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6202 lines
216 KiB
Python
6202 lines
216 KiB
Python
# Copyright (c) 2004 Python Software Foundation.
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# All rights reserved.
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# Written by Eric Price <eprice at tjhsst.edu>
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# and Facundo Batista <facundo at taniquetil.com.ar>
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# and Raymond Hettinger <python at rcn.com>
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# and Aahz <aahz at pobox.com>
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# and Tim Peters
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# This module is currently Py2.3 compatible and should be kept that way
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# unless a major compelling advantage arises. IOW, 2.3 compatibility is
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# strongly preferred, but not guaranteed.
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# Also, this module should be kept in sync with the latest updates of
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# the IBM specification as it evolves. Those updates will be treated
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# as bug fixes (deviation from the spec is a compatibility, usability
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# bug) and will be backported. At this point the spec is stabilizing
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# and the updates are becoming fewer, smaller, and less significant.
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"""
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This is a Py2.3 implementation of decimal floating point arithmetic based on
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the General Decimal Arithmetic Specification:
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http://speleotrove.com/decimal/decarith.html
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and IEEE standard 854-1987:
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http://en.wikipedia.org/wiki/IEEE_854-1987
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Decimal floating point has finite precision with arbitrarily large bounds.
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The purpose of this module is to support arithmetic using familiar
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"schoolhouse" rules and to avoid some of the tricky representation
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issues associated with binary floating point. The package is especially
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useful for financial applications or for contexts where users have
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expectations that are at odds with binary floating point (for instance,
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in binary floating point, 1.00 % 0.1 gives 0.09999999999999995 instead
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of the expected Decimal('0.00') returned by decimal floating point).
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Here are some examples of using the decimal module:
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>>> from decimal import *
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>>> setcontext(ExtendedContext)
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>>> Decimal(0)
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Decimal('0')
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>>> Decimal('1')
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Decimal('1')
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>>> Decimal('-.0123')
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Decimal('-0.0123')
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>>> Decimal(123456)
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Decimal('123456')
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>>> Decimal('123.45e12345678901234567890')
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Decimal('1.2345E+12345678901234567892')
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>>> Decimal('1.33') + Decimal('1.27')
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Decimal('2.60')
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>>> Decimal('12.34') + Decimal('3.87') - Decimal('18.41')
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Decimal('-2.20')
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>>> dig = Decimal(1)
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>>> print dig / Decimal(3)
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0.333333333
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>>> getcontext().prec = 18
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>>> print dig / Decimal(3)
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0.333333333333333333
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>>> print dig.sqrt()
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1
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>>> print Decimal(3).sqrt()
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1.73205080756887729
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>>> print Decimal(3) ** 123
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4.85192780976896427E+58
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>>> inf = Decimal(1) / Decimal(0)
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>>> print inf
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Infinity
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>>> neginf = Decimal(-1) / Decimal(0)
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>>> print neginf
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-Infinity
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>>> print neginf + inf
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NaN
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>>> print neginf * inf
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-Infinity
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>>> print dig / 0
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Infinity
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>>> getcontext().traps[DivisionByZero] = 1
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>>> print dig / 0
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Traceback (most recent call last):
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...
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...
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...
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DivisionByZero: x / 0
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>>> c = Context()
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>>> c.traps[InvalidOperation] = 0
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>>> print c.flags[InvalidOperation]
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0
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>>> c.divide(Decimal(0), Decimal(0))
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Decimal('NaN')
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>>> c.traps[InvalidOperation] = 1
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>>> print c.flags[InvalidOperation]
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1
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>>> c.flags[InvalidOperation] = 0
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>>> print c.flags[InvalidOperation]
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0
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>>> print c.divide(Decimal(0), Decimal(0))
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Traceback (most recent call last):
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...
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...
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...
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InvalidOperation: 0 / 0
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>>> print c.flags[InvalidOperation]
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1
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>>> c.flags[InvalidOperation] = 0
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>>> c.traps[InvalidOperation] = 0
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>>> print c.divide(Decimal(0), Decimal(0))
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NaN
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>>> print c.flags[InvalidOperation]
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1
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>>>
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"""
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__all__ = [
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# Two major classes
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'Decimal', 'Context',
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# Contexts
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'DefaultContext', 'BasicContext', 'ExtendedContext',
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# Exceptions
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'DecimalException', 'Clamped', 'InvalidOperation', 'DivisionByZero',
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'Inexact', 'Rounded', 'Subnormal', 'Overflow', 'Underflow',
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# Constants for use in setting up contexts
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'ROUND_DOWN', 'ROUND_HALF_UP', 'ROUND_HALF_EVEN', 'ROUND_CEILING',
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'ROUND_FLOOR', 'ROUND_UP', 'ROUND_HALF_DOWN', 'ROUND_05UP',
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# Functions for manipulating contexts
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'setcontext', 'getcontext', 'localcontext'
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]
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__version__ = '1.70' # Highest version of the spec this complies with
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import math as _math
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import numbers as _numbers
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try:
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from collections import namedtuple as _namedtuple
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DecimalTuple = _namedtuple('DecimalTuple', 'sign digits exponent')
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except ImportError:
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DecimalTuple = lambda *args: args
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# Rounding
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ROUND_DOWN = 'ROUND_DOWN'
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ROUND_HALF_UP = 'ROUND_HALF_UP'
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ROUND_HALF_EVEN = 'ROUND_HALF_EVEN'
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ROUND_CEILING = 'ROUND_CEILING'
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ROUND_FLOOR = 'ROUND_FLOOR'
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ROUND_UP = 'ROUND_UP'
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ROUND_HALF_DOWN = 'ROUND_HALF_DOWN'
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ROUND_05UP = 'ROUND_05UP'
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# Errors
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class DecimalException(ArithmeticError):
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"""Base exception class.
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Used exceptions derive from this.
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If an exception derives from another exception besides this (such as
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Underflow (Inexact, Rounded, Subnormal) that indicates that it is only
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called if the others are present. This isn't actually used for
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anything, though.
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handle -- Called when context._raise_error is called and the
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trap_enabler is not set. First argument is self, second is the
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context. More arguments can be given, those being after
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the explanation in _raise_error (For example,
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context._raise_error(NewError, '(-x)!', self._sign) would
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call NewError().handle(context, self._sign).)
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To define a new exception, it should be sufficient to have it derive
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from DecimalException.
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"""
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def handle(self, context, *args):
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pass
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class Clamped(DecimalException):
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"""Exponent of a 0 changed to fit bounds.
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This occurs and signals clamped if the exponent of a result has been
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altered in order to fit the constraints of a specific concrete
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representation. This may occur when the exponent of a zero result would
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be outside the bounds of a representation, or when a large normal
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number would have an encoded exponent that cannot be represented. In
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this latter case, the exponent is reduced to fit and the corresponding
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number of zero digits are appended to the coefficient ("fold-down").
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"""
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class InvalidOperation(DecimalException):
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"""An invalid operation was performed.
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Various bad things cause this:
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Something creates a signaling NaN
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-INF + INF
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0 * (+-)INF
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(+-)INF / (+-)INF
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x % 0
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(+-)INF % x
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x._rescale( non-integer )
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sqrt(-x) , x > 0
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0 ** 0
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x ** (non-integer)
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x ** (+-)INF
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An operand is invalid
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The result of the operation after these is a quiet positive NaN,
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except when the cause is a signaling NaN, in which case the result is
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also a quiet NaN, but with the original sign, and an optional
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diagnostic information.
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"""
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def handle(self, context, *args):
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if args:
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ans = _dec_from_triple(args[0]._sign, args[0]._int, 'n', True)
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return ans._fix_nan(context)
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return _NaN
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class ConversionSyntax(InvalidOperation):
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"""Trying to convert badly formed string.
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This occurs and signals invalid-operation if an string is being
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converted to a number and it does not conform to the numeric string
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syntax. The result is [0,qNaN].
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"""
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def handle(self, context, *args):
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return _NaN
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class DivisionByZero(DecimalException, ZeroDivisionError):
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"""Division by 0.
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This occurs and signals division-by-zero if division of a finite number
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by zero was attempted (during a divide-integer or divide operation, or a
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power operation with negative right-hand operand), and the dividend was
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not zero.
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The result of the operation is [sign,inf], where sign is the exclusive
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or of the signs of the operands for divide, or is 1 for an odd power of
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-0, for power.
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"""
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def handle(self, context, sign, *args):
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return _SignedInfinity[sign]
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class DivisionImpossible(InvalidOperation):
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"""Cannot perform the division adequately.
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This occurs and signals invalid-operation if the integer result of a
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divide-integer or remainder operation had too many digits (would be
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longer than precision). The result is [0,qNaN].
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"""
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def handle(self, context, *args):
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return _NaN
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class DivisionUndefined(InvalidOperation, ZeroDivisionError):
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"""Undefined result of division.
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This occurs and signals invalid-operation if division by zero was
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attempted (during a divide-integer, divide, or remainder operation), and
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the dividend is also zero. The result is [0,qNaN].
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"""
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def handle(self, context, *args):
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return _NaN
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class Inexact(DecimalException):
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"""Had to round, losing information.
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This occurs and signals inexact whenever the result of an operation is
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not exact (that is, it needed to be rounded and any discarded digits
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were non-zero), or if an overflow or underflow condition occurs. The
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result in all cases is unchanged.
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The inexact signal may be tested (or trapped) to determine if a given
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operation (or sequence of operations) was inexact.
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"""
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class InvalidContext(InvalidOperation):
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"""Invalid context. Unknown rounding, for example.
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This occurs and signals invalid-operation if an invalid context was
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detected during an operation. This can occur if contexts are not checked
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on creation and either the precision exceeds the capability of the
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underlying concrete representation or an unknown or unsupported rounding
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was specified. These aspects of the context need only be checked when
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the values are required to be used. The result is [0,qNaN].
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"""
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def handle(self, context, *args):
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return _NaN
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class Rounded(DecimalException):
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"""Number got rounded (not necessarily changed during rounding).
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This occurs and signals rounded whenever the result of an operation is
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rounded (that is, some zero or non-zero digits were discarded from the
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coefficient), or if an overflow or underflow condition occurs. The
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result in all cases is unchanged.
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The rounded signal may be tested (or trapped) to determine if a given
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operation (or sequence of operations) caused a loss of precision.
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"""
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class Subnormal(DecimalException):
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"""Exponent < Emin before rounding.
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This occurs and signals subnormal whenever the result of a conversion or
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operation is subnormal (that is, its adjusted exponent is less than
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Emin, before any rounding). The result in all cases is unchanged.
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The subnormal signal may be tested (or trapped) to determine if a given
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or operation (or sequence of operations) yielded a subnormal result.
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"""
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class Overflow(Inexact, Rounded):
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"""Numerical overflow.
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This occurs and signals overflow if the adjusted exponent of a result
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(from a conversion or from an operation that is not an attempt to divide
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by zero), after rounding, would be greater than the largest value that
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can be handled by the implementation (the value Emax).
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The result depends on the rounding mode:
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For round-half-up and round-half-even (and for round-half-down and
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round-up, if implemented), the result of the operation is [sign,inf],
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where sign is the sign of the intermediate result. For round-down, the
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result is the largest finite number that can be represented in the
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current precision, with the sign of the intermediate result. For
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round-ceiling, the result is the same as for round-down if the sign of
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the intermediate result is 1, or is [0,inf] otherwise. For round-floor,
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the result is the same as for round-down if the sign of the intermediate
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result is 0, or is [1,inf] otherwise. In all cases, Inexact and Rounded
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will also be raised.
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"""
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def handle(self, context, sign, *args):
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if context.rounding in (ROUND_HALF_UP, ROUND_HALF_EVEN,
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ROUND_HALF_DOWN, ROUND_UP):
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return _SignedInfinity[sign]
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if sign == 0:
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if context.rounding == ROUND_CEILING:
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return _SignedInfinity[sign]
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return _dec_from_triple(sign, '9'*context.prec,
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context.Emax-context.prec+1)
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if sign == 1:
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if context.rounding == ROUND_FLOOR:
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return _SignedInfinity[sign]
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return _dec_from_triple(sign, '9'*context.prec,
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context.Emax-context.prec+1)
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class Underflow(Inexact, Rounded, Subnormal):
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"""Numerical underflow with result rounded to 0.
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This occurs and signals underflow if a result is inexact and the
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adjusted exponent of the result would be smaller (more negative) than
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the smallest value that can be handled by the implementation (the value
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Emin). That is, the result is both inexact and subnormal.
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The result after an underflow will be a subnormal number rounded, if
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necessary, so that its exponent is not less than Etiny. This may result
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in 0 with the sign of the intermediate result and an exponent of Etiny.
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In all cases, Inexact, Rounded, and Subnormal will also be raised.
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"""
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# List of public traps and flags
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_signals = [Clamped, DivisionByZero, Inexact, Overflow, Rounded,
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Underflow, InvalidOperation, Subnormal]
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# Map conditions (per the spec) to signals
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_condition_map = {ConversionSyntax:InvalidOperation,
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DivisionImpossible:InvalidOperation,
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DivisionUndefined:InvalidOperation,
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InvalidContext:InvalidOperation}
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##### Context Functions ##################################################
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# The getcontext() and setcontext() function manage access to a thread-local
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# current context. Py2.4 offers direct support for thread locals. If that
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# is not available, use threading.currentThread() which is slower but will
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# work for older Pythons. If threads are not part of the build, create a
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# mock threading object with threading.local() returning the module namespace.
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try:
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import threading
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except ImportError:
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# Python was compiled without threads; create a mock object instead
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import sys
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class MockThreading(object):
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def local(self, sys=sys):
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return sys.modules[__name__]
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threading = MockThreading()
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del sys, MockThreading
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try:
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threading.local
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except AttributeError:
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# To fix reloading, force it to create a new context
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# Old contexts have different exceptions in their dicts, making problems.
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if hasattr(threading.currentThread(), '__decimal_context__'):
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del threading.currentThread().__decimal_context__
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def setcontext(context):
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"""Set this thread's context to context."""
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if context in (DefaultContext, BasicContext, ExtendedContext):
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context = context.copy()
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context.clear_flags()
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threading.currentThread().__decimal_context__ = context
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def getcontext():
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"""Returns this thread's context.
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If this thread does not yet have a context, returns
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a new context and sets this thread's context.
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New contexts are copies of DefaultContext.
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"""
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try:
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return threading.currentThread().__decimal_context__
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except AttributeError:
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context = Context()
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threading.currentThread().__decimal_context__ = context
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return context
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else:
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local = threading.local()
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if hasattr(local, '__decimal_context__'):
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del local.__decimal_context__
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def getcontext(_local=local):
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"""Returns this thread's context.
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If this thread does not yet have a context, returns
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a new context and sets this thread's context.
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New contexts are copies of DefaultContext.
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"""
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try:
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return _local.__decimal_context__
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except AttributeError:
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context = Context()
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_local.__decimal_context__ = context
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return context
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def setcontext(context, _local=local):
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"""Set this thread's context to context."""
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if context in (DefaultContext, BasicContext, ExtendedContext):
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context = context.copy()
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context.clear_flags()
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_local.__decimal_context__ = context
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del threading, local # Don't contaminate the namespace
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def localcontext(ctx=None):
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"""Return a context manager for a copy of the supplied context
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Uses a copy of the current context if no context is specified
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The returned context manager creates a local decimal context
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in a with statement:
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def sin(x):
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with localcontext() as ctx:
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ctx.prec += 2
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# Rest of sin calculation algorithm
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# uses a precision 2 greater than normal
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return +s # Convert result to normal precision
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def sin(x):
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with localcontext(ExtendedContext):
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# Rest of sin calculation algorithm
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# uses the Extended Context from the
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# General Decimal Arithmetic Specification
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return +s # Convert result to normal context
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>>> setcontext(DefaultContext)
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>>> print getcontext().prec
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28
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>>> with localcontext():
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... ctx = getcontext()
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... ctx.prec += 2
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... print ctx.prec
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...
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30
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>>> with localcontext(ExtendedContext):
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... print getcontext().prec
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...
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9
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>>> print getcontext().prec
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28
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"""
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if ctx is None: ctx = getcontext()
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return _ContextManager(ctx)
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##### Decimal class #######################################################
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class Decimal(object):
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"""Floating point class for decimal arithmetic."""
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__slots__ = ('_exp','_int','_sign', '_is_special')
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# Generally, the value of the Decimal instance is given by
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# (-1)**_sign * _int * 10**_exp
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# Special values are signified by _is_special == True
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# We're immutable, so use __new__ not __init__
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def __new__(cls, value="0", context=None):
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"""Create a decimal point instance.
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>>> Decimal('3.14') # string input
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Decimal('3.14')
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>>> Decimal((0, (3, 1, 4), -2)) # tuple (sign, digit_tuple, exponent)
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Decimal('3.14')
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>>> Decimal(314) # int or long
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Decimal('314')
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>>> Decimal(Decimal(314)) # another decimal instance
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Decimal('314')
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>>> Decimal(' 3.14 \\n') # leading and trailing whitespace okay
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Decimal('3.14')
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"""
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# 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.strip())
|
|
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') or ''
|
|
exp = int(m.group('exp') or '0')
|
|
self._int = str(int(intpart+fracpart))
|
|
self._exp = exp - len(fracpart)
|
|
self._is_special = False
|
|
else:
|
|
diag = m.group('diag')
|
|
if diag is not None:
|
|
# NaN
|
|
self._int = str(int(diag or '0')).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):
|
|
value = Decimal.from_float(value)
|
|
self._exp = value._exp
|
|
self._sign = value._sign
|
|
self._int = value._int
|
|
self._is_special = value._is_special
|
|
return self
|
|
|
|
raise TypeError("Cannot convert %r to Decimal" % value)
|
|
|
|
# @classmethod, but @decorator is not valid Python 2.3 syntax, so
|
|
# don't use it (see notes on Py2.3 compatibility at top of file)
|
|
def from_float(cls, f):
|
|
"""Converts a float to a decimal number, exactly.
|
|
|
|
Note that Decimal.from_float(0.1) is not the same as Decimal('0.1').
|
|
Since 0.1 is not exactly representable in binary floating point, the
|
|
value is stored as the nearest representable value which is
|
|
0x1.999999999999ap-4. The exact equivalent of the value in decimal
|
|
is 0.1000000000000000055511151231257827021181583404541015625.
|
|
|
|
>>> Decimal.from_float(0.1)
|
|
Decimal('0.1000000000000000055511151231257827021181583404541015625')
|
|
>>> Decimal.from_float(float('nan'))
|
|
Decimal('NaN')
|
|
>>> Decimal.from_float(float('inf'))
|
|
Decimal('Infinity')
|
|
>>> Decimal.from_float(-float('inf'))
|
|
Decimal('-Infinity')
|
|
>>> Decimal.from_float(-0.0)
|
|
Decimal('-0')
|
|
|
|
"""
|
|
if isinstance(f, (int, long)): # handle integer inputs
|
|
return cls(f)
|
|
if _math.isinf(f) or _math.isnan(f): # raises TypeError if not a float
|
|
return cls(repr(f))
|
|
if _math.copysign(1.0, f) == 1.0:
|
|
sign = 0
|
|
else:
|
|
sign = 1
|
|
n, d = abs(f).as_integer_ratio()
|
|
k = d.bit_length() - 1
|
|
result = _dec_from_triple(sign, str(n*5**k), -k)
|
|
if cls is Decimal:
|
|
return result
|
|
else:
|
|
return cls(result)
|
|
from_float = classmethod(from_float)
|
|
|
|
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 _compare_check_nans(self, other, context):
|
|
"""Version of _check_nans used for the signaling comparisons
|
|
compare_signal, __le__, __lt__, __ge__, __gt__.
|
|
|
|
Signal InvalidOperation if either self or other is a (quiet
|
|
or signaling) NaN. Signaling NaNs take precedence over quiet
|
|
NaNs.
|
|
|
|
Return 0 if neither operand is a NaN.
|
|
|
|
"""
|
|
if context is None:
|
|
context = getcontext()
|
|
|
|
if self._is_special or other._is_special:
|
|
if self.is_snan():
|
|
return context._raise_error(InvalidOperation,
|
|
'comparison involving sNaN',
|
|
self)
|
|
elif other.is_snan():
|
|
return context._raise_error(InvalidOperation,
|
|
'comparison involving sNaN',
|
|
other)
|
|
elif self.is_qnan():
|
|
return context._raise_error(InvalidOperation,
|
|
'comparison involving NaN',
|
|
self)
|
|
elif other.is_qnan():
|
|
return context._raise_error(InvalidOperation,
|
|
'comparison involving NaN',
|
|
other)
|
|
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):
|
|
"""Compare the two non-NaN decimal instances self and other.
|
|
|
|
Returns -1 if self < other, 0 if self == other and 1
|
|
if self > other. This routine is for internal use only."""
|
|
|
|
if self._is_special or other._is_special:
|
|
self_inf = self._isinfinity()
|
|
other_inf = other._isinfinity()
|
|
if self_inf == other_inf:
|
|
return 0
|
|
elif self_inf < other_inf:
|
|
return -1
|
|
else:
|
|
return 1
|
|
|
|
# check for zeros; Decimal('0') == Decimal('-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)
|
|
if self_padded == other_padded:
|
|
return 0
|
|
elif self_padded < other_padded:
|
|
return -(-1)**self._sign
|
|
else:
|
|
return (-1)**self._sign
|
|
elif self_adjusted > other_adjusted:
|
|
return (-1)**self._sign
|
|
else: # self_adjusted < other_adjusted
|
|
return -((-1)**self._sign)
|
|
|
|
# Note: The Decimal standard doesn't cover rich comparisons for
|
|
# Decimals. In particular, the specification is silent on the
|
|
# subject of what should happen for a comparison involving a NaN.
|
|
# We take the following approach:
|
|
#
|
|
# == comparisons involving a quiet NaN always return False
|
|
# != comparisons involving a quiet NaN always return True
|
|
# == or != comparisons involving a signaling NaN signal
|
|
# InvalidOperation, and return False or True as above if the
|
|
# InvalidOperation is not trapped.
|
|
# <, >, <= and >= comparisons involving a (quiet or signaling)
|
|
# NaN signal InvalidOperation, and return False if the
|
|
# InvalidOperation is not trapped.
|
|
#
|
|
# This behavior is designed to conform as closely as possible to
|
|
# that specified by IEEE 754.
|
|
|
|
def __eq__(self, other, context=None):
|
|
other = _convert_other(other, allow_float=True)
|
|
if other is NotImplemented:
|
|
return other
|
|
if self._check_nans(other, context):
|
|
return False
|
|
return self._cmp(other) == 0
|
|
|
|
def __ne__(self, other, context=None):
|
|
other = _convert_other(other, allow_float=True)
|
|
if other is NotImplemented:
|
|
return other
|
|
if self._check_nans(other, context):
|
|
return True
|
|
return self._cmp(other) != 0
|
|
|
|
def __lt__(self, other, context=None):
|
|
other = _convert_other(other, allow_float=True)
|
|
if other is NotImplemented:
|
|
return other
|
|
ans = self._compare_check_nans(other, context)
|
|
if ans:
|
|
return False
|
|
return self._cmp(other) < 0
|
|
|
|
def __le__(self, other, context=None):
|
|
other = _convert_other(other, allow_float=True)
|
|
if other is NotImplemented:
|
|
return other
|
|
ans = self._compare_check_nans(other, context)
|
|
if ans:
|
|
return False
|
|
return self._cmp(other) <= 0
|
|
|
|
def __gt__(self, other, context=None):
|
|
other = _convert_other(other, allow_float=True)
|
|
if other is NotImplemented:
|
|
return other
|
|
ans = self._compare_check_nans(other, context)
|
|
if ans:
|
|
return False
|
|
return self._cmp(other) > 0
|
|
|
|
def __ge__(self, other, context=None):
|
|
other = _convert_other(other, allow_float=True)
|
|
if other is NotImplemented:
|
|
return other
|
|
ans = self._compare_check_nans(other, context)
|
|
if ans:
|
|
return False
|
|
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')).
|
|
|
|
# Equality comparisons involving signaling nans can raise an
|
|
# exception; since equality checks are implicitly and
|
|
# unpredictably used when checking set and dict membership, we
|
|
# prevent signaling nans from being used as set elements or
|
|
# dict keys by making __hash__ raise an exception.
|
|
if self._is_special:
|
|
if self.is_snan():
|
|
raise TypeError('Cannot hash a signaling NaN value.')
|
|
elif self.is_nan():
|
|
# 0 to match hash(float('nan'))
|
|
return 0
|
|
else:
|
|
# values chosen to match hash(float('inf')) and
|
|
# hash(float('-inf')).
|
|
if self._sign:
|
|
return -271828
|
|
else:
|
|
return 314159
|
|
|
|
# In Python 2.7, we're allowing comparisons (but not
|
|
# arithmetic operations) between floats and Decimals; so if
|
|
# a Decimal instance is exactly representable as a float then
|
|
# its hash should match that of the float.
|
|
self_as_float = float(self)
|
|
if Decimal.from_float(self_as_float) == self:
|
|
return hash(self_as_float)
|
|
|
|
if self._isinteger():
|
|
op = _WorkRep(self.to_integral_value())
|
|
# to make computation feasible for Decimals with large
|
|
# exponent, we use the fact that hash(n) == hash(m) for
|
|
# any two nonzero integers n and m such that (i) n and m
|
|
# have the same sign, and (ii) n is congruent to m modulo
|
|
# 2**64-1. So we can replace hash((-1)**s*c*10**e) with
|
|
# hash((-1)**s*c*pow(10, e, 2**64-1).
|
|
return hash((-1)**op.sign*op.int*pow(10, op.exp, 2**64-1))
|
|
# 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 DecimalTuple(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 context is None:
|
|
context = getcontext()
|
|
|
|
if not self and context.rounding != ROUND_FLOOR:
|
|
# -Decimal('0') is Decimal('0'), not Decimal('-0'), except
|
|
# in ROUND_FLOOR rounding mode.
|
|
ans = self.copy_abs()
|
|
else:
|
|
ans = self.copy_negate()
|
|
|
|
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 context is None:
|
|
context = getcontext()
|
|
|
|
if not self and context.rounding != ROUND_FLOOR:
|
|
# + (-0) = 0, except in ROUND_FLOOR rounding mode.
|
|
ans = self.copy_abs()
|
|
else:
|
|
ans = Decimal(self)
|
|
|
|
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 _SignedInfinity[resultsign]
|
|
|
|
if other._isinfinity():
|
|
if not self:
|
|
return context._raise_error(InvalidOperation, '0 * (+-)INF')
|
|
return _SignedInfinity[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 __truediv__(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 _SignedInfinity[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)
|
|
|
|
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 __rtruediv__(self, other, context=None):
|
|
"""Swaps self/other and returns __truediv__."""
|
|
other = _convert_other(other)
|
|
if other is NotImplemented:
|
|
return other
|
|
return other.__truediv__(self, context=context)
|
|
|
|
__div__ = __truediv__
|
|
__rdiv__ = __rtruediv__
|
|
|
|
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 (_SignedInfinity[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 _SignedInfinity[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."""
|
|
if self._isnan():
|
|
if self.is_snan():
|
|
raise ValueError("Cannot convert signaling NaN to float")
|
|
s = "-nan" if self._sign else "nan"
|
|
else:
|
|
s = str(self)
|
|
return float(s)
|
|
|
|
def __int__(self):
|
|
"""Converts self to an int, truncating if necessary."""
|
|
if self._is_special:
|
|
if self._isnan():
|
|
raise ValueError("Cannot convert NaN to integer")
|
|
elif self._isinfinity():
|
|
raise OverflowError("Cannot convert infinity to integer")
|
|
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')
|
|
|
|
__trunc__ = __int__
|
|
|
|
def real(self):
|
|
return self
|
|
real = property(real)
|
|
|
|
def imag(self):
|
|
return Decimal(0)
|
|
imag = property(imag)
|
|
|
|
def conjugate(self):
|
|
return self
|
|
|
|
def __complex__(self):
|
|
return complex(float(self))
|
|
|
|
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
|
|
ans = context._raise_error(Overflow, 'above Emax', self._sign)
|
|
context._raise_error(Inexact)
|
|
context._raise_error(Rounded)
|
|
return ans
|
|
|
|
self_is_subnormal = exp_min < Etiny
|
|
if self_is_subnormal:
|
|
exp_min = Etiny
|
|
|
|
# round if self has too many digits
|
|
if self._exp < exp_min:
|
|
digits = len(self._int) + self._exp - exp_min
|
|
if digits < 0:
|
|
self = _dec_from_triple(self._sign, '1', exp_min-1)
|
|
digits = 0
|
|
rounding_method = self._pick_rounding_function[context.rounding]
|
|
changed = rounding_method(self, digits)
|
|
coeff = self._int[:digits] or '0'
|
|
if changed > 0:
|
|
coeff = str(int(coeff)+1)
|
|
if len(coeff) > context.prec:
|
|
coeff = coeff[:-1]
|
|
exp_min += 1
|
|
|
|
# check whether the rounding pushed the exponent out of range
|
|
if exp_min > Etop:
|
|
ans = context._raise_error(Overflow, 'above Emax', self._sign)
|
|
else:
|
|
ans = _dec_from_triple(self._sign, coeff, exp_min)
|
|
|
|
# raise the appropriate signals, taking care to respect
|
|
# the precedence described in the specification
|
|
if changed and self_is_subnormal:
|
|
context._raise_error(Underflow)
|
|
if self_is_subnormal:
|
|
context._raise_error(Subnormal)
|
|
if changed:
|
|
context._raise_error(Inexact)
|
|
context._raise_error(Rounded)
|
|
if not ans:
|
|
# raise Clamped on underflow to 0
|
|
context._raise_error(Clamped)
|
|
return ans
|
|
|
|
if self_is_subnormal:
|
|
context._raise_error(Subnormal)
|
|
|
|
# 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)
|
|
|
|
# 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)
|
|
|
|
_pick_rounding_function = dict(
|
|
ROUND_DOWN = _round_down,
|
|
ROUND_UP = _round_up,
|
|
ROUND_HALF_UP = _round_half_up,
|
|
ROUND_HALF_DOWN = _round_half_down,
|
|
ROUND_HALF_EVEN = _round_half_even,
|
|
ROUND_CEILING = _round_ceiling,
|
|
ROUND_FLOOR = _round_floor,
|
|
ROUND_05UP = _round_05up,
|
|
)
|
|
|
|
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 = _SignedInfinity[self._sign ^ other._sign]
|
|
elif other._exp == 'F':
|
|
if not self:
|
|
return context._raise_error(InvalidOperation,
|
|
'0 * INF in fma')
|
|
product = _SignedInfinity[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 abs(y) = yc*10**ye, with xc
|
|
# and yc positive integers not divisible by 10.
|
|
|
|
# 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. Express y as a rational number 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 y 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:
|
|
xe *= yc
|
|
# result is now 10**(xe * 10**ye); xe * 10**ye must be integral
|
|
while xe % 10 == 0:
|
|
xe //= 10
|
|
ye += 1
|
|
if ye < 0:
|
|
return None
|
|
exponent = xe * 10**ye
|
|
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
|
|
|
|
# We now have:
|
|
#
|
|
# x = 2**e * 10**xe, e > 0, and y < 0.
|
|
#
|
|
# The exact result is:
|
|
#
|
|
# x**y = 5**(-e*y) * 10**(e*y + xe*y)
|
|
#
|
|
# provided that both e*y and xe*y are integers. Note that if
|
|
# 5**(-e*y) >= 10**p, then the result can't be expressed
|
|
# exactly with p digits of precision.
|
|
#
|
|
# Using the above, we can guard against large values of ye.
|
|
# 93/65 is an upper bound for log(10)/log(5), so if
|
|
#
|
|
# ye >= len(str(93*p//65))
|
|
#
|
|
# then
|
|
#
|
|
# -e*y >= -y >= 10**ye > 93*p/65 > p*log(10)/log(5),
|
|
#
|
|
# so 5**(-e*y) >= 10**p, and the coefficient of the result
|
|
# can't be expressed in p digits.
|
|
|
|
# emax >= largest e such that 5**e < 10**p.
|
|
emax = p*93//65
|
|
if ye >= len(str(emax)):
|
|
return None
|
|
|
|
# Find -e*y and -xe*y; both must be integers
|
|
e = _decimal_lshift_exact(e * yc, ye)
|
|
xe = _decimal_lshift_exact(xe * yc, ye)
|
|
if e is None or xe is None:
|
|
return None
|
|
|
|
if e > emax:
|
|
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
|
|
|
|
# Guard against large values of ye, using the same logic as in
|
|
# the 'xc is a power of 2' branch. 10/3 is an upper bound for
|
|
# log(10)/log(2).
|
|
emax = p*10//3
|
|
if ye >= len(str(emax)):
|
|
return None
|
|
|
|
e = _decimal_lshift_exact(e * yc, ye)
|
|
xe = _decimal_lshift_exact(xe * yc, ye)
|
|
if e is None or xe is None:
|
|
return None
|
|
|
|
if e > emax:
|
|
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 _One
|
|
|
|
# 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 _SignedInfinity[result_sign]
|
|
|
|
# Inf**(+ve or Inf) = Inf; Inf**(-ve or -Inf) = 0
|
|
if self._isinfinity():
|
|
if other._sign == 0:
|
|
return _SignedInfinity[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 == _One:
|
|
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 _SignedInfinity[result_sign]
|
|
|
|
# from here on, the result always goes through the call
|
|
# to _fix at the end of this function.
|
|
ans = None
|
|
exact = False
|
|
|
|
# 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:
|
|
if result_sign == 1:
|
|
ans = _dec_from_triple(1, ans._int, ans._exp)
|
|
exact = True
|
|
|
|
# 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)
|
|
|
|
# unlike exp, ln and log10, the power function respects the
|
|
# rounding mode; no need to switch to ROUND_HALF_EVEN here
|
|
|
|
# There's a difficulty here when 'other' is not an integer and
|
|
# the result is exact. In this case, the specification
|
|
# requires that the Inexact flag be raised (in spite of
|
|
# exactness), but since the result is exact _fix won't do this
|
|
# for us. (Correspondingly, the Underflow signal should also
|
|
# be raised for subnormal results.) We can't directly raise
|
|
# these signals either before or after calling _fix, since
|
|
# that would violate the precedence for signals. So we wrap
|
|
# the ._fix call in a temporary context, and reraise
|
|
# afterwards.
|
|
if exact and not other._isinteger():
|
|
# pad with zeros up to length context.prec+1 if necessary; this
|
|
# ensures that the Rounded signal will be raised.
|
|
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)
|
|
|
|
# create a copy of the current context, with cleared flags/traps
|
|
newcontext = context.copy()
|
|
newcontext.clear_flags()
|
|
for exception in _signals:
|
|
newcontext.traps[exception] = 0
|
|
|
|
# round in the new context
|
|
ans = ans._fix(newcontext)
|
|
|
|
# raise Inexact, and if necessary, Underflow
|
|
newcontext._raise_error(Inexact)
|
|
if newcontext.flags[Subnormal]:
|
|
newcontext._raise_error(Underflow)
|
|
|
|
# propagate signals to the original context; _fix could
|
|
# have raised any of Overflow, Underflow, Subnormal,
|
|
# Inexact, Rounded, Clamped. Overflow needs the correct
|
|
# arguments. Note that the order of the exceptions is
|
|
# important here.
|
|
if newcontext.flags[Overflow]:
|
|
context._raise_error(Overflow, 'above Emax', ans._sign)
|
|
for exception in Underflow, Subnormal, Inexact, Rounded, Clamped:
|
|
if newcontext.flags[exception]:
|
|
context._raise_error(exception)
|
|
|
|
else:
|
|
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 and ans.adjusted() < context.Emin:
|
|
context._raise_error(Subnormal)
|
|
if ans._exp > self._exp:
|
|
if ans != self:
|
|
context._raise_error(Inexact)
|
|
context._raise_error(Rounded)
|
|
|
|
# call to fix takes care of any necessary folddown, and
|
|
# signals Clamped if necessary
|
|
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 = self._pick_rounding_function[rounding]
|
|
changed = this_function(self, digits)
|
|
coeff = self._int[:digits] or '0'
|
|
if changed == 1:
|
|
coeff = str(int(coeff)+1)
|
|
return _dec_from_triple(self._sign, coeff, exp)
|
|
|
|
def _round(self, places, rounding):
|
|
"""Round a nonzero, nonspecial Decimal to a fixed number of
|
|
significant figures, using the given rounding mode.
|
|
|
|
Infinities, NaNs and zeros are returned unaltered.
|
|
|
|
This operation is quiet: it raises no flags, and uses no
|
|
information from the context.
|
|
|
|
"""
|
|
if places <= 0:
|
|
raise ValueError("argument should be at least 1 in _round")
|
|
if self._is_special or not self:
|
|
return Decimal(self)
|
|
ans = self._rescale(self.adjusted()+1-places, rounding)
|
|
# it can happen that the rescale alters the adjusted exponent;
|
|
# for example when rounding 99.97 to 3 significant figures.
|
|
# When this happens we end up with an extra 0 at the end of
|
|
# the number; a second rescale fixes this.
|
|
if ans.adjusted() != self.adjusted():
|
|
ans = ans._rescale(ans.adjusted()+1-places, rounding)
|
|
return ans
|
|
|
|
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
|
|
ans = self._rescale(0, rounding)
|
|
if ans != self:
|
|
context._raise_error(Inexact)
|
|
context._raise_error(Rounded)
|
|
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 context is None:
|
|
context = getcontext()
|
|
|
|
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 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 == 0:
|
|
return self._fix(context)
|
|
if sn == 1 and on == 0:
|
|
return other._fix(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 == 0:
|
|
return self._fix(context)
|
|
if sn == 1 and on == 0:
|
|
return other._fix(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.
|
|
"""
|
|
other = _convert_other(other, raiseit = True)
|
|
ans = self._compare_check_nans(other, context)
|
|
if ans:
|
|
return ans
|
|
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.
|
|
"""
|
|
other = _convert_other(other, raiseit=True)
|
|
|
|
# if one is negative and the other is positive, it's easy
|
|
if self._sign and not other._sign:
|
|
return _NegativeOne
|
|
if not self._sign and other._sign:
|
|
return _One
|
|
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:
|
|
# compare payloads as though they're integers
|
|
self_key = len(self._int), self._int
|
|
other_key = len(other._int), other._int
|
|
if self_key < other_key:
|
|
if sign:
|
|
return _One
|
|
else:
|
|
return _NegativeOne
|
|
if self_key > other_key:
|
|
if sign:
|
|
return _NegativeOne
|
|
else:
|
|
return _One
|
|
return _Zero
|
|
|
|
if sign:
|
|
if self_nan == 1:
|
|
return _NegativeOne
|
|
if other_nan == 1:
|
|
return _One
|
|
if self_nan == 2:
|
|
return _NegativeOne
|
|
if other_nan == 2:
|
|
return _One
|
|
else:
|
|
if self_nan == 1:
|
|
return _One
|
|
if other_nan == 1:
|
|
return _NegativeOne
|
|
if self_nan == 2:
|
|
return _One
|
|
if other_nan == 2:
|
|
return _NegativeOne
|
|
|
|
if self < other:
|
|
return _NegativeOne
|
|
if self > other:
|
|
return _One
|
|
|
|
if self._exp < other._exp:
|
|
if sign:
|
|
return _One
|
|
else:
|
|
return _NegativeOne
|
|
if self._exp > other._exp:
|
|
if sign:
|
|
return _NegativeOne
|
|
else:
|
|
return _One
|
|
return _Zero
|
|
|
|
|
|
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.
|
|
"""
|
|
other = _convert_other(other, raiseit=True)
|
|
|
|
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."""
|
|
other = _convert_other(other, raiseit=True)
|
|
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 _Zero
|
|
|
|
# exp(0) = 1
|
|
if not self:
|
|
return _One
|
|
|
|
# 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()
|
|
|
|
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 _NegativeInfinity
|
|
|
|
# ln(Infinity) = Infinity
|
|
if self._isinfinity() == 1:
|
|
return _Infinity
|
|
|
|
# ln(1.0) == 0.0
|
|
if self == _One:
|
|
return _Zero
|
|
|
|
# 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 _NegativeInfinity
|
|
|
|
# log10(Infinity) = Infinity
|
|
if self._isinfinity() == 1:
|
|
return _Infinity
|
|
|
|
# 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 _Infinity
|
|
|
|
# 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.
|
|
ans = Decimal(self.adjusted())
|
|
return ans._fix(context)
|
|
|
|
def _islogical(self):
|
|
"""Return True if self is a logical operand.
|
|
|
|
For being logical, it must be a finite number 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()
|
|
|
|
other = _convert_other(other, raiseit=True)
|
|
|
|
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()
|
|
|
|
other = _convert_other(other, raiseit=True)
|
|
|
|
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()
|
|
|
|
other = _convert_other(other, raiseit=True)
|
|
|
|
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 == 0:
|
|
return self._fix(context)
|
|
if sn == 1 and on == 0:
|
|
return other._fix(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 == 0:
|
|
return self._fix(context)
|
|
if sn == 1 and on == 0:
|
|
return other._fix(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 _NegativeInfinity
|
|
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 _Infinity
|
|
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(Inexact)
|
|
context._raise_error(Rounded)
|
|
elif ans.adjusted() < context.Emin:
|
|
context._raise_error(Underflow)
|
|
context._raise_error(Subnormal)
|
|
context._raise_error(Inexact)
|
|
context._raise_error(Rounded)
|
|
# 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()
|
|
|
|
other = _convert_other(other, raiseit=True)
|
|
|
|
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 > 0:
|
|
rotdig = '0'*topad + rotdig
|
|
elif topad < 0:
|
|
rotdig = rotdig[-topad:]
|
|
|
|
# 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()
|
|
|
|
other = _convert_other(other, raiseit=True)
|
|
|
|
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()
|
|
|
|
other = _convert_other(other, raiseit=True)
|
|
|
|
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 > 0:
|
|
rotdig = '0'*topad + rotdig
|
|
elif topad < 0:
|
|
rotdig = rotdig[-topad:]
|
|
|
|
# let's shift!
|
|
if torot < 0:
|
|
shifted = rotdig[:torot]
|
|
else:
|
|
shifted = rotdig + '0'*torot
|
|
shifted = shifted[-context.prec:]
|
|
|
|
return _dec_from_triple(self._sign,
|
|
shifted.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) is Decimal:
|
|
return self # I'm immutable; therefore I am my own clone
|
|
return self.__class__(str(self))
|
|
|
|
def __deepcopy__(self, memo):
|
|
if type(self) is Decimal:
|
|
return self # My components are also immutable
|
|
return self.__class__(str(self))
|
|
|
|
# PEP 3101 support. the _localeconv keyword argument should be
|
|
# considered private: it's provided for ease of testing only.
|
|
def __format__(self, specifier, context=None, _localeconv=None):
|
|
"""Format a Decimal instance according to the given specifier.
|
|
|
|
The specifier should be a standard format specifier, with the
|
|
form described in PEP 3101. Formatting types 'e', 'E', 'f',
|
|
'F', 'g', 'G', 'n' and '%' are supported. If the formatting
|
|
type is omitted it defaults to 'g' or 'G', depending on the
|
|
value of context.capitals.
|
|
"""
|
|
|
|
# Note: PEP 3101 says that if the type is not present then
|
|
# there should be at least one digit after the decimal point.
|
|
# We take the liberty of ignoring this requirement for
|
|
# Decimal---it's presumably there to make sure that
|
|
# format(float, '') behaves similarly to str(float).
|
|
if context is None:
|
|
context = getcontext()
|
|
|
|
spec = _parse_format_specifier(specifier, _localeconv=_localeconv)
|
|
|
|
# special values don't care about the type or precision
|
|
if self._is_special:
|
|
sign = _format_sign(self._sign, spec)
|
|
body = str(self.copy_abs())
|
|
if spec['type'] == '%':
|
|
body += '%'
|
|
return _format_align(sign, body, spec)
|
|
|
|
# a type of None defaults to 'g' or 'G', depending on context
|
|
if spec['type'] is None:
|
|
spec['type'] = ['g', 'G'][context.capitals]
|
|
|
|
# if type is '%', adjust exponent of self accordingly
|
|
if spec['type'] == '%':
|
|
self = _dec_from_triple(self._sign, self._int, self._exp+2)
|
|
|
|
# round if necessary, taking rounding mode from the context
|
|
rounding = context.rounding
|
|
precision = spec['precision']
|
|
if precision is not None:
|
|
if spec['type'] in 'eE':
|
|
self = self._round(precision+1, rounding)
|
|
elif spec['type'] in 'fF%':
|
|
self = self._rescale(-precision, rounding)
|
|
elif spec['type'] in 'gG' and len(self._int) > precision:
|
|
self = self._round(precision, rounding)
|
|
# special case: zeros with a positive exponent can't be
|
|
# represented in fixed point; rescale them to 0e0.
|
|
if not self and self._exp > 0 and spec['type'] in 'fF%':
|
|
self = self._rescale(0, rounding)
|
|
|
|
# figure out placement of the decimal point
|
|
leftdigits = self._exp + len(self._int)
|
|
if spec['type'] in 'eE':
|
|
if not self and precision is not None:
|
|
dotplace = 1 - precision
|
|
else:
|
|
dotplace = 1
|
|
elif spec['type'] in 'fF%':
|
|
dotplace = leftdigits
|
|
elif spec['type'] in 'gG':
|
|
if self._exp <= 0 and leftdigits > -6:
|
|
dotplace = leftdigits
|
|
else:
|
|
dotplace = 1
|
|
|
|
# find digits before and after decimal point, and get exponent
|
|
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] or '0'
|
|
fracpart = self._int[dotplace:]
|
|
exp = leftdigits-dotplace
|
|
|
|
# done with the decimal-specific stuff; hand over the rest
|
|
# of the formatting to the _format_number function
|
|
return _format_number(self._sign, intpart, fracpart, exp, spec)
|
|
|
|
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
|
|
|
|
# Register Decimal as a kind of Number (an abstract base class).
|
|
# However, do not register it as Real (because Decimals are not
|
|
# interoperable with floats).
|
|
_numbers.Number.register(Decimal)
|
|
|
|
|
|
##### Context class #######################################################
|
|
|
|
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 set.
|
|
(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):
|
|
# Set defaults; for everything except flags and _ignored_flags,
|
|
# inherit from DefaultContext.
|
|
try:
|
|
dc = DefaultContext
|
|
except NameError:
|
|
pass
|
|
|
|
self.prec = prec if prec is not None else dc.prec
|
|
self.rounding = rounding if rounding is not None else dc.rounding
|
|
self.Emin = Emin if Emin is not None else dc.Emin
|
|
self.Emax = Emax if Emax is not None else dc.Emax
|
|
self.capitals = capitals if capitals is not None else dc.capitals
|
|
self._clamp = _clamp if _clamp is not None else dc._clamp
|
|
|
|
if _ignored_flags is None:
|
|
self._ignored_flags = []
|
|
else:
|
|
self._ignored_flags = _ignored_flags
|
|
|
|
if traps is None:
|
|
self.traps = dc.traps.copy()
|
|
elif not isinstance(traps, dict):
|
|
self.traps = dict((s, int(s in traps)) for s in _signals)
|
|
else:
|
|
self.traps = traps
|
|
|
|
if flags is None:
|
|
self.flags = dict.fromkeys(_signals, 0)
|
|
elif not isinstance(flags, dict):
|
|
self.flags = dict((s, int(s in flags)) for s in _signals)
|
|
else:
|
|
self.flags = flags
|
|
|
|
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 sets the flag, then, if the corresponding
|
|
trap_enabler is set, it reraises the exception. Otherwise, it returns
|
|
the default value after setting 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)
|
|
|
|
# We inherit object.__hash__, so we must deny this explicitly
|
|
__hash__ = None
|
|
|
|
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.
|
|
|
|
This method implements the to-number operation of the
|
|
IBM Decimal specification."""
|
|
|
|
if isinstance(num, basestring) and num != num.strip():
|
|
return self._raise_error(ConversionSyntax,
|
|
"no trailing or leading whitespace is "
|
|
"permitted.")
|
|
|
|
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)
|
|
|
|
def create_decimal_from_float(self, f):
|
|
"""Creates a new Decimal instance from a float but rounding using self
|
|
as the context.
|
|
|
|
>>> context = Context(prec=5, rounding=ROUND_DOWN)
|
|
>>> context.create_decimal_from_float(3.1415926535897932)
|
|
Decimal('3.1415')
|
|
>>> context = Context(prec=5, traps=[Inexact])
|
|
>>> context.create_decimal_from_float(3.1415926535897932)
|
|
Traceback (most recent call last):
|
|
...
|
|
Inexact: None
|
|
|
|
"""
|
|
d = Decimal.from_float(f) # An exact conversion
|
|
return d._fix(self) # Apply the context rounding
|
|
|
|
# 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')
|
|
>>> ExtendedContext.abs(-1)
|
|
Decimal('1')
|
|
"""
|
|
a = _convert_other(a, raiseit=True)
|
|
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')
|
|
>>> ExtendedContext.add(1, Decimal(2))
|
|
Decimal('3')
|
|
>>> ExtendedContext.add(Decimal(8), 5)
|
|
Decimal('13')
|
|
>>> ExtendedContext.add(5, 5)
|
|
Decimal('10')
|
|
"""
|
|
a = _convert_other(a, raiseit=True)
|
|
r = a.__add__(b, context=self)
|
|
if r is NotImplemented:
|
|
raise TypeError("Unable to convert %s to Decimal" % b)
|
|
else:
|
|
return r
|
|
|
|
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')
|
|
>>> ExtendedContext.compare(1, 2)
|
|
Decimal('-1')
|
|
>>> ExtendedContext.compare(Decimal(1), 2)
|
|
Decimal('-1')
|
|
>>> ExtendedContext.compare(1, Decimal(2))
|
|
Decimal('-1')
|
|
"""
|
|
a = _convert_other(a, raiseit=True)
|
|
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
|
|
>>> c.compare_signal(-1, 2)
|
|
Decimal('-1')
|
|
>>> c.compare_signal(Decimal(-1), 2)
|
|
Decimal('-1')
|
|
>>> c.compare_signal(-1, Decimal(2))
|
|
Decimal('-1')
|
|
"""
|
|
a = _convert_other(a, raiseit=True)
|
|
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')
|
|
>>> ExtendedContext.compare_total(1, 2)
|
|
Decimal('-1')
|
|
>>> ExtendedContext.compare_total(Decimal(1), 2)
|
|
Decimal('-1')
|
|
>>> ExtendedContext.compare_total(1, Decimal(2))
|
|
Decimal('-1')
|
|
"""
|
|
a = _convert_other(a, raiseit=True)
|
|
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.
|
|
"""
|
|
a = _convert_other(a, raiseit=True)
|
|
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')
|
|
>>> ExtendedContext.copy_abs(-1)
|
|
Decimal('1')
|
|
"""
|
|
a = _convert_other(a, raiseit=True)
|
|
return a.copy_abs()
|
|
|
|
def copy_decimal(self, a):
|
|
"""Returns a copy of the decimal object.
|
|
|
|
>>> ExtendedContext.copy_decimal(Decimal('2.1'))
|
|
Decimal('2.1')
|
|
>>> ExtendedContext.copy_decimal(Decimal('-1.00'))
|
|
Decimal('-1.00')
|
|
>>> ExtendedContext.copy_decimal(1)
|
|
Decimal('1')
|
|
"""
|
|
a = _convert_other(a, raiseit=True)
|
|
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')
|
|
>>> ExtendedContext.copy_negate(1)
|
|
Decimal('-1')
|
|
"""
|
|
a = _convert_other(a, raiseit=True)
|
|
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')
|
|
>>> ExtendedContext.copy_sign(1, -2)
|
|
Decimal('-1')
|
|
>>> ExtendedContext.copy_sign(Decimal(1), -2)
|
|
Decimal('-1')
|
|
>>> ExtendedContext.copy_sign(1, Decimal(-2))
|
|
Decimal('-1')
|
|
"""
|
|
a = _convert_other(a, raiseit=True)
|
|
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')
|
|
>>> ExtendedContext.divide(5, 5)
|
|
Decimal('1')
|
|
>>> ExtendedContext.divide(Decimal(5), 5)
|
|
Decimal('1')
|
|
>>> ExtendedContext.divide(5, Decimal(5))
|
|
Decimal('1')
|
|
"""
|
|
a = _convert_other(a, raiseit=True)
|
|
r = a.__div__(b, context=self)
|
|
if r is NotImplemented:
|
|
raise TypeError("Unable to convert %s to Decimal" % b)
|
|
else:
|
|
return r
|
|
|
|
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')
|
|
>>> ExtendedContext.divide_int(10, 3)
|
|
Decimal('3')
|
|
>>> ExtendedContext.divide_int(Decimal(10), 3)
|
|
Decimal('3')
|
|
>>> ExtendedContext.divide_int(10, Decimal(3))
|
|
Decimal('3')
|
|
"""
|
|
a = _convert_other(a, raiseit=True)
|
|
r = a.__floordiv__(b, context=self)
|
|
if r is NotImplemented:
|
|
raise TypeError("Unable to convert %s to Decimal" % b)
|
|
else:
|
|
return r
|
|
|
|
def divmod(self, a, b):
|
|
"""Return (a // b, a % b).
|
|
|
|
>>> ExtendedContext.divmod(Decimal(8), Decimal(3))
|
|
(Decimal('2'), Decimal('2'))
|
|
>>> ExtendedContext.divmod(Decimal(8), Decimal(4))
|
|
(Decimal('2'), Decimal('0'))
|
|
>>> ExtendedContext.divmod(8, 4)
|
|
(Decimal('2'), Decimal('0'))
|
|
>>> ExtendedContext.divmod(Decimal(8), 4)
|
|
(Decimal('2'), Decimal('0'))
|
|
>>> ExtendedContext.divmod(8, Decimal(4))
|
|
(Decimal('2'), Decimal('0'))
|
|
"""
|
|
a = _convert_other(a, raiseit=True)
|
|
r = a.__divmod__(b, context=self)
|
|
if r is NotImplemented:
|
|
raise TypeError("Unable to convert %s to Decimal" % b)
|
|
else:
|
|
return r
|
|
|
|
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')
|
|
>>> c.exp(10)
|
|
Decimal('22026.4658')
|
|
"""
|
|
a =_convert_other(a, raiseit=True)
|
|
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')
|
|
>>> ExtendedContext.fma(1, 3, 4)
|
|
Decimal('7')
|
|
>>> ExtendedContext.fma(1, Decimal(3), 4)
|
|
Decimal('7')
|
|
>>> ExtendedContext.fma(1, 3, Decimal(4))
|
|
Decimal('7')
|
|
"""
|
|
a = _convert_other(a, raiseit=True)
|
|
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
|
|
>>> ExtendedContext.is_finite(1)
|
|
True
|
|
"""
|
|
a = _convert_other(a, raiseit=True)
|
|
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
|
|
>>> ExtendedContext.is_infinite(1)
|
|
False
|
|
"""
|
|
a = _convert_other(a, raiseit=True)
|
|
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
|
|
>>> ExtendedContext.is_nan(1)
|
|
False
|
|
"""
|
|
a = _convert_other(a, raiseit=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
|
|
>>> c.is_normal(1)
|
|
True
|
|
"""
|
|
a = _convert_other(a, raiseit=True)
|
|
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
|
|
>>> ExtendedContext.is_qnan(1)
|
|
False
|
|
"""
|
|
a = _convert_other(a, raiseit=True)
|
|
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
|
|
>>> ExtendedContext.is_signed(8)
|
|
False
|
|
>>> ExtendedContext.is_signed(-8)
|
|
True
|
|
"""
|
|
a = _convert_other(a, raiseit=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
|
|
>>> ExtendedContext.is_snan(1)
|
|
False
|
|
"""
|
|
a = _convert_other(a, raiseit=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
|
|
>>> c.is_subnormal(1)
|
|
False
|
|
"""
|
|
a = _convert_other(a, raiseit=True)
|
|
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
|
|
>>> ExtendedContext.is_zero(1)
|
|
False
|
|
>>> ExtendedContext.is_zero(0)
|
|
True
|
|
"""
|
|
a = _convert_other(a, raiseit=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')
|
|
>>> c.ln(1)
|
|
Decimal('0')
|
|
"""
|
|
a = _convert_other(a, raiseit=True)
|
|
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')
|
|
>>> c.log10(0)
|
|
Decimal('-Infinity')
|
|
>>> c.log10(1)
|
|
Decimal('0')
|
|
"""
|
|
a = _convert_other(a, raiseit=True)
|
|
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')
|
|
>>> ExtendedContext.logb(1)
|
|
Decimal('0')
|
|
>>> ExtendedContext.logb(10)
|
|
Decimal('1')
|
|
>>> ExtendedContext.logb(100)
|
|
Decimal('2')
|
|
"""
|
|
a = _convert_other(a, raiseit=True)
|
|
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')
|
|
>>> ExtendedContext.logical_and(110, 1101)
|
|
Decimal('100')
|
|
>>> ExtendedContext.logical_and(Decimal(110), 1101)
|
|
Decimal('100')
|
|
>>> ExtendedContext.logical_and(110, Decimal(1101))
|
|
Decimal('100')
|
|
"""
|
|
a = _convert_other(a, raiseit=True)
|
|
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')
|
|
>>> ExtendedContext.logical_invert(1101)
|
|
Decimal('111110010')
|
|
"""
|
|
a = _convert_other(a, raiseit=True)
|
|
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')
|
|
>>> ExtendedContext.logical_or(110, 1101)
|
|
Decimal('1111')
|
|
>>> ExtendedContext.logical_or(Decimal(110), 1101)
|
|
Decimal('1111')
|
|
>>> ExtendedContext.logical_or(110, Decimal(1101))
|
|
Decimal('1111')
|
|
"""
|
|
a = _convert_other(a, raiseit=True)
|
|
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')
|
|
>>> ExtendedContext.logical_xor(110, 1101)
|
|
Decimal('1011')
|
|
>>> ExtendedContext.logical_xor(Decimal(110), 1101)
|
|
Decimal('1011')
|
|
>>> ExtendedContext.logical_xor(110, Decimal(1101))
|
|
Decimal('1011')
|
|
"""
|
|
a = _convert_other(a, raiseit=True)
|
|
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 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')
|
|
>>> ExtendedContext.max(1, 2)
|
|
Decimal('2')
|
|
>>> ExtendedContext.max(Decimal(1), 2)
|
|
Decimal('2')
|
|
>>> ExtendedContext.max(1, Decimal(2))
|
|
Decimal('2')
|
|
"""
|
|
a = _convert_other(a, raiseit=True)
|
|
return a.max(b, context=self)
|
|
|
|
def max_mag(self, a, b):
|
|
"""Compares the values numerically with their sign ignored.
|
|
|
|
>>> ExtendedContext.max_mag(Decimal('7'), Decimal('NaN'))
|
|
Decimal('7')
|
|
>>> ExtendedContext.max_mag(Decimal('7'), Decimal('-10'))
|
|
Decimal('-10')
|
|
>>> ExtendedContext.max_mag(1, -2)
|
|
Decimal('-2')
|
|
>>> ExtendedContext.max_mag(Decimal(1), -2)
|
|
Decimal('-2')
|
|
>>> ExtendedContext.max_mag(1, Decimal(-2))
|
|
Decimal('-2')
|
|
"""
|
|
a = _convert_other(a, raiseit=True)
|
|
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 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')
|
|
>>> ExtendedContext.min(1, 2)
|
|
Decimal('1')
|
|
>>> ExtendedContext.min(Decimal(1), 2)
|
|
Decimal('1')
|
|
>>> ExtendedContext.min(1, Decimal(29))
|
|
Decimal('1')
|
|
"""
|
|
a = _convert_other(a, raiseit=True)
|
|
return a.min(b, context=self)
|
|
|
|
def min_mag(self, a, b):
|
|
"""Compares the values numerically with their sign ignored.
|
|
|
|
>>> ExtendedContext.min_mag(Decimal('3'), Decimal('-2'))
|
|
Decimal('-2')
|
|
>>> ExtendedContext.min_mag(Decimal('-3'), Decimal('NaN'))
|
|
Decimal('-3')
|
|
>>> ExtendedContext.min_mag(1, -2)
|
|
Decimal('1')
|
|
>>> ExtendedContext.min_mag(Decimal(1), -2)
|
|
Decimal('1')
|
|
>>> ExtendedContext.min_mag(1, Decimal(-2))
|
|
Decimal('1')
|
|
"""
|
|
a = _convert_other(a, raiseit=True)
|
|
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')
|
|
>>> ExtendedContext.minus(1)
|
|
Decimal('-1')
|
|
"""
|
|
a = _convert_other(a, raiseit=True)
|
|
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')
|
|
>>> ExtendedContext.multiply(7, 7)
|
|
Decimal('49')
|
|
>>> ExtendedContext.multiply(Decimal(7), 7)
|
|
Decimal('49')
|
|
>>> ExtendedContext.multiply(7, Decimal(7))
|
|
Decimal('49')
|
|
"""
|
|
a = _convert_other(a, raiseit=True)
|
|
r = a.__mul__(b, context=self)
|
|
if r is NotImplemented:
|
|
raise TypeError("Unable to convert %s to Decimal" % b)
|
|
else:
|
|
return r
|
|
|
|
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')
|
|
>>> c.next_minus(1)
|
|
Decimal('0.999999999')
|
|
"""
|
|
a = _convert_other(a, raiseit=True)
|
|
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')
|
|
>>> c.next_plus(1)
|
|
Decimal('1.00000001')
|
|
"""
|
|
a = _convert_other(a, raiseit=True)
|
|
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')
|
|
>>> c.next_toward(0, 1)
|
|
Decimal('1E-1007')
|
|
>>> c.next_toward(Decimal(0), 1)
|
|
Decimal('1E-1007')
|
|
>>> c.next_toward(0, Decimal(1))
|
|
Decimal('1E-1007')
|
|
"""
|
|
a = _convert_other(a, raiseit=True)
|
|
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')
|
|
>>> ExtendedContext.normalize(6)
|
|
Decimal('6')
|
|
"""
|
|
a = _convert_other(a, raiseit=True)
|
|
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'
|
|
>>> c.number_class(123)
|
|
'+Normal'
|
|
"""
|
|
a = _convert_other(a, raiseit=True)
|
|
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')
|
|
>>> ExtendedContext.plus(-1)
|
|
Decimal('-1')
|
|
"""
|
|
a = _convert_other(a, raiseit=True)
|
|
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')
|
|
>>> ExtendedContext.power(7, 7)
|
|
Decimal('823543')
|
|
>>> ExtendedContext.power(Decimal(7), 7)
|
|
Decimal('823543')
|
|
>>> ExtendedContext.power(7, Decimal(7), 2)
|
|
Decimal('1')
|
|
"""
|
|
a = _convert_other(a, raiseit=True)
|
|
r = a.__pow__(b, modulo, context=self)
|
|
if r is NotImplemented:
|
|
raise TypeError("Unable to convert %s to Decimal" % b)
|
|
else:
|
|
return r
|
|
|
|
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')
|
|
>>> ExtendedContext.quantize(1, 2)
|
|
Decimal('1')
|
|
>>> ExtendedContext.quantize(Decimal(1), 2)
|
|
Decimal('1')
|
|
>>> ExtendedContext.quantize(1, Decimal(2))
|
|
Decimal('1')
|
|
"""
|
|
a = _convert_other(a, raiseit=True)
|
|
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')
|
|
>>> ExtendedContext.remainder(22, 6)
|
|
Decimal('4')
|
|
>>> ExtendedContext.remainder(Decimal(22), 6)
|
|
Decimal('4')
|
|
>>> ExtendedContext.remainder(22, Decimal(6))
|
|
Decimal('4')
|
|
"""
|
|
a = _convert_other(a, raiseit=True)
|
|
r = a.__mod__(b, context=self)
|
|
if r is NotImplemented:
|
|
raise TypeError("Unable to convert %s to Decimal" % b)
|
|
else:
|
|
return r
|
|
|
|
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')
|
|
>>> ExtendedContext.remainder_near(3, 11)
|
|
Decimal('3')
|
|
>>> ExtendedContext.remainder_near(Decimal(3), 11)
|
|
Decimal('3')
|
|
>>> ExtendedContext.remainder_near(3, Decimal(11))
|
|
Decimal('3')
|
|
"""
|
|
a = _convert_other(a, raiseit=True)
|
|
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')
|
|
>>> ExtendedContext.rotate(1333333, 1)
|
|
Decimal('13333330')
|
|
>>> ExtendedContext.rotate(Decimal(1333333), 1)
|
|
Decimal('13333330')
|
|
>>> ExtendedContext.rotate(1333333, Decimal(1))
|
|
Decimal('13333330')
|
|
"""
|
|
a = _convert_other(a, raiseit=True)
|
|
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
|
|
>>> ExtendedContext.same_quantum(10000, -1)
|
|
True
|
|
>>> ExtendedContext.same_quantum(Decimal(10000), -1)
|
|
True
|
|
>>> ExtendedContext.same_quantum(10000, Decimal(-1))
|
|
True
|
|
"""
|
|
a = _convert_other(a, raiseit=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')
|
|
>>> ExtendedContext.scaleb(1, 4)
|
|
Decimal('1E+4')
|
|
>>> ExtendedContext.scaleb(Decimal(1), 4)
|
|
Decimal('1E+4')
|
|
>>> ExtendedContext.scaleb(1, Decimal(4))
|
|
Decimal('1E+4')
|
|
"""
|
|
a = _convert_other(a, raiseit=True)
|
|
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')
|
|
>>> ExtendedContext.shift(88888888, 2)
|
|
Decimal('888888800')
|
|
>>> ExtendedContext.shift(Decimal(88888888), 2)
|
|
Decimal('888888800')
|
|
>>> ExtendedContext.shift(88888888, Decimal(2))
|
|
Decimal('888888800')
|
|
"""
|
|
a = _convert_other(a, raiseit=True)
|
|
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.sqrt(2)
|
|
Decimal('1.41421356')
|
|
>>> ExtendedContext.prec
|
|
9
|
|
"""
|
|
a = _convert_other(a, raiseit=True)
|
|
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')
|
|
>>> ExtendedContext.subtract(8, 5)
|
|
Decimal('3')
|
|
>>> ExtendedContext.subtract(Decimal(8), 5)
|
|
Decimal('3')
|
|
>>> ExtendedContext.subtract(8, Decimal(5))
|
|
Decimal('3')
|
|
"""
|
|
a = _convert_other(a, raiseit=True)
|
|
r = a.__sub__(b, context=self)
|
|
if r is NotImplemented:
|
|
raise TypeError("Unable to convert %s to Decimal" % b)
|
|
else:
|
|
return r
|
|
|
|
def to_eng_string(self, a):
|
|
"""Converts a number to a string, using scientific notation.
|
|
|
|
The operation is not affected by the context.
|
|
"""
|
|
a = _convert_other(a, raiseit=True)
|
|
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.
|
|
"""
|
|
a = _convert_other(a, raiseit=True)
|
|
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')
|
|
"""
|
|
a = _convert_other(a, raiseit=True)
|
|
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')
|
|
"""
|
|
a = _convert_other(a, raiseit=True)
|
|
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 _decimal_lshift_exact(n, e):
|
|
""" Given integers n and e, return n * 10**e if it's an integer, else None.
|
|
|
|
The computation is designed to avoid computing large powers of 10
|
|
unnecessarily.
|
|
|
|
>>> _decimal_lshift_exact(3, 4)
|
|
30000
|
|
>>> _decimal_lshift_exact(300, -999999999) # returns None
|
|
|
|
"""
|
|
if n == 0:
|
|
return 0
|
|
elif e >= 0:
|
|
return n * 10**e
|
|
else:
|
|
# val_n = largest power of 10 dividing n.
|
|
str_n = str(abs(n))
|
|
val_n = len(str_n) - len(str_n.rstrip('0'))
|
|
return None if val_n < -e else n // 10**-e
|
|
|
|
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, allow_float=False):
|
|
"""Convert other to Decimal.
|
|
|
|
Verifies that it's ok to use in an implicit construction.
|
|
If allow_float is true, allow conversion from float; this
|
|
is used in the comparison methods (__eq__ and friends).
|
|
|
|
"""
|
|
if isinstance(other, Decimal):
|
|
return other
|
|
if isinstance(other, (int, long)):
|
|
return Decimal(other)
|
|
if allow_float and isinstance(other, float):
|
|
return Decimal.from_float(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 #############################################
|
|
#
|
|
# 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.
|
|
|
|
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*) # having 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 info.
|
|
)
|
|
# \s*
|
|
\Z
|
|
""", re.VERBOSE | re.IGNORECASE | re.UNICODE).match
|
|
|
|
_all_zeros = re.compile('0*$').match
|
|
_exact_half = re.compile('50*$').match
|
|
|
|
##### PEP3101 support functions ##############################################
|
|
# The functions in this section have little to do with the Decimal
|
|
# class, and could potentially be reused or adapted for other pure
|
|
# Python numeric classes that want to implement __format__
|
|
#
|
|
# A format specifier for Decimal looks like:
|
|
#
|
|
# [[fill]align][sign][0][minimumwidth][,][.precision][type]
|
|
|
|
_parse_format_specifier_regex = re.compile(r"""\A
|
|
(?:
|
|
(?P<fill>.)?
|
|
(?P<align>[<>=^])
|
|
)?
|
|
(?P<sign>[-+ ])?
|
|
(?P<zeropad>0)?
|
|
(?P<minimumwidth>(?!0)\d+)?
|
|
(?P<thousands_sep>,)?
|
|
(?:\.(?P<precision>0|(?!0)\d+))?
|
|
(?P<type>[eEfFgGn%])?
|
|
\Z
|
|
""", re.VERBOSE)
|
|
|
|
del re
|
|
|
|
# The locale module is only needed for the 'n' format specifier. The
|
|
# rest of the PEP 3101 code functions quite happily without it, so we
|
|
# don't care too much if locale isn't present.
|
|
try:
|
|
import locale as _locale
|
|
except ImportError:
|
|
pass
|
|
|
|
def _parse_format_specifier(format_spec, _localeconv=None):
|
|
"""Parse and validate a format specifier.
|
|
|
|
Turns a standard numeric format specifier into a dict, with the
|
|
following entries:
|
|
|
|
fill: fill character to pad field to minimum width
|
|
align: alignment type, either '<', '>', '=' or '^'
|
|
sign: either '+', '-' or ' '
|
|
minimumwidth: nonnegative integer giving minimum width
|
|
zeropad: boolean, indicating whether to pad with zeros
|
|
thousands_sep: string to use as thousands separator, or ''
|
|
grouping: grouping for thousands separators, in format
|
|
used by localeconv
|
|
decimal_point: string to use for decimal point
|
|
precision: nonnegative integer giving precision, or None
|
|
type: one of the characters 'eEfFgG%', or None
|
|
unicode: boolean (always True for Python 3.x)
|
|
|
|
"""
|
|
m = _parse_format_specifier_regex.match(format_spec)
|
|
if m is None:
|
|
raise ValueError("Invalid format specifier: " + format_spec)
|
|
|
|
# get the dictionary
|
|
format_dict = m.groupdict()
|
|
|
|
# zeropad; defaults for fill and alignment. If zero padding
|
|
# is requested, the fill and align fields should be absent.
|
|
fill = format_dict['fill']
|
|
align = format_dict['align']
|
|
format_dict['zeropad'] = (format_dict['zeropad'] is not None)
|
|
if format_dict['zeropad']:
|
|
if fill is not None:
|
|
raise ValueError("Fill character conflicts with '0'"
|
|
" in format specifier: " + format_spec)
|
|
if align is not None:
|
|
raise ValueError("Alignment conflicts with '0' in "
|
|
"format specifier: " + format_spec)
|
|
format_dict['fill'] = fill or ' '
|
|
# PEP 3101 originally specified that the default alignment should
|
|
# be left; it was later agreed that right-aligned makes more sense
|
|
# for numeric types. See http://bugs.python.org/issue6857.
|
|
format_dict['align'] = align or '>'
|
|
|
|
# default sign handling: '-' for negative, '' for positive
|
|
if format_dict['sign'] is None:
|
|
format_dict['sign'] = '-'
|
|
|
|
# minimumwidth defaults to 0; precision remains None if not given
|
|
format_dict['minimumwidth'] = int(format_dict['minimumwidth'] or '0')
|
|
if format_dict['precision'] is not None:
|
|
format_dict['precision'] = int(format_dict['precision'])
|
|
|
|
# if format type is 'g' or 'G' then a precision of 0 makes little
|
|
# sense; convert it to 1. Same if format type is unspecified.
|
|
if format_dict['precision'] == 0:
|
|
if format_dict['type'] is None or format_dict['type'] in 'gG':
|
|
format_dict['precision'] = 1
|
|
|
|
# determine thousands separator, grouping, and decimal separator, and
|
|
# add appropriate entries to format_dict
|
|
if format_dict['type'] == 'n':
|
|
# apart from separators, 'n' behaves just like 'g'
|
|
format_dict['type'] = 'g'
|
|
if _localeconv is None:
|
|
_localeconv = _locale.localeconv()
|
|
if format_dict['thousands_sep'] is not None:
|
|
raise ValueError("Explicit thousands separator conflicts with "
|
|
"'n' type in format specifier: " + format_spec)
|
|
format_dict['thousands_sep'] = _localeconv['thousands_sep']
|
|
format_dict['grouping'] = _localeconv['grouping']
|
|
format_dict['decimal_point'] = _localeconv['decimal_point']
|
|
else:
|
|
if format_dict['thousands_sep'] is None:
|
|
format_dict['thousands_sep'] = ''
|
|
format_dict['grouping'] = [3, 0]
|
|
format_dict['decimal_point'] = '.'
|
|
|
|
# record whether return type should be str or unicode
|
|
try:
|
|
format_dict['unicode'] = isinstance(format_spec, unicode)
|
|
except NameError:
|
|
format_dict['unicode'] = False
|
|
|
|
return format_dict
|
|
|
|
def _format_align(sign, body, spec):
|
|
"""Given an unpadded, non-aligned numeric string 'body' and sign
|
|
string 'sign', add padding and alignment conforming to the given
|
|
format specifier dictionary 'spec' (as produced by
|
|
parse_format_specifier).
|
|
|
|
Also converts result to unicode if necessary.
|
|
|
|
"""
|
|
# how much extra space do we have to play with?
|
|
minimumwidth = spec['minimumwidth']
|
|
fill = spec['fill']
|
|
padding = fill*(minimumwidth - len(sign) - len(body))
|
|
|
|
align = spec['align']
|
|
if align == '<':
|
|
result = sign + body + padding
|
|
elif align == '>':
|
|
result = padding + sign + body
|
|
elif align == '=':
|
|
result = sign + padding + body
|
|
elif align == '^':
|
|
half = len(padding)//2
|
|
result = padding[:half] + sign + body + padding[half:]
|
|
else:
|
|
raise ValueError('Unrecognised alignment field')
|
|
|
|
# make sure that result is unicode if necessary
|
|
if spec['unicode']:
|
|
result = unicode(result)
|
|
|
|
return result
|
|
|
|
def _group_lengths(grouping):
|
|
"""Convert a localeconv-style grouping into a (possibly infinite)
|
|
iterable of integers representing group lengths.
|
|
|
|
"""
|
|
# The result from localeconv()['grouping'], and the input to this
|
|
# function, should be a list of integers in one of the
|
|
# following three forms:
|
|
#
|
|
# (1) an empty list, or
|
|
# (2) nonempty list of positive integers + [0]
|
|
# (3) list of positive integers + [locale.CHAR_MAX], or
|
|
|
|
from itertools import chain, repeat
|
|
if not grouping:
|
|
return []
|
|
elif grouping[-1] == 0 and len(grouping) >= 2:
|
|
return chain(grouping[:-1], repeat(grouping[-2]))
|
|
elif grouping[-1] == _locale.CHAR_MAX:
|
|
return grouping[:-1]
|
|
else:
|
|
raise ValueError('unrecognised format for grouping')
|
|
|
|
def _insert_thousands_sep(digits, spec, min_width=1):
|
|
"""Insert thousands separators into a digit string.
|
|
|
|
spec is a dictionary whose keys should include 'thousands_sep' and
|
|
'grouping'; typically it's the result of parsing the format
|
|
specifier using _parse_format_specifier.
|
|
|
|
The min_width keyword argument gives the minimum length of the
|
|
result, which will be padded on the left with zeros if necessary.
|
|
|
|
If necessary, the zero padding adds an extra '0' on the left to
|
|
avoid a leading thousands separator. For example, inserting
|
|
commas every three digits in '123456', with min_width=8, gives
|
|
'0,123,456', even though that has length 9.
|
|
|
|
"""
|
|
|
|
sep = spec['thousands_sep']
|
|
grouping = spec['grouping']
|
|
|
|
groups = []
|
|
for l in _group_lengths(grouping):
|
|
if l <= 0:
|
|
raise ValueError("group length should be positive")
|
|
# max(..., 1) forces at least 1 digit to the left of a separator
|
|
l = min(max(len(digits), min_width, 1), l)
|
|
groups.append('0'*(l - len(digits)) + digits[-l:])
|
|
digits = digits[:-l]
|
|
min_width -= l
|
|
if not digits and min_width <= 0:
|
|
break
|
|
min_width -= len(sep)
|
|
else:
|
|
l = max(len(digits), min_width, 1)
|
|
groups.append('0'*(l - len(digits)) + digits[-l:])
|
|
return sep.join(reversed(groups))
|
|
|
|
def _format_sign(is_negative, spec):
|
|
"""Determine sign character."""
|
|
|
|
if is_negative:
|
|
return '-'
|
|
elif spec['sign'] in ' +':
|
|
return spec['sign']
|
|
else:
|
|
return ''
|
|
|
|
def _format_number(is_negative, intpart, fracpart, exp, spec):
|
|
"""Format a number, given the following data:
|
|
|
|
is_negative: true if the number is negative, else false
|
|
intpart: string of digits that must appear before the decimal point
|
|
fracpart: string of digits that must come after the point
|
|
exp: exponent, as an integer
|
|
spec: dictionary resulting from parsing the format specifier
|
|
|
|
This function uses the information in spec to:
|
|
insert separators (decimal separator and thousands separators)
|
|
format the sign
|
|
format the exponent
|
|
add trailing '%' for the '%' type
|
|
zero-pad if necessary
|
|
fill and align if necessary
|
|
"""
|
|
|
|
sign = _format_sign(is_negative, spec)
|
|
|
|
if fracpart:
|
|
fracpart = spec['decimal_point'] + fracpart
|
|
|
|
if exp != 0 or spec['type'] in 'eE':
|
|
echar = {'E': 'E', 'e': 'e', 'G': 'E', 'g': 'e'}[spec['type']]
|
|
fracpart += "{0}{1:+}".format(echar, exp)
|
|
if spec['type'] == '%':
|
|
fracpart += '%'
|
|
|
|
if spec['zeropad']:
|
|
min_width = spec['minimumwidth'] - len(fracpart) - len(sign)
|
|
else:
|
|
min_width = 0
|
|
intpart = _insert_thousands_sep(intpart, spec, min_width)
|
|
|
|
return _format_align(sign, intpart+fracpart, spec)
|
|
|
|
|
|
##### Useful Constants (internal use only) ################################
|
|
|
|
# Reusable defaults
|
|
_Infinity = Decimal('Inf')
|
|
_NegativeInfinity = Decimal('-Inf')
|
|
_NaN = Decimal('NaN')
|
|
_Zero = Decimal(0)
|
|
_One = Decimal(1)
|
|
_NegativeOne = Decimal(-1)
|
|
|
|
# _SignedInfinity[sign] is infinity w/ that sign
|
|
_SignedInfinity = (_Infinity, _NegativeInfinity)
|
|
|
|
|
|
|
|
if __name__ == '__main__':
|
|
import doctest, sys
|
|
doctest.testmod(sys.modules[__name__])
|