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+# Copyright (c) 2004 Python Software Foundation.
+# All rights reserved.
+# Written by Eric Price <eprice at tjhsst.edu>
+#    and Facundo Batista <facundo at taniquetil.com.ar>
+#    and Raymond Hettinger <python at rcn.com>
+#    and Aahz <aahz at pobox.com>
+#    and Tim Peters
+# This module is currently Py2.3 compatible and should be kept that way
+# unless a major compelling advantage arises.  IOW, 2.3 compatibility is
+# strongly preferred, but not guaranteed.
+# Also, this module should be kept in sync with the latest updates of
+# the IBM specification as it evolves.  Those updates will be treated
+# as bug fixes (deviation from the spec is a compatibility, usability
+# bug) and will be backported.  At this point the spec is stabilizing
+# and the updates are becoming fewer, smaller, and less significant.
+"""
+This is a Py2.3 implementation of decimal floating point arithmetic based on
+the General Decimal Arithmetic Specification:
+www2.hursley.ibm.com/decimal/decarith.html
+and IEEE standard 854-1987:
+www.cs.berkeley.edu/~ejr/projects/754/private/drafts/854-1987/dir.html
+Decimal floating point has finite precision with arbitrarily large bounds.
+The purpose of this module is to support arithmetic using familiar
+"schoolhouse" rules and to avoid some of the tricky representation
+issues associated with binary floating point.  The package is especially
+useful for financial applications or for contexts where users have
+expectations that are at odds with binary floating point (for instance,
+in binary floating point, 1.00 % 0.1 gives 0.09999999999999995 instead
+of the expected Decimal("0.00") returned by decimal floating point).
+Here are some examples of using the decimal module:
+>>> from decimal import *
+>>> setcontext(ExtendedContext)
+>>> Decimal(0)
+Decimal("0")
+>>> Decimal("1")
+Decimal("1")
+>>> Decimal("-.0123")
+Decimal("-0.0123")
+>>> Decimal(123456)
+Decimal("123456")
+>>> Decimal("123.45e12345678901234567890")
+Decimal("1.2345E+12345678901234567892")
+>>> Decimal("1.33") + Decimal("1.27")
+Decimal("2.60")
+>>> Decimal("12.34") + Decimal("3.87") - Decimal("18.41")
+Decimal("-2.20")
+>>> dig = Decimal(1)
+>>> print dig / Decimal(3)
+0.333333333
+>>> getcontext().prec = 18
+>>> print dig / Decimal(3)
+0.333333333333333333
+>>> print dig.sqrt()
+1
+>>> print Decimal(3).sqrt()
+1.73205080756887729
+>>> print Decimal(3) ** 123
+4.85192780976896427E+58
+>>> inf = Decimal(1) / Decimal(0)
+>>> print inf
+Infinity
+>>> neginf = Decimal(-1) / Decimal(0)
+>>> print neginf
+-Infinity
+>>> print neginf + inf
+NaN
+>>> print neginf * inf
+-Infinity
+>>> print dig / 0
+Infinity
+>>> getcontext().traps[DivisionByZero] = 1
+>>> print dig / 0
+Traceback (most recent call last):
+...
+...
+...
+DivisionByZero: x / 0
+>>> c = Context()
+>>> c.traps[InvalidOperation] = 0
+>>> print c.flags[InvalidOperation]
+0
+>>> c.divide(Decimal(0), Decimal(0))
+Decimal("NaN")
+>>> c.traps[InvalidOperation] = 1
+>>> print c.flags[InvalidOperation]
+1
+>>> c.flags[InvalidOperation] = 0
+>>> print c.flags[InvalidOperation]
+0
+>>> print c.divide(Decimal(0), Decimal(0))
+Traceback (most recent call last):
+...
+...
+...
+InvalidOperation: 0 / 0
+>>> print c.flags[InvalidOperation]
+1
+>>> c.flags[InvalidOperation] = 0
+>>> c.traps[InvalidOperation] = 0
+>>> print c.divide(Decimal(0), Decimal(0))
+NaN
+>>> print c.flags[InvalidOperation]
+1
+>>>
+"""
+__all__ = [
+# Two major classes
+'Decimal', 'Context',
+# Contexts
+'DefaultContext', 'BasicContext', 'ExtendedContext',
+# Exceptions
+'DecimalException', 'Clamped', 'InvalidOperation', 'DivisionByZero',
+'Inexact', 'Rounded', 'Subnormal', 'Overflow', 'Underflow',
+# Constants for use in setting up contexts
+'ROUND_DOWN', 'ROUND_HALF_UP', 'ROUND_HALF_EVEN', 'ROUND_CEILING',
+'ROUND_FLOOR', 'ROUND_UP', 'ROUND_HALF_DOWN', 'ROUND_05UP',
+# Functions for manipulating contexts
+'setcontext', 'getcontext', 'localcontext'
+]
+import copy as _copy
+# Rounding
+ROUND_DOWN = 'ROUND_DOWN'
+ROUND_HALF_UP = 'ROUND_HALF_UP'
+ROUND_HALF_EVEN = 'ROUND_HALF_EVEN'
+ROUND_CEILING = 'ROUND_CEILING'
+ROUND_FLOOR = 'ROUND_FLOOR'
+ROUND_UP = 'ROUND_UP'
+ROUND_HALF_DOWN = 'ROUND_HALF_DOWN'
+ROUND_05UP = 'ROUND_05UP'
+# Errors
+class DecimalException(ArithmeticError):
+"""Base exception class.
+Used exceptions derive from this.
+If an exception derives from another exception besides this (such as
+Underflow (Inexact, Rounded, Subnormal) that indicates that it is only
+called if the others are present.  This isn't actually used for
+anything, though.
+handle  -- Called when context._raise_error is called and the
+trap_enabler is set.  First argument is self, second is the
+context.  More arguments can be given, those being after
+the explanation in _raise_error (For example,
+context._raise_error(NewError, '(-x)!', self._sign) would
+call NewError().handle(context, self._sign).)
+To define a new exception, it should be sufficient to have it derive
+from DecimalException.
+"""
+def handle(self, context, *args):
+pass
+class Clamped(DecimalException):
+"""Exponent of a 0 changed to fit bounds.
+This occurs and signals clamped if the exponent of a result has been
+altered in order to fit the constraints of a specific concrete
+representation.  This may occur when the exponent of a zero result would
+be outside the bounds of a representation, or when a large normal
+number would have an encoded exponent that cannot be represented.  In
+this latter case, the exponent is reduced to fit and the corresponding
+number of zero digits are appended to the coefficient ("fold-down").
+"""
+class InvalidOperation(DecimalException):
+"""An invalid operation was performed.
+Various bad things cause this:
+Something creates a signaling NaN
+-INF + INF
+0 * (+-)INF
+(+-)INF / (+-)INF
+x % 0
+(+-)INF % x
+x._rescale( non-integer )
+sqrt(-x) , x > 0
+0 ** 0
+x ** (non-integer)
+x ** (+-)INF
+An operand is invalid
+The result of the operation after these is a quiet positive NaN,
+except when the cause is a signaling NaN, in which case the result is
+also a quiet NaN, but with the original sign, and an optional
+diagnostic information.
+"""
+def handle(self, context, *args):
+if args:
+ans = _dec_from_triple(args[0]._sign, args[0]._int, 'n', True)
+return ans._fix_nan(context)
+return NaN
+class ConversionSyntax(InvalidOperation):
+"""Trying to convert badly formed string.
+This occurs and signals invalid-operation if an string is being
+converted to a number and it does not conform to the numeric string
+syntax.  The result is [0,qNaN].
+"""
+def handle(self, context, *args):
+return NaN
+class DivisionByZero(DecimalException, ZeroDivisionError):
+"""Division by 0.
+This occurs and signals division-by-zero if division of a finite number
+by zero was attempted (during a divide-integer or divide operation, or a
+power operation with negative right-hand operand), and the dividend was
+not zero.
+The result of the operation is [sign,inf], where sign is the exclusive
+or of the signs of the operands for divide, or is 1 for an odd power of
+-0, for power.
+"""
+def handle(self, context, sign, *args):
+return Infsign[sign]
+class DivisionImpossible(InvalidOperation):
+"""Cannot perform the division adequately.
+This occurs and signals invalid-operation if the integer result of a
+divide-integer or remainder operation had too many digits (would be
+longer than precision).  The result is [0,qNaN].
+"""
+def handle(self, context, *args):
+return NaN
+class DivisionUndefined(InvalidOperation, ZeroDivisionError):
+"""Undefined result of division.
+This occurs and signals invalid-operation if division by zero was
+attempted (during a divide-integer, divide, or remainder operation), and
+the dividend is also zero.  The result is [0,qNaN].
+"""
+def handle(self, context, *args):
+return NaN
+class Inexact(DecimalException):
+"""Had to round, losing information.
+This occurs and signals inexact whenever the result of an operation is
+not exact (that is, it needed to be rounded and any discarded digits
+were non-zero), or if an overflow or underflow condition occurs.  The
+result in all cases is unchanged.
+The inexact signal may be tested (or trapped) to determine if a given
+operation (or sequence of operations) was inexact.
+"""
+class InvalidContext(InvalidOperation):
+"""Invalid context.  Unknown rounding, for example.
+This occurs and signals invalid-operation if an invalid context was
+detected during an operation.  This can occur if contexts are not checked
+on creation and either the precision exceeds the capability of the
+underlying concrete representation or an unknown or unsupported rounding
+was specified.  These aspects of the context need only be checked when
+the values are required to be used.  The result is [0,qNaN].
+"""
+def handle(self, context, *args):
+return NaN
+class Rounded(DecimalException):
+"""Number got rounded (not  necessarily changed during rounding).
+This occurs and signals rounded whenever the result of an operation is
+rounded (that is, some zero or non-zero digits were discarded from the
+coefficient), or if an overflow or underflow condition occurs.  The
+result in all cases is unchanged.
+The rounded signal may be tested (or trapped) to determine if a given
+operation (or sequence of operations) caused a loss of precision.
+"""
+class Subnormal(DecimalException):
+"""Exponent < Emin before rounding.
+This occurs and signals subnormal whenever the result of a conversion or
+operation is subnormal (that is, its adjusted exponent is less than
+Emin, before any rounding).  The result in all cases is unchanged.
+The subnormal signal may be tested (or trapped) to determine if a given
+or operation (or sequence of operations) yielded a subnormal result.
+"""
+class Overflow(Inexact, Rounded):
+"""Numerical overflow.
+This occurs and signals overflow if the adjusted exponent of a result
+(from a conversion or from an operation that is not an attempt to divide
+by zero), after rounding, would be greater than the largest value that
+can be handled by the implementation (the value Emax).
+The result depends on the rounding mode:
+For round-half-up and round-half-even (and for round-half-down and
+round-up, if implemented), the result of the operation is [sign,inf],
+where sign is the sign of the intermediate result.  For round-down, the
+result is the largest finite number that can be represented in the
+current precision, with the sign of the intermediate result.  For
+round-ceiling, the result is the same as for round-down if the sign of
+the intermediate result is 1, or is [0,inf] otherwise.  For round-floor,
+the result is the same as for round-down if the sign of the intermediate
+result is 0, or is [1,inf] otherwise.  In all cases, Inexact and Rounded
+will also be raised.
+"""
+def handle(self, context, sign, *args):
+if context.rounding in (ROUND_HALF_UP, ROUND_HALF_EVEN,
+ROUND_HALF_DOWN, ROUND_UP):
+return Infsign[sign]
+if sign == 0:
+if context.rounding == ROUND_CEILING:
+return Infsign[sign]
+return _dec_from_triple(sign, '9'*context.prec,
+context.Emax-context.prec+1)
+if sign == 1:
+if context.rounding == ROUND_FLOOR:
+return Infsign[sign]
+return _dec_from_triple(sign, '9'*context.prec,
+context.Emax-context.prec+1)
+class Underflow(Inexact, Rounded, Subnormal):
+"""Numerical underflow with result rounded to 0.
+This occurs and signals underflow if a result is inexact and the
+adjusted exponent of the result would be smaller (more negative) than
+the smallest value that can be handled by the implementation (the value
+Emin).  That is, the result is both inexact and subnormal.
+The result after an underflow will be a subnormal number rounded, if
+necessary, so that its exponent is not less than Etiny.  This may result
+in 0 with the sign of the intermediate result and an exponent of Etiny.
+In all cases, Inexact, Rounded, and Subnormal will also be raised.
+"""
+# List of public traps and flags
+_signals = [Clamped, DivisionByZero, Inexact, Overflow, Rounded,
+Underflow, InvalidOperation, Subnormal]
+# Map conditions (per the spec) to signals
+_condition_map = {ConversionSyntax:InvalidOperation,
+DivisionImpossible:InvalidOperation,
+DivisionUndefined:InvalidOperation,
+InvalidContext:InvalidOperation}
+##### Context Functions ##################################################
+# The getcontext() and setcontext() function manage access to a thread-local
+# current context.  Py2.4 offers direct support for thread locals.  If that
+# is not available, use threading.currentThread() which is slower but will
+# work for older Pythons.  If threads are not part of the build, create a
+# mock threading object with threading.local() returning the module namespace.
+try:
+import threading
+except ImportError:
+# Python was compiled without threads; create a mock object instead
+import sys
+class MockThreading(object):
+def local(self, sys=sys):
+return sys.modules[__name__]
+threading = MockThreading()
+del sys, MockThreading
+try:
+threading.local
+except AttributeError:
+# To fix reloading, force it to create a new context
+# Old contexts have different exceptions in their dicts, making problems.
+if hasattr(threading.currentThread(), '__decimal_context__'):
+del threading.currentThread().__decimal_context__
+def setcontext(context):
+"""Set this thread's context to context."""
+if context in (DefaultContext, BasicContext, ExtendedContext):
+context = context.copy()
+context.clear_flags()
+threading.currentThread().__decimal_context__ = context
+def getcontext():
+"""Returns this thread's context.
+If this thread does not yet have a context, returns
+a new context and sets this thread's context.
+New contexts are copies of DefaultContext.
+"""
+try:
+return threading.currentThread().__decimal_context__
+except AttributeError:
+context = Context()
+threading.currentThread().__decimal_context__ = context
+return context
+else:
+local = threading.local()
+if hasattr(local, '__decimal_context__'):
+del local.__decimal_context__
+def getcontext(_local=local):
+"""Returns this thread's context.
+If this thread does not yet have a context, returns
+a new context and sets this thread's context.
+New contexts are copies of DefaultContext.
+"""
+try:
+return _local.__decimal_context__
+except AttributeError:
+context = Context()
+_local.__decimal_context__ = context
+return context
+def setcontext(context, _local=local):
+"""Set this thread's context to context."""
+if context in (DefaultContext, BasicContext, ExtendedContext):
+context = context.copy()
+context.clear_flags()
+_local.__decimal_context__ = context
+del threading, local        # Don't contaminate the namespace
+def localcontext(ctx=None):
+"""Return a context manager for a copy of the supplied context
+Uses a copy of the current context if no context is specified
+The returned context manager creates a local decimal context
+in a with statement:
+def sin(x):
+with localcontext() as ctx:
+ctx.prec += 2
+# Rest of sin calculation algorithm
+# uses a precision 2 greater than normal
+return +s  # Convert result to normal precision
+def sin(x):
+with localcontext(ExtendedContext):
+# Rest of sin calculation algorithm
+# uses the Extended Context from the
+# General Decimal Arithmetic Specification
+return +s  # Convert result to normal context
+"""
+# The string below can't be included in the docstring until Python 2.6
+# as the doctest module doesn't understand __future__ statements
+"""
+>>> from __future__ import with_statement
+>>> print getcontext().prec
+28
+>>> with localcontext():
+...     ctx = getcontext()
+...     ctx.prec += 2
+...     print ctx.prec
+...
+30
+>>> with localcontext(ExtendedContext):
+...     print getcontext().prec
+...
+9
+>>> print getcontext().prec
+28
+"""
+if ctx is None: ctx = getcontext()
+return _ContextManager(ctx)
+##### Decimal class #######################################################
+class Decimal(object):
+"""Floating point class for decimal arithmetic."""
+__slots__ = ('_exp','_int','_sign', '_is_special')
+# Generally, the value of the Decimal instance is given by
+#  (-1)**_sign * _int * 10**_exp
+# Special values are signified by _is_special == True
+# We're immutable, so use __new__ not __init__
+def __new__(cls, value="0", context=None):
+"""Create a decimal point instance.
+>>> Decimal('3.14')              # string input
+Decimal("3.14")
+>>> Decimal((0, (3, 1, 4), -2))  # tuple (sign, digit_tuple, exponent)
+Decimal("3.14")
+>>> Decimal(314)                 # int or long
+Decimal("314")
+>>> Decimal(Decimal(314))        # another decimal instance
+Decimal("314")
+"""
+# Note that the coefficient, self._int, is actually stored as
+# a string rather than as a tuple of digits.  This speeds up
+# the "digits to integer" and "integer to digits" conversions
+# that are used in almost every arithmetic operation on
+# Decimals.  This is an internal detail: the as_tuple function
+# and the Decimal constructor still deal with tuples of
+# digits.
+self = object.__new__(cls)
+# From a string
+# REs insist on real strings, so we can too.
+if isinstance(value, basestring):
+m = _parser(value)
+if m is None:
+if context is None:
+context = getcontext()
+return context._raise_error(ConversionSyntax,
+"Invalid literal for Decimal: %r" % value)
+if m.group('sign') == "-":
+self._sign = 1
+else:
+self._sign = 0
+intpart = m.group('int')
+if intpart is not None:
+# finite number
+fracpart = m.group('frac')
+exp = int(m.group('exp') or '0')
+if fracpart is not None:
+self._int = (intpart+fracpart).lstrip('0') or '0'
+self._exp = exp - len(fracpart)
+else:
+self._int = intpart.lstrip('0') or '0'
+self._exp = exp
+self._is_special = False
+else:
+diag = m.group('diag')
+if diag is not None:
+# NaN
+self._int = diag.lstrip('0')
+if m.group('signal'):
+self._exp = 'N'
+else:
+self._exp = 'n'
+else:
+# infinity
+self._int = '0'
+self._exp = 'F'
+self._is_special = True
+return self
+# From an integer
+if isinstance(value, (int,long)):
+if value >= 0:
+self._sign = 0
+else:
+self._sign = 1
+self._exp = 0
+self._int = str(abs(value))
+self._is_special = False
+return self
+# From another decimal
+if isinstance(value, Decimal):
+self._exp  = value._exp
+self._sign = value._sign
+self._int  = value._int
+self._is_special  = value._is_special
+return self
+# From an internal working value
+if isinstance(value, _WorkRep):
+self._sign = value.sign
+self._int = str(value.int)
+self._exp = int(value.exp)
+self._is_special = False
+return self
+# tuple/list conversion (possibly from as_tuple())
+if isinstance(value, (list,tuple)):
+if len(value) != 3:
+raise ValueError('Invalid tuple size in creation of Decimal '
+'from list or tuple.  The list or tuple '
+'should have exactly three elements.')
+# process sign.  The isinstance test rejects floats
+if not (isinstance(value[0], (int, long)) and value[0] in (0,1)):
+raise ValueError("Invalid sign.  The first value in the tuple "
+"should be an integer; either 0 for a "
+"positive number or 1 for a negative number.")
+self._sign = value[0]
+if value[2] == 'F':
+# infinity: value[1] is ignored
+self._int = '0'
+self._exp = value[2]
+self._is_special = True
+else:
+# process and validate the digits in value[1]
+digits = []
+for digit in value[1]:
+if isinstance(digit, (int, long)) and 0 <= digit <= 9:
+# skip leading zeros
+if digits or digit != 0:
+digits.append(digit)
+else:
+raise ValueError("The second value in the tuple must "
+"be composed of integers in the range "
+"0 through 9.")
+if value[2] in ('n', 'N'):
+# NaN: digits form the diagnostic
+self._int = ''.join(map(str, digits))
+self._exp = value[2]
+self._is_special = True
+elif isinstance(value[2], (int, long)):
+# finite number: digits give the coefficient
+self._int = ''.join(map(str, digits or [0]))
+self._exp = value[2]
+self._is_special = False
+else:
+raise ValueError("The third value in the tuple must "
+"be an integer, or one of the "
+"strings 'F', 'n', 'N'.")
+return self
+if isinstance(value, float):
+raise TypeError("Cannot convert float to Decimal.  " +
+"First convert the float to a string")
+raise TypeError("Cannot convert %r to Decimal" % value)
+def _isnan(self):
+"""Returns whether the number is not actually one.
+0 if a number
+1 if NaN
+2 if sNaN
+"""
+if self._is_special:
+exp = self._exp
+if exp == 'n':
+return 1
+elif exp == 'N':
+return 2
+return 0
+def _isinfinity(self):
+"""Returns whether the number is infinite
+0 if finite or not a number
+1 if +INF
+-1 if -INF
+"""
+if self._exp == 'F':
+if self._sign:
+return -1
+return 1
+return 0
+def _check_nans(self, other=None, context=None):
+"""Returns whether the number is not actually one.
+if self, other are sNaN, signal
+if self, other are NaN return nan
+return 0
+Done before operations.
+"""
+self_is_nan = self._isnan()
+if other is None:
+other_is_nan = False
+else:
+other_is_nan = other._isnan()
+if self_is_nan or other_is_nan:
+if context is None:
+context = getcontext()
+if self_is_nan == 2:
+return context._raise_error(InvalidOperation, 'sNaN',
+self)
+if other_is_nan == 2:
+return context._raise_error(InvalidOperation, 'sNaN',
+other)
+if self_is_nan:
+return self._fix_nan(context)
+return other._fix_nan(context)
+return 0
+def __nonzero__(self):
+"""Return True if self is nonzero; otherwise return False.
+NaNs and infinities are considered nonzero.
+"""
+return self._is_special or self._int != '0'
+def __cmp__(self, other):
+other = _convert_other(other)
+if other is NotImplemented:
+# Never return NotImplemented
+return 1
+if self._is_special or other._is_special:
+# check for nans, without raising on a signaling nan
+if self._isnan() or other._isnan():
+return 1  # Comparison involving NaN's always reports self > other
+# INF = INF
+return cmp(self._isinfinity(), other._isinfinity())
+# check for zeros;  note that cmp(0, -0) should return 0
+if not self:
+if not other:
+return 0
+else:
+return -((-1)**other._sign)
+if not other:
+return (-1)**self._sign
+# If different signs, neg one is less
+if other._sign < self._sign:
+return -1
+if self._sign < other._sign:
+return 1
+self_adjusted = self.adjusted()
+other_adjusted = other.adjusted()
+if self_adjusted == other_adjusted:
+self_padded = self._int + '0'*(self._exp - other._exp)
+other_padded = other._int + '0'*(other._exp - self._exp)
+return cmp(self_padded, other_padded) * (-1)**self._sign
+elif self_adjusted > other_adjusted:
+return (-1)**self._sign
+else: # self_adjusted < other_adjusted
+return -((-1)**self._sign)
+def __eq__(self, other):
+if not isinstance(other, (Decimal, int, long)):
+return NotImplemented
+return self.__cmp__(other) == 0
+def __ne__(self, other):
+if not isinstance(other, (Decimal, int, long)):
+return NotImplemented
+return self.__cmp__(other) != 0
+def compare(self, other, context=None):
+"""Compares one to another.
+-1 => a < b
+0  => a = b
+1  => a > b
+NaN => one is NaN
+Like __cmp__, but returns Decimal instances.
+"""
+other = _convert_other(other, raiseit=True)
+# Compare(NaN, NaN) = NaN
+if (self._is_special or other and other._is_special):
+ans = self._check_nans(other, context)
+if ans:
+return ans
+return Decimal(self.__cmp__(other))
+def __hash__(self):
+"""x.__hash__() <==> hash(x)"""
+# Decimal integers must hash the same as the ints
+#
+# The hash of a nonspecial noninteger Decimal must depend only
+# on the value of that Decimal, and not on its representation.
+# For example: hash(Decimal("100E-1")) == hash(Decimal("10")).
+if self._is_special:
+if self._isnan():
+raise TypeError('Cannot hash a NaN value.')
+return hash(str(self))
+if not self:
+return 0
+if self._isinteger():
+op = _WorkRep(self.to_integral_value())
+return hash((-1)**op.sign*op.int*10**op.exp)
+# The value of a nonzero nonspecial Decimal instance is
+# faithfully represented by the triple consisting of its sign,
+# its adjusted exponent, and its coefficient with trailing
+# zeros removed.
+return hash((self._sign,
+self._exp+len(self._int),
+self._int.rstrip('0')))
+def as_tuple(self):
+"""Represents the number as a triple tuple.
+To show the internals exactly as they are.
+"""
+return (self._sign, tuple(map(int, self._int)), self._exp)
+def __repr__(self):
+"""Represents the number as an instance of Decimal."""
+# Invariant:  eval(repr(d)) == d
+return 'Decimal("%s")' % str(self)
+def __str__(self, eng=False, context=None):
+"""Return string representation of the number in scientific notation.
+Captures all of the information in the underlying representation.
+"""
+sign = ['', '-'][self._sign]
+if self._is_special:
+if self._exp == 'F':
+return sign + 'Infinity'
+elif self._exp == 'n':
+return sign + 'NaN' + self._int
+else: # self._exp == 'N'
+return sign + 'sNaN' + self._int
+# number of digits of self._int to left of decimal point
+leftdigits = self._exp + len(self._int)
+# dotplace is number of digits of self._int to the left of the
+# decimal point in the mantissa of the output string (that is,
+# after adjusting the exponent)
+if self._exp <= 0 and leftdigits > -6:
+# no exponent required
+dotplace = leftdigits
+elif not eng:
+# usual scientific notation: 1 digit on left of the point
+dotplace = 1
+elif self._int == '0':
+# engineering notation, zero
+dotplace = (leftdigits + 1) % 3 - 1
+else:
+# engineering notation, nonzero
+dotplace = (leftdigits - 1) % 3 + 1
+if dotplace <= 0:
+intpart = '0'
+fracpart = '.' + '0'*(-dotplace) + self._int
+elif dotplace >= len(self._int):
+intpart = self._int+'0'*(dotplace-len(self._int))
+fracpart = ''
+else:
+intpart = self._int[:dotplace]
+fracpart = '.' + self._int[dotplace:]
+if leftdigits == dotplace:
+exp = ''
+else:
+if context is None:
+context = getcontext()
+exp = ['e', 'E'][context.capitals] + "%+d" % (leftdigits-dotplace)
+return sign + intpart + fracpart + exp
+def to_eng_string(self, context=None):
+"""Convert to engineering-type string.
+Engineering notation has an exponent which is a multiple of 3, so there
+are up to 3 digits left of the decimal place.
+Same rules for when in exponential and when as a value as in __str__.
+"""
+return self.__str__(eng=True, context=context)
+def __neg__(self, context=None):
+"""Returns a copy with the sign switched.
+Rounds, if it has reason.
+"""
+if self._is_special:
+ans = self._check_nans(context=context)
+if ans:
+return ans
+if not self:
+# -Decimal('0') is Decimal('0'), not Decimal('-0')
+ans = self.copy_abs()
+else:
+ans = self.copy_negate()
+if context is None:
+context = getcontext()
+return ans._fix(context)
+def __pos__(self, context=None):
+"""Returns a copy, unless it is a sNaN.
+Rounds the number (if more then precision digits)
+"""
+if self._is_special:
+ans = self._check_nans(context=context)
+if ans:
+return ans
+if not self:
+# + (-0) = 0
+ans = self.copy_abs()
+else:
+ans = Decimal(self)
+if context is None:
+context = getcontext()
+return ans._fix(context)
+def __abs__(self, round=True, context=None):
+"""Returns the absolute value of self.
+If the keyword argument 'round' is false, do not round.  The
+expression self.__abs__(round=False) is equivalent to
+self.copy_abs().
+"""
+if not round:
+return self.copy_abs()
+if self._is_special:
+ans = self._check_nans(context=context)
+if ans:
+return ans
+if self._sign:
+ans = self.__neg__(context=context)
+else:
+ans = self.__pos__(context=context)
+return ans
+def __add__(self, other, context=None):
+"""Returns self + other.
+-INF + INF (or the reverse) cause InvalidOperation errors.
+"""
+other = _convert_other(other)
+if other is NotImplemented:
+return other
+if context is None:
+context = getcontext()
+if self._is_special or other._is_special:
+ans = self._check_nans(other, context)
+if ans:
+return ans
+if self._isinfinity():
+# If both INF, same sign => same as both, opposite => error.
+if self._sign != other._sign and other._isinfinity():
+return context._raise_error(InvalidOperation, '-INF + INF')
+return Decimal(self)
+if other._isinfinity():
+return Decimal(other)  # Can't both be infinity here
+exp = min(self._exp, other._exp)
+negativezero = 0
+if context.rounding == ROUND_FLOOR and self._sign != other._sign:
+# If the answer is 0, the sign should be negative, in this case.
+negativezero = 1
+if not self and not other:
+sign = min(self._sign, other._sign)
+if negativezero:
+sign = 1
+ans = _dec_from_triple(sign, '0', exp)
+ans = ans._fix(context)
+return ans
+if not self:
+exp = max(exp, other._exp - context.prec-1)
+ans = other._rescale(exp, context.rounding)
+ans = ans._fix(context)
+return ans
+if not other:
+exp = max(exp, self._exp - context.prec-1)
+ans = self._rescale(exp, context.rounding)
+ans = ans._fix(context)
+return ans
+op1 = _WorkRep(self)
+op2 = _WorkRep(other)
+op1, op2 = _normalize(op1, op2, context.prec)
+result = _WorkRep()
+if op1.sign != op2.sign:
+# Equal and opposite
+if op1.int == op2.int:
+ans = _dec_from_triple(negativezero, '0', exp)
+ans = ans._fix(context)
+return ans
+if op1.int < op2.int:
+op1, op2 = op2, op1
+# OK, now abs(op1) > abs(op2)
+if op1.sign == 1:
+result.sign = 1
+op1.sign, op2.sign = op2.sign, op1.sign
+else:
+result.sign = 0
+# So we know the sign, and op1 > 0.
+elif op1.sign == 1:
+result.sign = 1
+op1.sign, op2.sign = (0, 0)
+else:
+result.sign = 0
+# Now, op1 > abs(op2) > 0
+if op2.sign == 0:
+result.int = op1.int + op2.int
+else:
+result.int = op1.int - op2.int
+result.exp = op1.exp
+ans = Decimal(result)
+ans = ans._fix(context)
+return ans
+__radd__ = __add__
+def __sub__(self, other, context=None):
+"""Return self - other"""
+other = _convert_other(other)
+if other is NotImplemented:
+return other
+if self._is_special or other._is_special:
+ans = self._check_nans(other, context=context)
+if ans:
+return ans
+# self - other is computed as self + other.copy_negate()
+return self.__add__(other.copy_negate(), context=context)
+def __rsub__(self, other, context=None):
+"""Return other - self"""
+other = _convert_other(other)
+if other is NotImplemented:
+return other
+return other.__sub__(self, context=context)
+def __mul__(self, other, context=None):
+"""Return self * other.
+(+-) INF * 0 (or its reverse) raise InvalidOperation.
+"""
+other = _convert_other(other)
+if other is NotImplemented:
+return other
+if context is None:
+context = getcontext()
+resultsign = self._sign ^ other._sign
+if self._is_special or other._is_special:
+ans = self._check_nans(other, context)
+if ans:
+return ans
+if self._isinfinity():
+if not other:
+return context._raise_error(InvalidOperation, '(+-)INF * 0')
+return Infsign[resultsign]
+if other._isinfinity():
+if not self:
+return context._raise_error(InvalidOperation, '0 * (+-)INF')
+return Infsign[resultsign]
+resultexp = self._exp + other._exp
+# Special case for multiplying by zero
+if not self or not other:
+ans = _dec_from_triple(resultsign, '0', resultexp)
+# Fixing in case the exponent is out of bounds
+ans = ans._fix(context)
+return ans
+# Special case for multiplying by power of 10
+if self._int == '1':
+ans = _dec_from_triple(resultsign, other._int, resultexp)
+ans = ans._fix(context)
+return ans
+if other._int == '1':
+ans = _dec_from_triple(resultsign, self._int, resultexp)
+ans = ans._fix(context)
+return ans
+op1 = _WorkRep(self)
+op2 = _WorkRep(other)
+ans = _dec_from_triple(resultsign, str(op1.int * op2.int), resultexp)
+ans = ans._fix(context)
+return ans
+__rmul__ = __mul__
+def __div__(self, other, context=None):
+"""Return self / other."""
+other = _convert_other(other)
+if other is NotImplemented:
+return NotImplemented
+if context is None:
+context = getcontext()
+sign = self._sign ^ other._sign
+if self._is_special or other._is_special:
+ans = self._check_nans(other, context)
+if ans:
+return ans
+if self._isinfinity() and other._isinfinity():
+return context._raise_error(InvalidOperation, '(+-)INF/(+-)INF')
+if self._isinfinity():
+return Infsign[sign]
+if other._isinfinity():
+context._raise_error(Clamped, 'Division by infinity')
+return _dec_from_triple(sign, '0', context.Etiny())
+# Special cases for zeroes
+if not other:
+if not self:
+return context._raise_error(DivisionUndefined, '0 / 0')
+return context._raise_error(DivisionByZero, 'x / 0', sign)
+if not self:
+exp = self._exp - other._exp
+coeff = 0
+else:
+# OK, so neither = 0, INF or NaN
+shift = len(other._int) - len(self._int) + context.prec + 1
+exp = self._exp - other._exp - shift
+op1 = _WorkRep(self)
+op2 = _WorkRep(other)
+if shift >= 0:
+coeff, remainder = divmod(op1.int * 10**shift, op2.int)
+else:
+coeff, remainder = divmod(op1.int, op2.int * 10**-shift)
+if remainder:
+# result is not exact; adjust to ensure correct rounding
+if coeff % 5 == 0:
+coeff += 1
+else:
+# result is exact; get as close to ideal exponent as possible
+ideal_exp = self._exp - other._exp
+while exp < ideal_exp and coeff % 10 == 0:
+coeff //= 10
+exp += 1
+ans = _dec_from_triple(sign, str(coeff), exp)
+return ans._fix(context)
+__truediv__ = __div__
+def _divide(self, other, context):
+"""Return (self // other, self % other), to context.prec precision.
+Assumes that neither self nor other is a NaN, that self is not
+infinite and that other is nonzero.
+"""
+sign = self._sign ^ other._sign
+if other._isinfinity():
+ideal_exp = self._exp
+else:
+ideal_exp = min(self._exp, other._exp)
+expdiff = self.adjusted() - other.adjusted()
+if not self or other._isinfinity() or expdiff <= -2:
+return (_dec_from_triple(sign, '0', 0),
+self._rescale(ideal_exp, context.rounding))
+if expdiff <= context.prec:
+op1 = _WorkRep(self)
+op2 = _WorkRep(other)
+if op1.exp >= op2.exp:
+op1.int *= 10**(op1.exp - op2.exp)
+else:
+op2.int *= 10**(op2.exp - op1.exp)
+q, r = divmod(op1.int, op2.int)
+if q < 10**context.prec:
+return (_dec_from_triple(sign, str(q), 0),
+_dec_from_triple(self._sign, str(r), ideal_exp))
+# Here the quotient is too large to be representable
+ans = context._raise_error(DivisionImpossible,
+'quotient too large in //, % or divmod')
+return ans, ans
+def __rdiv__(self, other, context=None):
+"""Swaps self/other and returns __div__."""
+other = _convert_other(other)
+if other is NotImplemented:
+return other
+return other.__div__(self, context=context)
+__rtruediv__ = __rdiv__
+def __divmod__(self, other, context=None):
+"""
+Return (self // other, self % other)
+"""
+other = _convert_other(other)
+if other is NotImplemented:
+return other
+if context is None:
+context = getcontext()
+ans = self._check_nans(other, context)
+if ans:
+return (ans, ans)
+sign = self._sign ^ other._sign
+if self._isinfinity():
+if other._isinfinity():
+ans = context._raise_error(InvalidOperation, 'divmod(INF, INF)')
+return ans, ans
+else:
+return (Infsign[sign],
+context._raise_error(InvalidOperation, 'INF % x'))
+if not other:
+if not self:
+ans = context._raise_error(DivisionUndefined, 'divmod(0, 0)')
+return ans, ans
+else:
+return (context._raise_error(DivisionByZero, 'x // 0', sign),
+context._raise_error(InvalidOperation, 'x % 0'))
+quotient, remainder = self._divide(other, context)
+remainder = remainder._fix(context)
+return quotient, remainder
+def __rdivmod__(self, other, context=None):
+"""Swaps self/other and returns __divmod__."""
+other = _convert_other(other)
+if other is NotImplemented:
+return other
+return other.__divmod__(self, context=context)
+def __mod__(self, other, context=None):
+"""
+self % other
+"""
+other = _convert_other(other)
+if other is NotImplemented:
+return other
+if context is None:
+context = getcontext()
+ans = self._check_nans(other, context)
+if ans:
+return ans
+if self._isinfinity():
+return context._raise_error(InvalidOperation, 'INF % x')
+elif not other:
+if self:
+return context._raise_error(InvalidOperation, 'x % 0')
+else:
+return context._raise_error(DivisionUndefined, '0 % 0')
+remainder = self._divide(other, context)[1]
+remainder = remainder._fix(context)
+return remainder
+def __rmod__(self, other, context=None):
+"""Swaps self/other and returns __mod__."""
+other = _convert_other(other)
+if other is NotImplemented:
+return other
+return other.__mod__(self, context=context)
+def remainder_near(self, other, context=None):
+"""
+Remainder nearest to 0-  abs(remainder-near) <= other/2
+"""
+if context is None:
+context = getcontext()
+other = _convert_other(other, raiseit=True)
+ans = self._check_nans(other, context)
+if ans:
+return ans
+# self == +/-infinity -> InvalidOperation
+if self._isinfinity():
+return context._raise_error(InvalidOperation,
+'remainder_near(infinity, x)')
+# other == 0 -> either InvalidOperation or DivisionUndefined
+if not other:
+if self:
+return context._raise_error(InvalidOperation,
+'remainder_near(x, 0)')
+else:
+return context._raise_error(DivisionUndefined,
+'remainder_near(0, 0)')
+# other = +/-infinity -> remainder = self
+if other._isinfinity():
+ans = Decimal(self)
+return ans._fix(context)
+# self = 0 -> remainder = self, with ideal exponent
+ideal_exponent = min(self._exp, other._exp)
+if not self:
+ans = _dec_from_triple(self._sign, '0', ideal_exponent)
+return ans._fix(context)
+# catch most cases of large or small quotient
+expdiff = self.adjusted() - other.adjusted()
+if expdiff >= context.prec + 1:
+# expdiff >= prec+1 => abs(self/other) > 10**prec
+return context._raise_error(DivisionImpossible)
+if expdiff <= -2:
+# expdiff <= -2 => abs(self/other) < 0.1
+ans = self._rescale(ideal_exponent, context.rounding)
+return ans._fix(context)
+# adjust both arguments to have the same exponent, then divide
+op1 = _WorkRep(self)
+op2 = _WorkRep(other)
+if op1.exp >= op2.exp:
+op1.int *= 10**(op1.exp - op2.exp)
+else:
+op2.int *= 10**(op2.exp - op1.exp)
+q, r = divmod(op1.int, op2.int)
+# remainder is r*10**ideal_exponent; other is +/-op2.int *
+# 10**ideal_exponent.   Apply correction to ensure that
+# abs(remainder) <= abs(other)/2
+if 2*r + (q&1) > op2.int:
+r -= op2.int
+q += 1
+if q >= 10**context.prec:
+return context._raise_error(DivisionImpossible)
+# result has same sign as self unless r is negative
+sign = self._sign
+if r < 0:
+sign = 1-sign
+r = -r
+ans = _dec_from_triple(sign, str(r), ideal_exponent)
+return ans._fix(context)
+def __floordiv__(self, other, context=None):
+"""self // other"""
+other = _convert_other(other)
+if other is NotImplemented:
+return other
+if context is None:
+context = getcontext()
+ans = self._check_nans(other, context)
+if ans:
+return ans
+if self._isinfinity():
+if other._isinfinity():
+return context._raise_error(InvalidOperation, 'INF // INF')
+else:
+return Infsign[self._sign ^ other._sign]
+if not other:
+if self:
+return context._raise_error(DivisionByZero, 'x // 0',
+self._sign ^ other._sign)
+else:
+return context._raise_error(DivisionUndefined, '0 // 0')
+return self._divide(other, context)[0]
+def __rfloordiv__(self, other, context=None):
+"""Swaps self/other and returns __floordiv__."""
+other = _convert_other(other)
+if other is NotImplemented:
+return other
+return other.__floordiv__(self, context=context)
+def __float__(self):
+"""Float representation."""
+return float(str(self))
+def __int__(self):
+"""Converts self to an int, truncating if necessary."""
+if self._is_special:
+if self._isnan():
+context = getcontext()
+return context._raise_error(InvalidContext)
+elif self._isinfinity():
+raise OverflowError("Cannot convert infinity to long")
+s = (-1)**self._sign
+if self._exp >= 0:
+return s*int(self._int)*10**self._exp
+else:
+return s*int(self._int[:self._exp] or '0')
+def __long__(self):
+"""Converts to a long.
+Equivalent to long(int(self))
+"""
+return long(self.__int__())
+def _fix_nan(self, context):
+"""Decapitate the payload of a NaN to fit the context"""
+payload = self._int
+# maximum length of payload is precision if _clamp=0,
+# precision-1 if _clamp=1.
+max_payload_len = context.prec - context._clamp
+if len(payload) > max_payload_len:
+payload = payload[len(payload)-max_payload_len:].lstrip('0')
+return _dec_from_triple(self._sign, payload, self._exp, True)
+return Decimal(self)
+def _fix(self, context):
+"""Round if it is necessary to keep self within prec precision.
+Rounds and fixes the exponent.  Does not raise on a sNaN.
+Arguments:
+self - Decimal instance
+context - context used.
+"""
+if self._is_special:
+if self._isnan():
+# decapitate payload if necessary
+return self._fix_nan(context)
+else:
+# self is +/-Infinity; return unaltered
+return Decimal(self)
+# if self is zero then exponent should be between Etiny and
+# Emax if _clamp==0, and between Etiny and Etop if _clamp==1.
+Etiny = context.Etiny()
+Etop = context.Etop()
+if not self:
+exp_max = [context.Emax, Etop][context._clamp]
+new_exp = min(max(self._exp, Etiny), exp_max)
+if new_exp != self._exp:
+context._raise_error(Clamped)
+return _dec_from_triple(self._sign, '0', new_exp)
+else:
+return Decimal(self)
+# exp_min is the smallest allowable exponent of the result,
+# equal to max(self.adjusted()-context.prec+1, Etiny)
+exp_min = len(self._int) + self._exp - context.prec
+if exp_min > Etop:
+# overflow: exp_min > Etop iff self.adjusted() > Emax
+context._raise_error(Inexact)
+context._raise_error(Rounded)
+return context._raise_error(Overflow, 'above Emax', self._sign)
+self_is_subnormal = exp_min < Etiny
+if self_is_subnormal:
+context._raise_error(Subnormal)
+exp_min = Etiny
+# round if self has too many digits
+if self._exp < exp_min:
+context._raise_error(Rounded)
+digits = len(self._int) + self._exp - exp_min
+if digits < 0:
+self = _dec_from_triple(self._sign, '1', exp_min-1)
+digits = 0
+this_function = getattr(self, self._pick_rounding_function[context.rounding])
+changed = this_function(digits)
+coeff = self._int[:digits] or '0'
+if changed == 1:
+coeff = str(int(coeff)+1)
+ans = _dec_from_triple(self._sign, coeff, exp_min)
+if changed:
+context._raise_error(Inexact)
+if self_is_subnormal:
+context._raise_error(Underflow)
+if not ans:
+# raise Clamped on underflow to 0
+context._raise_error(Clamped)
+elif len(ans._int) == context.prec+1:
+# we get here only if rescaling rounds the
+# cofficient up to exactly 10**context.prec
+if ans._exp < Etop:
+ans = _dec_from_triple(ans._sign,
+ans._int[:-1], ans._exp+1)
+else:
+# Inexact and Rounded have already been raised
+ans = context._raise_error(Overflow, 'above Emax',
+self._sign)
+return ans
+# fold down if _clamp == 1 and self has too few digits
+if context._clamp == 1 and self._exp > Etop:
+context._raise_error(Clamped)
+self_padded = self._int + '0'*(self._exp - Etop)
+return _dec_from_triple(self._sign, self_padded, Etop)
+# here self was representable to begin with; return unchanged
+return Decimal(self)
+_pick_rounding_function = {}
+# for each of the rounding functions below:
+#   self is a finite, nonzero Decimal
+#   prec is an integer satisfying 0 <= prec < len(self._int)
+#
+# each function returns either -1, 0, or 1, as follows:
+#   1 indicates that self should be rounded up (away from zero)
+#   0 indicates that self should be truncated, and that all the
+#     digits to be truncated are zeros (so the value is unchanged)
+#  -1 indicates that there are nonzero digits to be truncated
+def _round_down(self, prec):
+"""Also known as round-towards-0, truncate."""
+if _all_zeros(self._int, prec):
+return 0
+else:
+return -1
+def _round_up(self, prec):
+"""Rounds away from 0."""
+return -self._round_down(prec)
+def _round_half_up(self, prec):
+"""Rounds 5 up (away from 0)"""
+if self._int[prec] in '56789':
+return 1
+elif _all_zeros(self._int, prec):
+return 0
+else:
+return -1
+def _round_half_down(self, prec):
+"""Round 5 down"""
+if _exact_half(self._int, prec):
+return -1
+else:
+return self._round_half_up(prec)
+def _round_half_even(self, prec):
+"""Round 5 to even, rest to nearest."""
+if _exact_half(self._int, prec) and \
+(prec == 0 or self._int[prec-1] in '02468'):
+return -1
+else:
+return self._round_half_up(prec)
+def _round_ceiling(self, prec):
+"""Rounds up (not away from 0 if negative.)"""
+if self._sign:
+return self._round_down(prec)
+else:
+return -self._round_down(prec)
+def _round_floor(self, prec):
+"""Rounds down (not towards 0 if negative)"""
+if not self._sign:
+return self._round_down(prec)
+else:
+return -self._round_down(prec)
+def _round_05up(self, prec):
+"""Round down unless digit prec-1 is 0 or 5."""
+if prec and self._int[prec-1] not in '05':
+return self._round_down(prec)
+else:
+return -self._round_down(prec)
+def fma(self, other, third, context=None):
+"""Fused multiply-add.
+Returns self*other+third with no rounding of the intermediate
+product self*other.
+self and other are multiplied together, with no rounding of
+the result.  The third operand is then added to the result,
+and a single final rounding is performed.
+"""
+other = _convert_other(other, raiseit=True)
+# compute product; raise InvalidOperation if either operand is
+# a signaling NaN or if the product is zero times infinity.
+if self._is_special or other._is_special:
+if context is None:
+context = getcontext()
+if self._exp == 'N':
+return context._raise_error(InvalidOperation, 'sNaN', self)
+if other._exp == 'N':
+return context._raise_error(InvalidOperation, 'sNaN', other)
+if self._exp == 'n':
+product = self
+elif other._exp == 'n':
+product = other
+elif self._exp == 'F':
+if not other:
+return context._raise_error(InvalidOperation,
+'INF * 0 in fma')
+product = Infsign[self._sign ^ other._sign]
+elif other._exp == 'F':
+if not self:
+return context._raise_error(InvalidOperation,
+'0 * INF in fma')
+product = Infsign[self._sign ^ other._sign]
+else:
+product = _dec_from_triple(self._sign ^ other._sign,
+str(int(self._int) * int(other._int)),
+self._exp + other._exp)
+third = _convert_other(third, raiseit=True)
+return product.__add__(third, context)
+def _power_modulo(self, other, modulo, context=None):
+"""Three argument version of __pow__"""
+# if can't convert other and modulo to Decimal, raise
+# TypeError; there's no point returning NotImplemented (no
+# equivalent of __rpow__ for three argument pow)
+other = _convert_other(other, raiseit=True)
+modulo = _convert_other(modulo, raiseit=True)
+if context is None:
+context = getcontext()
+# deal with NaNs: if there are any sNaNs then first one wins,
+# (i.e. behaviour for NaNs is identical to that of fma)
+self_is_nan = self._isnan()
+other_is_nan = other._isnan()
+modulo_is_nan = modulo._isnan()
+if self_is_nan or other_is_nan or modulo_is_nan:
+if self_is_nan == 2:
+return context._raise_error(InvalidOperation, 'sNaN',
+self)
+if other_is_nan == 2:
+return context._raise_error(InvalidOperation, 'sNaN',
+other)
+if modulo_is_nan == 2:
+return context._raise_error(InvalidOperation, 'sNaN',
+modulo)
+if self_is_nan:
+return self._fix_nan(context)
+if other_is_nan:
+return other._fix_nan(context)
+return modulo._fix_nan(context)
+# check inputs: we apply same restrictions as Python's pow()
+if not (self._isinteger() and
+other._isinteger() and
+modulo._isinteger()):
+return context._raise_error(InvalidOperation,
+'pow() 3rd argument not allowed '
+'unless all arguments are integers')
+if other < 0:
+return context._raise_error(InvalidOperation,
+'pow() 2nd argument cannot be '
+'negative when 3rd argument specified')
+if not modulo:
+return context._raise_error(InvalidOperation,
+'pow() 3rd argument cannot be 0')
+# additional restriction for decimal: the modulus must be less
+# than 10**prec in absolute value
+if modulo.adjusted() >= context.prec:
+return context._raise_error(InvalidOperation,
+'insufficient precision: pow() 3rd '
+'argument must not have more than '
+'precision digits')
+# define 0**0 == NaN, for consistency with two-argument pow
+# (even though it hurts!)
+if not other and not self:
+return context._raise_error(InvalidOperation,
+'at least one of pow() 1st argument '
+'and 2nd argument must be nonzero ;'
+'0**0 is not defined')
+# compute sign of result
+if other._iseven():
+sign = 0
+else:
+sign = self._sign
+# convert modulo to a Python integer, and self and other to
+# Decimal integers (i.e. force their exponents to be >= 0)
+modulo = abs(int(modulo))
+base = _WorkRep(self.to_integral_value())
+exponent = _WorkRep(other.to_integral_value())
+# compute result using integer pow()
+base = (base.int % modulo * pow(10, base.exp, modulo)) % modulo
+for i in xrange(exponent.exp):
+base = pow(base, 10, modulo)
+base = pow(base, exponent.int, modulo)
+return _dec_from_triple(sign, str(base), 0)
+def _power_exact(self, other, p):
+"""Attempt to compute self**other exactly.
+Given Decimals self and other and an integer p, attempt to
+compute an exact result for the power self**other, with p
+digits of precision.  Return None if self**other is not
+exactly representable in p digits.
+Assumes that elimination of special cases has already been
+performed: self and other must both be nonspecial; self must
+be positive and not numerically equal to 1; other must be
+nonzero.  For efficiency, other._exp should not be too large,
+so that 10**abs(other._exp) is a feasible calculation."""
+# In the comments below, we write x for the value of self and
+# y for the value of other.  Write x = xc*10**xe and y =
+# yc*10**ye.
+# The main purpose of this method is to identify the *failure*
+# of x**y to be exactly representable with as little effort as
+# possible.  So we look for cheap and easy tests that
+# eliminate the possibility of x**y being exact.  Only if all
+# these tests are passed do we go on to actually compute x**y.
+# Here's the main idea.  First normalize both x and y.  We
+# express y as a rational m/n, with m and n relatively prime
+# and n>0.  Then for x**y to be exactly representable (at
+# *any* precision), xc must be the nth power of a positive
+# integer and xe must be divisible by n.  If m is negative
+# then additionally xc must be a power of either 2 or 5, hence
+# a power of 2**n or 5**n.
+#
+# There's a limit to how small |y| can be: if y=m/n as above
+# then:
+#
+#  (1) if xc != 1 then for the result to be representable we
+#      need xc**(1/n) >= 2, and hence also xc**|y| >= 2.  So
+#      if |y| <= 1/nbits(xc) then xc < 2**nbits(xc) <=
+#      2**(1/|y|), hence xc**|y| < 2 and the result is not
+#      representable.
+#
+#  (2) if xe != 0, |xe|*(1/n) >= 1, so |xe|*|y| >= 1.  Hence if
+#      |y| < 1/|xe| then the result is not representable.
+#
+# Note that since x is not equal to 1, at least one of (1) and
+# (2) must apply.  Now |y| < 1/nbits(xc) iff |yc|*nbits(xc) <
+# 10**-ye iff len(str(|yc|*nbits(xc)) <= -ye.
+#
+# There's also a limit to how large y can be, at least if it's
+# positive: the normalized result will have coefficient xc**y,
+# so if it's representable then xc**y < 10**p, and y <
+# p/log10(xc).  Hence if y*log10(xc) >= p then the result is
+# not exactly representable.
+# if len(str(abs(yc*xe)) <= -ye then abs(yc*xe) < 10**-ye,
+# so |y| < 1/xe and the result is not representable.
+# Similarly, len(str(abs(yc)*xc_bits)) <= -ye implies |y|
+# < 1/nbits(xc).
+x = _WorkRep(self)
+xc, xe = x.int, x.exp
+while xc % 10 == 0:
+xc //= 10
+xe += 1
+y = _WorkRep(other)
+yc, ye = y.int, y.exp
+while yc % 10 == 0:
+yc //= 10
+ye += 1
+# case where xc == 1: result is 10**(xe*y), with xe*y
+# required to be an integer
+if xc == 1:
+if ye >= 0:
+exponent = xe*yc*10**ye
+else:
+exponent, remainder = divmod(xe*yc, 10**-ye)
+if remainder:
+return None
+if y.sign == 1:
+exponent = -exponent
+# if other is a nonnegative integer, use ideal exponent
+if other._isinteger() and other._sign == 0:
+ideal_exponent = self._exp*int(other)
+zeros = min(exponent-ideal_exponent, p-1)
+else:
+zeros = 0
+return _dec_from_triple(0, '1' + '0'*zeros, exponent-zeros)
+# case where y is negative: xc must be either a power
+# of 2 or a power of 5.
+if y.sign == 1:
+last_digit = xc % 10
+if last_digit in (2,4,6,8):
+# quick test for power of 2
+if xc & -xc != xc:
+return None
+# now xc is a power of 2; e is its exponent
+e = _nbits(xc)-1
+# find e*y and xe*y; both must be integers
+if ye >= 0:
+y_as_int = yc*10**ye
+e = e*y_as_int
+xe = xe*y_as_int
+else:
+ten_pow = 10**-ye
+e, remainder = divmod(e*yc, ten_pow)
+if remainder:
+return None
+xe, remainder = divmod(xe*yc, ten_pow)
+if remainder:
+return None
+if e*65 >= p*93: # 93/65 > log(10)/log(5)
+return None
+xc = 5**e
+elif last_digit == 5:
+# e >= log_5(xc) if xc is a power of 5; we have
+# equality all the way up to xc=5**2658
+e = _nbits(xc)*28//65
+xc, remainder = divmod(5**e, xc)
+if remainder:
+return None
+while xc % 5 == 0:
+xc //= 5
+e -= 1
+if ye >= 0:
+y_as_integer = yc*10**ye
+e = e*y_as_integer
+xe = xe*y_as_integer
+else:
+ten_pow = 10**-ye
+e, remainder = divmod(e*yc, ten_pow)
+if remainder:
+return None
+xe, remainder = divmod(xe*yc, ten_pow)
+if remainder:
+return None
+if e*3 >= p*10: # 10/3 > log(10)/log(2)
+return None
+xc = 2**e
+else:
+return None
+if xc >= 10**p:
+return None
+xe = -e-xe
+return _dec_from_triple(0, str(xc), xe)
+# now y is positive; find m and n such that y = m/n
+if ye >= 0:
+m, n = yc*10**ye, 1
+else:
+if xe != 0 and len(str(abs(yc*xe))) <= -ye:
+return None
+xc_bits = _nbits(xc)
+if xc != 1 and len(str(abs(yc)*xc_bits)) <= -ye:
+return None
+m, n = yc, 10**(-ye)
+while m % 2 == n % 2 == 0:
+m //= 2
+n //= 2
+while m % 5 == n % 5 == 0:
+m //= 5
+n //= 5
+# compute nth root of xc*10**xe
+if n > 1:
+# if 1 < xc < 2**n then xc isn't an nth power
+if xc != 1 and xc_bits <= n:
+return None
+xe, rem = divmod(xe, n)
+if rem != 0:
+return None
+# compute nth root of xc using Newton's method
+a = 1L << -(-_nbits(xc)//n) # initial estimate
+while True:
+q, r = divmod(xc, a**(n-1))
+if a <= q:
+break
+else:
+a = (a*(n-1) + q)//n
+if not (a == q and r == 0):
+return None
+xc = a
+# now xc*10**xe is the nth root of the original xc*10**xe
+# compute mth power of xc*10**xe
+# if m > p*100//_log10_lb(xc) then m > p/log10(xc), hence xc**m >
+# 10**p and the result is not representable.
+if xc > 1 and m > p*100//_log10_lb(xc):
+return None
+xc = xc**m
+xe *= m
+if xc > 10**p:
+return None
+# by this point the result *is* exactly representable
+# adjust the exponent to get as close as possible to the ideal
+# exponent, if necessary
+str_xc = str(xc)
+if other._isinteger() and other._sign == 0:
+ideal_exponent = self._exp*int(other)
+zeros = min(xe-ideal_exponent, p-len(str_xc))
+else:
+zeros = 0
+return _dec_from_triple(0, str_xc+'0'*zeros, xe-zeros)
+def __pow__(self, other, modulo=None, context=None):
+"""Return self ** other [ % modulo].
+With two arguments, compute self**other.
+With three arguments, compute (self**other) % modulo.  For the
+three argument form, the following restrictions on the
+arguments hold:
+- all three arguments must be integral
+- other must be nonnegative
+- either self or other (or both) must be nonzero
+- modulo must be nonzero and must have at most p digits,
+where p is the context precision.
+If any of these restrictions is violated the InvalidOperation
+flag is raised.
+The result of pow(self, other, modulo) is identical to the
+result that would be obtained by computing (self**other) %
+modulo with unbounded precision, but is computed more
+efficiently.  It is always exact.
+"""
+if modulo is not None:
+return self._power_modulo(other, modulo, context)
+other = _convert_other(other)
+if other is NotImplemented:
+return other
+if context is None:
+context = getcontext()
+# either argument is a NaN => result is NaN
+ans = self._check_nans(other, context)
+if ans:
+return ans
+# 0**0 = NaN (!), x**0 = 1 for nonzero x (including +/-Infinity)
+if not other:
+if not self:
+return context._raise_error(InvalidOperation, '0 ** 0')
+else:
+return Dec_p1
+# result has sign 1 iff self._sign is 1 and other is an odd integer
+result_sign = 0
+if self._sign == 1:
+if other._isinteger():
+if not other._iseven():
+result_sign = 1
+else:
+# -ve**noninteger = NaN
+# (-0)**noninteger = 0**noninteger
+if self:
+return context._raise_error(InvalidOperation,
+'x ** y with x negative and y not an integer')
+# negate self, without doing any unwanted rounding
+self = self.copy_negate()
+# 0**(+ve or Inf)= 0; 0**(-ve or -Inf) = Infinity
+if not self:
+if other._sign == 0:
+return _dec_from_triple(result_sign, '0', 0)
+else:
+return Infsign[result_sign]
+# Inf**(+ve or Inf) = Inf; Inf**(-ve or -Inf) = 0
+if self._isinfinity():
+if other._sign == 0:
+return Infsign[result_sign]
+else:
+return _dec_from_triple(result_sign, '0', 0)
+# 1**other = 1, but the choice of exponent and the flags
+# depend on the exponent of self, and on whether other is a
+# positive integer, a negative integer, or neither
+if self == Dec_p1:
+if other._isinteger():
+# exp = max(self._exp*max(int(other), 0),
+# 1-context.prec) but evaluating int(other) directly
+# is dangerous until we know other is small (other
+# could be 1e999999999)
+if other._sign == 1:
+multiplier = 0
+elif other > context.prec:
+multiplier = context.prec
+else:
+multiplier = int(other)
+exp = self._exp * multiplier
+if exp < 1-context.prec:
+exp = 1-context.prec
+context._raise_error(Rounded)
+else:
+context._raise_error(Inexact)
+context._raise_error(Rounded)
+exp = 1-context.prec
+return _dec_from_triple(result_sign, '1'+'0'*-exp, exp)
+# compute adjusted exponent of self
+self_adj = self.adjusted()
+# self ** infinity is infinity if self > 1, 0 if self < 1
+# self ** -infinity is infinity if self < 1, 0 if self > 1
+if other._isinfinity():
+if (other._sign == 0) == (self_adj < 0):
+return _dec_from_triple(result_sign, '0', 0)
+else:
+return Infsign[result_sign]
+# from here on, the result always goes through the call
+# to _fix at the end of this function.
+ans = None
+# crude test to catch cases of extreme overflow/underflow.  If
+# log10(self)*other >= 10**bound and bound >= len(str(Emax))
+# then 10**bound >= 10**len(str(Emax)) >= Emax+1 and hence
+# self**other >= 10**(Emax+1), so overflow occurs.  The test
+# for underflow is similar.
+bound = self._log10_exp_bound() + other.adjusted()
+if (self_adj >= 0) == (other._sign == 0):
+# self > 1 and other +ve, or self < 1 and other -ve
+# possibility of overflow
+if bound >= len(str(context.Emax)):
+ans = _dec_from_triple(result_sign, '1', context.Emax+1)
+else:
+# self > 1 and other -ve, or self < 1 and other +ve
+# possibility of underflow to 0
+Etiny = context.Etiny()
+if bound >= len(str(-Etiny)):
+ans = _dec_from_triple(result_sign, '1', Etiny-1)
+# try for an exact result with precision +1
+if ans is None:
+ans = self._power_exact(other, context.prec + 1)
+if ans is not None and result_sign == 1:
+ans = _dec_from_triple(1, ans._int, ans._exp)
+# usual case: inexact result, x**y computed directly as exp(y*log(x))
+if ans is None:
+p = context.prec
+x = _WorkRep(self)
+xc, xe = x.int, x.exp
+y = _WorkRep(other)
+yc, ye = y.int, y.exp
+if y.sign == 1:
+yc = -yc
+# compute correctly rounded result:  start with precision +3,
+# then increase precision until result is unambiguously roundable
+extra = 3
+while True:
+coeff, exp = _dpower(xc, xe, yc, ye, p+extra)
+if coeff % (5*10**(len(str(coeff))-p-1)):
+break
+extra += 3
+ans = _dec_from_triple(result_sign, str(coeff), exp)
+# the specification says that for non-integer other we need to
+# raise Inexact, even when the result is actually exact.  In
+# the same way, we need to raise Underflow here if the result
+# is subnormal.  (The call to _fix will take care of raising
+# Rounded and Subnormal, as usual.)
+if not other._isinteger():
+context._raise_error(Inexact)
+# pad with zeros up to length context.prec+1 if necessary
+if len(ans._int) <= context.prec:
+expdiff = context.prec+1 - len(ans._int)
+ans = _dec_from_triple(ans._sign, ans._int+'0'*expdiff,
+ans._exp-expdiff)
+if ans.adjusted() < context.Emin:
+context._raise_error(Underflow)
+# unlike exp, ln and log10, the power function respects the
+# rounding mode; no need to use ROUND_HALF_EVEN here
+ans = ans._fix(context)
+return ans
+def __rpow__(self, other, context=None):
+"""Swaps self/other and returns __pow__."""
+other = _convert_other(other)
+if other is NotImplemented:
+return other
+return other.__pow__(self, context=context)
+def normalize(self, context=None):
+"""Normalize- strip trailing 0s, change anything equal to 0 to 0e0"""
+if context is None:
+context = getcontext()
+if self._is_special:
+ans = self._check_nans(context=context)
+if ans:
+return ans
+dup = self._fix(context)
+if dup._isinfinity():
+return dup
+if not dup:
+return _dec_from_triple(dup._sign, '0', 0)
+exp_max = [context.Emax, context.Etop()][context._clamp]
+end = len(dup._int)
+exp = dup._exp
+while dup._int[end-1] == '0' and exp < exp_max:
+exp += 1
+end -= 1
+return _dec_from_triple(dup._sign, dup._int[:end], exp)
+def quantize(self, exp, rounding=None, context=None, watchexp=True):
+"""Quantize self so its exponent is the same as that of exp.
+Similar to self._rescale(exp._exp) but with error checking.
+"""
+exp = _convert_other(exp, raiseit=True)
+if context is None:
+context = getcontext()
+if rounding is None:
+rounding = context.rounding
+if self._is_special or exp._is_special:
+ans = self._check_nans(exp, context)
+if ans:
+return ans
+if exp._isinfinity() or self._isinfinity():
+if exp._isinfinity() and self._isinfinity():
+return Decimal(self)  # if both are inf, it is OK
+return context._raise_error(InvalidOperation,
+'quantize with one INF')
+# if we're not watching exponents, do a simple rescale
+if not watchexp:
+ans = self._rescale(exp._exp, rounding)
+# raise Inexact and Rounded where appropriate
+if ans._exp > self._exp:
+context._raise_error(Rounded)
+if ans != self:
+context._raise_error(Inexact)
+return ans
+# exp._exp should be between Etiny and Emax
+if not (context.Etiny() <= exp._exp <= context.Emax):
+return context._raise_error(InvalidOperation,
+'target exponent out of bounds in quantize')
+if not self:
+ans = _dec_from_triple(self._sign, '0', exp._exp)
+return ans._fix(context)
+self_adjusted = self.adjusted()
+if self_adjusted > context.Emax:
+return context._raise_error(InvalidOperation,
+'exponent of quantize result too large for current context')
+if self_adjusted - exp._exp + 1 > context.prec:
+return context._raise_error(InvalidOperation,
+'quantize result has too many digits for current context')
+ans = self._rescale(exp._exp, rounding)
+if ans.adjusted() > context.Emax:
+return context._raise_error(InvalidOperation,
+'exponent of quantize result too large for current context')
+if len(ans._int) > context.prec:
+return context._raise_error(InvalidOperation,
+'quantize result has too many digits for current context')
+# raise appropriate flags
+if ans._exp > self._exp:
+context._raise_error(Rounded)
+if ans != self:
+context._raise_error(Inexact)
+if ans and ans.adjusted() < context.Emin:
+context._raise_error(Subnormal)
+# call to fix takes care of any necessary folddown
+ans = ans._fix(context)
+return ans
+def same_quantum(self, other):
+"""Return True if self and other have the same exponent; otherwise
+return False.
+If either operand is a special value, the following rules are used:
+* return True if both operands are infinities
+* return True if both operands are NaNs
+* otherwise, return False.
+"""
+other = _convert_other(other, raiseit=True)
+if self._is_special or other._is_special:
+return (self.is_nan() and other.is_nan() or
+self.is_infinite() and other.is_infinite())
+return self._exp == other._exp
+def _rescale(self, exp, rounding):
+"""Rescale self so that the exponent is exp, either by padding with zeros
+or by truncating digits, using the given rounding mode.
+Specials are returned without change.  This operation is
+quiet: it raises no flags, and uses no information from the
+context.
+exp = exp to scale to (an integer)
+rounding = rounding mode
+"""
+if self._is_special:
+return Decimal(self)
+if not self:
+return _dec_from_triple(self._sign, '0', exp)
+if self._exp >= exp:
+# pad answer with zeros if necessary
+return _dec_from_triple(self._sign,
+self._int + '0'*(self._exp - exp), exp)
+# too many digits; round and lose data.  If self.adjusted() <
+# exp-1, replace self by 10**(exp-1) before rounding
+digits = len(self._int) + self._exp - exp
+if digits < 0:
+self = _dec_from_triple(self._sign, '1', exp-1)
+digits = 0
+this_function = getattr(self, self._pick_rounding_function[rounding])
+changed = this_function(digits)
+coeff = self._int[:digits] or '0'
+if changed == 1:
+coeff = str(int(coeff)+1)
+return _dec_from_triple(self._sign, coeff, exp)
+def to_integral_exact(self, rounding=None, context=None):
+"""Rounds to a nearby integer.
+If no rounding mode is specified, take the rounding mode from
+the context.  This method raises the Rounded and Inexact flags
+when appropriate.
+See also: to_integral_value, which does exactly the same as
+this method except that it doesn't raise Inexact or Rounded.
+"""
+if self._is_special:
+ans = self._check_nans(context=context)
+if ans:
+return ans
+return Decimal(self)
+if self._exp >= 0:
+return Decimal(self)
+if not self:
+return _dec_from_triple(self._sign, '0', 0)
+if context is None:
+context = getcontext()
+if rounding is None:
+rounding = context.rounding
+context._raise_error(Rounded)
+ans = self._rescale(0, rounding)
+if ans != self:
+context._raise_error(Inexact)
+return ans
+def to_integral_value(self, rounding=None, context=None):
+"""Rounds to the nearest integer, without raising inexact, rounded."""
+if context is None:
+context = getcontext()
+if rounding is None:
+rounding = context.rounding
+if self._is_special:
+ans = self._check_nans(context=context)
+if ans:
+return ans
+return Decimal(self)
+if self._exp >= 0:
+return Decimal(self)
+else:
+return self._rescale(0, rounding)
+# the method name changed, but we provide also the old one, for compatibility
+to_integral = to_integral_value
+def sqrt(self, context=None):
+"""Return the square root of self."""
+if self._is_special:
+ans = self._check_nans(context=context)
+if ans:
+return ans
+if self._isinfinity() and self._sign == 0:
+return Decimal(self)
+if not self:
+# exponent = self._exp // 2.  sqrt(-0) = -0
+ans = _dec_from_triple(self._sign, '0', self._exp // 2)
+return ans._fix(context)
+if context is None:
+context = getcontext()
+if self._sign == 1:
+return context._raise_error(InvalidOperation, 'sqrt(-x), x > 0')
+# At this point self represents a positive number.  Let p be
+# the desired precision and express self in the form c*100**e
+# with c a positive real number and e an integer, c and e
+# being chosen so that 100**(p-1) <= c < 100**p.  Then the
+# (exact) square root of self is sqrt(c)*10**e, and 10**(p-1)
+# <= sqrt(c) < 10**p, so the closest representable Decimal at
+# precision p is n*10**e where n = round_half_even(sqrt(c)),
+# the closest integer to sqrt(c) with the even integer chosen
+# in the case of a tie.
+#
+# To ensure correct rounding in all cases, we use the
+# following trick: we compute the square root to an extra
+# place (precision p+1 instead of precision p), rounding down.
+# Then, if the result is inexact and its last digit is 0 or 5,
+# we increase the last digit to 1 or 6 respectively; if it's
+# exact we leave the last digit alone.  Now the final round to
+# p places (or fewer in the case of underflow) will round
+# correctly and raise the appropriate flags.
+# use an extra digit of precision
+prec = context.prec+1
+# write argument in the form c*100**e where e = self._exp//2
+# is the 'ideal' exponent, to be used if the square root is
+# exactly representable.  l is the number of 'digits' of c in
+# base 100, so that 100**(l-1) <= c < 100**l.
+op = _WorkRep(self)
+e = op.exp >> 1
+if op.exp & 1:
+c = op.int * 10
+l = (len(self._int) >> 1) + 1
+else:
+c = op.int
+l = len(self._int)+1 >> 1
+# rescale so that c has exactly prec base 100 'digits'
+shift = prec-l
+if shift >= 0:
+c *= 100**shift
+exact = True
+else:
+c, remainder = divmod(c, 100**-shift)
+exact = not remainder
+e -= shift
+# find n = floor(sqrt(c)) using Newton's method
+n = 10**prec
+while True:
+q = c//n
+if n <= q:
+break
+else:
+n = n + q >> 1
+exact = exact and n*n == c
+if exact:
+# result is exact; rescale to use ideal exponent e
+if shift >= 0:
+# assert n % 10**shift == 0
+n //= 10**shift
+else:
+n *= 10**-shift
+e += shift
+else:
+# result is not exact; fix last digit as described above
+if n % 5 == 0:
+n += 1
+ans = _dec_from_triple(0, str(n), e)
+# round, and fit to current context
+context = context._shallow_copy()
+rounding = context._set_rounding(ROUND_HALF_EVEN)
+ans = ans._fix(context)
+context.rounding = rounding
+return ans
+def max(self, other, context=None):
+"""Returns the larger value.
+Like max(self, other) except if one is not a number, returns
+NaN (and signals if one is sNaN).  Also rounds.
+"""
+other = _convert_other(other, raiseit=True)
+if context is None:
+context = getcontext()
+if self._is_special or other._is_special:
+# If one operand is a quiet NaN and the other is number, then the
+# number is always returned
+sn = self._isnan()
+on = other._isnan()
+if sn or on:
+if on == 1 and sn != 2:
+return self._fix_nan(context)
+if sn == 1 and on != 2:
+return other._fix_nan(context)
+return self._check_nans(other, context)
+c = self.__cmp__(other)
+if c == 0:
+# If both operands are finite and equal in numerical value
+# then an ordering is applied:
+#
+# If the signs differ then max returns the operand with the
+# positive sign and min returns the operand with the negative sign
+#
+# If the signs are the same then the exponent is used to select
+# the result.  This is exactly the ordering used in compare_total.
+c = self.compare_total(other)
+if c == -1:
+ans = other
+else:
+ans = self
+return ans._fix(context)
+def min(self, other, context=None):
+"""Returns the smaller value.
+Like min(self, other) except if one is not a number, returns
+NaN (and signals if one is sNaN).  Also rounds.
+"""
+other = _convert_other(other, raiseit=True)
+if context is None:
+context = getcontext()
+if self._is_special or other._is_special:
+# If one operand is a quiet NaN and the other is number, then the
+# number is always returned
+sn = self._isnan()
+on = other._isnan()
+if sn or on:
+if on == 1 and sn != 2:
+return self._fix_nan(context)
+if sn == 1 and on != 2:
+return other._fix_nan(context)
+return self._check_nans(other, context)
+c = self.__cmp__(other)
+if c == 0:
+c = self.compare_total(other)
+if c == -1:
+ans = self
+else:
+ans = other
+return ans._fix(context)
+def _isinteger(self):
+"""Returns whether self is an integer"""
+if self._is_special:
+return False
+if self._exp >= 0:
+return True
+rest = self._int[self._exp:]
+return rest == '0'*len(rest)
+def _iseven(self):
+"""Returns True if self is even.  Assumes self is an integer."""
+if not self or self._exp > 0:
+return True
+return self._int[-1+self._exp] in '02468'
+def adjusted(self):
+"""Return the adjusted exponent of self"""
+try:
+return self._exp + len(self._int) - 1
+# If NaN or Infinity, self._exp is string
+except TypeError:
+return 0
+def canonical(self, context=None):
+"""Returns the same Decimal object.
+As we do not have different encodings for the same number, the
+received object already is in its canonical form.
+"""
+return self
+def compare_signal(self, other, context=None):
+"""Compares self to the other operand numerically.
+It's pretty much like compare(), but all NaNs signal, with signaling
+NaNs taking precedence over quiet NaNs.
+"""
+if context is None:
+context = getcontext()
+self_is_nan = self._isnan()
+other_is_nan = other._isnan()
+if self_is_nan == 2:
+return context._raise_error(InvalidOperation, 'sNaN',
+self)
+if other_is_nan == 2:
+return context._raise_error(InvalidOperation, 'sNaN',
+other)
+if self_is_nan:
+return context._raise_error(InvalidOperation, 'NaN in compare_signal',
+self)
+if other_is_nan:
+return context._raise_error(InvalidOperation, 'NaN in compare_signal',
+other)
+return self.compare(other, context=context)
+def compare_total(self, other):
+"""Compares self to other using the abstract representations.
+This is not like the standard compare, which use their numerical
+value. Note that a total ordering is defined for all possible abstract
+representations.
+"""
+# if one is negative and the other is positive, it's easy
+if self._sign and not other._sign:
+return Dec_n1
+if not self._sign and other._sign:
+return Dec_p1
+sign = self._sign
+# let's handle both NaN types
+self_nan = self._isnan()
+other_nan = other._isnan()
+if self_nan or other_nan:
+if self_nan == other_nan:
+if self._int < other._int:
+if sign:
+return Dec_p1
+else:
+return Dec_n1
+if self._int > other._int:
+if sign:
+return Dec_n1
+else:
+return Dec_p1
+return Dec_0
+if sign:
+if self_nan == 1:
+return Dec_n1
+if other_nan == 1:
+return Dec_p1
+if self_nan == 2:
+return Dec_n1
+if other_nan == 2:
+return Dec_p1
+else:
+if self_nan == 1:
+return Dec_p1
+if other_nan == 1:
+return Dec_n1
+if self_nan == 2:
+return Dec_p1
+if other_nan == 2:
+return Dec_n1
+if self < other:
+return Dec_n1
+if self > other:
+return Dec_p1
+if self._exp < other._exp:
+if sign:
+return Dec_p1
+else:
+return Dec_n1
+if self._exp > other._exp:
+if sign:
+return Dec_n1
+else:
+return Dec_p1
+return Dec_0
+def compare_total_mag(self, other):
+"""Compares self to other using abstract repr., ignoring sign.
+Like compare_total, but with operand's sign ignored and assumed to be 0.
+"""
+s = self.copy_abs()
+o = other.copy_abs()
+return s.compare_total(o)
+def copy_abs(self):
+"""Returns a copy with the sign set to 0. """
+return _dec_from_triple(0, self._int, self._exp, self._is_special)
+def copy_negate(self):
+"""Returns a copy with the sign inverted."""
+if self._sign:
+return _dec_from_triple(0, self._int, self._exp, self._is_special)
+else:
+return _dec_from_triple(1, self._int, self._exp, self._is_special)
+def copy_sign(self, other):
+"""Returns self with the sign of other."""
+return _dec_from_triple(other._sign, self._int,
+self._exp, self._is_special)
+def exp(self, context=None):
+"""Returns e ** self."""
+if context is None:
+context = getcontext()
+# exp(NaN) = NaN
+ans = self._check_nans(context=context)
+if ans:
+return ans
+# exp(-Infinity) = 0
+if self._isinfinity() == -1:
+return Dec_0
+# exp(0) = 1
+if not self:
+return Dec_p1
+# exp(Infinity) = Infinity
+if self._isinfinity() == 1:
+return Decimal(self)
+# the result is now guaranteed to be inexact (the true
+# mathematical result is transcendental). There's no need to
+# raise Rounded and Inexact here---they'll always be raised as
+# a result of the call to _fix.
+p = context.prec
+adj = self.adjusted()
+# we only need to do any computation for quite a small range
+# of adjusted exponents---for example, -29 <= adj <= 10 for
+# the default context.  For smaller exponent the result is
+# indistinguishable from 1 at the given precision, while for
+# larger exponent the result either overflows or underflows.
+if self._sign == 0 and adj > len(str((context.Emax+1)*3)):
+# overflow
+ans = _dec_from_triple(0, '1', context.Emax+1)
+elif self._sign == 1 and adj > len(str((-context.Etiny()+1)*3)):
+# underflow to 0
+ans = _dec_from_triple(0, '1', context.Etiny()-1)
+elif self._sign == 0 and adj < -p:
+# p+1 digits; final round will raise correct flags
+ans = _dec_from_triple(0, '1' + '0'*(p-1) + '1', -p)
+elif self._sign == 1 and adj < -p-1:
+# p+1 digits; final round will raise correct flags
+ans = _dec_from_triple(0, '9'*(p+1), -p-1)
+# general case
+else:
+op = _WorkRep(self)
+c, e = op.int, op.exp
+if op.sign == 1:
+c = -c
+# compute correctly rounded result: increase precision by
+# 3 digits at a time until we get an unambiguously
+# roundable result
+extra = 3
+while True:
+coeff, exp = _dexp(c, e, p+extra)
+if coeff % (5*10**(len(str(coeff))-p-1)):
+break
+extra += 3
+ans = _dec_from_triple(0, str(coeff), exp)
+# at this stage, ans should round correctly with *any*
+# rounding mode, not just with ROUND_HALF_EVEN
+context = context._shallow_copy()
+rounding = context._set_rounding(ROUND_HALF_EVEN)
+ans = ans._fix(context)
+context.rounding = rounding
+return ans
+def is_canonical(self):
+"""Return True if self is canonical; otherwise return False.
+Currently, the encoding of a Decimal instance is always
+canonical, so this method returns True for any Decimal.
+"""
+return True
+def is_finite(self):
+"""Return True if self is finite; otherwise return False.
+A Decimal instance is considered finite if it is neither
+infinite nor a NaN.
+"""
+return not self._is_special
+def is_infinite(self):
+"""Return True if self is infinite; otherwise return False."""
+return self._exp == 'F'
+def is_nan(self):
+"""Return True if self is a qNaN or sNaN; otherwise return False."""
+return self._exp in ('n', 'N')
+def is_normal(self, context=None):
+"""Return True if self is a normal number; otherwise return False."""
+if self._is_special or not self:
+return False
+if context is None:
+context = getcontext()
+return context.Emin <= self.adjusted() <= context.Emax
+def is_qnan(self):
+"""Return True if self is a quiet NaN; otherwise return False."""
+return self._exp == 'n'
+def is_signed(self):
+"""Return True if self is negative; otherwise return False."""
+return self._sign == 1
+def is_snan(self):
+"""Return True if self is a signaling NaN; otherwise return False."""
+return self._exp == 'N'
+def is_subnormal(self, context=None):
+"""Return True if self is subnormal; otherwise return False."""
+if self._is_special or not self:
+return False
+if context is None:
+context = getcontext()
+return self.adjusted() < context.Emin
+def is_zero(self):
+"""Return True if self is a zero; otherwise return False."""
+return not self._is_special and self._int == '0'
+def _ln_exp_bound(self):
+"""Compute a lower bound for the adjusted exponent of self.ln().
+In other words, compute r such that self.ln() >= 10**r.  Assumes
+that self is finite and positive and that self != 1.
+"""
+# for 0.1 <= x <= 10 we use the inequalities 1-1/x <= ln(x) <= x-1
+adj = self._exp + len(self._int) - 1
+if adj >= 1:
+# argument >= 10; we use 23/10 = 2.3 as a lower bound for ln(10)
+return len(str(adj*23//10)) - 1
+if adj <= -2:
+# argument <= 0.1
+return len(str((-1-adj)*23//10)) - 1
+op = _WorkRep(self)
+c, e = op.int, op.exp
+if adj == 0:
+# 1 < self < 10
+num = str(c-10**-e)
+den = str(c)
+return len(num) - len(den) - (num < den)
+# adj == -1, 0.1 <= self < 1
+return e + len(str(10**-e - c)) - 1
+def ln(self, context=None):
+"""Returns the natural (base e) logarithm of self."""
+if context is None:
+context = getcontext()
+# ln(NaN) = NaN
+ans = self._check_nans(context=context)
+if ans:
+return ans
+# ln(0.0) == -Infinity
+if not self:
+return negInf
+# ln(Infinity) = Infinity
+if self._isinfinity() == 1:
+return Inf
+# ln(1.0) == 0.0
+if self == Dec_p1:
+return Dec_0
+# ln(negative) raises InvalidOperation
+if self._sign == 1:
+return context._raise_error(InvalidOperation,
+'ln of a negative value')
+# result is irrational, so necessarily inexact
+op = _WorkRep(self)
+c, e = op.int, op.exp
+p = context.prec
+# correctly rounded result: repeatedly increase precision by 3
+# until we get an unambiguously roundable result
+places = p - self._ln_exp_bound() + 2 # at least p+3 places
+while True:
+coeff = _dlog(c, e, places)
+# assert len(str(abs(coeff)))-p >= 1
+if coeff % (5*10**(len(str(abs(coeff)))-p-1)):
+break
+places += 3
+ans = _dec_from_triple(int(coeff<0), str(abs(coeff)), -places)
+context = context._shallow_copy()
+rounding = context._set_rounding(ROUND_HALF_EVEN)
+ans = ans._fix(context)
+context.rounding = rounding
+return ans
+def _log10_exp_bound(self):
+"""Compute a lower bound for the adjusted exponent of self.log10().
+In other words, find r such that self.log10() >= 10**r.
+Assumes that self is finite and positive and that self != 1.
+"""
+# For x >= 10 or x < 0.1 we only need a bound on the integer
+# part of log10(self), and this comes directly from the
+# exponent of x.  For 0.1 <= x <= 10 we use the inequalities
+# 1-1/x <= log(x) <= x-1. If x > 1 we have |log10(x)| >
+# (1-1/x)/2.31 > 0.  If x < 1 then |log10(x)| > (1-x)/2.31 > 0
+adj = self._exp + len(self._int) - 1
+if adj >= 1:
+# self >= 10
+return len(str(adj))-1
+if adj <= -2:
+# self < 0.1
+return len(str(-1-adj))-1
+op = _WorkRep(self)
+c, e = op.int, op.exp
+if adj == 0:
+# 1 < self < 10
+num = str(c-10**-e)
+den = str(231*c)
+return len(num) - len(den) - (num < den) + 2
+# adj == -1, 0.1 <= self < 1
+num = str(10**-e-c)
+return len(num) + e - (num < "231") - 1
+def log10(self, context=None):
+"""Returns the base 10 logarithm of self."""
+if context is None:
+context = getcontext()
+# log10(NaN) = NaN
+ans = self._check_nans(context=context)
+if ans:
+return ans
+# log10(0.0) == -Infinity
+if not self:
+return negInf
+# log10(Infinity) = Infinity
+if self._isinfinity() == 1:
+return Inf
+# log10(negative or -Infinity) raises InvalidOperation
+if self._sign == 1:
+return context._raise_error(InvalidOperation,
+'log10 of a negative value')
+# log10(10**n) = n
+if self._int[0] == '1' and self._int[1:] == '0'*(len(self._int) - 1):
+# answer may need rounding
+ans = Decimal(self._exp + len(self._int) - 1)
+else:
+# result is irrational, so necessarily inexact
+op = _WorkRep(self)
+c, e = op.int, op.exp
+p = context.prec
+# correctly rounded result: repeatedly increase precision
+# until result is unambiguously roundable
+places = p-self._log10_exp_bound()+2
+while True:
+coeff = _dlog10(c, e, places)
+# assert len(str(abs(coeff)))-p >= 1
+if coeff % (5*10**(len(str(abs(coeff)))-p-1)):
+break
+places += 3
+ans = _dec_from_triple(int(coeff<0), str(abs(coeff)), -places)
+context = context._shallow_copy()
+rounding = context._set_rounding(ROUND_HALF_EVEN)
+ans = ans._fix(context)
+context.rounding = rounding
+return ans
+def logb(self, context=None):
+""" Returns the exponent of the magnitude of self's MSD.
+The result is the integer which is the exponent of the magnitude
+of the most significant digit of self (as though it were truncated
+to a single digit while maintaining the value of that digit and
+without limiting the resulting exponent).
+"""
+# logb(NaN) = NaN
+ans = self._check_nans(context=context)
+if ans:
+return ans
+if context is None:
+context = getcontext()
+# logb(+/-Inf) = +Inf
+if self._isinfinity():
+return Inf
+# logb(0) = -Inf, DivisionByZero
+if not self:
+return context._raise_error(DivisionByZero, 'logb(0)', 1)
+# otherwise, simply return the adjusted exponent of self, as a
+# Decimal.  Note that no attempt is made to fit the result
+# into the current context.
+return Decimal(self.adjusted())
+def _islogical(self):
+"""Return True if self is a logical operand.
+For being logical, it must be a finite numbers with a sign of 0,
+an exponent of 0, and a coefficient whose digits must all be
+either 0 or 1.
+"""
+if self._sign != 0 or self._exp != 0:
+return False
+for dig in self._int:
+if dig not in '01':
+return False
+return True
+def _fill_logical(self, context, opa, opb):
+dif = context.prec - len(opa)
+if dif > 0:
+opa = '0'*dif + opa
+elif dif < 0:
+opa = opa[-context.prec:]
+dif = context.prec - len(opb)
+if dif > 0:
+opb = '0'*dif + opb
+elif dif < 0:
+opb = opb[-context.prec:]
+return opa, opb
+def logical_and(self, other, context=None):
+"""Applies an 'and' operation between self and other's digits."""
+if context is None:
+context = getcontext()
+if not self._islogical() or not other._islogical():
+return context._raise_error(InvalidOperation)
+# fill to context.prec
+(opa, opb) = self._fill_logical(context, self._int, other._int)
+# make the operation, and clean starting zeroes
+result = "".join([str(int(a)&int(b)) for a,b in zip(opa,opb)])
+return _dec_from_triple(0, result.lstrip('0') or '0', 0)
+def logical_invert(self, context=None):
+"""Invert all its digits."""
+if context is None:
+context = getcontext()
+return self.logical_xor(_dec_from_triple(0,'1'*context.prec,0),
+context)
+def logical_or(self, other, context=None):
+"""Applies an 'or' operation between self and other's digits."""
+if context is None:
+context = getcontext()
+if not self._islogical() or not other._islogical():
+return context._raise_error(InvalidOperation)
+# fill to context.prec
+(opa, opb) = self._fill_logical(context, self._int, other._int)
+# make the operation, and clean starting zeroes
+result = "".join(str(int(a)|int(b)) for a,b in zip(opa,opb))
+return _dec_from_triple(0, result.lstrip('0') or '0', 0)
+def logical_xor(self, other, context=None):
+"""Applies an 'xor' operation between self and other's digits."""
+if context is None:
+context = getcontext()
+if not self._islogical() or not other._islogical():
+return context._raise_error(InvalidOperation)
+# fill to context.prec
+(opa, opb) = self._fill_logical(context, self._int, other._int)
+# make the operation, and clean starting zeroes
+result = "".join(str(int(a)^int(b)) for a,b in zip(opa,opb))
+return _dec_from_triple(0, result.lstrip('0') or '0', 0)
+def max_mag(self, other, context=None):
+"""Compares the values numerically with their sign ignored."""
+other = _convert_other(other, raiseit=True)
+if context is None:
+context = getcontext()
+if self._is_special or other._is_special:
+# If one operand is a quiet NaN and the other is number, then the
+# number is always returned
+sn = self._isnan()
+on = other._isnan()
+if sn or on:
+if on == 1 and sn != 2:
+return self._fix_nan(context)
+if sn == 1 and on != 2:
+return other._fix_nan(context)
+return self._check_nans(other, context)
+c = self.copy_abs().__cmp__(other.copy_abs())
+if c == 0:
+c = self.compare_total(other)
+if c == -1:
+ans = other
+else:
+ans = self
+return ans._fix(context)
+def min_mag(self, other, context=None):
+"""Compares the values numerically with their sign ignored."""
+other = _convert_other(other, raiseit=True)
+if context is None:
+context = getcontext()
+if self._is_special or other._is_special:
+# If one operand is a quiet NaN and the other is number, then the
+# number is always returned
+sn = self._isnan()
+on = other._isnan()
+if sn or on:
+if on == 1 and sn != 2:
+return self._fix_nan(context)
+if sn == 1 and on != 2:
+return other._fix_nan(context)
+return self._check_nans(other, context)
+c = self.copy_abs().__cmp__(other.copy_abs())
+if c == 0:
+c = self.compare_total(other)
+if c == -1:
+ans = self
+else:
+ans = other
+return ans._fix(context)
+def next_minus(self, context=None):
+"""Returns the largest representable number smaller than itself."""
+if context is None:
+context = getcontext()
+ans = self._check_nans(context=context)
+if ans:
+return ans
+if self._isinfinity() == -1:
+return negInf
+if self._isinfinity() == 1:
+return _dec_from_triple(0, '9'*context.prec, context.Etop())
+context = context.copy()
+context._set_rounding(ROUND_FLOOR)
+context._ignore_all_flags()
+new_self = self._fix(context)
+if new_self != self:
+return new_self
+return self.__sub__(_dec_from_triple(0, '1', context.Etiny()-1),
+context)
+def next_plus(self, context=None):
+"""Returns the smallest representable number larger than itself."""
+if context is None:
+context = getcontext()
+ans = self._check_nans(context=context)
+if ans:
+return ans
+if self._isinfinity() == 1:
+return Inf
+if self._isinfinity() == -1:
+return _dec_from_triple(1, '9'*context.prec, context.Etop())
+context = context.copy()
+context._set_rounding(ROUND_CEILING)
+context._ignore_all_flags()
+new_self = self._fix(context)
+if new_self != self:
+return new_self
+return self.__add__(_dec_from_triple(0, '1', context.Etiny()-1),
+context)
+def next_toward(self, other, context=None):
+"""Returns the number closest to self, in the direction towards other.
+The result is the closest representable number to self
+(excluding self) that is in the direction towards other,
+unless both have the same value.  If the two operands are
+numerically equal, then the result is a copy of self with the
+sign set to be the same as the sign of other.
+"""
+other = _convert_other(other, raiseit=True)
+if context is None:
+context = getcontext()
+ans = self._check_nans(other, context)
+if ans:
+return ans
+comparison = self.__cmp__(other)
+if comparison == 0:
+return self.copy_sign(other)
+if comparison == -1:
+ans = self.next_plus(context)
+else: # comparison == 1
+ans = self.next_minus(context)
+# decide which flags to raise using value of ans
+if ans._isinfinity():
+context._raise_error(Overflow,
+'Infinite result from next_toward',
+ans._sign)
+context._raise_error(Rounded)
+context._raise_error(Inexact)
+elif ans.adjusted() < context.Emin:
+context._raise_error(Underflow)
+context._raise_error(Subnormal)
+context._raise_error(Rounded)
+context._raise_error(Inexact)
+# if precision == 1 then we don't raise Clamped for a
+# result 0E-Etiny.
+if not ans:
+context._raise_error(Clamped)
+return ans
+def number_class(self, context=None):
+"""Returns an indication of the class of self.
+The class is one of the following strings:
+sNaN
+NaN
+-Infinity
+-Normal
+-Subnormal
+-Zero
++Zero
++Subnormal
++Normal
++Infinity
+"""
+if self.is_snan():
+return "sNaN"
+if self.is_qnan():
+return "NaN"
+inf = self._isinfinity()
+if inf == 1:
+return "+Infinity"
+if inf == -1:
+return "-Infinity"
+if self.is_zero():
+if self._sign:
+return "-Zero"
+else:
+return "+Zero"
+if context is None:
+context = getcontext()
+if self.is_subnormal(context=context):
+if self._sign:
+return "-Subnormal"
+else:
+return "+Subnormal"
+# just a normal, regular, boring number, :)
+if self._sign:
+return "-Normal"
+else:
+return "+Normal"
+def radix(self):
+"""Just returns 10, as this is Decimal, :)"""
+return Decimal(10)
+def rotate(self, other, context=None):
+"""Returns a rotated copy of self, value-of-other times."""
+if context is None:
+context = getcontext()
+ans = self._check_nans(other, context)
+if ans:
+return ans
+if other._exp != 0:
+return context._raise_error(InvalidOperation)
+if not (-context.prec <= int(other) <= context.prec):
+return context._raise_error(InvalidOperation)
+if self._isinfinity():
+return Decimal(self)
+# get values, pad if necessary
+torot = int(other)
+rotdig = self._int
+topad = context.prec - len(rotdig)
+if topad:
+rotdig = '0'*topad + rotdig
+# let's rotate!
+rotated = rotdig[torot:] + rotdig[:torot]
+return _dec_from_triple(self._sign,
+rotated.lstrip('0') or '0', self._exp)
+def scaleb (self, other, context=None):
+"""Returns self operand after adding the second value to its exp."""
+if context is None:
+context = getcontext()
+ans = self._check_nans(other, context)
+if ans:
+return ans
+if other._exp != 0:
+return context._raise_error(InvalidOperation)
+liminf = -2 * (context.Emax + context.prec)
+limsup =  2 * (context.Emax + context.prec)
+if not (liminf <= int(other) <= limsup):
+return context._raise_error(InvalidOperation)
+if self._isinfinity():
+return Decimal(self)
+d = _dec_from_triple(self._sign, self._int, self._exp + int(other))
+d = d._fix(context)
+return d
+def shift(self, other, context=None):
+"""Returns a shifted copy of self, value-of-other times."""
+if context is None:
+context = getcontext()
+ans = self._check_nans(other, context)
+if ans:
+return ans
+if other._exp != 0:
+return context._raise_error(InvalidOperation)
+if not (-context.prec <= int(other) <= context.prec):
+return context._raise_error(InvalidOperation)
+if self._isinfinity():
+return Decimal(self)
+# get values, pad if necessary
+torot = int(other)
+if not torot:
+return Decimal(self)
+rotdig = self._int
+topad = context.prec - len(rotdig)
+if topad:
+rotdig = '0'*topad + rotdig
+# let's shift!
+if torot < 0:
+rotated = rotdig[:torot]
+else:
+rotated = rotdig + '0'*torot
+rotated = rotated[-context.prec:]
+return _dec_from_triple(self._sign,
+rotated.lstrip('0') or '0', self._exp)
+# Support for pickling, copy, and deepcopy
+def __reduce__(self):
+return (self.__class__, (str(self),))
+def __copy__(self):
+if type(self) == Decimal:
+return self     # I'm immutable; therefore I am my own clone
+return self.__class__(str(self))
+def __deepcopy__(self, memo):
+if type(self) == Decimal:
+return self     # My components are also immutable
+return self.__class__(str(self))
+def _dec_from_triple(sign, coefficient, exponent, special=False):
+"""Create a decimal instance directly, without any validation,
+normalization (e.g. removal of leading zeros) or argument
+conversion.
+This function is for *internal use only*.
+"""
+self = object.__new__(Decimal)
+self._sign = sign
+self._int = coefficient
+self._exp = exponent
+self._is_special = special
+return self
+##### Context class #######################################################
+# get rounding method function:
+rounding_functions = [name for name in Decimal.__dict__.keys()
+if name.startswith('_round_')]
+for name in rounding_functions:
+# name is like _round_half_even, goes to the global ROUND_HALF_EVEN value.
+globalname = name[1:].upper()
+val = globals()[globalname]
+Decimal._pick_rounding_function[val] = name
+del name, val, globalname, rounding_functions
+class _ContextManager(object):
+"""Context manager class to support localcontext().
+Sets a copy of the supplied context in __enter__() and restores
+the previous decimal context in __exit__()
+"""
+def __init__(self, new_context):
+self.new_context = new_context.copy()
+def __enter__(self):
+self.saved_context = getcontext()
+setcontext(self.new_context)
+return self.new_context
+def __exit__(self, t, v, tb):
+setcontext(self.saved_context)
+class Context(object):
+"""Contains the context for a Decimal instance.
+Contains:
+prec - precision (for use in rounding, division, square roots..)
+rounding - rounding type (how you round)
+traps - If traps[exception] = 1, then the exception is
+raised when it is caused.  Otherwise, a value is
+substituted in.
+flags  - When an exception is caused, flags[exception] is incremented.
+(Whether or not the trap_enabler is set)
+Should be reset by user of Decimal instance.
+Emin -   Minimum exponent
+Emax -   Maximum exponent
+capitals -      If 1, 1*10^1 is printed as 1E+1.
+If 0, printed as 1e1
+_clamp - If 1, change exponents if too high (Default 0)
+"""
+def __init__(self, prec=None, rounding=None,
+traps=None, flags=None,
+Emin=None, Emax=None,
+capitals=None, _clamp=0,
+_ignored_flags=None):
+if flags is None:
+flags = []
+if _ignored_flags is None:
+_ignored_flags = []
+if not isinstance(flags, dict):
+flags = dict([(s,s in flags) for s in _signals])
+del s
+if traps is not None and not isinstance(traps, dict):
+traps = dict([(s,s in traps) for s in _signals])
+del s
+for name, val in locals().items():
+if val is None:
+setattr(self, name, _copy.copy(getattr(DefaultContext, name)))
+else:
+setattr(self, name, val)
+del self.self
+def __repr__(self):
+"""Show the current context."""
+s = []
+s.append('Context(prec=%(prec)d, rounding=%(rounding)s, '
+'Emin=%(Emin)d, Emax=%(Emax)d, capitals=%(capitals)d'
+% vars(self))
+names = [f.__name__ for f, v in self.flags.items() if v]
+s.append('flags=[' + ', '.join(names) + ']')
+names = [t.__name__ for t, v in self.traps.items() if v]
+s.append('traps=[' + ', '.join(names) + ']')
+return ', '.join(s) + ')'
+def clear_flags(self):
+"""Reset all flags to zero"""
+for flag in self.flags:
+self.flags[flag] = 0
+def _shallow_copy(self):
+"""Returns a shallow copy from self."""
+nc = Context(self.prec, self.rounding, self.traps,
+self.flags, self.Emin, self.Emax,
+self.capitals, self._clamp, self._ignored_flags)
+return nc
+def copy(self):
+"""Returns a deep copy from self."""
+nc = Context(self.prec, self.rounding, self.traps.copy(),
+self.flags.copy(), self.Emin, self.Emax,
+self.capitals, self._clamp, self._ignored_flags)
+return nc
+__copy__ = copy
+def _raise_error(self, condition, explanation = None, *args):
+"""Handles an error
+If the flag is in _ignored_flags, returns the default response.
+Otherwise, it increments the flag, then, if the corresponding
+trap_enabler is set, it reaises the exception.  Otherwise, it returns
+the default value after incrementing the flag.
+"""
+error = _condition_map.get(condition, condition)
+if error in self._ignored_flags:
+# Don't touch the flag
+return error().handle(self, *args)
+self.flags[error] += 1
+if not self.traps[error]:
+# The errors define how to handle themselves.
+return condition().handle(self, *args)
+# Errors should only be risked on copies of the context
+# self._ignored_flags = []
+raise error, explanation
+def _ignore_all_flags(self):
+"""Ignore all flags, if they are raised"""
+return self._ignore_flags(*_signals)
+def _ignore_flags(self, *flags):
+"""Ignore the flags, if they are raised"""
+# Do not mutate-- This way, copies of a context leave the original
+# alone.
+self._ignored_flags = (self._ignored_flags + list(flags))
+return list(flags)
+def _regard_flags(self, *flags):
+"""Stop ignoring the flags, if they are raised"""
+if flags and isinstance(flags[0], (tuple,list)):
+flags = flags[0]
+for flag in flags:
+self._ignored_flags.remove(flag)
+def __hash__(self):
+"""A Context cannot be hashed."""
+# We inherit object.__hash__, so we must deny this explicitly
+raise TypeError("Cannot hash a Context.")
+def Etiny(self):
+"""Returns Etiny (= Emin - prec + 1)"""
+return int(self.Emin - self.prec + 1)
+def Etop(self):
+"""Returns maximum exponent (= Emax - prec + 1)"""
+return int(self.Emax - self.prec + 1)
+def _set_rounding(self, type):
+"""Sets the rounding type.
+Sets the rounding type, and returns the current (previous)
+rounding type.  Often used like:
+context = context.copy()
+# so you don't change the calling context
+# if an error occurs in the middle.
+rounding = context._set_rounding(ROUND_UP)
+val = self.__sub__(other, context=context)
+context._set_rounding(rounding)
+This will make it round up for that operation.
+"""
+rounding = self.rounding
+self.rounding= type
+return rounding
+def create_decimal(self, num='0'):
+"""Creates a new Decimal instance but using self as context."""
+d = Decimal(num, context=self)
+if d._isnan() and len(d._int) > self.prec - self._clamp:
+return self._raise_error(ConversionSyntax,
+"diagnostic info too long in NaN")
+return d._fix(self)
+# Methods
+def abs(self, a):
+"""Returns the absolute value of the operand.
+If the operand is negative, the result is the same as using the minus
+operation on the operand.  Otherwise, the result is the same as using
+the plus operation on the operand.
+>>> ExtendedContext.abs(Decimal('2.1'))
+Decimal("2.1")
+>>> ExtendedContext.abs(Decimal('-100'))
+Decimal("100")
+>>> ExtendedContext.abs(Decimal('101.5'))
+Decimal("101.5")
+>>> ExtendedContext.abs(Decimal('-101.5'))
+Decimal("101.5")
+"""
+return a.__abs__(context=self)
+def add(self, a, b):
+"""Return the sum of the two operands.
+>>> ExtendedContext.add(Decimal('12'), Decimal('7.00'))
+Decimal("19.00")
+>>> ExtendedContext.add(Decimal('1E+2'), Decimal('1.01E+4'))
+Decimal("1.02E+4")
+"""
+return a.__add__(b, context=self)
+def _apply(self, a):
+return str(a._fix(self))
+def canonical(self, a):
+"""Returns the same Decimal object.
+As we do not have different encodings for the same number, the
+received object already is in its canonical form.
+>>> ExtendedContext.canonical(Decimal('2.50'))
+Decimal("2.50")
+"""
+return a.canonical(context=self)
+def compare(self, a, b):
+"""Compares values numerically.
+If the signs of the operands differ, a value representing each operand
+('-1' if the operand is less than zero, '0' if the operand is zero or
+negative zero, or '1' if the operand is greater than zero) is used in
+place of that operand for the comparison instead of the actual
+operand.
+The comparison is then effected by subtracting the second operand from
+the first and then returning a value according to the result of the
+subtraction: '-1' if the result is less than zero, '0' if the result is
+zero or negative zero, or '1' if the result is greater than zero.
+>>> ExtendedContext.compare(Decimal('2.1'), Decimal('3'))
+Decimal("-1")
+>>> ExtendedContext.compare(Decimal('2.1'), Decimal('2.1'))
+Decimal("0")
+>>> ExtendedContext.compare(Decimal('2.1'), Decimal('2.10'))
+Decimal("0")
+>>> ExtendedContext.compare(Decimal('3'), Decimal('2.1'))
+Decimal("1")
+>>> ExtendedContext.compare(Decimal('2.1'), Decimal('-3'))
+Decimal("1")
+>>> ExtendedContext.compare(Decimal('-3'), Decimal('2.1'))
+Decimal("-1")
+"""
+return a.compare(b, context=self)
+def compare_signal(self, a, b):
+"""Compares the values of the two operands numerically.
+It's pretty much like compare(), but all NaNs signal, with signaling
+NaNs taking precedence over quiet NaNs.
+>>> c = ExtendedContext
+>>> c.compare_signal(Decimal('2.1'), Decimal('3'))
+Decimal("-1")
+>>> c.compare_signal(Decimal('2.1'), Decimal('2.1'))
+Decimal("0")
+>>> c.flags[InvalidOperation] = 0
+>>> print c.flags[InvalidOperation]
+0
+>>> c.compare_signal(Decimal('NaN'), Decimal('2.1'))
+Decimal("NaN")
+>>> print c.flags[InvalidOperation]
+1
+>>> c.flags[InvalidOperation] = 0
+>>> print c.flags[InvalidOperation]
+0
+>>> c.compare_signal(Decimal('sNaN'), Decimal('2.1'))
+Decimal("NaN")
+>>> print c.flags[InvalidOperation]
+1
+"""
+return a.compare_signal(b, context=self)
+def compare_total(self, a, b):
+"""Compares two operands using their abstract representation.
+This is not like the standard compare, which use their numerical
+value. Note that a total ordering is defined for all possible abstract
+representations.
+>>> ExtendedContext.compare_total(Decimal('12.73'), Decimal('127.9'))
+Decimal("-1")
+>>> ExtendedContext.compare_total(Decimal('-127'),  Decimal('12'))
+Decimal("-1")
+>>> ExtendedContext.compare_total(Decimal('12.30'), Decimal('12.3'))
+Decimal("-1")
+>>> ExtendedContext.compare_total(Decimal('12.30'), Decimal('12.30'))
+Decimal("0")
+>>> ExtendedContext.compare_total(Decimal('12.3'),  Decimal('12.300'))
+Decimal("1")
+>>> ExtendedContext.compare_total(Decimal('12.3'),  Decimal('NaN'))
+Decimal("-1")
+"""
+return a.compare_total(b)
+def compare_total_mag(self, a, b):
+"""Compares two operands using their abstract representation ignoring sign.
+Like compare_total, but with operand's sign ignored and assumed to be 0.
+"""
+return a.compare_total_mag(b)
+def copy_abs(self, a):
+"""Returns a copy of the operand with the sign set to 0.
+>>> ExtendedContext.copy_abs(Decimal('2.1'))
+Decimal("2.1")
+>>> ExtendedContext.copy_abs(Decimal('-100'))
+Decimal("100")
+"""
+return a.copy_abs()
+def copy_decimal(self, a):
+"""Returns a copy of the decimal objet.
+>>> ExtendedContext.copy_decimal(Decimal('2.1'))
+Decimal("2.1")
+>>> ExtendedContext.copy_decimal(Decimal('-1.00'))
+Decimal("-1.00")
+"""
+return Decimal(a)
+def copy_negate(self, a):
+"""Returns a copy of the operand with the sign inverted.
+>>> ExtendedContext.copy_negate(Decimal('101.5'))
+Decimal("-101.5")
+>>> ExtendedContext.copy_negate(Decimal('-101.5'))
+Decimal("101.5")
+"""
+return a.copy_negate()
+def copy_sign(self, a, b):
+"""Copies the second operand's sign to the first one.
+In detail, it returns a copy of the first operand with the sign
+equal to the sign of the second operand.
+>>> ExtendedContext.copy_sign(Decimal( '1.50'), Decimal('7.33'))
+Decimal("1.50")
+>>> ExtendedContext.copy_sign(Decimal('-1.50'), Decimal('7.33'))
+Decimal("1.50")
+>>> ExtendedContext.copy_sign(Decimal( '1.50'), Decimal('-7.33'))
+Decimal("-1.50")
+>>> ExtendedContext.copy_sign(Decimal('-1.50'), Decimal('-7.33'))
+Decimal("-1.50")
+"""
+return a.copy_sign(b)
+def divide(self, a, b):
+"""Decimal division in a specified context.
+>>> ExtendedContext.divide(Decimal('1'), Decimal('3'))
+Decimal("0.333333333")
+>>> ExtendedContext.divide(Decimal('2'), Decimal('3'))
+Decimal("0.666666667")
+>>> ExtendedContext.divide(Decimal('5'), Decimal('2'))
+Decimal("2.5")
+>>> ExtendedContext.divide(Decimal('1'), Decimal('10'))
+Decimal("0.1")
+>>> ExtendedContext.divide(Decimal('12'), Decimal('12'))
+Decimal("1")
+>>> ExtendedContext.divide(Decimal('8.00'), Decimal('2'))
+Decimal("4.00")
+>>> ExtendedContext.divide(Decimal('2.400'), Decimal('2.0'))
+Decimal("1.20")
+>>> ExtendedContext.divide(Decimal('1000'), Decimal('100'))
+Decimal("10")
+>>> ExtendedContext.divide(Decimal('1000'), Decimal('1'))
+Decimal("1000")
+>>> ExtendedContext.divide(Decimal('2.40E+6'), Decimal('2'))
+Decimal("1.20E+6")
+"""
+return a.__div__(b, context=self)
+def divide_int(self, a, b):
+"""Divides two numbers and returns the integer part of the result.
+>>> ExtendedContext.divide_int(Decimal('2'), Decimal('3'))
+Decimal("0")
+>>> ExtendedContext.divide_int(Decimal('10'), Decimal('3'))
+Decimal("3")
+>>> ExtendedContext.divide_int(Decimal('1'), Decimal('0.3'))
+Decimal("3")
+"""
+return a.__floordiv__(b, context=self)
+def divmod(self, a, b):
+return a.__divmod__(b, context=self)
+def exp(self, a):
+"""Returns e ** a.
+>>> c = ExtendedContext.copy()
+>>> c.Emin = -999
+>>> c.Emax = 999
+>>> c.exp(Decimal('-Infinity'))
+Decimal("0")
+>>> c.exp(Decimal('-1'))
+Decimal("0.367879441")
+>>> c.exp(Decimal('0'))
+Decimal("1")
+>>> c.exp(Decimal('1'))
+Decimal("2.71828183")
+>>> c.exp(Decimal('0.693147181'))
+Decimal("2.00000000")
+>>> c.exp(Decimal('+Infinity'))
+Decimal("Infinity")
+"""
+return a.exp(context=self)
+def fma(self, a, b, c):
+"""Returns a multiplied by b, plus c.
+The first two operands are multiplied together, using multiply,
+the third operand is then added to the result of that
+multiplication, using add, all with only one final rounding.
+>>> ExtendedContext.fma(Decimal('3'), Decimal('5'), Decimal('7'))
+Decimal("22")
+>>> ExtendedContext.fma(Decimal('3'), Decimal('-5'), Decimal('7'))
+Decimal("-8")
+>>> ExtendedContext.fma(Decimal('888565290'), Decimal('1557.96930'), Decimal('-86087.7578'))
+Decimal("1.38435736E+12")
+"""
+return a.fma(b, c, context=self)
+def is_canonical(self, a):
+"""Return True if the operand is canonical; otherwise return False.
+Currently, the encoding of a Decimal instance is always
+canonical, so this method returns True for any Decimal.
+>>> ExtendedContext.is_canonical(Decimal('2.50'))
+True
+"""
+return a.is_canonical()
+def is_finite(self, a):
+"""Return True if the operand is finite; otherwise return False.
+A Decimal instance is considered finite if it is neither
+infinite nor a NaN.
+>>> ExtendedContext.is_finite(Decimal('2.50'))
+True
+>>> ExtendedContext.is_finite(Decimal('-0.3'))
+True
+>>> ExtendedContext.is_finite(Decimal('0'))
+True
+>>> ExtendedContext.is_finite(Decimal('Inf'))
+False
+>>> ExtendedContext.is_finite(Decimal('NaN'))
+False
+"""
+return a.is_finite()
+def is_infinite(self, a):
+"""Return True if the operand is infinite; otherwise return False.
+>>> ExtendedContext.is_infinite(Decimal('2.50'))
+False
+>>> ExtendedContext.is_infinite(Decimal('-Inf'))
+True
+>>> ExtendedContext.is_infinite(Decimal('NaN'))
+False
+"""
+return a.is_infinite()
+def is_nan(self, a):
+"""Return True if the operand is a qNaN or sNaN;
+otherwise return False.
+>>> ExtendedContext.is_nan(Decimal('2.50'))
+False
+>>> ExtendedContext.is_nan(Decimal('NaN'))
+True
+>>> ExtendedContext.is_nan(Decimal('-sNaN'))
+True
+"""
+return a.is_nan()
+def is_normal(self, a):
+"""Return True if the operand is a normal number;
+otherwise return False.
+>>> c = ExtendedContext.copy()
+>>> c.Emin = -999
+>>> c.Emax = 999
+>>> c.is_normal(Decimal('2.50'))
+True
+>>> c.is_normal(Decimal('0.1E-999'))
+False
+>>> c.is_normal(Decimal('0.00'))
+False
+>>> c.is_normal(Decimal('-Inf'))
+False
+>>> c.is_normal(Decimal('NaN'))
+False
+"""
+return a.is_normal(context=self)
+def is_qnan(self, a):
+"""Return True if the operand is a quiet NaN; otherwise return False.
+>>> ExtendedContext.is_qnan(Decimal('2.50'))
+False
+>>> ExtendedContext.is_qnan(Decimal('NaN'))
+True
+>>> ExtendedContext.is_qnan(Decimal('sNaN'))
+False
+"""
+return a.is_qnan()
+def is_signed(self, a):
+"""Return True if the operand is negative; otherwise return False.
+>>> ExtendedContext.is_signed(Decimal('2.50'))
+False
+>>> ExtendedContext.is_signed(Decimal('-12'))
+True
+>>> ExtendedContext.is_signed(Decimal('-0'))
+True
+"""
+return a.is_signed()
+def is_snan(self, a):
+"""Return True if the operand is a signaling NaN;
+otherwise return False.
+>>> ExtendedContext.is_snan(Decimal('2.50'))
+False
+>>> ExtendedContext.is_snan(Decimal('NaN'))
+False
+>>> ExtendedContext.is_snan(Decimal('sNaN'))
+True
+"""
+return a.is_snan()
+def is_subnormal(self, a):
+"""Return True if the operand is subnormal; otherwise return False.
+>>> c = ExtendedContext.copy()
+>>> c.Emin = -999
+>>> c.Emax = 999
+>>> c.is_subnormal(Decimal('2.50'))
+False
+>>> c.is_subnormal(Decimal('0.1E-999'))
+True
+>>> c.is_subnormal(Decimal('0.00'))
+False
+>>> c.is_subnormal(Decimal('-Inf'))
+False
+>>> c.is_subnormal(Decimal('NaN'))
+False
+"""
+return a.is_subnormal(context=self)
+def is_zero(self, a):
+"""Return True if the operand is a zero; otherwise return False.
+>>> ExtendedContext.is_zero(Decimal('0'))
+True
+>>> ExtendedContext.is_zero(Decimal('2.50'))
+False
+>>> ExtendedContext.is_zero(Decimal('-0E+2'))
+True
+"""
+return a.is_zero()
+def ln(self, a):
+"""Returns the natural (base e) logarithm of the operand.
+>>> c = ExtendedContext.copy()
+>>> c.Emin = -999
+>>> c.Emax = 999
+>>> c.ln(Decimal('0'))
+Decimal("-Infinity")
+>>> c.ln(Decimal('1.000'))
+Decimal("0")
+>>> c.ln(Decimal('2.71828183'))
+Decimal("1.00000000")
+>>> c.ln(Decimal('10'))
+Decimal("2.30258509")
+>>> c.ln(Decimal('+Infinity'))
+Decimal("Infinity")
+"""
+return a.ln(context=self)
+def log10(self, a):
+"""Returns the base 10 logarithm of the operand.
+>>> c = ExtendedContext.copy()
+>>> c.Emin = -999
+>>> c.Emax = 999
+>>> c.log10(Decimal('0'))
+Decimal("-Infinity")
+>>> c.log10(Decimal('0.001'))
+Decimal("-3")
+>>> c.log10(Decimal('1.000'))
+Decimal("0")
+>>> c.log10(Decimal('2'))
+Decimal("0.301029996")
+>>> c.log10(Decimal('10'))
+Decimal("1")
+>>> c.log10(Decimal('70'))
+Decimal("1.84509804")
+>>> c.log10(Decimal('+Infinity'))
+Decimal("Infinity")
+"""
+return a.log10(context=self)
+def logb(self, a):
+""" Returns the exponent of the magnitude of the operand's MSD.
+The result is the integer which is the exponent of the magnitude
+of the most significant digit of the operand (as though the
+operand were truncated to a single digit while maintaining the
+value of that digit and without limiting the resulting exponent).
+>>> ExtendedContext.logb(Decimal('250'))
+Decimal("2")
+>>> ExtendedContext.logb(Decimal('2.50'))
+Decimal("0")
+>>> ExtendedContext.logb(Decimal('0.03'))
+Decimal("-2")
+>>> ExtendedContext.logb(Decimal('0'))
+Decimal("-Infinity")
+"""
+return a.logb(context=self)
+def logical_and(self, a, b):
+"""Applies the logical operation 'and' between each operand's digits.
+The operands must be both logical numbers.
+>>> ExtendedContext.logical_and(Decimal('0'), Decimal('0'))
+Decimal("0")
+>>> ExtendedContext.logical_and(Decimal('0'), Decimal('1'))
+Decimal("0")
+>>> ExtendedContext.logical_and(Decimal('1'), Decimal('0'))
+Decimal("0")
+>>> ExtendedContext.logical_and(Decimal('1'), Decimal('1'))
+Decimal("1")
+>>> ExtendedContext.logical_and(Decimal('1100'), Decimal('1010'))
+Decimal("1000")
+>>> ExtendedContext.logical_and(Decimal('1111'), Decimal('10'))
+Decimal("10")
+"""
+return a.logical_and(b, context=self)
+def logical_invert(self, a):
+"""Invert all the digits in the operand.
+The operand must be a logical number.
+>>> ExtendedContext.logical_invert(Decimal('0'))
+Decimal("111111111")
+>>> ExtendedContext.logical_invert(Decimal('1'))
+Decimal("111111110")
+>>> ExtendedContext.logical_invert(Decimal('111111111'))
+Decimal("0")
+>>> ExtendedContext.logical_invert(Decimal('101010101'))
+Decimal("10101010")
+"""
+return a.logical_invert(context=self)
+def logical_or(self, a, b):
+"""Applies the logical operation 'or' between each operand's digits.
+The operands must be both logical numbers.
+>>> ExtendedContext.logical_or(Decimal('0'), Decimal('0'))
+Decimal("0")
+>>> ExtendedContext.logical_or(Decimal('0'), Decimal('1'))
+Decimal("1")
+>>> ExtendedContext.logical_or(Decimal('1'), Decimal('0'))
+Decimal("1")
+>>> ExtendedContext.logical_or(Decimal('1'), Decimal('1'))
+Decimal("1")
+>>> ExtendedContext.logical_or(Decimal('1100'), Decimal('1010'))
+Decimal("1110")
+>>> ExtendedContext.logical_or(Decimal('1110'), Decimal('10'))
+Decimal("1110")
+"""
+return a.logical_or(b, context=self)
+def logical_xor(self, a, b):
+"""Applies the logical operation 'xor' between each operand's digits.
+The operands must be both logical numbers.
+>>> ExtendedContext.logical_xor(Decimal('0'), Decimal('0'))
+Decimal("0")
+>>> ExtendedContext.logical_xor(Decimal('0'), Decimal('1'))
+Decimal("1")
+>>> ExtendedContext.logical_xor(Decimal('1'), Decimal('0'))
+Decimal("1")
+>>> ExtendedContext.logical_xor(Decimal('1'), Decimal('1'))
+Decimal("0")
+>>> ExtendedContext.logical_xor(Decimal('1100'), Decimal('1010'))
+Decimal("110")
+>>> ExtendedContext.logical_xor(Decimal('1111'), Decimal('10'))
+Decimal("1101")
+"""
+return a.logical_xor(b, context=self)
+def max(self, a,b):
+"""max compares two values numerically and returns the maximum.
+If either operand is a NaN then the general rules apply.
+Otherwise, the operands are compared as as though by the compare
+operation.  If they are numerically equal then the left-hand operand
+is chosen as the result.  Otherwise the maximum (closer to positive
+infinity) of the two operands is chosen as the result.
+>>> ExtendedContext.max(Decimal('3'), Decimal('2'))
+Decimal("3")
+>>> ExtendedContext.max(Decimal('-10'), Decimal('3'))
+Decimal("3")
+>>> ExtendedContext.max(Decimal('1.0'), Decimal('1'))
+Decimal("1")
+>>> ExtendedContext.max(Decimal('7'), Decimal('NaN'))
+Decimal("7")
+"""
+return a.max(b, context=self)
+def max_mag(self, a, b):
+"""Compares the values numerically with their sign ignored."""
+return a.max_mag(b, context=self)
+def min(self, a,b):
+"""min compares two values numerically and returns the minimum.
+If either operand is a NaN then the general rules apply.
+Otherwise, the operands are compared as as though by the compare
+operation.  If they are numerically equal then the left-hand operand
+is chosen as the result.  Otherwise the minimum (closer to negative
+infinity) of the two operands is chosen as the result.
+>>> ExtendedContext.min(Decimal('3'), Decimal('2'))
+Decimal("2")
+>>> ExtendedContext.min(Decimal('-10'), Decimal('3'))
+Decimal("-10")
+>>> ExtendedContext.min(Decimal('1.0'), Decimal('1'))
+Decimal("1.0")
+>>> ExtendedContext.min(Decimal('7'), Decimal('NaN'))
+Decimal("7")
+"""
+return a.min(b, context=self)
+def min_mag(self, a, b):
+"""Compares the values numerically with their sign ignored."""
+return a.min_mag(b, context=self)
+def minus(self, a):
+"""Minus corresponds to unary prefix minus in Python.
+The operation is evaluated using the same rules as subtract; the
+operation minus(a) is calculated as subtract('0', a) where the '0'
+has the same exponent as the operand.
+>>> ExtendedContext.minus(Decimal('1.3'))
+Decimal("-1.3")
+>>> ExtendedContext.minus(Decimal('-1.3'))
+Decimal("1.3")
+"""
+return a.__neg__(context=self)
+def multiply(self, a, b):
+"""multiply multiplies two operands.
+If either operand is a special value then the general rules apply.
+Otherwise, the operands are multiplied together ('long multiplication'),
+resulting in a number which may be as long as the sum of the lengths
+of the two operands.
+>>> ExtendedContext.multiply(Decimal('1.20'), Decimal('3'))
+Decimal("3.60")
+>>> ExtendedContext.multiply(Decimal('7'), Decimal('3'))
+Decimal("21")
+>>> ExtendedContext.multiply(Decimal('0.9'), Decimal('0.8'))
+Decimal("0.72")
+>>> ExtendedContext.multiply(Decimal('0.9'), Decimal('-0'))
+Decimal("-0.0")
+>>> ExtendedContext.multiply(Decimal('654321'), Decimal('654321'))
+Decimal("4.28135971E+11")
+"""
+return a.__mul__(b, context=self)
+def next_minus(self, a):
+"""Returns the largest representable number smaller than a.
+>>> c = ExtendedContext.copy()
+>>> c.Emin = -999
+>>> c.Emax = 999
+>>> ExtendedContext.next_minus(Decimal('1'))
+Decimal("0.999999999")
+>>> c.next_minus(Decimal('1E-1007'))
+Decimal("0E-1007")
+>>> ExtendedContext.next_minus(Decimal('-1.00000003'))
+Decimal("-1.00000004")
+>>> c.next_minus(Decimal('Infinity'))
+Decimal("9.99999999E+999")
+"""
+return a.next_minus(context=self)
+def next_plus(self, a):
+"""Returns the smallest representable number larger than a.
+>>> c = ExtendedContext.copy()
+>>> c.Emin = -999
+>>> c.Emax = 999
+>>> ExtendedContext.next_plus(Decimal('1'))
+Decimal("1.00000001")
+>>> c.next_plus(Decimal('-1E-1007'))
+Decimal("-0E-1007")
+>>> ExtendedContext.next_plus(Decimal('-1.00000003'))
+Decimal("-1.00000002")
+>>> c.next_plus(Decimal('-Infinity'))
+Decimal("-9.99999999E+999")
+"""
+return a.next_plus(context=self)
+def next_toward(self, a, b):
+"""Returns the number closest to a, in direction towards b.
+The result is the closest representable number from the first
+operand (but not the first operand) that is in the direction
+towards the second operand, unless the operands have the same
+value.
+>>> c = ExtendedContext.copy()
+>>> c.Emin = -999
+>>> c.Emax = 999
+>>> c.next_toward(Decimal('1'), Decimal('2'))
+Decimal("1.00000001")
+>>> c.next_toward(Decimal('-1E-1007'), Decimal('1'))
+Decimal("-0E-1007")
+>>> c.next_toward(Decimal('-1.00000003'), Decimal('0'))
+Decimal("-1.00000002")
+>>> c.next_toward(Decimal('1'), Decimal('0'))
+Decimal("0.999999999")
+>>> c.next_toward(Decimal('1E-1007'), Decimal('-100'))
+Decimal("0E-1007")
+>>> c.next_toward(Decimal('-1.00000003'), Decimal('-10'))
+Decimal("-1.00000004")
+>>> c.next_toward(Decimal('0.00'), Decimal('-0.0000'))
+Decimal("-0.00")
+"""
+return a.next_toward(b, context=self)
+def normalize(self, a):
+"""normalize reduces an operand to its simplest form.
+Essentially a plus operation with all trailing zeros removed from the
+result.
+>>> ExtendedContext.normalize(Decimal('2.1'))
+Decimal("2.1")
+>>> ExtendedContext.normalize(Decimal('-2.0'))
+Decimal("-2")
+>>> ExtendedContext.normalize(Decimal('1.200'))
+Decimal("1.2")
+>>> ExtendedContext.normalize(Decimal('-120'))
+Decimal("-1.2E+2")
+>>> ExtendedContext.normalize(Decimal('120.00'))
+Decimal("1.2E+2")
+>>> ExtendedContext.normalize(Decimal('0.00'))
+Decimal("0")
+"""
+return a.normalize(context=self)
+def number_class(self, a):
+"""Returns an indication of the class of the operand.
+The class is one of the following strings:
+-sNaN
+-NaN
+-Infinity
+-Normal
+-Subnormal
+-Zero
++Zero
++Subnormal
++Normal
++Infinity
+>>> c = Context(ExtendedContext)
+>>> c.Emin = -999
+>>> c.Emax = 999
+>>> c.number_class(Decimal('Infinity'))
+'+Infinity'
+>>> c.number_class(Decimal('1E-10'))
+'+Normal'
+>>> c.number_class(Decimal('2.50'))
+'+Normal'
+>>> c.number_class(Decimal('0.1E-999'))
+'+Subnormal'
+>>> c.number_class(Decimal('0'))
+'+Zero'
+>>> c.number_class(Decimal('-0'))
+'-Zero'
+>>> c.number_class(Decimal('-0.1E-999'))
+'-Subnormal'
+>>> c.number_class(Decimal('-1E-10'))
+'-Normal'
+>>> c.number_class(Decimal('-2.50'))
+'-Normal'
+>>> c.number_class(Decimal('-Infinity'))
+'-Infinity'
+>>> c.number_class(Decimal('NaN'))
+'NaN'
+>>> c.number_class(Decimal('-NaN'))
+'NaN'
+>>> c.number_class(Decimal('sNaN'))
+'sNaN'
+"""
+return a.number_class(context=self)
+def plus(self, a):
+"""Plus corresponds to unary prefix plus in Python.
+The operation is evaluated using the same rules as add; the
+operation plus(a) is calculated as add('0', a) where the '0'
+has the same exponent as the operand.
+>>> ExtendedContext.plus(Decimal('1.3'))
+Decimal("1.3")
+>>> ExtendedContext.plus(Decimal('-1.3'))
+Decimal("-1.3")
+"""
+return a.__pos__(context=self)
+def power(self, a, b, modulo=None):
+"""Raises a to the power of b, to modulo if given.
+With two arguments, compute a**b.  If a is negative then b
+must be integral.  The result will be inexact unless b is
+integral and the result is finite and can be expressed exactly
+in 'precision' digits.
+With three arguments, compute (a**b) % modulo.  For the
+three argument form, the following restrictions on the
+arguments hold:
+- all three arguments must be integral
+- b must be nonnegative
+- at least one of a or b must be nonzero
+- modulo must be nonzero and have at most 'precision' digits
+The result of pow(a, b, modulo) is identical to the result
+that would be obtained by computing (a**b) % modulo with
+unbounded precision, but is computed more efficiently.  It is
+always exact.
+>>> c = ExtendedContext.copy()
+>>> c.Emin = -999
+>>> c.Emax = 999
+>>> c.power(Decimal('2'), Decimal('3'))
+Decimal("8")
+>>> c.power(Decimal('-2'), Decimal('3'))
+Decimal("-8")
+>>> c.power(Decimal('2'), Decimal('-3'))
+Decimal("0.125")
+>>> c.power(Decimal('1.7'), Decimal('8'))
+Decimal("69.7575744")
+>>> c.power(Decimal('10'), Decimal('0.301029996'))
+Decimal("2.00000000")
+>>> c.power(Decimal('Infinity'), Decimal('-1'))
+Decimal("0")
+>>> c.power(Decimal('Infinity'), Decimal('0'))
+Decimal("1")
+>>> c.power(Decimal('Infinity'), Decimal('1'))
+Decimal("Infinity")
+>>> c.power(Decimal('-Infinity'), Decimal('-1'))
+Decimal("-0")
+>>> c.power(Decimal('-Infinity'), Decimal('0'))
+Decimal("1")
+>>> c.power(Decimal('-Infinity'), Decimal('1'))
+Decimal("-Infinity")
+>>> c.power(Decimal('-Infinity'), Decimal('2'))
+Decimal("Infinity")
+>>> c.power(Decimal('0'), Decimal('0'))
+Decimal("NaN")
+>>> c.power(Decimal('3'), Decimal('7'), Decimal('16'))
+Decimal("11")
+>>> c.power(Decimal('-3'), Decimal('7'), Decimal('16'))
+Decimal("-11")
+>>> c.power(Decimal('-3'), Decimal('8'), Decimal('16'))
+Decimal("1")
+>>> c.power(Decimal('3'), Decimal('7'), Decimal('-16'))
+Decimal("11")
+>>> c.power(Decimal('23E12345'), Decimal('67E189'), Decimal('123456789'))
+Decimal("11729830")
+>>> c.power(Decimal('-0'), Decimal('17'), Decimal('1729'))
+Decimal("-0")
+>>> c.power(Decimal('-23'), Decimal('0'), Decimal('65537'))
+Decimal("1")
+"""
+return a.__pow__(b, modulo, context=self)
+def quantize(self, a, b):
+"""Returns a value equal to 'a' (rounded), having the exponent of 'b'.
+The coefficient of the result is derived from that of the left-hand
+operand.  It may be rounded using the current rounding setting (if the
+exponent is being increased), multiplied by a positive power of ten (if
+the exponent is being decreased), or is unchanged (if the exponent is
+already equal to that of the right-hand operand).
+Unlike other operations, if the length of the coefficient after the
+quantize operation would be greater than precision then an Invalid
+operation condition is raised.  This guarantees that, unless there is
+an error condition, the exponent of the result of a quantize is always
+equal to that of the right-hand operand.
+Also unlike other operations, quantize will never raise Underflow, even
+if the result is subnormal and inexact.
+>>> ExtendedContext.quantize(Decimal('2.17'), Decimal('0.001'))
+Decimal("2.170")
+>>> ExtendedContext.quantize(Decimal('2.17'), Decimal('0.01'))
+Decimal("2.17")
+>>> ExtendedContext.quantize(Decimal('2.17'), Decimal('0.1'))
+Decimal("2.2")
+>>> ExtendedContext.quantize(Decimal('2.17'), Decimal('1e+0'))
+Decimal("2")
+>>> ExtendedContext.quantize(Decimal('2.17'), Decimal('1e+1'))
+Decimal("0E+1")
+>>> ExtendedContext.quantize(Decimal('-Inf'), Decimal('Infinity'))
+Decimal("-Infinity")
+>>> ExtendedContext.quantize(Decimal('2'), Decimal('Infinity'))
+Decimal("NaN")
+>>> ExtendedContext.quantize(Decimal('-0.1'), Decimal('1'))
+Decimal("-0")
+>>> ExtendedContext.quantize(Decimal('-0'), Decimal('1e+5'))
+Decimal("-0E+5")
+>>> ExtendedContext.quantize(Decimal('+35236450.6'), Decimal('1e-2'))
+Decimal("NaN")
+>>> ExtendedContext.quantize(Decimal('-35236450.6'), Decimal('1e-2'))
+Decimal("NaN")
+>>> ExtendedContext.quantize(Decimal('217'), Decimal('1e-1'))
+Decimal("217.0")
+>>> ExtendedContext.quantize(Decimal('217'), Decimal('1e-0'))
+Decimal("217")
+>>> ExtendedContext.quantize(Decimal('217'), Decimal('1e+1'))
+Decimal("2.2E+2")
+>>> ExtendedContext.quantize(Decimal('217'), Decimal('1e+2'))
+Decimal("2E+2")
+"""
+return a.quantize(b, context=self)
+def radix(self):
+"""Just returns 10, as this is Decimal, :)
+>>> ExtendedContext.radix()
+Decimal("10")
+"""
+return Decimal(10)
+def remainder(self, a, b):
+"""Returns the remainder from integer division.
+The result is the residue of the dividend after the operation of
+calculating integer division as described for divide-integer, rounded
+to precision digits if necessary.  The sign of the result, if
+non-zero, is the same as that of the original dividend.
+This operation will fail under the same conditions as integer division
+(that is, if integer division on the same two operands would fail, the
+remainder cannot be calculated).
+>>> ExtendedContext.remainder(Decimal('2.1'), Decimal('3'))
+Decimal("2.1")
+>>> ExtendedContext.remainder(Decimal('10'), Decimal('3'))
+Decimal("1")
+>>> ExtendedContext.remainder(Decimal('-10'), Decimal('3'))
+Decimal("-1")
+>>> ExtendedContext.remainder(Decimal('10.2'), Decimal('1'))
+Decimal("0.2")
+>>> ExtendedContext.remainder(Decimal('10'), Decimal('0.3'))
+Decimal("0.1")
+>>> ExtendedContext.remainder(Decimal('3.6'), Decimal('1.3'))
+Decimal("1.0")
+"""
+return a.__mod__(b, context=self)
+def remainder_near(self, a, b):
+"""Returns to be "a - b * n", where n is the integer nearest the exact
+value of "x / b" (if two integers are equally near then the even one
+is chosen).  If the result is equal to 0 then its sign will be the
+sign of a.
+This operation will fail under the same conditions as integer division
+(that is, if integer division on the same two operands would fail, the
+remainder cannot be calculated).
+>>> ExtendedContext.remainder_near(Decimal('2.1'), Decimal('3'))
+Decimal("-0.9")
+>>> ExtendedContext.remainder_near(Decimal('10'), Decimal('6'))
+Decimal("-2")
+>>> ExtendedContext.remainder_near(Decimal('10'), Decimal('3'))
+Decimal("1")
+>>> ExtendedContext.remainder_near(Decimal('-10'), Decimal('3'))
+Decimal("-1")
+>>> ExtendedContext.remainder_near(Decimal('10.2'), Decimal('1'))
+Decimal("0.2")
+>>> ExtendedContext.remainder_near(Decimal('10'), Decimal('0.3'))
+Decimal("0.1")
+>>> ExtendedContext.remainder_near(Decimal('3.6'), Decimal('1.3'))
+Decimal("-0.3")
+"""
+return a.remainder_near(b, context=self)
+def rotate(self, a, b):
+"""Returns a rotated copy of a, b times.
+The coefficient of the result is a rotated copy of the digits in
+the coefficient of the first operand.  The number of places of
+rotation is taken from the absolute value of the second operand,
+with the rotation being to the left if the second operand is
+positive or to the right otherwise.
+>>> ExtendedContext.rotate(Decimal('34'), Decimal('8'))
+Decimal("400000003")
+>>> ExtendedContext.rotate(Decimal('12'), Decimal('9'))
+Decimal("12")
+>>> ExtendedContext.rotate(Decimal('123456789'), Decimal('-2'))
+Decimal("891234567")
+>>> ExtendedContext.rotate(Decimal('123456789'), Decimal('0'))
+Decimal("123456789")
+>>> ExtendedContext.rotate(Decimal('123456789'), Decimal('+2'))
+Decimal("345678912")
+"""
+return a.rotate(b, context=self)
+def same_quantum(self, a, b):
+"""Returns True if the two operands have the same exponent.
+The result is never affected by either the sign or the coefficient of
+either operand.
+>>> ExtendedContext.same_quantum(Decimal('2.17'), Decimal('0.001'))
+False
+>>> ExtendedContext.same_quantum(Decimal('2.17'), Decimal('0.01'))
+True
+>>> ExtendedContext.same_quantum(Decimal('2.17'), Decimal('1'))
+False
+>>> ExtendedContext.same_quantum(Decimal('Inf'), Decimal('-Inf'))
+True
+"""
+return a.same_quantum(b)
+def scaleb (self, a, b):
+"""Returns the first operand after adding the second value its exp.
+>>> ExtendedContext.scaleb(Decimal('7.50'), Decimal('-2'))
+Decimal("0.0750")
+>>> ExtendedContext.scaleb(Decimal('7.50'), Decimal('0'))
+Decimal("7.50")
+>>> ExtendedContext.scaleb(Decimal('7.50'), Decimal('3'))
+Decimal("7.50E+3")
+"""
+return a.scaleb (b, context=self)
+def shift(self, a, b):
+"""Returns a shifted copy of a, b times.
+The coefficient of the result is a shifted copy of the digits
+in the coefficient of the first operand.  The number of places
+to shift is taken from the absolute value of the second operand,
+with the shift being to the left if the second operand is
+positive or to the right otherwise.  Digits shifted into the
+coefficient are zeros.
+>>> ExtendedContext.shift(Decimal('34'), Decimal('8'))
+Decimal("400000000")
+>>> ExtendedContext.shift(Decimal('12'), Decimal('9'))
+Decimal("0")
+>>> ExtendedContext.shift(Decimal('123456789'), Decimal('-2'))
+Decimal("1234567")
+>>> ExtendedContext.shift(Decimal('123456789'), Decimal('0'))
+Decimal("123456789")
+>>> ExtendedContext.shift(Decimal('123456789'), Decimal('+2'))
+Decimal("345678900")
+"""
+return a.shift(b, context=self)
+def sqrt(self, a):
+"""Square root of a non-negative number to context precision.
+If the result must be inexact, it is rounded using the round-half-even
+algorithm.
+>>> ExtendedContext.sqrt(Decimal('0'))
+Decimal("0")
+>>> ExtendedContext.sqrt(Decimal('-0'))
+Decimal("-0")
+>>> ExtendedContext.sqrt(Decimal('0.39'))
+Decimal("0.624499800")
+>>> ExtendedContext.sqrt(Decimal('100'))
+Decimal("10")
+>>> ExtendedContext.sqrt(Decimal('1'))
+Decimal("1")
+>>> ExtendedContext.sqrt(Decimal('1.0'))
+Decimal("1.0")
+>>> ExtendedContext.sqrt(Decimal('1.00'))
+Decimal("1.0")
+>>> ExtendedContext.sqrt(Decimal('7'))
+Decimal("2.64575131")
+>>> ExtendedContext.sqrt(Decimal('10'))
+Decimal("3.16227766")
+>>> ExtendedContext.prec
+9
+"""
+return a.sqrt(context=self)
+def subtract(self, a, b):
+"""Return the difference between the two operands.
+>>> ExtendedContext.subtract(Decimal('1.3'), Decimal('1.07'))
+Decimal("0.23")
+>>> ExtendedContext.subtract(Decimal('1.3'), Decimal('1.30'))
+Decimal("0.00")
+>>> ExtendedContext.subtract(Decimal('1.3'), Decimal('2.07'))
+Decimal("-0.77")
+"""
+return a.__sub__(b, context=self)
+def to_eng_string(self, a):
+"""Converts a number to a string, using scientific notation.
+The operation is not affected by the context.
+"""
+return a.to_eng_string(context=self)
+def to_sci_string(self, a):
+"""Converts a number to a string, using scientific notation.
+The operation is not affected by the context.
+"""
+return a.__str__(context=self)
+def to_integral_exact(self, a):
+"""Rounds to an integer.
+When the operand has a negative exponent, the result is the same
+as using the quantize() operation using the given operand as the
+left-hand-operand, 1E+0 as the right-hand-operand, and the precision
+of the operand as the precision setting; Inexact and Rounded flags
+are allowed in this operation.  The rounding mode is taken from the
+context.
+>>> ExtendedContext.to_integral_exact(Decimal('2.1'))
+Decimal("2")
+>>> ExtendedContext.to_integral_exact(Decimal('100'))
+Decimal("100")
+>>> ExtendedContext.to_integral_exact(Decimal('100.0'))
+Decimal("100")
+>>> ExtendedContext.to_integral_exact(Decimal('101.5'))
+Decimal("102")
+>>> ExtendedContext.to_integral_exact(Decimal('-101.5'))
+Decimal("-102")
+>>> ExtendedContext.to_integral_exact(Decimal('10E+5'))
+Decimal("1.0E+6")
+>>> ExtendedContext.to_integral_exact(Decimal('7.89E+77'))
+Decimal("7.89E+77")
+>>> ExtendedContext.to_integral_exact(Decimal('-Inf'))
+Decimal("-Infinity")
+"""
+return a.to_integral_exact(context=self)
+def to_integral_value(self, a):
+"""Rounds to an integer.
+When the operand has a negative exponent, the result is the same
+as using the quantize() operation using the given operand as the
+left-hand-operand, 1E+0 as the right-hand-operand, and the precision
+of the operand as the precision setting, except that no flags will
+be set.  The rounding mode is taken from the context.
+>>> ExtendedContext.to_integral_value(Decimal('2.1'))
+Decimal("2")
+>>> ExtendedContext.to_integral_value(Decimal('100'))
+Decimal("100")
+>>> ExtendedContext.to_integral_value(Decimal('100.0'))
+Decimal("100")
+>>> ExtendedContext.to_integral_value(Decimal('101.5'))
+Decimal("102")
+>>> ExtendedContext.to_integral_value(Decimal('-101.5'))
+Decimal("-102")
+>>> ExtendedContext.to_integral_value(Decimal('10E+5'))
+Decimal("1.0E+6")
+>>> ExtendedContext.to_integral_value(Decimal('7.89E+77'))
+Decimal("7.89E+77")
+>>> ExtendedContext.to_integral_value(Decimal('-Inf'))
+Decimal("-Infinity")
+"""
+return a.to_integral_value(context=self)
+# the method name changed, but we provide also the old one, for compatibility
+to_integral = to_integral_value
+class _WorkRep(object):
+__slots__ = ('sign','int','exp')
+# sign: 0 or 1
+# int:  int or long
+# exp:  None, int, or string
+def __init__(self, value=None):
+if value is None:
+self.sign = None
+self.int = 0
+self.exp = None
+elif isinstance(value, Decimal):
+self.sign = value._sign
+self.int = int(value._int)
+self.exp = value._exp
+else:
+# assert isinstance(value, tuple)
+self.sign = value[0]
+self.int = value[1]
+self.exp = value[2]
+def __repr__(self):
+return "(%r, %r, %r)" % (self.sign, self.int, self.exp)
+__str__ = __repr__
+def _normalize(op1, op2, prec = 0):
+"""Normalizes op1, op2 to have the same exp and length of coefficient.
+Done during addition.
+"""
+if op1.exp < op2.exp:
+tmp = op2
+other = op1
+else:
+tmp = op1
+other = op2
+# Let exp = min(tmp.exp - 1, tmp.adjusted() - precision - 1).
+# Then adding 10**exp to tmp has the same effect (after rounding)
+# as adding any positive quantity smaller than 10**exp; similarly
+# for subtraction.  So if other is smaller than 10**exp we replace
+# it with 10**exp.  This avoids tmp.exp - other.exp getting too large.
+tmp_len = len(str(tmp.int))
+other_len = len(str(other.int))
+exp = tmp.exp + min(-1, tmp_len - prec - 2)
+if other_len + other.exp - 1 < exp:
+other.int = 1
+other.exp = exp
+tmp.int *= 10 ** (tmp.exp - other.exp)
+tmp.exp = other.exp
+return op1, op2
+##### Integer arithmetic functions used by ln, log10, exp and __pow__ #####
+# This function from Tim Peters was taken from here:
+# http://mail.python.org/pipermail/python-list/1999-July/007758.html
+# The correction being in the function definition is for speed, and
+# the whole function is not resolved with math.log because of avoiding
+# the use of floats.
+def _nbits(n, correction = {
+'0': 4, '1': 3, '2': 2, '3': 2,
+'4': 1, '5': 1, '6': 1, '7': 1,
+'8': 0, '9': 0, 'a': 0, 'b': 0,
+'c': 0, 'd': 0, 'e': 0, 'f': 0}):
+"""Number of bits in binary representation of the positive integer n,
+or 0 if n == 0.
+"""
+if n < 0:
+raise ValueError("The argument to _nbits should be nonnegative.")
+hex_n = "%x" % n
+return 4*len(hex_n) - correction[hex_n[0]]
+def _sqrt_nearest(n, a):
+"""Closest integer to the square root of the positive integer n.  a is
+an initial approximation to the square root.  Any positive integer
+will do for a, but the closer a is to the square root of n the
+faster convergence will be.
+"""
+if n <= 0 or a <= 0:
+raise ValueError("Both arguments to _sqrt_nearest should be positive.")
+b=0
+while a != b:
+b, a = a, a--n//a>>1
+return a
+def _rshift_nearest(x, shift):
+"""Given an integer x and a nonnegative integer shift, return closest
+integer to x / 2**shift; use round-to-even in case of a tie.
+"""
+b, q = 1L << shift, x >> shift
+return q + (2*(x & (b-1)) + (q&1) > b)
+def _div_nearest(a, b):
+"""Closest integer to a/b, a and b positive integers; rounds to even
+in the case of a tie.
+"""
+q, r = divmod(a, b)
+return q + (2*r + (q&1) > b)
+def _ilog(x, M, L = 8):
+"""Integer approximation to M*log(x/M), with absolute error boundable
+in terms only of x/M.
+Given positive integers x and M, return an integer approximation to
+M * log(x/M).  For L = 8 and 0.1 <= x/M <= 10 the difference
+between the approximation and the exact result is at most 22.  For
+L = 8 and 1.0 <= x/M <= 10.0 the difference is at most 15.  In
+both cases these are upper bounds on the error; it will usually be
+much smaller."""
+# The basic algorithm is the following: let log1p be the function
+# log1p(x) = log(1+x).  Then log(x/M) = log1p((x-M)/M).  We use
+# the reduction
+#
+#    log1p(y) = 2*log1p(y/(1+sqrt(1+y)))
+#
+# repeatedly until the argument to log1p is small (< 2**-L in
+# absolute value).  For small y we can use the Taylor series
+# expansion
+#
+#    log1p(y) ~ y - y**2/2 + y**3/3 - ... - (-y)**T/T
+#
+# truncating at T such that y**T is small enough.  The whole
+# computation is carried out in a form of fixed-point arithmetic,
+# with a real number z being represented by an integer
+# approximation to z*M.  To avoid loss of precision, the y below
+# is actually an integer approximation to 2**R*y*M, where R is the
+# number of reductions performed so far.
+y = x-M
+# argument reduction; R = number of reductions performed
+R = 0
+while (R <= L and long(abs(y)) << L-R >= M or
+R > L and abs(y) >> R-L >= M):
+y = _div_nearest(long(M*y) << 1,
+M + _sqrt_nearest(M*(M+_rshift_nearest(y, R)), M))
+R += 1
+# Taylor series with T terms
+T = -int(-10*len(str(M))//(3*L))
+yshift = _rshift_nearest(y, R)
+w = _div_nearest(M, T)
+for k in xrange(T-1, 0, -1):
+w = _div_nearest(M, k) - _div_nearest(yshift*w, M)
+return _div_nearest(w*y, M)
+def _dlog10(c, e, p):
+"""Given integers c, e and p with c > 0, p >= 0, compute an integer
+approximation to 10**p * log10(c*10**e), with an absolute error of
+at most 1.  Assumes that c*10**e is not exactly 1."""
+# increase precision by 2; compensate for this by dividing
+# final result by 100
+p += 2
+# write c*10**e as d*10**f with either:
+#   f >= 0 and 1 <= d <= 10, or
+#   f <= 0 and 0.1 <= d <= 1.
+# Thus for c*10**e close to 1, f = 0
+l = len(str(c))
+f = e+l - (e+l >= 1)
+if p > 0:
+M = 10**p
+k = e+p-f
+if k >= 0:
+c *= 10**k
+else:
+c = _div_nearest(c, 10**-k)
+log_d = _ilog(c, M) # error < 5 + 22 = 27
+log_10 = _log10_digits(p) # error < 1
+log_d = _div_nearest(log_d*M, log_10)
+log_tenpower = f*M # exact
+else:
+log_d = 0  # error < 2.31
+log_tenpower = div_nearest(f, 10**-p) # error < 0.5
+return _div_nearest(log_tenpower+log_d, 100)
+def _dlog(c, e, p):
+"""Given integers c, e and p with c > 0, compute an integer
+approximation to 10**p * log(c*10**e), with an absolute error of
+at most 1.  Assumes that c*10**e is not exactly 1."""
+# Increase precision by 2. The precision increase is compensated
+# for at the end with a division by 100.
+p += 2
+# rewrite c*10**e as d*10**f with either f >= 0 and 1 <= d <= 10,
+# or f <= 0 and 0.1 <= d <= 1.  Then we can compute 10**p * log(c*10**e)
+# as 10**p * log(d) + 10**p*f * log(10).
+l = len(str(c))
+f = e+l - (e+l >= 1)
+# compute approximation to 10**p*log(d), with error < 27
+if p > 0:
+k = e+p-f
+if k >= 0:
+c *= 10**k
+else:
+c = _div_nearest(c, 10**-k)  # error of <= 0.5 in c
+# _ilog magnifies existing error in c by a factor of at most 10
+log_d = _ilog(c, 10**p) # error < 5 + 22 = 27
+else:
+# p <= 0: just approximate the whole thing by 0; error < 2.31
+log_d = 0
+# compute approximation to f*10**p*log(10), with error < 11.
+if f:
+extra = len(str(abs(f)))-1
+if p + extra >= 0:
+# error in f * _log10_digits(p+extra) < |f| * 1 = |f|
+# after division, error < |f|/10**extra + 0.5 < 10 + 0.5 < 11
+f_log_ten = _div_nearest(f*_log10_digits(p+extra), 10**extra)
+else:
+f_log_ten = 0
+else:
+f_log_ten = 0
+# error in sum < 11+27 = 38; error after division < 0.38 + 0.5 < 1
+return _div_nearest(f_log_ten + log_d, 100)
+class _Log10Memoize(object):
+"""Class to compute, store, and allow retrieval of, digits of the
+constant log(10) = 2.302585....  This constant is needed by
+Decimal.ln, Decimal.log10, Decimal.exp and Decimal.__pow__."""
+def __init__(self):
+self.digits = "23025850929940456840179914546843642076011014886"
+def getdigits(self, p):
+"""Given an integer p >= 0, return floor(10**p)*log(10).
+For example, self.getdigits(3) returns 2302.
+"""
+# digits are stored as a string, for quick conversion to
+# integer in the case that we've already computed enough
+# digits; the stored digits should always be correct
+# (truncated, not rounded to nearest).
+if p < 0:
+raise ValueError("p should be nonnegative")
+if p >= len(self.digits):
+# compute p+3, p+6, p+9, ... digits; continue until at
+# least one of the extra digits is nonzero
+extra = 3
+while True:
+# compute p+extra digits, correct to within 1ulp
+M = 10**(p+extra+2)
+digits = str(_div_nearest(_ilog(10*M, M), 100))
+if digits[-extra:] != '0'*extra:
+break
+extra += 3
+# keep all reliable digits so far; remove trailing zeros
+# and next nonzero digit
+self.digits = digits.rstrip('0')[:-1]
+return int(self.digits[:p+1])
+_log10_digits = _Log10Memoize().getdigits
+def _iexp(x, M, L=8):
+"""Given integers x and M, M > 0, such that x/M is small in absolute
+value, compute an integer approximation to M*exp(x/M).  For 0 <=
+x/M <= 2.4, the absolute error in the result is bounded by 60 (and
+is usually much smaller)."""
+# Algorithm: to compute exp(z) for a real number z, first divide z
+# by a suitable power R of 2 so that |z/2**R| < 2**-L.  Then
+# compute expm1(z/2**R) = exp(z/2**R) - 1 using the usual Taylor
+# series
+#
+#     expm1(x) = x + x**2/2! + x**3/3! + ...
+#
+# Now use the identity
+#
+#     expm1(2x) = expm1(x)*(expm1(x)+2)
+#
+# R times to compute the sequence expm1(z/2**R),
+# expm1(z/2**(R-1)), ... , exp(z/2), exp(z).
+# Find R such that x/2**R/M <= 2**-L
+R = _nbits((long(x)<<L)//M)
+# Taylor series.  (2**L)**T > M
+T = -int(-10*len(str(M))//(3*L))
+y = _div_nearest(x, T)
+Mshift = long(M)<<R
+for i in xrange(T-1, 0, -1):
+y = _div_nearest(x*(Mshift + y), Mshift * i)
+# Expansion
+for k in xrange(R-1, -1, -1):
+Mshift = long(M)<<(k+2)
+y = _div_nearest(y*(y+Mshift), Mshift)
+return M+y
+def _dexp(c, e, p):
+"""Compute an approximation to exp(c*10**e), with p decimal places of
+precision.
+Returns integers d, f such that:
+10**(p-1) <= d <= 10**p, and
+(d-1)*10**f < exp(c*10**e) < (d+1)*10**f
+In other words, d*10**f is an approximation to exp(c*10**e) with p
+digits of precision, and with an error in d of at most 1.  This is
+almost, but not quite, the same as the error being < 1ulp: when d
+= 10**(p-1) the error could be up to 10 ulp."""
+# we'll call iexp with M = 10**(p+2), giving p+3 digits of precision
+p += 2
+# compute log(10) with extra precision = adjusted exponent of c*10**e
+extra = max(0, e + len(str(c)) - 1)
+q = p + extra
+# compute quotient c*10**e/(log(10)) = c*10**(e+q)/(log(10)*10**q),
+# rounding down
+shift = e+q
+if shift >= 0:
+cshift = c*10**shift
+else:
+cshift = c//10**-shift
+quot, rem = divmod(cshift, _log10_digits(q))
+# reduce remainder back to original precision
+rem = _div_nearest(rem, 10**extra)
+# error in result of _iexp < 120;  error after division < 0.62
+return _div_nearest(_iexp(rem, 10**p), 1000), quot - p + 3
+def _dpower(xc, xe, yc, ye, p):
+"""Given integers xc, xe, yc and ye representing Decimals x = xc*10**xe and
+y = yc*10**ye, compute x**y.  Returns a pair of integers (c, e) such that:
+10**(p-1) <= c <= 10**p, and
+(c-1)*10**e < x**y < (c+1)*10**e
+in other words, c*10**e is an approximation to x**y with p digits
+of precision, and with an error in c of at most 1.  (This is
+almost, but not quite, the same as the error being < 1ulp: when c
+== 10**(p-1) we can only guarantee error < 10ulp.)
+We assume that: x is positive and not equal to 1, and y is nonzero.
+"""
+# Find b such that 10**(b-1) <= |y| <= 10**b
+b = len(str(abs(yc))) + ye
+# log(x) = lxc*10**(-p-b-1), to p+b+1 places after the decimal point
+lxc = _dlog(xc, xe, p+b+1)
+# compute product y*log(x) = yc*lxc*10**(-p-b-1+ye) = pc*10**(-p-1)
+shift = ye-b
+if shift >= 0:
+pc = lxc*yc*10**shift
+else:
+pc = _div_nearest(lxc*yc, 10**-shift)
+if pc == 0:
+# we prefer a result that isn't exactly 1; this makes it
+# easier to compute a correctly rounded result in __pow__
+if ((len(str(xc)) + xe >= 1) == (yc > 0)): # if x**y > 1:
+coeff, exp = 10**(p-1)+1, 1-p
+else:
+coeff, exp = 10**p-1, -p
+else:
+coeff, exp = _dexp(pc, -(p+1), p+1)
+coeff = _div_nearest(coeff, 10)
+exp += 1
+return coeff, exp
+def _log10_lb(c, correction = {
+'1': 100, '2': 70, '3': 53, '4': 40, '5': 31,
+'6': 23, '7': 16, '8': 10, '9': 5}):
+"""Compute a lower bound for 100*log10(c) for a positive integer c."""
+if c <= 0:
+raise ValueError("The argument to _log10_lb should be nonnegative.")
+str_c = str(c)
+return 100*len(str_c) - correction[str_c[0]]
+##### Helper Functions ####################################################
+def _convert_other(other, raiseit=False):
+"""Convert other to Decimal.
+Verifies that it's ok to use in an implicit construction.
+"""
+if isinstance(other, Decimal):
+return other
+if isinstance(other, (int, long)):
+return Decimal(other)
+if raiseit:
+raise TypeError("Unable to convert %s to Decimal" % other)
+return NotImplemented
+##### Setup Specific Contexts ############################################
+# The default context prototype used by Context()
+# Is mutable, so that new contexts can have different default values
+DefaultContext = Context(
+prec=28, rounding=ROUND_HALF_EVEN,
+traps=[DivisionByZero, Overflow, InvalidOperation],
+flags=[],
+Emax=999999999,
+Emin=-999999999,
+capitals=1
+)
+# Pre-made alternate contexts offered by the specification
+# Don't change these; the user should be able to select these
+# contexts and be able to reproduce results from other implementations
+# of the spec.
+BasicContext = Context(
+prec=9, rounding=ROUND_HALF_UP,
+traps=[DivisionByZero, Overflow, InvalidOperation, Clamped, Underflow],
+flags=[],
+)
+ExtendedContext = Context(
+prec=9, rounding=ROUND_HALF_EVEN,
+traps=[],
+flags=[],
+)
+##### crud for parsing strings #############################################
+import re
+# Regular expression used for parsing numeric strings.  Additional
+# comments:
+#
+# 1. Uncomment the two '\s*' lines to allow leading and/or trailing
+# whitespace.  But note that the specification disallows whitespace in
+# a numeric string.
+#
+# 2. For finite numbers (not infinities and NaNs) the body of the
+# number between the optional sign and the optional exponent must have
+# at least one decimal digit, possibly after the decimal point.  The
+# lookahead expression '(?=\d|\.\d)' checks this.
+#
+# As the flag UNICODE is not enabled here, we're explicitly avoiding any
+# other meaning for \d than the numbers [0-9].
+import re
+_parser = re.compile(r"""     # A numeric string consists of:
+#    \s*
+(?P<sign>[-+])?           # an optional sign, followed by either...
+(
+(?=\d|\.\d)           # ...a number (with at least one digit)
+(?P<int>\d*)          # consisting of a (possibly empty) integer part
+(\.(?P<frac>\d*))?    # followed by an optional fractional part
+(E(?P<exp>[-+]?\d+))? # followed by an optional exponent, or...
+|
+Inf(inity)?           # ...an infinity, or...
+|
+(?P<signal>s)?        # ...an (optionally signaling)
+NaN                   # NaN
+(?P<diag>\d*)         # with (possibly empty) diagnostic information.
+)
+#    \s*
+$
+""", re.VERBOSE | re.IGNORECASE).match
+_all_zeros = re.compile('0*$').match
+_exact_half = re.compile('50*$').match
+del re
+##### Useful Constants (internal use only) ################################
+# Reusable defaults
+Inf = Decimal('Inf')
+negInf = Decimal('-Inf')
+NaN = Decimal('NaN')
+Dec_0 = Decimal(0)
+Dec_p1 = Decimal(1)
+Dec_n1 = Decimal(-1)
+# Infsign[sign] is infinity w/ that sign
+Infsign = (Inf, negInf)
+if __name__ == '__main__':
+import doctest, sys
+doctest.testmod(sys.modules[__name__])