('-Infinity')
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If the :exc:`FloatOperation` signal is trapped, accidental mixing of
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decimals and floats in constructors or ordering comparisons raises
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an exception::
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>>> c = getcontext()
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>>> c.traps[FloatOperation] = True
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>>> Decimal(3.14)
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Traceback (most recent call last):
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File "<stdin>", line 1, in <module>
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decimal.FloatOperation: [<class 'decimal.FloatOperation'>]
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>>> Decimal('3.5') < 3.7
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Traceback (most recent call last):
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File "<stdin>", line 1, in <module>
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decimal.FloatOperation: [<class 'decimal.FloatOperation'>]
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>>> Decimal('3.5') == 3.5
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True
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.. versionadded:: 3.3
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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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operations.
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.. doctest:: newcontext
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>>> getcontext().prec = 6
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>>> Decimal('3.0')
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Decimal('3.0')
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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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Decimal('5.85987')
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>>> getcontext().rounding = ROUND_UP
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>>> Decimal('3.1415926535') + Decimal('2.7182818285')
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Decimal('5.85988')
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If the internal limits of the C version are exceeded, constructing
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a decimal raises :class:`InvalidOperation`::
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>>> Decimal("1e9999999999999999999")
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Traceback (most recent call last):
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File "<stdin>", line 1, in <module>
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decimal.InvalidOperation: [<class 'decimal.InvalidOperation'>]
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.. versionchanged:: 3.3
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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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.. doctest::
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:options: +NORMALIZE_WHITESPACE
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>>> data = list(map(Decimal, '1.34 1.87 3.45 2.35 1.00 0.03 9.25'.split()))
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>>> max(data)
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Decimal('9.25')
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>>> min(data)
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Decimal('0.03')
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>>> sorted(data)
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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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>>> sum(data)
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Decimal('19.29')
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>>> a,b,c = data[:3]
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>>> str(a)
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'1.34'
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>>> float(a)
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1.34
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>>> round(a, 1)
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Decimal('1.3')
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>>> int(a)
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1
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>>> a * 5
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Decimal('6.70')
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>>> a * b
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Decimal('2.5058')
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>>> c % a
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Decimal('0.77')
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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(1).exp()
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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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Decimal('1')
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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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places:
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>>> Decimal('7.325').quantize(Decimal('.01'), rounding=ROUND_DOWN)
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Decimal('7.32')
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>>> Decimal('7.325').quantize(Decimal('1.'), rounding=ROUND_UP)
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Decimal('8')
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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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applications.
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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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function.
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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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enabled:
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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)
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>>> setcontext(myothercontext)
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>>> Decimal(1) / Decimal(7)
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Decimal('0.142857142857142857142857142857142857142857142857142857142857')
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>>> ExtendedContext
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Context(prec=9, rounding=ROUND_HALF_EVEN, Emin=-999999, Emax=999999,
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capitals=1, clamp=0, 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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Decimal('Infinity')
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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)
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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)
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>>> getcontext().clear_flags()
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>>> Decimal(355) / Decimal(113)
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Decimal('3.14159292')
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>>> getcontext()
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Context(prec=9, rounding=ROUND_HALF_EVEN, Emin=-999999, Emax=999999,
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capitals=1, clamp=0, flags=[Inexact, Rounded], traps=[])
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The *flags* entry shows that the rational approximation to :const:`Pi` was
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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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context:
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.. doctest:: newcontext
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>>> setcontext(ExtendedContext)
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>>> Decimal(1) / Decimal(0)
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Decimal('Infinity')
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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
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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
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types.
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.. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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.. _decimal-decimal:
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Decimal objects
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---------------
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.. class:: Decimal(value="0", context=None)
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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`
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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, as well as underscores throughout, are removed::
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sign ::= '+' | '-'
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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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Other Unicode decimal digits are also permitted where ``digit``
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appears above. These include decimal digits from various other
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alphabets (for example, Arabic-Indic and Devanāgarī digits) along
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with the fullwidth digits ``'\uff10'`` through ``'\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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converts to
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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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only three.
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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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:const:`NaN`.
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Once constructed, :class:`Decimal` objects are immutable.
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.. versionchanged:: 3.2
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The argument to the constructor is now permitted to be a :class:`float`
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instance.
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.. versionchanged:: 3.3
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:class:`float` arguments raise an exception if the :exc:`FloatOperation`
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trap is set. By default the trap is off.
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.. versionchanged:: 3.6
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Underscores are allowed for grouping, as with integral and floating-point
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literals in code.
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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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:class:`int`).
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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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>>> (-7) % 4
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1
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>>> Decimal(-7) % Decimal(4)
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Decimal('-3')
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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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>>> -7 // 4
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-2
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>>> Decimal(-7) // Decimal(4)
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Decimal('-1')
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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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specification.
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Decimal objects cannot generally be combined with floats or
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instances of :class:`fractions.Fraction` in arithmetic operations:
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an attempt to add a :class:`Decimal` to a :class:`float`, for
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example, will raise a :exc:`TypeError`. However, it is possible to
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use Python's comparison operators to compare a :class:`Decimal`
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instance ``x`` with another number ``y``. This avoids confusing results
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when doing equality comparisons between numbers of different types.
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.. versionchanged:: 3.2
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Mixed-type comparisons between :class:`Decimal` instances and other
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numeric types are now fully supported.
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In addition to the standard numeric properties, decimal floating point
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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
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position of the most significant digit with respect to the decimal point.
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.. method:: as_integer_ratio()
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Return a pair ``(n, d)`` of integers that represent the given
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:class:`Decimal` instance as a fraction, in lowest terms and
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with a positive denominator::
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>>> Decimal('-3.14').as_integer_ratio()
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(-157, 50)
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The conversion is exact. Raise OverflowError on infinities and ValueError
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on NaNs.
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.. versionadded:: 3.6
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.. method:: as_tuple()
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Return a :term:`named tuple` representation of the number:
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``DecimalTuple(sign, digits, exponent)``.
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.. method:: canonical()
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|
|
Return the canonical encoding of the argument. Currently, the encoding of
|
|
a :class:`Decimal` instance is always canonical, so this operation returns
|
|
its argument unchanged.
|
|
|
|
.. method:: compare(other, context=None)
|
|
|
|
Compare the values of two Decimal instances. :meth:`compare` returns a
|
|
Decimal instance, 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')
|
|
|
|
.. method:: compare_signal(other, context=None)
|
|
|
|
This operation is identical to the :meth:`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.
|
|
|
|
.. method:: compare_total(other, context=None)
|
|
|
|
Compare two operands using their abstract representation rather than their
|
|
numerical value. Similar to the :meth:`compare` method, but the result
|
|
gives a total ordering on :class:`Decimal` instances. Two
|
|
:class:`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.
|
|
|
|
This operation is unaffected by context and is quiet: no flags are changed
|
|
and no rounding is performed. As an exception, the C version may raise
|
|
InvalidOperation if the second operand cannot be converted exactly.
|
|
|
|
.. method:: compare_total_mag(other, context=None)
|
|
|
|
Compare two operands using their abstract representation rather than their
|
|
value as in :meth:`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())``.
|
|
|
|
This operation is unaffected by context and is quiet: no flags are changed
|
|
and no rounding is performed. As an exception, the C version may raise
|
|
InvalidOperation if the second operand cannot be converted exactly.
|
|
|
|
.. method:: conjugate()
|
|
|
|
Just returns self, this method is only to comply with the Decimal
|
|
Specification.
|
|
|
|
.. method:: 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.
|
|
|
|
.. method:: 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.
|
|
|
|
.. method:: copy_sign(other, context=None)
|
|
|
|
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 context and is quiet: no flags are changed
|
|
and no rounding is performed. As an exception, the C version may raise
|
|
InvalidOperation if the second operand cannot be converted exactly.
|
|
|
|
.. method:: exp(context=None)
|
|
|
|
Return the value of the (natural) exponential function ``e**x`` at the
|
|
given number. The result is correctly rounded using the
|
|
:const:`ROUND_HALF_EVEN` rounding mode.
|
|
|
|
>>> Decimal(1).exp()
|
|
Decimal('2.718281828459045235360287471')
|
|
>>> Decimal(321).exp()
|
|
Decimal('2.561702493119680037517373933E+139')
|
|
|
|
.. method:: from_float(f)
|
|
|
|
Classmethod that converts a float to a decimal number, exactly.
|
|
|
|
Note `Decimal.from_float(0.1)` is not the same as `Decimal('0.1')`.
|
|
Since 0.1 is not exactly representable in binary floating point, the
|
|
value is stored as the nearest representable value which is
|
|
`0x1.999999999999ap-4`. That equivalent value in decimal is
|
|
`0.1000000000000000055511151231257827021181583404541015625`.
|
|
|
|
.. note:: From Python 3.2 onwards, a :class:`Decimal` instance
|
|
can also be constructed directly from a :class:`float`.
|
|
|
|
.. doctest::
|
|
|
|
>>> Decimal.from_float(0.1)
|
|
Decimal('0.1000000000000000055511151231257827021181583404541015625')
|
|
>>> Decimal.from_float(float('nan'))
|
|
Decimal('NaN')
|
|
>>> Decimal.from_float(float('inf'))
|
|
Decimal('Infinity')
|
|
>>> Decimal.from_float(float('-inf'))
|
|
Decimal('-Infinity')
|
|
|
|
.. versionadded:: 3.1
|
|
|
|
.. method:: fma(other, third, context=None)
|
|
|
|
Fused multiply-add. Return self*other+third with no rounding of the
|
|
intermediate product self*other.
|
|
|
|
>>> Decimal(2).fma(3, 5)
|
|
Decimal('11')
|
|
|
|
.. method:: is_canonical()
|
|
|
|
Return :const:`True` if the argument is canonical and :const:`False`
|
|
otherwise. Currently, a :class:`Decimal` instance is always canonical, so
|
|
this operation always returns :const:`True`.
|
|
|
|
.. method:: is_finite()
|
|
|
|
Return :const:`True` if the argument is a finite number, and
|
|
:const:`False` if the argument is an infinity or a NaN.
|
|
|
|
.. method:: is_infinite()
|
|
|
|
Return :const:`True` if the argument is either positive or negative
|
|
infinity and :const:`False` otherwise.
|
|
|
|
.. method:: is_integer()
|
|
|
|
Return :const:`True` if the argument is a finite integral value and
|
|
:const:`False` otherwise.
|
|
|
|
.. versionadded:: 3.10
|
|
|
|
.. method:: is_nan()
|
|
|
|
Return :const:`True` if the argument is a (quiet or signaling) NaN and
|
|
:const:`False` otherwise.
|
|
|
|
.. method:: is_normal(context=None)
|
|
|
|
Return :const:`True` if the argument is a *normal* finite number. Return
|
|
:const:`False` if the argument is zero, subnormal, infinite or a NaN.
|
|
|
|
.. method:: is_qnan()
|
|
|
|
Return :const:`True` if the argument is a quiet NaN, and
|
|
:const:`False` otherwise.
|
|
|
|
.. method:: is_signed()
|
|
|
|
Return :const:`True` if the argument has a negative sign and
|
|
:const:`False` otherwise. Note that zeros and NaNs can both carry signs.
|
|
|
|
.. method:: is_snan()
|
|
|
|
Return :const:`True` if the argument is a signaling NaN and :const:`False`
|
|
otherwise.
|
|
|
|
.. method:: is_subnormal(context=None)
|
|
|
|
Return :const:`True` if the argument is subnormal, and :const:`False`
|
|
otherwise.
|
|
|
|
.. method:: is_zero()
|
|
|
|
Return :const:`True` if the argument is a (positive or negative) zero and
|
|
:const:`False` otherwise.
|
|
|
|
.. method:: ln(context=None)
|
|
|
|
Return the natural (base e) logarithm of the operand. The result is
|
|
correctly rounded using the :const:`ROUND_HALF_EVEN` rounding mode.
|
|
|
|
.. method:: log10(context=None)
|
|
|
|
Return the base ten logarithm of the operand. The result is correctly
|
|
rounded using the :const:`ROUND_HALF_EVEN` rounding mode.
|
|
|
|
.. method:: logb(context=None)
|
|
|
|
For a nonzero number, return the adjusted exponent of its operand as a
|
|
:class:`Decimal` instance. If the operand is a zero then
|
|
``Decimal('-Infinity')`` is returned and the :const:`DivisionByZero` flag
|
|
is raised. If the operand is an infinity then ``Decimal('Infinity')`` is
|
|
returned.
|
|
|
|
.. method:: logical_and(other, context=None)
|
|
|
|
:meth:`logical_and` is a logical operation which takes two *logical
|
|
operands* (see :ref:`logical_operands_label`). The result is the
|
|
digit-wise ``and`` of the two operands.
|
|
|
|
.. method:: logical_invert(context=None)
|
|
|
|
:meth:`logical_invert` is a logical operation. The
|
|
result is the digit-wise inversion of the operand.
|
|
|
|
.. method:: logical_or(other, context=None)
|
|
|
|
:meth:`logical_or` is a logical operation which takes two *logical
|
|
operands* (see :ref:`logical_operands_label`). The result is the
|
|
digit-wise ``or`` of the two operands.
|
|
|
|
.. method:: logical_xor(other, context=None)
|
|
|
|
:meth:`logical_xor` is a logical operation which takes two *logical
|
|
operands* (see :ref:`logical_operands_label`). The result is the
|
|
digit-wise exclusive or of the two operands.
|
|
|
|
.. method:: max(other, context=None)
|
|
|
|
Like ``max(self, other)`` except that the context rounding rule is applied
|
|
before returning and that :const:`NaN` values are either signaled or
|
|
ignored (depending on the context and whether they are signaling or
|
|
quiet).
|
|
|
|
.. method:: max_mag(other, context=None)
|
|
|
|
Similar to the :meth:`.max` method, but the comparison is done using the
|
|
absolute values of the operands.
|
|
|
|
.. method:: min(other, context=None)
|
|
|
|
Like ``min(self, other)`` except that the context rounding rule is applied
|
|
before returning and that :const:`NaN` values are either signaled or
|
|
ignored (depending on the context and whether they are signaling or
|
|
quiet).
|
|
|
|
.. method:: min_mag(other, context=None)
|
|
|
|
Similar to the :meth:`.min` method, but the comparison is done using the
|
|
absolute values of the operands.
|
|
|
|
.. method:: next_minus(context=None)
|
|
|
|
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.
|
|
|
|
.. method:: next_plus(context=None)
|
|
|
|
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.
|
|
|
|
.. method:: next_toward(other, context=None)
|
|
|
|
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.
|
|
|
|
.. method:: normalize(context=None)
|
|
|
|
Normalize the number by stripping the rightmost trailing zeros and
|
|
converting any result equal to :const:`Decimal('0')` to
|
|
:const:`Decimal('0e0')`. Used for producing canonical values for attributes
|
|
of an equivalence class. For example, ``Decimal('32.100')`` and
|
|
``Decimal('0.321000e+2')`` both normalize to the equivalent value
|
|
``Decimal('32.1')``.
|
|
|
|
.. method:: number_class(context=None)
|
|
|
|
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.
|
|
|
|
.. method:: quantize(exp, rounding=None, context=None)
|
|
|
|
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
|
|
:const:`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.
|
|
|
|
An error is returned whenever the resulting exponent is greater than
|
|
:attr:`Emax` or less than :attr:`Etiny`.
|
|
|
|
.. method:: radix()
|
|
|
|
Return ``Decimal(10)``, the radix (base) in which the :class:`Decimal`
|
|
class does all its arithmetic. Included for compatibility with the
|
|
specification.
|
|
|
|
.. method:: remainder_near(other, context=None)
|
|
|
|
Return the remainder from dividing *self* by *other*. This differs from
|
|
``self % other`` in that the sign of the remainder is chosen so as to
|
|
minimize its absolute value. More precisely, the return value is
|
|
``self - n * other`` where ``n`` is the integer nearest to the exact
|
|
value of ``self / other``, and if two integers are equally near then the
|
|
even one is chosen.
|
|
|
|
If the result is zero then its sign will be the sign of *self*.
|
|
|
|
>>> Decimal(18).remainder_near(Decimal(10))
|
|
Decimal('-2')
|
|
>>> Decimal(25).remainder_near(Decimal(10))
|
|
Decimal('5')
|
|
>>> Decimal(35).remainder_near(Decimal(10))
|
|
Decimal('-5')
|
|
|
|
.. method:: rotate(other, context=None)
|
|
|
|
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.
|
|
|
|
.. method:: same_quantum(other, context=None)
|
|
|
|
Test whether self and other have the same exponent or whether both are
|
|
:const:`NaN`.
|
|
|
|
This operation is unaffected by context and is quiet: no flags are changed
|
|
and no rounding is performed. As an exception, the C version may raise
|
|
InvalidOperation if the second operand cannot be converted exactly.
|
|
|
|
.. method:: scaleb(other, context=None)
|
|
|
|
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.
|
|
|
|
.. method:: shift(other, context=None)
|
|
|
|
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.
|
|
|
|
.. method:: sqrt(context=None)
|
|
|
|
Return the square root of the argument to full precision.
|
|
|
|
|
|
.. method:: to_eng_string(context=None)
|
|
|
|
Convert to a string, using engineering notation if an exponent is needed.
|
|
|
|
Engineering notation has an exponent which is a multiple of 3. This
|
|
can leave up to 3 digits to the left of the decimal place and may
|
|
require the addition of either one or two trailing zeros.
|
|
|
|
For example, this converts ``Decimal('123E+1')`` to ``Decimal('1.23E+3')``.
|
|
|
|
.. method:: to_integral(rounding=None, context=None)
|
|
|
|
Identical to the :meth:`to_integral_value` method. The ``to_integral``
|
|
name has been kept for compatibility with older versions.
|
|
|
|
.. method:: to_integral_exact(rounding=None, context=None)
|
|
|
|
Round to the nearest integer, signaling :const:`Inexact` or
|
|
:const:`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.
|
|
|
|
.. method:: to_integral_value(rounding=None, context=None)
|
|
|
|
Round to the nearest integer without signaling :const:`Inexact` or
|
|
:const:`Rounded`. If given, applies *rounding*; otherwise, uses the
|
|
rounding method in either the supplied *context* or the current context.
|
|
|
|
|
|
.. _logical_operands_label:
|
|
|
|
Logical operands
|
|
^^^^^^^^^^^^^^^^
|
|
|
|
The :meth:`logical_and`, :meth:`logical_invert`, :meth:`logical_or`,
|
|
and :meth:`logical_xor` methods expect their arguments to be *logical
|
|
operands*. A *logical operand* is a :class:`Decimal` instance whose
|
|
exponent and sign are both zero, and whose digits are all either
|
|
:const:`0` or :const:`1`.
|
|
|
|
.. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
|
|
.. _decimal-context:
|
|
|
|
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
|
|
:func:`getcontext` and :func:`setcontext` functions:
|
|
|
|
|
|
.. function:: getcontext()
|
|
|
|
Return the current context for the active thread.
|
|
|
|
|
|
.. function:: setcontext(c)
|
|
|
|
Set the current context for the active thread to *c*.
|
|
|
|
You can also use the :keyword:`with` statement and the :func:`localcontext`
|
|
function to temporarily change the active context.
|
|
|
|
.. function:: localcontext(ctx=None)
|
|
|
|
Return a context manager that will set the current context for the active thread
|
|
to a copy of *ctx* 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.
|
|
|
|
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 :class:`Context` constructor
|
|
described below. In addition, the module provides three pre-made contexts:
|
|
|
|
|
|
.. class:: BasicContext
|
|
|
|
This is a standard context defined by the General Decimal Arithmetic
|
|
Specification. Precision is set to nine. Rounding is set to
|
|
:const:`ROUND_HALF_UP`. All flags are cleared. All traps are enabled (treated
|
|
as exceptions) except :const:`Inexact`, :const:`Rounded`, and
|
|
:const:`Subnormal`.
|
|
|
|
Because many of the traps are enabled, this context is useful for debugging.
|
|
|
|
|
|
.. class:: ExtendedContext
|
|
|
|
This is a standard context defined by the General Decimal Arithmetic
|
|
Specification. Precision is set to nine. Rounding is set to
|
|
:const:`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 :const:`NaN` or :const:`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:: DefaultContext
|
|
|
|
This context is used by the :class:`Context` constructor as a prototype for new
|
|
contexts. Changing a field (such a precision) has the effect of changing the
|
|
default for new contexts created by the :class:`Context` constructor.
|
|
|
|
This context is most useful in multi-threaded environments. Changing one of the
|
|
fields before threads are started has the effect of setting system-wide
|
|
defaults. Changing the fields after threads have started is not recommended as
|
|
it would require thread synchronization to prevent race conditions.
|
|
|
|
In single threaded environments, it is preferable to not use this context at
|
|
all. Instead, simply create contexts explicitly as described below.
|
|
|
|
The default values are :attr:`prec`\ =\ :const:`28`,
|
|
:attr:`rounding`\ =\ :const:`ROUND_HALF_EVEN`,
|
|
and enabled traps for :class:`Overflow`, :class:`InvalidOperation`, and
|
|
:class:`DivisionByZero`.
|
|
|
|
In addition to the three supplied contexts, new contexts can be created with the
|
|
:class:`Context` constructor.
|
|
|
|
|
|
.. class:: Context(prec=None, rounding=None, Emin=None, Emax=None, capitals=None, clamp=None, flags=None, traps=None)
|
|
|
|
Creates a new context. If a field is not specified or is :const:`None`, the
|
|
default values are copied from the :const:`DefaultContext`. If the *flags*
|
|
field is not specified or is :const:`None`, all flags are cleared.
|
|
|
|
*prec* is an integer in the range [:const:`1`, :const:`MAX_PREC`] that sets
|
|
the precision for arithmetic operations in the context.
|
|
|
|
The *rounding* option is one of the constants listed in the section
|
|
`Rounding Modes`_.
|
|
|
|
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. *Emin* must be in the range [:const:`MIN_EMIN`, :const:`0`],
|
|
*Emax* in the range [:const:`0`, :const:`MAX_EMAX`].
|
|
|
|
The *capitals* field is either :const:`0` or :const:`1` (the default). If set to
|
|
:const:`1`, exponents are printed with a capital :const:`E`; otherwise, a
|
|
lowercase :const:`e` is used: :const:`Decimal('6.02e+23')`.
|
|
|
|
The *clamp* field is either :const:`0` (the default) or :const:`1`.
|
|
If set to :const:`1`, the exponent ``e`` of a :class:`Decimal`
|
|
instance representable in this context is strictly limited to the
|
|
range ``Emin - prec + 1 <= e <= Emax - prec + 1``. If *clamp* is
|
|
:const:`0` then a weaker condition holds: the adjusted exponent of
|
|
the :class:`Decimal` instance is at most ``Emax``. When *clamp* is
|
|
:const:`1`, a large normal number will, where possible, have its
|
|
exponent reduced and a corresponding number of zeros added to its
|
|
coefficient, in order to fit the exponent constraints; this
|
|
preserves the value of the number but loses information about
|
|
significant trailing zeros. For example::
|
|
|
|
>>> Context(prec=6, Emax=999, clamp=1).create_decimal('1.23e999')
|
|
Decimal('1.23000E+999')
|
|
|
|
A *clamp* value of :const:`1` allows compatibility with the
|
|
fixed-width decimal interchange formats specified in IEEE 754.
|
|
|
|
The :class:`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 :class:`Decimal` methods described above (with
|
|
the exception of the :meth:`adjusted` and :meth:`as_tuple` methods) there is
|
|
a corresponding :class:`Context` method. For example, for a :class:`Context`
|
|
instance ``C`` and :class:`Decimal` instance ``x``, ``C.exp(x)`` is
|
|
equivalent to ``x.exp(context=C)``. Each :class:`Context` method accepts a
|
|
Python integer (an instance of :class:`int`) anywhere that a
|
|
Decimal instance is accepted.
|
|
|
|
|
|
.. method:: clear_flags()
|
|
|
|
Resets all of the flags to :const:`0`.
|
|
|
|
.. method:: clear_traps()
|
|
|
|
Resets all of the traps to :const:`0`.
|
|
|
|
.. versionadded:: 3.3
|
|
|
|
.. method:: copy()
|
|
|
|
Return a duplicate of the context.
|
|
|
|
.. method:: copy_decimal(num)
|
|
|
|
Return a copy of the Decimal instance num.
|
|
|
|
.. method:: create_decimal(num)
|
|
|
|
Creates a new Decimal instance from *num* but using *self* as
|
|
context. Unlike the :class:`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:
|
|
|
|
.. doctest:: newcontext
|
|
|
|
>>> 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 or
|
|
underscores are permitted.
|
|
|
|
.. method:: create_decimal_from_float(f)
|
|
|
|
Creates a new Decimal instance from a float *f* but rounding using *self*
|
|
as the context. Unlike the :meth:`Decimal.from_float` class method,
|
|
the context precision, rounding method, flags, and traps are applied to
|
|
the conversion.
|
|
|
|
.. doctest::
|
|
|
|
>>> context = Context(prec=5, rounding=ROUND_DOWN)
|
|
>>> context.create_decimal_from_float(math.pi)
|
|
Decimal('3.1415')
|
|
>>> context = Context(prec=5, traps=[Inexact])
|
|
>>> context.create_decimal_from_float(math.pi)
|
|
Traceback (most recent call last):
|
|
...
|
|
decimal.Inexact: None
|
|
|
|
.. versionadded:: 3.1
|
|
|
|
.. method:: 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 :const:`Etiny`.
|
|
|
|
.. method:: Etop()
|
|
|
|
Returns a value equal to ``Emax - prec + 1``.
|
|
|
|
The usual approach to working with decimals is to create :class:`Decimal`
|
|
instances and then apply arithmetic operations which take place within the
|
|
current context for the active thread. An alternative approach is to use
|
|
context methods for calculating within a specific context. The methods are
|
|
similar to those for the :class:`Decimal` class and are only briefly
|
|
recounted here.
|
|
|
|
|
|
.. method:: abs(x)
|
|
|
|
Returns the absolute value of *x*.
|
|
|
|
|
|
.. method:: add(x, y)
|
|
|
|
Return the sum of *x* and *y*.
|
|
|
|
|
|
.. method:: canonical(x)
|
|
|
|
Returns the same Decimal object *x*.
|
|
|
|
|
|
.. method:: compare(x, y)
|
|
|
|
Compares *x* and *y* numerically.
|
|
|
|
|
|
.. method:: compare_signal(x, y)
|
|
|
|
Compares the values of the two operands numerically.
|
|
|
|
|
|
.. method:: compare_total(x, y)
|
|
|
|
Compares two operands using their abstract representation.
|
|
|
|
|
|
.. method:: compare_total_mag(x, y)
|
|
|
|
Compares two operands using their abstract representation, ignoring sign.
|
|
|
|
|
|
.. method:: copy_abs(x)
|
|
|
|
Returns a copy of *x* with the sign set to 0.
|
|
|
|
|
|
.. method:: copy_negate(x)
|
|
|
|
Returns a copy of *x* with the sign inverted.
|
|
|
|
|
|
.. method:: copy_sign(x, y)
|
|
|
|
Copies the sign from *y* to *x*.
|
|
|
|
|
|
.. method:: divide(x, y)
|
|
|
|
Return *x* divided by *y*.
|
|
|
|
|
|
.. method:: divide_int(x, y)
|
|
|
|
Return *x* divided by *y*, truncated to an integer.
|
|
|
|
|
|
.. method:: divmod(x, y)
|
|
|
|
Divides two numbers and returns the integer part of the result.
|
|
|
|
|
|
.. method:: exp(x)
|
|
|
|
Returns `e ** x`.
|
|
|
|
|
|
.. method:: fma(x, y, z)
|
|
|
|
Returns *x* multiplied by *y*, plus *z*.
|
|
|
|
|
|
.. method:: is_canonical(x)
|
|
|
|
Returns ``True`` if *x* is canonical; otherwise returns ``False``.
|
|
|
|
|
|
.. method:: is_finite(x)
|
|
|
|
Returns ``True`` if *x* is finite; otherwise returns ``False``.
|
|
|
|
|
|
.. method:: is_infinite(x)
|
|
|
|
Returns ``True`` if *x* is infinite; otherwise returns ``False``.
|
|
|
|
|
|
.. method:: is_integer(x)
|
|
|
|
Returns ``True`` if *x* is finite and integral; otherwise
|
|
returns ``False``.
|
|
|
|
.. versionadded:: 3.10
|
|
|
|
.. method:: is_nan(x)
|
|
|
|
Returns ``True`` if *x* is a qNaN or sNaN; otherwise returns ``False``.
|
|
|
|
|
|
.. method:: is_normal(x)
|
|
|
|
Returns ``True`` if *x* is a normal number; otherwise returns ``False``.
|
|
|
|
|
|
.. method:: is_qnan(x)
|
|
|
|
Returns ``True`` if *x* is a quiet NaN; otherwise returns ``False``.
|
|
|
|
|
|
.. method:: is_signed(x)
|
|
|
|
Returns ``True`` if *x* is negative; otherwise returns ``False``.
|
|
|
|
|
|
.. method:: is_snan(x)
|
|
|
|
Returns ``True`` if *x* is a signaling NaN; otherwise returns ``False``.
|
|
|
|
|
|
.. method:: is_subnormal(x)
|
|
|
|
Returns ``True`` if *x* is subnormal; otherwise returns ``False``.
|
|
|
|
|
|
.. method:: is_zero(x)
|
|
|
|
Returns ``True`` if *x* is a zero; otherwise returns ``False``.
|
|
|
|
|
|
.. method:: ln(x)
|
|
|
|
Returns the natural (base e) logarithm of *x*.
|
|
|
|
|
|
.. method:: log10(x)
|
|
|
|
Returns the base 10 logarithm of *x*.
|
|
|
|
|
|
.. method:: logb(x)
|
|
|
|
Returns the exponent of the magnitude of the operand's MSD.
|
|
|
|
|
|
.. method:: logical_and(x, y)
|
|
|
|
Applies the logical operation *and* between each operand's digits.
|
|
|
|
|
|
.. method:: logical_invert(x)
|
|
|
|
Invert all the digits in *x*.
|
|
|
|
|
|
.. method:: logical_or(x, y)
|
|
|
|
Applies the logical operation *or* between each operand's digits.
|
|
|
|
|
|
.. method:: logical_xor(x, y)
|
|
|
|
Applies the logical operation *xor* between each operand's digits.
|
|
|
|
|
|
.. method:: max(x, y)
|
|
|
|
Compares two values numerically and returns the maximum.
|
|
|
|
|
|
.. method:: max_mag(x, y)
|
|
|
|
Compares the values numerically with their sign ignored.
|
|
|
|
|
|
.. method:: min(x, y)
|
|
|
|
Compares two values numerically and returns the minimum.
|
|
|
|
|
|
.. method:: min_mag(x, y)
|
|
|
|
Compares the values numerically with their sign ignored.
|
|
|
|
|
|
.. method:: minus(x)
|
|
|
|
Minus corresponds to the unary prefix minus operator in Python.
|
|
|
|
|
|
.. method:: multiply(x, y)
|
|
|
|
Return the product of *x* and *y*.
|
|
|
|
|
|
.. method:: next_minus(x)
|
|
|
|
Returns the largest representable number smaller than *x*.
|
|
|
|
|
|
.. method:: next_plus(x)
|
|
|
|
Returns the smallest representable number larger than *x*.
|
|
|
|
|
|
.. method:: next_toward(x, y)
|
|
|
|
Returns the number closest to *x*, in direction towards *y*.
|
|
|
|
|
|
.. method:: normalize(x)
|
|
|
|
Reduces *x* to its simplest form.
|
|
|
|
|
|
.. method:: number_class(x)
|
|
|
|
Returns an indication of the class of *x*.
|
|
|
|
|
|
.. method:: 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.
|
|
|
|
|
|
.. method:: power(x, y, modulo=None)
|
|
|
|
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 rounding mode of the context is used. Results are always correctly-rounded
|
|
in the Python version.
|
|
|
|
``Decimal(0) ** Decimal(0)`` results in ``InvalidOperation``, and if ``InvalidOperation``
|
|
is not trapped, then results in ``Decimal('NaN')``.
|
|
|
|
.. versionchanged:: 3.3
|
|
The C module computes :meth:`power` in terms of the correctly-rounded
|
|
:meth:`exp` and :meth:`ln` functions. The result is well-defined but
|
|
only "almost always correctly-rounded".
|
|
|
|
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.
|
|
|
|
|
|
.. method:: quantize(x, y)
|
|
|
|
Returns a value equal to *x* (rounded), having the exponent of *y*.
|
|
|
|
|
|
.. method:: radix()
|
|
|
|
Just returns 10, as this is Decimal, :)
|
|
|
|
|
|
.. method:: 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.
|
|
|
|
|
|
.. method:: 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*).
|
|
|
|
|
|
.. method:: rotate(x, y)
|
|
|
|
Returns a rotated copy of *x*, *y* times.
|
|
|
|
|
|
.. method:: same_quantum(x, y)
|
|
|
|
Returns ``True`` if the two operands have the same exponent.
|
|
|
|
|
|
.. method:: scaleb (x, y)
|
|
|
|
Returns the first operand after adding the second value its exp.
|
|
|
|
|
|
.. method:: shift(x, y)
|
|
|
|
Returns a shifted copy of *x*, *y* times.
|
|
|
|
|
|
.. method:: sqrt(x)
|
|
|
|
Square root of a non-negative number to context precision.
|
|
|
|
|
|
.. method:: subtract(x, y)
|
|
|
|
Return the difference between *x* and *y*.
|
|
|
|
|
|
.. method:: to_eng_string(x)
|
|
|
|
Convert to a string, using engineering notation if an exponent is needed.
|
|
|
|
Engineering notation has an exponent which is a multiple of 3. This
|
|
can leave up to 3 digits to the left of the decimal place and may
|
|
require the addition of either one or two trailing zeros.
|
|
|
|
|
|
.. method:: to_integral_exact(x)
|
|
|
|
Rounds to an integer.
|
|
|
|
|
|
.. method:: to_sci_string(x)
|
|
|
|
Converts a number to a string using scientific notation.
|
|
|
|
.. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
.. _decimal-rounding-modes:
|
|
|
|
Constants
|
|
---------
|
|
|
|
The constants in this section are only relevant for the C module. They
|
|
are also included in the pure Python version for compatibility.
|
|
|
|
+---------------------+---------------------+-------------------------------+
|
|
| | 32-bit | 64-bit |
|
|
+=====================+=====================+===============================+
|
|
| .. data:: MAX_PREC | :const:`425000000` | :const:`999999999999999999` |
|
|
+---------------------+---------------------+-------------------------------+
|
|
| .. data:: MAX_EMAX | :const:`425000000` | :const:`999999999999999999` |
|
|
+---------------------+---------------------+-------------------------------+
|
|
| .. data:: MIN_EMIN | :const:`-425000000` | :const:`-999999999999999999` |
|
|
+---------------------+---------------------+-------------------------------+
|
|
| .. data:: MIN_ETINY | :const:`-849999999` | :const:`-1999999999999999997` |
|
|
+---------------------+---------------------+-------------------------------+
|
|
|
|
|
|
.. data:: HAVE_THREADS
|
|
|
|
The value is ``True``. Deprecated, because Python now always has threads.
|
|
|
|
.. deprecated:: 3.9
|
|
|
|
.. data:: HAVE_CONTEXTVAR
|
|
|
|
The default value is ``True``. If Python is compiled ``--without-decimal-contextvar``,
|
|
the C version uses a thread-local rather than a coroutine-local context and the value
|
|
is ``False``. This is slightly faster in some nested context scenarios.
|
|
|
|
.. versionadded:: 3.9 backported to 3.7 and 3.8.
|
|
|
|
|
|
Rounding modes
|
|
--------------
|
|
|
|
.. data:: ROUND_CEILING
|
|
|
|
Round towards :const:`Infinity`.
|
|
|
|
.. data:: ROUND_DOWN
|
|
|
|
Round towards zero.
|
|
|
|
.. data:: ROUND_FLOOR
|
|
|
|
Round towards :const:`-Infinity`.
|
|
|
|
.. data:: ROUND_HALF_DOWN
|
|
|
|
Round to nearest with ties going towards zero.
|
|
|
|
.. data:: ROUND_HALF_EVEN
|
|
|
|
Round to nearest with ties going to nearest even integer.
|
|
|
|
.. data:: ROUND_HALF_UP
|
|
|
|
Round to nearest with ties going away from zero.
|
|
|
|
.. data:: ROUND_UP
|
|
|
|
Round away from zero.
|
|
|
|
.. data:: ROUND_05UP
|
|
|
|
Round away from zero if last digit after rounding towards zero would have
|
|
been 0 or 5; otherwise round towards zero.
|
|
|
|
|
|
.. _decimal-signals:
|
|
|
|
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 :class:`DivisionByZero` trap
|
|
is set, then a :exc:`DivisionByZero` exception is raised upon encountering the
|
|
condition.
|
|
|
|
|
|
.. class:: Clamped
|
|
|
|
Altered an exponent to fit representation constraints.
|
|
|
|
Typically, clamping occurs when an exponent falls outside the context's
|
|
:attr:`Emin` and :attr:`Emax` limits. If possible, the exponent is reduced to
|
|
fit by adding zeros to the coefficient.
|
|
|
|
|
|
.. class:: DecimalException
|
|
|
|
Base class for other signals and a subclass of :exc:`ArithmeticError`.
|
|
|
|
|
|
.. class:: 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 :const:`Infinity` or
|
|
:const:`-Infinity` with the sign determined by the inputs to the calculation.
|
|
|
|
|
|
.. class:: 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:: InvalidOperation
|
|
|
|
An invalid operation was performed.
|
|
|
|
Indicates that an operation was requested that does not make sense. If not
|
|
trapped, returns :const:`NaN`. Possible causes include::
|
|
|
|
Infinity - Infinity
|
|
0 * Infinity
|
|
Infinity / Infinity
|
|
x % 0
|
|
Infinity % x
|
|
sqrt(-x) and x > 0
|
|
0 ** 0
|
|
x ** (non-integer)
|
|
x ** Infinity
|
|
|
|
|
|
.. class:: Overflow
|
|
|
|
Numerical overflow.
|
|
|
|
Indicates the exponent is larger than :attr:`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 :const:`Infinity`. In either case, :class:`Inexact` and :class:`Rounded`
|
|
are also signaled.
|
|
|
|
|
|
.. class:: Rounded
|
|
|
|
Rounding occurred though possibly no information was lost.
|
|
|
|
Signaled whenever rounding discards digits; even if those digits are zero
|
|
(such as rounding :const:`5.00` to :const:`5.0`). If not trapped, returns
|
|
the result unchanged. This signal is used to detect loss of significant
|
|
digits.
|
|
|
|
|
|
.. class:: Subnormal
|
|
|
|
Exponent was lower than :attr:`Emin` prior to rounding.
|
|
|
|
Occurs when an operation result is subnormal (the exponent is too small). If
|
|
not trapped, returns the result unchanged.
|
|
|
|
|
|
.. class:: Underflow
|
|
|
|
Numerical underflow with result rounded to zero.
|
|
|
|
Occurs when a subnormal result is pushed to zero by rounding. :class:`Inexact`
|
|
and :class:`Subnormal` are also signaled.
|
|
|
|
|
|
.. class:: FloatOperation
|
|
|
|
Enable stricter semantics for mixing floats and Decimals.
|
|
|
|
If the signal is not trapped (default), mixing floats and Decimals is
|
|
permitted in the :class:`~decimal.Decimal` constructor,
|
|
:meth:`~decimal.Context.create_decimal` and all comparison operators.
|
|
Both conversion and comparisons are exact. Any occurrence of a mixed
|
|
operation is silently recorded by setting :exc:`FloatOperation` in the
|
|
context flags. Explicit conversions with :meth:`~decimal.Decimal.from_float`
|
|
or :meth:`~decimal.Context.create_decimal_from_float` do not set the flag.
|
|
|
|
Otherwise (the signal is trapped), only equality comparisons and explicit
|
|
conversions are silent. All other mixed operations raise :exc:`FloatOperation`.
|
|
|
|
|
|
The following table summarizes the hierarchy of signals::
|
|
|
|
exceptions.ArithmeticError(exceptions.Exception)
|
|
DecimalException
|
|
Clamped
|
|
DivisionByZero(DecimalException, exceptions.ZeroDivisionError)
|
|
Inexact
|
|
Overflow(Inexact, Rounded)
|
|
Underflow(Inexact, Rounded, Subnormal)
|
|
InvalidOperation
|
|
Rounded
|
|
Subnormal
|
|
FloatOperation(DecimalException, exceptions.TypeError)
|
|
|
|
.. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
|
|
|
|
.. _decimal-notes:
|
|
|
|
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 :const:`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:
|
|
|
|
.. doctest:: newcontext
|
|
|
|
# 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 :mod:`decimal` module makes it possible to restore the identities by
|
|
expanding the precision sufficiently to avoid loss of significance:
|
|
|
|
.. doctest:: newcontext
|
|
|
|
>>> 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 :mod:`decimal` module provides special values
|
|
including :const:`NaN`, :const:`sNaN`, :const:`-Infinity`, :const:`Infinity`,
|
|
and two zeros, :const:`+0` and :const:`-0`.
|
|
|
|
Infinities can be constructed directly with: ``Decimal('Infinity')``. Also,
|
|
they can arise from dividing by zero when the :exc:`DivisionByZero` signal is
|
|
not trapped. Likewise, when the :exc:`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 :const:`NaN`, or if the
|
|
:exc:`InvalidOperation` signal is trapped, raise an exception. For example,
|
|
``0/0`` returns :const:`NaN` which means "not a number". This variety of
|
|
:const:`NaN` is quiet and, once created, will flow through other computations
|
|
always resulting in another :const:`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 :const:`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
|
|
:const:`NaN` is involved. A test for equality where one of the operands is a
|
|
quiet or signaling :const:`NaN` always returns :const:`False` (even when doing
|
|
``Decimal('NaN')==Decimal('NaN')``), while a test for inequality always returns
|
|
:const:`True`. An attempt to compare two Decimals using any of the ``<``,
|
|
``<=``, ``>`` or ``>=`` operators will raise the :exc:`InvalidOperation` signal
|
|
if either operand is a :const:`NaN`, and return :const:`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 :const:`NaN` were taken from the IEEE 854 standard (see Table 3 in
|
|
section 5.7). To ensure strict standards-compliance, use the :meth:`compare`
|
|
and :meth:`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-1000026')
|
|
|
|
.. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
|
|
.. _decimal-threads:
|
|
|
|
Working with threads
|
|
--------------------
|
|
|
|
The :func:`getcontext` function accesses a different :class:`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 :func:`setcontext` function automatically assigns its target to
|
|
the current thread.
|
|
|
|
If :func:`setcontext` has not been called before :func:`getcontext`, then
|
|
:func:`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 :func:`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()
|
|
. . .
|
|
|
|
.. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
|
|
.. _decimal-recipes:
|
|
|
|
Recipes
|
|
-------
|
|
|
|
Here are a few recipes that serve as utility functions and that demonstrate ways
|
|
to work with the :class:`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 = list(map(str, digits))
|
|
build, next = result.append, digits.pop
|
|
if sign:
|
|
build(trailneg)
|
|
for i in range(places):
|
|
build(next() if digits else '0')
|
|
if places:
|
|
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.
|
|
|
|
The Taylor series approximation works best for a small value of x.
|
|
For larger values, first compute x = x % (2 * pi).
|
|
|
|
>>> 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.
|
|
|
|
The Taylor series approximation works best for a small value of x.
|
|
For larger values, first compute x = x % (2 * pi).
|
|
|
|
>>> 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:
|
|
|
|
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 :meth:`quantize` method rounds to a fixed number of decimal places. If
|
|
the :const:`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 :meth:`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 :meth:`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 :const:`200`,
|
|
:const:`200.000`, :const:`2E2`, and :const:`.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 :meth:`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
|
|
:const:`5.0E+3` as :const:`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 :class:`Decimal`?
|
|
|
|
A. Yes, any binary floating point number can be exactly expressed as a
|
|
Decimal though an exact conversion may take more precision than intuition would
|
|
suggest:
|
|
|
|
.. doctest::
|
|
|
|
>>> Decimal(math.pi)
|
|
Decimal('3.141592653589793115997963468544185161590576171875')
|
|
|
|
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
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re-run calculations using greater precision and with various rounding modes.
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Widely differing results indicate insufficient precision, rounding mode issues,
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ill-conditioned inputs, or a numerically unstable algorithm.
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Q. I noticed that context precision is applied to the results of operations but
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not to the inputs. Is there anything to watch out for when mixing values of
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different precisions?
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A. Yes. The principle is that all values are considered to be exact and so is
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the arithmetic on those values. Only the results are rounded. The advantage
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for inputs is that "what you type is what you get". A disadvantage is that the
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results can look odd if you forget that the inputs haven't been rounded:
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.. doctest:: newcontext
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>>> getcontext().prec = 3
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>>> Decimal('3.104') + Decimal('2.104')
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Decimal('5.21')
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>>> Decimal('3.104') + Decimal('0.000') + Decimal('2.104')
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Decimal('5.20')
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The solution is either to increase precision or to force rounding of inputs
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using the unary plus operation:
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.. doctest:: newcontext
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>>> getcontext().prec = 3
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>>> +Decimal('1.23456789') # unary plus triggers rounding
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Decimal('1.23')
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Alternatively, inputs can be rounded upon creation using the
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:meth:`Context.create_decimal` method:
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>>> Context(prec=5, rounding=ROUND_DOWN).create_decimal('1.2345678')
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Decimal('1.2345')
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Q. Is the CPython implementation fast for large numbers?
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A. Yes. In the CPython and PyPy3 implementations, the C/CFFI versions of
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the decimal module integrate the high speed `libmpdec
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<https://www.bytereef.org/mpdecimal/doc/libmpdec/index.html>`_ library for
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arbitrary precision correctly-rounded decimal floating point arithmetic [#]_.
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``libmpdec`` uses `Karatsuba multiplication
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<https://en.wikipedia.org/wiki/Karatsuba_algorithm>`_
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for medium-sized numbers and the `Number Theoretic Transform
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<https://en.wikipedia.org/wiki/Discrete_Fourier_transform_(general)#Number-theoretic_transform>`_
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for very large numbers.
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The context must be adapted for exact arbitrary precision arithmetic. :attr:`Emin`
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and :attr:`Emax` should always be set to the maximum values, :attr:`clamp`
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should always be 0 (the default). Setting :attr:`prec` requires some care.
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The easiest approach for trying out bignum arithmetic is to use the maximum
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value for :attr:`prec` as well [#]_::
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>>> setcontext(Context(prec=MAX_PREC, Emax=MAX_EMAX, Emin=MIN_EMIN))
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>>> x = Decimal(2) ** 256
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>>> x / 128
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Decimal('904625697166532776746648320380374280103671755200316906558262375061821325312')
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For inexact results, :attr:`MAX_PREC` is far too large on 64-bit platforms and
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the available memory will be insufficient::
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>>> Decimal(1) / 3
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Traceback (most recent call last):
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File "<stdin>", line 1, in <module>
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MemoryError
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On systems with overallocation (e.g. Linux), a more sophisticated approach is to
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adjust :attr:`prec` to the amount of available RAM. Suppose that you have 8GB of
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RAM and expect 10 simultaneous operands using a maximum of 500MB each::
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>>> import sys
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>>>
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>>> # Maximum number of digits for a single operand using 500MB in 8-byte words
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>>> # with 19 digits per word (4-byte and 9 digits for the 32-bit build):
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>>> maxdigits = 19 * ((500 * 1024**2) // 8)
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>>>
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>>> # Check that this works:
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>>> c = Context(prec=maxdigits, Emax=MAX_EMAX, Emin=MIN_EMIN)
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>>> c.traps[Inexact] = True
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>>> setcontext(c)
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>>>
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>>> # Fill the available precision with nines:
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>>> x = Decimal(0).logical_invert() * 9
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>>> sys.getsizeof(x)
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524288112
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>>> x + 2
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Traceback (most recent call last):
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File "<stdin>", line 1, in <module>
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decimal.Inexact: [<class 'decimal.Inexact'>]
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In general (and especially on systems without overallocation), it is recommended
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to estimate even tighter bounds and set the :attr:`Inexact` trap if all calculations
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are expected to be exact.
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.. [#]
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.. versionadded:: 3.3
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.. [#]
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.. versionchanged:: 3.9
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This approach now works for all exact results except for non-integer powers.
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