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# Copyright (c) 2004 Python Software Foundation.
# All rights reserved.
# Written by Eric Price <eprice at tjhsst.edu>
# and Facundo Batista <facundo at taniquetil.com.ar>
# and Raymond Hettinger <python at rcn.com>
# and Aahz <aahz at pobox.com>
# and Tim Peters
# This module should be kept in sync with the latest updates of the
# IBM specification as it evolves. Those updates will be treated
# as bug fixes (deviation from the spec is a compatibility, usability
# bug) and will be backported. At this point the spec is stabilizing
# and the updates are becoming fewer, smaller, and less significant.
"""
This is an implementation of decimal floating point arithmetic based on
the General Decimal Arithmetic Specification:
http://speleotrove.com/decimal/decarith.html
and IEEE standard 854-1987:
http://en.wikipedia.org/wiki/IEEE_854-1987
Decimal floating point has finite precision with arbitrarily large bounds.
The purpose of this module is to support arithmetic using familiar
"schoolhouse" rules and to avoid some of the tricky representation
issues associated with binary floating point. The package is especially
useful for financial applications or for contexts where users have
expectations that are at odds with binary floating point (for instance,
in binary floating point, 1.00 % 0.1 gives 0.09999999999999995 instead
of 0.0; Decimal('1.00') % Decimal('0.1') returns the expected
Decimal('0.00')).
Here are some examples of using the decimal module:
>>> from decimal import *
>>> setcontext(ExtendedContext)
>>> Decimal(0)
Decimal('0')
>>> Decimal('1')
Decimal('1')
>>> Decimal('-.0123')
Decimal('-0.0123')
>>> Decimal(123456)
Decimal('123456')
>>> Decimal('123.45e12345678')
Decimal('1.2345E+12345680')
>>> Decimal('1.33') + Decimal('1.27')
Decimal('2.60')
>>> Decimal('12.34') + Decimal('3.87') - Decimal('18.41')
Decimal('-2.20')
>>> dig = Decimal(1)
>>> print(dig / Decimal(3))
0.333333333
>>> getcontext().prec = 18
>>> print(dig / Decimal(3))
0.333333333333333333
>>> print(dig.sqrt())
1
>>> print(Decimal(3).sqrt())
1.73205080756887729
>>> print(Decimal(3) ** 123)
4.85192780976896427E+58
>>> inf = Decimal(1) / Decimal(0)
>>> print(inf)
Infinity
>>> neginf = Decimal(-1) / Decimal(0)
>>> print(neginf)
-Infinity
>>> print(neginf + inf)
NaN
>>> print(neginf * inf)
-Infinity
>>> print(dig / 0)
Infinity
>>> getcontext().traps[DivisionByZero] = 1
>>> print(dig / 0)
Traceback (most recent call last):
...
...
...
decimal.DivisionByZero: x / 0
>>> c = Context()
>>> c.traps[InvalidOperation] = 0
>>> print(c.flags[InvalidOperation])
0
>>> c.divide(Decimal(0), Decimal(0))
Decimal('NaN')
>>> c.traps[InvalidOperation] = 1
>>> print(c.flags[InvalidOperation])
1
>>> c.flags[InvalidOperation] = 0
>>> print(c.flags[InvalidOperation])
0
>>> print(c.divide(Decimal(0), Decimal(0)))
Traceback (most recent call last):
...
...
...
decimal.InvalidOperation: 0 / 0
>>> print(c.flags[InvalidOperation])
1
>>> c.flags[InvalidOperation] = 0
>>> c.traps[InvalidOperation] = 0
>>> print(c.divide(Decimal(0), Decimal(0)))
NaN
>>> print(c.flags[InvalidOperation])
1
>>>
"""
__all__ = [
# Two major classes
'Decimal', 'Context',
# Named tuple representation
'DecimalTuple',
# Contexts
'DefaultContext', 'BasicContext', 'ExtendedContext',
# Exceptions
'DecimalException', 'Clamped', 'InvalidOperation', 'DivisionByZero',
'Inexact', 'Rounded', 'Subnormal', 'Overflow', 'Underflow',
'FloatOperation',
# Exceptional conditions that trigger InvalidOperation
'DivisionImpossible', 'InvalidContext', 'ConversionSyntax', 'DivisionUndefined',
# Constants for use in setting up contexts
'ROUND_DOWN', 'ROUND_HALF_UP', 'ROUND_HALF_EVEN', 'ROUND_CEILING',
'ROUND_FLOOR', 'ROUND_UP', 'ROUND_HALF_DOWN', 'ROUND_05UP',
# Functions for manipulating contexts
'setcontext', 'getcontext', 'localcontext',
# Limits for the C version for compatibility
'MAX_PREC', 'MAX_EMAX', 'MIN_EMIN', 'MIN_ETINY',
# C version: compile time choice that enables the thread local context
'HAVE_THREADS'
]
__xname__ = __name__ # sys.modules lookup (--without-threads)
__name__ = 'decimal' # For pickling
__version__ = '1.70' # Highest version of the spec this complies with
# See http://speleotrove.com/decimal/
__libmpdec_version__ = "2.4.2" # compatible libmpdec version
import math as _math
import numbers as _numbers
import sys
try:
from collections import namedtuple as _namedtuple
DecimalTuple = _namedtuple('DecimalTuple', 'sign digits exponent')
except ImportError:
DecimalTuple = lambda *args: args
# Rounding
ROUND_DOWN = 'ROUND_DOWN'
ROUND_HALF_UP = 'ROUND_HALF_UP'
ROUND_HALF_EVEN = 'ROUND_HALF_EVEN'
ROUND_CEILING = 'ROUND_CEILING'
ROUND_FLOOR = 'ROUND_FLOOR'
ROUND_UP = 'ROUND_UP'
ROUND_HALF_DOWN = 'ROUND_HALF_DOWN'
ROUND_05UP = 'ROUND_05UP'
# Compatibility with the C version
HAVE_THREADS = True
if sys.maxsize == 2**63-1:
MAX_PREC = 999999999999999999
MAX_EMAX = 999999999999999999
MIN_EMIN = -999999999999999999
else:
MAX_PREC = 425000000
MAX_EMAX = 425000000
MIN_EMIN = -425000000
MIN_ETINY = MIN_EMIN - (MAX_PREC-1)
# Errors
class DecimalException(ArithmeticError):
"""Base exception class.
Used exceptions derive from this.
If an exception derives from another exception besides this (such as
Underflow (Inexact, Rounded, Subnormal) that indicates that it is only
called if the others are present. This isn't actually used for
anything, though.
handle -- Called when context._raise_error is called and the
trap_enabler is not set. First argument is self, second is the
context. More arguments can be given, those being after
the explanation in _raise_error (For example,
context._raise_error(NewError, '(-x)!', self._sign) would
call NewError().handle(context, self._sign).)
To define a new exception, it should be sufficient to have it derive
from DecimalException.
"""
def handle(self, context, *args):
pass
class Clamped(DecimalException):
"""Exponent of a 0 changed to fit bounds.
This occurs and signals clamped if the exponent of a result has been
altered in order to fit the constraints of a specific concrete
representation. This may occur when the exponent of a zero result would
be outside the bounds of a representation, or when a large normal
number would have an encoded exponent that cannot be represented. In
this latter case, the exponent is reduced to fit and the corresponding
number of zero digits are appended to the coefficient ("fold-down").
"""
class InvalidOperation(DecimalException):
"""An invalid operation was performed.
Various bad things cause this:
Something creates a signaling NaN
-INF + INF
0 * (+-)INF
(+-)INF / (+-)INF
x % 0
(+-)INF % x
x._rescale( non-integer )
sqrt(-x) , x > 0
0 ** 0
x ** (non-integer)
x ** (+-)INF
An operand is invalid
The result of the operation after these is a quiet positive NaN,
except when the cause is a signaling NaN, in which case the result is
also a quiet NaN, but with the original sign, and an optional
diagnostic information.
"""
def handle(self, context, *args):
if args:
ans = _dec_from_triple(args[0]._sign, args[0]._int, 'n', True)
return ans._fix_nan(context)
return _NaN
class ConversionSyntax(InvalidOperation):
"""Trying to convert badly formed string.
This occurs and signals invalid-operation if a string is being
converted to a number and it does not conform to the numeric string
syntax. The result is [0,qNaN].
"""
def handle(self, context, *args):
return _NaN
class DivisionByZero(DecimalException, ZeroDivisionError):
"""Division by 0.
This occurs and signals division-by-zero if division of a finite number
by zero was attempted (during a divide-integer or divide operation, or a
power operation with negative right-hand operand), and the dividend was
not zero.
The result of the operation is [sign,inf], where sign is the exclusive
or of the signs of the operands for divide, or is 1 for an odd power of
-0, for power.
"""
def handle(self, context, sign, *args):
return _SignedInfinity[sign]
class DivisionImpossible(InvalidOperation):
"""Cannot perform the division adequately.
This occurs and signals invalid-operation if the integer result of a
divide-integer or remainder operation had too many digits (would be
longer than precision). The result is [0,qNaN].
"""
def handle(self, context, *args):
return _NaN
class DivisionUndefined(InvalidOperation, ZeroDivisionError):
"""Undefined result of division.
This occurs and signals invalid-operation if division by zero was
attempted (during a divide-integer, divide, or remainder operation), and
the dividend is also zero. The result is [0,qNaN].
"""
def handle(self, context, *args):
return _NaN
class Inexact(DecimalException):
"""Had to round, losing information.
This occurs and signals inexact whenever the result of an operation is
not exact (that is, it needed to be rounded and any discarded digits
were non-zero), or if an overflow or underflow condition occurs. The
result in all cases is unchanged.
The inexact signal may be tested (or trapped) to determine if a given
operation (or sequence of operations) was inexact.
"""
class InvalidContext(InvalidOperation):
"""Invalid context. Unknown rounding, for example.
This occurs and signals invalid-operation if an invalid context was
detected during an operation. This can occur if contexts are not checked
on creation and either the precision exceeds the capability of the
underlying concrete representation or an unknown or unsupported rounding
was specified. These aspects of the context need only be checked when
the values are required to be used. The result is [0,qNaN].
"""
def handle(self, context, *args):
return _NaN
class Rounded(DecimalException):
"""Number got rounded (not necessarily changed during rounding).
This occurs and signals rounded whenever the result of an operation is
rounded (that is, some zero or non-zero digits were discarded from the
coefficient), or if an overflow or underflow condition occurs. The
result in all cases is unchanged.
The rounded signal may be tested (or trapped) to determine if a given
operation (or sequence of operations) caused a loss of precision.
"""
class Subnormal(DecimalException):
"""Exponent < Emin before rounding.
This occurs and signals subnormal whenever the result of a conversion or
operation is subnormal (that is, its adjusted exponent is less than
Emin, before any rounding). The result in all cases is unchanged.
The subnormal signal may be tested (or trapped) to determine if a given
or operation (or sequence of operations) yielded a subnormal result.
"""
class Overflow(Inexact, Rounded):
"""Numerical overflow.
This occurs and signals overflow if the adjusted exponent of a result
(from a conversion or from an operation that is not an attempt to divide
by zero), after rounding, would be greater than the largest value that
can be handled by the implementation (the value Emax).
The result depends on the rounding mode:
For round-half-up and round-half-even (and for round-half-down and
round-up, if implemented), the result of the operation is [sign,inf],
where sign is the sign of the intermediate result. For round-down, the
result is the largest finite number that can be represented in the
current precision, with the sign of the intermediate result. For
round-ceiling, the result is the same as for round-down if the sign of
the intermediate result is 1, or is [0,inf] otherwise. For round-floor,
the result is the same as for round-down if the sign of the intermediate
result is 0, or is [1,inf] otherwise. In all cases, Inexact and Rounded
will also be raised.
"""
def handle(self, context, sign, *args):
if context.rounding in (ROUND_HALF_UP, ROUND_HALF_EVEN,
ROUND_HALF_DOWN, ROUND_UP):
return _SignedInfinity[sign]
if sign == 0:
if context.rounding == ROUND_CEILING:
return _SignedInfinity[sign]
return _dec_from_triple(sign, '9'*context.prec,
context.Emax-context.prec+1)
if sign == 1:
if context.rounding == ROUND_FLOOR:
return _SignedInfinity[sign]
return _dec_from_triple(sign, '9'*context.prec,
context.Emax-context.prec+1)
class Underflow(Inexact, Rounded, Subnormal):
"""Numerical underflow with result rounded to 0.
This occurs and signals underflow if a result is inexact and the
adjusted exponent of the result would be smaller (more negative) than
the smallest value that can be handled by the implementation (the value
Emin). That is, the result is both inexact and subnormal.
The result after an underflow will be a subnormal number rounded, if
necessary, so that its exponent is not less than Etiny. This may result
in 0 with the sign of the intermediate result and an exponent of Etiny.
In all cases, Inexact, Rounded, and Subnormal will also be raised.
"""
class FloatOperation(DecimalException, TypeError):
"""Enable stricter semantics for mixing floats and Decimals.
If the signal is not trapped (default), mixing floats and Decimals is
permitted in the Decimal() constructor, context.create_decimal() and
all comparison operators. Both conversion and comparisons are exact.
Any occurrence of a mixed operation is silently recorded by setting
FloatOperation in the context flags. Explicit conversions with
Decimal.from_float() or context.create_decimal_from_float() do not
set the flag.
Otherwise (the signal is trapped), only equality comparisons and explicit
conversions are silent. All other mixed operations raise FloatOperation.
"""
# List of public traps and flags
_signals = [Clamped, DivisionByZero, Inexact, Overflow, Rounded,
Underflow, InvalidOperation, Subnormal, FloatOperation]
# Map conditions (per the spec) to signals
_condition_map = {ConversionSyntax:InvalidOperation,
DivisionImpossible:InvalidOperation,
DivisionUndefined:InvalidOperation,
InvalidContext:InvalidOperation}
# Valid rounding modes
_rounding_modes = (ROUND_DOWN, ROUND_HALF_UP, ROUND_HALF_EVEN, ROUND_CEILING,
ROUND_FLOOR, ROUND_UP, ROUND_HALF_DOWN, ROUND_05UP)
##### Context Functions ##################################################
# The getcontext() and setcontext() function manage access to a thread-local
# current context.
import contextvars
_current_context_var = contextvars.ContextVar('decimal_context')
def getcontext():
"""Returns this thread's context.
If this thread does not yet have a context, returns
a new context and sets this thread's context.
New contexts are copies of DefaultContext.
"""
try:
return _current_context_var.get()
except LookupError:
context = Context()
_current_context_var.set(context)
return context
def setcontext(context):
"""Set this thread's context to context."""
if context in (DefaultContext, BasicContext, ExtendedContext):
context = context.copy()
context.clear_flags()
_current_context_var.set(context)
del contextvars # Don't contaminate the namespace
def localcontext(ctx=None):
"""Return a context manager for a copy of the supplied context
Uses a copy of the current context if no context is specified
The returned context manager creates a local decimal context
in a with statement:
def sin(x):
with localcontext() as ctx:
ctx.prec += 2
# Rest of sin calculation algorithm
# uses a precision 2 greater than normal
return +s # Convert result to normal precision
def sin(x):
with localcontext(ExtendedContext):
# Rest of sin calculation algorithm
# uses the Extended Context from the
# General Decimal Arithmetic Specification
return +s # Convert result to normal context
>>> setcontext(DefaultContext)
>>> print(getcontext().prec)
28
>>> with localcontext():
... ctx = getcontext()
... ctx.prec += 2
... print(ctx.prec)
...
30
>>> with localcontext(ExtendedContext):
... print(getcontext().prec)
...
9
>>> print(getcontext().prec)
28
"""
if ctx is None: ctx = getcontext()
return _ContextManager(ctx)
##### Decimal class #######################################################
# Do not subclass Decimal from numbers.Real and do not register it as such
# (because Decimals are not interoperable with floats). See the notes in
# numbers.py for more detail.
class Decimal(object):
"""Floating point class for decimal arithmetic."""
__slots__ = ('_exp','_int','_sign', '_is_special')
# Generally, the value of the Decimal instance is given by
# (-1)**_sign * _int * 10**_exp
# Special values are signified by _is_special == True
# We're immutable, so use __new__ not __init__
def __new__(cls, value="0", context=None):
"""Create a decimal point instance.
>>> Decimal('3.14') # string input
Decimal('3.14')
>>> Decimal((0, (3, 1, 4), -2)) # tuple (sign, digit_tuple, exponent)
Decimal('3.14')
>>> Decimal(314) # int
Decimal('314')
>>> Decimal(Decimal(314)) # another decimal instance
Decimal('314')
>>> Decimal(' 3.14 \\n') # leading and trailing whitespace okay
Decimal('3.14')
"""
# 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, str):
m = _parser(value.strip().replace("_", ""))
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):
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) 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) 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):
# 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):
if context is None:
context = getcontext()
context._raise_error(FloatOperation,
"strict semantics for mixing floats and Decimals are "
"enabled")
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
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): # handle integer inputs
sign = 0 if f >= 0 else 1
k = 0
coeff = str(abs(f))
elif isinstance(f, float):
if _math.isinf(f) or _math.isnan(f):
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
coeff = str(n*5**k)
else:
raise TypeError("argument must be int or float.")
result = _dec_from_triple(sign, coeff, -k)
if cls is Decimal:
return result
else:
return cls(result)
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 __bool__(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):
self, other = _convert_for_comparison(self, other, equality_op=True)
if other is NotImplemented:
return other
if self._check_nans(other, context):
return False
return self._cmp(other) == 0
def __lt__(self, other, context=None):
self, other = _convert_for_comparison(self, other)
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):
self, other = _convert_for_comparison(self, other)
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):
self, other = _convert_for_comparison(self, other)
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):
self, other = _convert_for_comparison(self, other)
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):
"""Compare self to other. Return a decimal value:
a or b is a NaN ==> Decimal('NaN')
a < b ==> Decimal('-1')
a == b ==> Decimal('0')
a > b ==> Decimal('1')
"""
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)"""
# In order to make sure that the hash of a Decimal instance
# agrees with the hash of a numerically equal integer, float
# or Fraction, we follow the rules for numeric hashes outlined
# in the documentation. (See library docs, 'Built-in Types').
if self._is_special:
if self.is_snan():
raise TypeError('Cannot hash a signaling NaN value.')
elif self.is_nan():
return _PyHASH_NAN
else:
if self._sign:
return -_PyHASH_INF
else:
return _PyHASH_INF
if self._exp >= 0:
exp_hash = pow(10, self._exp, _PyHASH_MODULUS)
else:
exp_hash = pow(_PyHASH_10INV, -self._exp, _PyHASH_MODULUS)
hash_ = int(self._int) * exp_hash % _PyHASH_MODULUS
ans = hash_ if self >= 0 else -hash_
return -2 if ans == -1 else ans
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 as_integer_ratio(self):
"""Express a finite Decimal instance in the form n / d.
Returns a pair (n, d) of integers. When called on an infinity
or NaN, raises OverflowError or ValueError respectively.
>>> Decimal('3.14').as_integer_ratio()
(157, 50)
>>> Decimal('-123e5').as_integer_ratio()
(-12300000, 1)
>>> Decimal('0.00').as_integer_ratio()
(0, 1)
"""
if self._is_special:
if self.is_nan():
raise ValueError("cannot convert NaN to integer ratio")
else:
raise OverflowError("cannot convert Infinity to integer ratio")
if not self:
return 0, 1
# Find n, d in lowest terms such that abs(self) == n / d;
# we'll deal with the sign later.
n = int(self._int)
if self._exp >= 0:
# self is an integer.
n, d = n * 10**self._exp, 1
else:
# Find d2, d5 such that abs(self) = n / (2**d2 * 5**d5).
d5 = -self._exp
while d5 > 0 and n % 5 == 0:
n //= 5
d5 -= 1
# (n & -n).bit_length() - 1 counts trailing zeros in binary
# representation of n (provided n is nonzero).
d2 = -self._exp
shift2 = min((n & -n).bit_length() - 1, d2)
if shift2:
n >>= shift2
d2 -= shift2
d = 5**d5 << d2
if self._sign:
n = -n
return n, d
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 a string, using engineering notation if an exponent is needed.
Engineering notation has an exponent which is a multiple of 3. This
can leave up to 3 digits to the left of the decimal place and may
require the addition of either one or two trailing zeros.
"""
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 than 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)
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__
@property
def real(self):
return self
@property
def imag(self):
return Decimal(0)
def conjugate(self):
return self
def __complex__(self):
return complex(float(self))
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 __round__(self, n=None):
"""Round self to the nearest integer, or to a given precision.
If only one argument is supplied, round a finite Decimal
instance self to the nearest integer. If self is infinite or
a NaN then a Python exception is raised. If self is finite
and lies exactly halfway between two integers then it is
rounded to the integer with even last digit.
>>> round(Decimal('123.456'))
123
>>> round(Decimal('-456.789'))
-457
>>> round(Decimal('-3.0'))
-3
>>> round(Decimal('2.5'))
2
>>> round(Decimal('3.5'))
4
>>> round(Decimal('Inf'))
Traceback (most recent call last):
...
OverflowError: cannot round an infinity
>>> round(Decimal('NaN'))
Traceback (most recent call last):
...
ValueError: cannot round a NaN
If a second argument n is supplied, self is rounded to n
decimal places using the rounding mode for the current
context.
For an integer n, round(self, -n) is exactly equivalent to
self.quantize(Decimal('1En')).
>>> round(Decimal('123.456'), 0)
Decimal('123')
>>> round(Decimal('123.456'), 2)
Decimal('123.46')
>>> round(Decimal('123.456'), -2)
Decimal('1E+2')
>>> round(Decimal('-Infinity'), 37)
Decimal('NaN')
>>> round(Decimal('sNaN123'), 0)
Decimal('NaN123')
"""
if n is not None:
# two-argument form: use the equivalent quantize call
if not isinstance(n, int):
raise TypeError('Second argument to round should be integral')
exp = _dec_from_triple(0, '1', -n)
return self.quantize(exp)
# one-argument form
if self._is_special:
if self.is_nan():
raise ValueError("cannot round a NaN")
else:
raise OverflowError("cannot round an infinity")
return int(self._rescale(0, ROUND_HALF_EVEN))
def __floor__(self):
"""Return the floor of self, as an integer.
For a finite Decimal instance self, return the greatest
integer n such that n <= self. If self is infinite or a NaN
then a Python exception is raised.
"""
if self._is_special:
if self.is_nan():
raise ValueError("cannot round a NaN")
else:
raise OverflowError("cannot round an infinity")
return int(self._rescale(0, ROUND_FLOOR))
def __ceil__(self):
"""Return the ceiling of self, as an integer.
For a finite Decimal instance self, return the least integer n
such that n >= self. If self is infinite or a NaN then a
Python exception is raised.
"""
if self._is_special:
if self.is_nan():
raise ValueError("cannot round a NaN")
else:
raise OverflowError("cannot round an infinity")
return int(self._rescale(0, ROUND_CEILING))
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)
third = _convert_other(third, 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)
return product.__add__(third, context)
def _power_modulo(self, other, modulo, context=None):
"""Three argument version of __pow__"""
other = _convert_other(other)
if other is NotImplemented:
return other
modulo = _convert_other(modulo)
if modulo is NotImplemented:
return modulo
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 range(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 = 1 << -(-_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):
"""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')
# 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, context=None):
"""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):
"""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, context=None):
"""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, context=None):
"""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, context=None):
"""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: