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| author | Stefan Krah <skrah@bytereef.org> | 2014-09-10 15:58:15 (GMT) |
|---|---|---|
| committer | Stefan Krah <skrah@bytereef.org> | 2014-09-10 15:58:15 (GMT) |
| commit | b578f8a1efdb8ad5b6671427d3a49f4eef5e5b70 (patch) | |
| tree | 4406bd8c0c25536ea07bdac4712c05766e40ab5c /Lib/decimal.py | |
| parent | 082a9b1274ae61011e97b5c7e424e0e438114123 (diff) | |
| download | cpython-b578f8a1efdb8ad5b6671427d3a49f4eef5e5b70.zip cpython-b578f8a1efdb8ad5b6671427d3a49f4eef5e5b70.tar.gz cpython-b578f8a1efdb8ad5b6671427d3a49f4eef5e5b70.tar.bz2 | |
Issue #19232: Speed up decimal import. Additionally, since _decimal is
self-contained, this change facilitates maintenance and the Python version
can be easily imported for experimentation.
Diffstat (limited to 'Lib/decimal.py')
| -rw-r--r-- | Lib/decimal.py | 6406 |
1 files changed, 8 insertions, 6398 deletions
diff --git a/Lib/decimal.py b/Lib/decimal.py index b6d0a34..3f798e4 100644 --- a/Lib/decimal.py +++ b/Lib/decimal.py @@ -1,6404 +1,14 @@ -# 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' -] - -__version__ = '1.70' # Highest version of the spec this complies with - # See http://speleotrove.com/decimal/ -__libmpdec_version__ = "2.4.1" # 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 an string is being - converted to a number and it does not conform to the numeric string - syntax. The result is [0,qNaN]. - """ - def handle(self, context, *args): - return _NaN - -class DivisionByZero(DecimalException, ZeroDivisionError): - """Division by 0. - - This occurs and signals division-by-zero if division of a finite number - by zero was attempted (during a divide-integer or divide operation, or a - power operation with negative right-hand operand), and the dividend was - not zero. - - The result of the operation is [sign,inf], where sign is the exclusive - or of the signs of the operands for divide, or is 1 for an odd power of - -0, for power. - """ - - def handle(self, context, sign, *args): - return _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. Py2.4 offers direct support for thread locals. If that -# is not available, use threading.current_thread() which is slower but will -# work for older Pythons. If threads are not part of the build, create a -# mock threading object with threading.local() returning the module namespace. try: - import threading -except ImportError: - # Python was compiled without threads; create a mock object instead - class MockThreading(object): - def local(self, sys=sys): - return sys.modules[__name__] - threading = MockThreading() - del MockThreading - -try: - threading.local - -except AttributeError: - - # To fix reloading, force it to create a new context - # Old contexts have different exceptions in their dicts, making problems. - if hasattr(threading.current_thread(), '__decimal_context__'): - del threading.current_thread().__decimal_context__ - - def setcontext(context): - """Set this thread's context to context.""" - if context in (DefaultContext, BasicContext, ExtendedContext): - context = context.copy() - context.clear_flags() - threading.current_thread().__decimal_context__ = context - - def getcontext(): - """Returns this thread's context. - - If this thread does not yet have a context, returns - a new context and sets this thread's context. - New contexts are copies of DefaultContext. - """ - try: - return threading.current_thread().__decimal_context__ - except AttributeError: - context = Context() - threading.current_thread().__decimal_context__ = context - return context - -else: - - local = threading.local() - if hasattr(local, '__decimal_context__'): - del local.__decimal_context__ - - def getcontext(_local=local): - """Returns this thread's context. - - If this thread does not yet have a context, returns - a new context and sets this thread's context. - New contexts are copies of DefaultContext. - """ - try: - return _local.__decimal_context__ - except AttributeError: - context = Context() - _local.__decimal_context__ = context - return context - - def setcontext(context, _local=local): - """Set this thread's context to context.""" - if context in (DefaultContext, BasicContext, ExtendedContext): - context = context.copy() - context.clear_flags() - _local.__decimal_context__ = context - - del threading, local # Don't contaminate the namespace - -def localcontext(ctx=None): - """Return a context manager for a copy of the supplied context - - Uses a copy of the current context if no context is specified - The returned context manager creates a local decimal context - in a with statement: - def sin(x): - with localcontext() as ctx: - ctx.prec += 2 - # Rest of sin calculation algorithm - # uses a precision 2 greater than normal - return +s # Convert result to normal precision - - def sin(x): - with localcontext(ExtendedContext): - # Rest of sin calculation algorithm - # uses the Extended Context from the - # General Decimal Arithmetic Specification - return +s # Convert result to normal context - - >>> 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()) - 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 - return cls(f) - if not isinstance(f, float): - raise TypeError("argument must be int or 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 - result = _dec_from_triple(sign, str(n*5**k), -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 __ne__(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 True - 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): - """Compares one to another. - - -1 => a < b - 0 => a = b - 1 => a > b - NaN => one is NaN - Like __cmp__, but returns Decimal instances. - """ - other = _convert_other(other, raiseit=True) - - # Compare(NaN, NaN) = NaN - if (self._is_special or other and other._is_special): - ans = self._check_nans(other, context) - if ans: - return ans - - return Decimal(self._cmp(other)) - - def __hash__(self): - """x.__hash__() <==> hash(x)""" - - # 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 __repr__(self): - """Represents the number as an instance of Decimal.""" - # Invariant: eval(repr(d)) == d - return "Decimal('%s')" % str(self) - - def __str__(self, eng=False, context=None): - """Return string representation of the number in scientific notation. - - Captures all of the information in the underlying representation. - """ - - sign = ['', '-'][self._sign] - if self._is_special: - if self._exp == 'F': - return sign + 'Infinity' - elif self._exp == 'n': - return sign + 'NaN' + self._int - else: # self._exp == 'N' - return sign + 'sNaN' + self._int - - # number of digits of self._int to left of decimal point - leftdigits = self._exp + len(self._int) - - # dotplace is number of digits of self._int to the left of the - # decimal point in the mantissa of the output string (that is, - # after adjusting the exponent) - if self._exp <= 0 and leftdigits > -6: - # no exponent required - dotplace = leftdigits - elif not eng: - # usual scientific notation: 1 digit on left of the point - dotplace = 1 - elif self._int == '0': - # engineering notation, zero - dotplace = (leftdigits + 1) % 3 - 1 - else: - # engineering notation, nonzero - dotplace = (leftdigits - 1) % 3 + 1 - - if dotplace <= 0: - intpart = '0' - fracpart = '.' + '0'*(-dotplace) + self._int - elif dotplace >= len(self._int): - intpart = self._int+'0'*(dotplace-len(self._int)) - fracpart = '' - else: - intpart = self._int[:dotplace] - fracpart = '.' + self._int[dotplace:] - if leftdigits == dotplace: - exp = '' - else: - if context is None: - context = getcontext() - exp = ['e', 'E'][context.capitals] + "%+d" % (leftdigits-dotplace) - - return sign + intpart + fracpart + exp - - def to_eng_string(self, context=None): - """Convert to engineering-type string. - - Engineering notation has an exponent which is a multiple of 3, so there - are up to 3 digits left of the decimal place. - - Same rules for when in exponential and when as a value as in __str__. - """ - return self.__str__(eng=True, context=context) - - def __neg__(self, context=None): - """Returns a copy with the sign switched. - - Rounds, if it has reason. - """ - if self._is_special: - ans = self._check_nans(context=context) - if ans: - return ans - - if context is None: - context = getcontext() - - if not self and context.rounding != ROUND_FLOOR: - # -Decimal('0') is Decimal('0'), not Decimal('-0'), except - # in ROUND_FLOOR rounding mode. - ans = self.copy_abs() - else: - ans = self.copy_negate() - - return ans._fix(context) - - def __pos__(self, context=None): - """Returns a copy, unless it is a sNaN. - - Rounds the number (if more then precision digits) - """ - if self._is_special: - ans = self._check_nans(context=context) - if ans: - return ans - - if context is None: - context = getcontext() - - if not self and context.rounding != ROUND_FLOOR: - # + (-0) = 0, except in ROUND_FLOOR rounding mode. - ans = self.copy_abs() - else: - ans = Decimal(self) - - return ans._fix(context) - - def __abs__(self, round=True, context=None): - """Returns the absolute value of self. - - If the keyword argument 'round' is false, do not round. The - expression self.__abs__(round=False) is equivalent to - self.copy_abs(). - """ - if not round: - return self.copy_abs() - - if self._is_special: - ans = self._check_nans(context=context) - if ans: - return ans - - if self._sign: - ans = self.__neg__(context=context) - else: - ans = self.__pos__(context=context) - - return ans - - def __add__(self, other, context=None): - """Returns self + other. - - -INF + INF (or the reverse) cause InvalidOperation errors. - """ - other = _convert_other(other) - if other is NotImplemented: - return other - - if context is None: - context = getcontext() - - if self._is_special or other._is_special: - ans = self._check_nans(other, context) - if ans: - return ans - - if self._isinfinity(): - # If both INF, same sign => same as both, opposite => error. - if self._sign != other._sign and other._isinfinity(): - return context._raise_error(InvalidOperation, '-INF + INF') - return Decimal(self) - if other._isinfinity(): - return Decimal(other) # Can't both be infinity here - - exp = min(self._exp, other._exp) - negativezero = 0 - if context.rounding == ROUND_FLOOR and self._sign != other._sign: - # If the answer is 0, the sign should be negative, in this case. - negativezero = 1 - - if not self and not other: - sign = min(self._sign, other._sign) - if negativezero: - sign = 1 - ans = _dec_from_triple(sign, '0', exp) - ans = ans._fix(context) - return ans - if not self: - exp = max(exp, other._exp - context.prec-1) - ans = other._rescale(exp, context.rounding) - ans = ans._fix(context) - return ans - if not other: - exp = max(exp, self._exp - context.prec-1) - ans = self._rescale(exp, context.rounding) - ans = ans._fix(context) - return ans - - op1 = _WorkRep(self) - op2 = _WorkRep(other) - op1, op2 = _normalize(op1, op2, context.prec) - - result = _WorkRep() - if op1.sign != op2.sign: - # Equal and opposite - if op1.int == op2.int: - ans = _dec_from_triple(negativezero, '0', exp) - ans = ans._fix(context) - return ans - if op1.int < op2.int: - op1, op2 = op2, op1 - # OK, now abs(op1) > abs(op2) - if op1.sign == 1: - result.sign = 1 - op1.sign, op2.sign = op2.sign, op1.sign - else: - result.sign = 0 - # So we know the sign, and op1 > 0. - elif op1.sign == 1: - result.sign = 1 - op1.sign, op2.sign = (0, 0) - else: - result.sign = 0 - # Now, op1 > abs(op2) > 0 - - if op2.sign == 0: - result.int = op1.int + op2.int - else: - result.int = op1.int - op2.int - - result.exp = op1.exp - ans = Decimal(result) - ans = ans._fix(context) - return ans - - __radd__ = __add__ - - def __sub__(self, other, context=None): - """Return self - other""" - other = _convert_other(other) - if other is NotImplemented: - return other - - if self._is_special or other._is_special: - ans = self._check_nans(other, context=context) - if ans: - return ans - - # self - other is computed as self + other.copy_negate() - return self.__add__(other.copy_negate(), context=context) - - def __rsub__(self, other, context=None): - """Return other - self""" - other = _convert_other(other) - if other is NotImplemented: - return other - - return other.__sub__(self, context=context) - - def __mul__(self, other, context=None): - """Return self * other. - - (+-) INF * 0 (or its reverse) raise InvalidOperation. - """ - other = _convert_other(other) - if other is NotImplemented: - return other - - if context is None: - context = getcontext() - - resultsign = self._sign ^ other._sign - - if self._is_special or other._is_special: - ans = self._check_nans(other, context) - if ans: - return ans - - if self._isinfinity(): - if not other: - return context._raise_error(InvalidOperation, '(+-)INF * 0') - return _SignedInfinity[resultsign] - - if other._isinfinity(): - if not self: - return context._raise_error(InvalidOperation, '0 * (+-)INF') - return _SignedInfinity[resultsign] - - resultexp = self._exp + other._exp - - # Special case for multiplying by zero - if not self or not other: - ans = _dec_from_triple(resultsign, '0', resultexp) - # Fixing in case the exponent is out of bounds - ans = ans._fix(context) - return ans - - # Special case for multiplying by power of 10 - if self._int == '1': - ans = _dec_from_triple(resultsign, other._int, resultexp) - ans = ans._fix(context) - return ans - if other._int == '1': - ans = _dec_from_triple(resultsign, self._int, resultexp) - ans = ans._fix(context) - return ans - - op1 = _WorkRep(self) - op2 = _WorkRep(other) - - ans = _dec_from_triple(resultsign, str(op1.int * op2.int), resultexp) - ans = ans._fix(context) - - return ans - __rmul__ = __mul__ - - def __truediv__(self, other, context=None): - """Return self / other.""" - other = _convert_other(other) - if other is NotImplemented: - return NotImplemented - - if context is None: - context = getcontext() - - sign = self._sign ^ other._sign - - if self._is_special or other._is_special: - ans = self._check_nans(other, context) - if ans: - return ans - - if self._isinfinity() and other._isinfinity(): - return context._raise_error(InvalidOperation, '(+-)INF/(+-)INF') - - if self._isinfinity(): - return _SignedInfinity[sign] - - if other._isinfinity(): - context._raise_error(Clamped, 'Division by infinity') - return _dec_from_triple(sign, '0', context.Etiny()) - - # Special cases for zeroes - if not other: - if not self: - return context._raise_error(DivisionUndefined, '0 / 0') - return context._raise_error(DivisionByZero, 'x / 0', sign) - - if not self: - exp = self._exp - other._exp - coeff = 0 - else: - # OK, so neither = 0, INF or NaN - shift = len(other._int) - len(self._int) + context.prec + 1 - exp = self._exp - other._exp - shift - op1 = _WorkRep(self) - op2 = _WorkRep(other) - if shift >= 0: - coeff, remainder = divmod(op1.int * 10**shift, op2.int) - else: - coeff, remainder = divmod(op1.int, op2.int * 10**-shift) - if remainder: - # result is not exact; adjust to ensure correct rounding - if coeff % 5 == 0: - coeff += 1 - else: - # result is exact; get as close to ideal exponent as possible - ideal_exp = self._exp - other._exp - while exp < ideal_exp and coeff % 10 == 0: - coeff //= 10 - exp += 1 - - ans = _dec_from_triple(sign, str(coeff), exp) - return ans._fix(context) - - def _divide(self, other, context): - """Return (self // other, self % other), to context.prec precision. - - Assumes that neither self nor other is a NaN, that self is not - infinite and that other is nonzero. - """ - sign = self._sign ^ other._sign - if other._isinfinity(): - ideal_exp = self._exp - else: - ideal_exp = min(self._exp, other._exp) - - expdiff = self.adjusted() - other.adjusted() - if not self or other._isinfinity() or expdiff <= -2: - return (_dec_from_triple(sign, '0', 0), - self._rescale(ideal_exp, context.rounding)) - if expdiff <= context.prec: - op1 = _WorkRep(self) - op2 = _WorkRep(other) - if op1.exp >= op2.exp: - op1.int *= 10**(op1.exp - op2.exp) - else: - op2.int *= 10**(op2.exp - op1.exp) - q, r = divmod(op1.int, op2.int) - if q < 10**context.prec: - return (_dec_from_triple(sign, str(q), 0), - _dec_from_triple(self._sign, str(r), ideal_exp)) - - # Here the quotient is too large to be representable - ans = context._raise_error(DivisionImpossible, - 'quotient too large in //, % or divmod') - return ans, ans - - def __rtruediv__(self, other, context=None): - """Swaps self/other and returns __truediv__.""" - other = _convert_other(other) - if other is NotImplemented: - return other - return other.__truediv__(self, context=context) - - def __divmod__(self, other, context=None): - """ - Return (self // other, self % other) - """ - other = _convert_other(other) - if other is NotImplemented: - return other - - if context is None: - context = getcontext() - - ans = self._check_nans(other, context) - if ans: - return (ans, ans) - - sign = self._sign ^ other._sign - if self._isinfinity(): - if other._isinfinity(): - ans = context._raise_error(InvalidOperation, 'divmod(INF, INF)') - return ans, ans - else: - return (_SignedInfinity[sign], - context._raise_error(InvalidOperation, 'INF % x')) - - if not other: - if not self: - ans = context._raise_error(DivisionUndefined, 'divmod(0, 0)') - return ans, ans - else: - return (context._raise_error(DivisionByZero, 'x // 0', sign), - context._raise_error(InvalidOperation, 'x % 0')) - - quotient, remainder = self._divide(other, context) - remainder = remainder._fix(context) - return quotient, remainder - - def __rdivmod__(self, other, context=None): - """Swaps self/other and returns __divmod__.""" - other = _convert_other(other) - if other is NotImplemented: - return other - return other.__divmod__(self, context=context) - - def __mod__(self, other, context=None): - """ - self % other - """ - other = _convert_other(other) - if other is NotImplemented: - return other - - if context is None: - context = getcontext() - - ans = self._check_nans(other, context) - if ans: - return ans - - if self._isinfinity(): - return context._raise_error(InvalidOperation, 'INF % x') - elif not other: - if self: - return context._raise_error(InvalidOperation, 'x % 0') - else: - return context._raise_error(DivisionUndefined, '0 % 0') - - remainder = self._divide(other, context)[1] - remainder = remainder._fix(context) - return remainder - - def __rmod__(self, other, context=None): - """Swaps self/other and returns __mod__.""" - other = _convert_other(other) - if other is NotImplemented: - return other - return other.__mod__(self, context=context) - - def remainder_near(self, other, context=None): - """ - Remainder nearest to 0- abs(remainder-near) <= other/2 - """ - if context is None: - context = getcontext() - - other = _convert_other(other, raiseit=True) - - ans = self._check_nans(other, context) - if ans: - return ans - - # self == +/-infinity -> InvalidOperation - if self._isinfinity(): - return context._raise_error(InvalidOperation, - 'remainder_near(infinity, x)') - - # other == 0 -> either InvalidOperation or DivisionUndefined - if not other: - if self: - return context._raise_error(InvalidOperation, - 'remainder_near(x, 0)') - else: - return context._raise_error(DivisionUndefined, - 'remainder_near(0, 0)') - - # other = +/-infinity -> remainder = self - if other._isinfinity(): - ans = Decimal(self) - return ans._fix(context) - - # self = 0 -> remainder = self, with ideal exponent - ideal_exponent = min(self._exp, other._exp) - if not self: - ans = _dec_from_triple(self._sign, '0', ideal_exponent) - return ans._fix(context) - - # catch most cases of large or small quotient - expdiff = self.adjusted() - other.adjusted() - if expdiff >= context.prec + 1: - # expdiff >= prec+1 => abs(self/other) > 10**prec - return context._raise_error(DivisionImpossible) - if expdiff <= -2: - # expdiff <= -2 => abs(self/other) < 0.1 - ans = self._rescale(ideal_exponent, context.rounding) - return ans._fix(context) - - # adjust both arguments to have the same exponent, then divide - op1 = _WorkRep(self) - op2 = _WorkRep(other) - if op1.exp >= op2.exp: - op1.int *= 10**(op1.exp - op2.exp) - else: - op2.int *= 10**(op2.exp - op1.exp) - q, r = divmod(op1.int, op2.int) - # remainder is r*10**ideal_exponent; other is +/-op2.int * - # 10**ideal_exponent. Apply correction to ensure that - # abs(remainder) <= abs(other)/2 - if 2*r + (q&1) > op2.int: - r -= op2.int - q += 1 - - if q >= 10**context.prec: - return context._raise_error(DivisionImpossible) - - # result has same sign as self unless r is negative - sign = self._sign - if r < 0: - sign = 1-sign - r = -r - - ans = _dec_from_triple(sign, str(r), ideal_exponent) - return ans._fix(context) - - def __floordiv__(self, other, context=None): - """self // other""" - other = _convert_other(other) - if other is NotImplemented: - return other - - if context is None: - context = getcontext() - - ans = self._check_nans(other, context) - if ans: - return ans - - if self._isinfinity(): - if other._isinfinity(): - return context._raise_error(InvalidOperation, 'INF // INF') - else: - return _SignedInfinity[self._sign ^ other._sign] - - if not other: - if self: - return context._raise_error(DivisionByZero, 'x // 0', - self._sign ^ other._sign) - else: - return context._raise_error(DivisionUndefined, '0 // 0') - - return self._divide(other, context)[0] - - def __rfloordiv__(self, other, context=None): - """Swaps self/other and returns __floordiv__.""" - other = _convert_other(other) - if other is NotImplemented: - return other - return other.__floordiv__(self, context=context) - - def __float__(self): - """Float representation.""" - if self._isnan(): - if self.is_snan(): - raise ValueError("Cannot convert signaling NaN to float") - s = "-nan" if self._sign else "nan" - else: - s = str(self) - return float(s) - - def __int__(self): - """Converts self to an int, truncating if necessary.""" - if self._is_special: - if self._isnan(): - raise ValueError("Cannot convert NaN to integer") - elif self._isinfinity(): - raise OverflowError("Cannot convert infinity to integer") - s = (-1)**self._sign - if self._exp >= 0: - return s*int(self._int)*10**self._exp - else: - return s*int(self._int[:self._exp] or '0') - - __trunc__ = __int__ - - def real(self): - return self - real = property(real) - - def imag(self): - return Decimal(0) - imag = property(imag) - - def conjugate(self): - return self - - def __complex__(self): - return complex(float(self)) - - def _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: - return self.copy_sign(other) - - if comparison == -1: - ans = self.next_plus(context) - else: # comparison == 1 - ans = self.next_minus(context) - - # decide which flags to raise using value of ans - if ans._isinfinity(): - context._raise_error(Overflow, - 'Infinite result from next_toward', - ans._sign) - context._raise_error(Inexact) - context._raise_error(Rounded) - elif ans.adjusted() < context.Emin: - context._raise_error(Underflow) - context._raise_error(Subnormal) - context._raise_error(Inexact) - context._raise_error(Rounded) - # if precision == 1 then we don't raise Clamped for a - # result 0E-Etiny. - if not ans: - context._raise_error(Clamped) - - return ans - - def number_class(self, context=None): - """Returns an indication of the class of self. - - The class is one of the following strings: - sNaN - NaN - -Infinity - -Normal - -Subnormal - -Zero - +Zero - +Subnormal - +Normal - +Infinity - """ - if self.is_snan(): - return "sNaN" - if self.is_qnan(): - return "NaN" - inf = self._isinfinity() - if inf == 1: - return "+Infinity" - if inf == -1: - return "-Infinity" - if self.is_zero(): - if self._sign: - return "-Zero" - else: - return "+Zero" - if context is None: - context = getcontext() - if self.is_subnormal(context=context): - if self._sign: - return "-Subnormal" - else: - return "+Subnormal" - # just a normal, regular, boring number, :) - if self._sign: - return "-Normal" - else: - return "+Normal" - - def radix(self): - """Just returns 10, as this is Decimal, :)""" - return Decimal(10) - - def rotate(self, other, context=None): - """Returns a rotated copy of self, value-of-other times.""" - if context is None: - context = getcontext() - - other = _convert_other(other, raiseit=True) - - ans = self._check_nans(other, context) - if ans: - return ans - - if other._exp != 0: - return context._raise_error(InvalidOperation) - if not (-context.prec <= int(other) <= context.prec): - return context._raise_error(InvalidOperation) - - if self._isinfinity(): - return Decimal(self) - - # get values, pad if necessary - torot = int(other) - rotdig = self._int - topad = context.prec - len(rotdig) - if topad > 0: - rotdig = '0'*topad + rotdig - elif topad < 0: - rotdig = rotdig[-topad:] - - # let's rotate! - rotated = rotdig[torot:] + rotdig[:torot] - return _dec_from_triple(self._sign, - rotated.lstrip('0') or '0', self._exp) - - def scaleb(self, other, context=None): - """Returns self operand after adding the second value to its exp.""" - if context is None: - context = getcontext() - - other = _convert_other(other, raiseit=True) - - ans = self._check_nans(other, context) - if ans: - return ans - - if other._exp != 0: - return context._raise_error(InvalidOperation) - liminf = -2 * (context.Emax + context.prec) - limsup = 2 * (context.Emax + context.prec) - if not (liminf <= int(other) <= limsup): - return context._raise_error(InvalidOperation) - - if self._isinfinity(): - return Decimal(self) - - d = _dec_from_triple(self._sign, self._int, self._exp + int(other)) - d = d._fix(context) - return d - - def shift(self, other, context=None): - """Returns a shifted copy of self, value-of-other times.""" - if context is None: - context = getcontext() - - other = _convert_other(other, raiseit=True) - - ans = self._check_nans(other, context) - if ans: - return ans - - if other._exp != 0: - return context._raise_error(InvalidOperation) - if not (-context.prec <= int(other) <= context.prec): - return context._raise_error(InvalidOperation) - - if self._isinfinity(): - return Decimal(self) - - # get values, pad if necessary - torot = int(other) - rotdig = self._int - topad = context.prec - len(rotdig) - if topad > 0: - rotdig = '0'*topad + rotdig - elif topad < 0: - rotdig = rotdig[-topad:] - - # let's shift! - if torot < 0: - shifted = rotdig[:torot] - else: - shifted = rotdig + '0'*torot - shifted = shifted[-context.prec:] - - return _dec_from_triple(self._sign, - shifted.lstrip('0') or '0', self._exp) - - # Support for pickling, copy, and deepcopy - def __reduce__(self): - return (self.__class__, (str(self),)) - - def __copy__(self): - if type(self) is Decimal: - return self # I'm immutable; therefore I am my own clone - return self.__class__(str(self)) - - def __deepcopy__(self, memo): - if type(self) is Decimal: - return self # My components are also immutable - return self.__class__(str(self)) - - # PEP 3101 support. the _localeconv keyword argument should be - # considered private: it's provided for ease of testing only. - def __format__(self, specifier, context=None, _localeconv=None): - """Format a Decimal instance according to the given specifier. - - The specifier should be a standard format specifier, with the - form described in PEP 3101. Formatting types 'e', 'E', 'f', - 'F', 'g', 'G', 'n' and '%' are supported. If the formatting - type is omitted it defaults to 'g' or 'G', depending on the - value of context.capitals. - """ - - # Note: PEP 3101 says that if the type is not present then - # there should be at least one digit after the decimal point. - # We take the liberty of ignoring this requirement for - # Decimal---it's presumably there to make sure that - # format(float, '') behaves similarly to str(float). - if context is None: - context = getcontext() - - spec = _parse_format_specifier(specifier, _localeconv=_localeconv) - - # special values don't care about the type or precision - if self._is_special: - sign = _format_sign(self._sign, spec) - body = str(self.copy_abs()) - if spec['type'] == '%': - body += '%' - return _format_align(sign, body, spec) - - # a type of None defaults to 'g' or 'G', depending on context - if spec['type'] is None: - spec['type'] = ['g', 'G'][context.capitals] - - # if type is '%', adjust exponent of self accordingly - if spec['type'] == '%': - self = _dec_from_triple(self._sign, self._int, self._exp+2) - - # round if necessary, taking rounding mode from the context - rounding = context.rounding - precision = spec['precision'] - if precision is not None: - if spec['type'] in 'eE': - self = self._round(precision+1, rounding) - elif spec['type'] in 'fF%': - self = self._rescale(-precision, rounding) - elif spec['type'] in 'gG' and len(self._int) > precision: - self = self._round(precision, rounding) - # special case: zeros with a positive exponent can't be - # represented in fixed point; rescale them to 0e0. - if not self and self._exp > 0 and spec['type'] in 'fF%': - self = self._rescale(0, rounding) - - # figure out placement of the decimal point - leftdigits = self._exp + len(self._int) - if spec['type'] in 'eE': - if not self and precision is not None: - dotplace = 1 - precision - else: - dotplace = 1 - elif spec['type'] in 'fF%': - dotplace = leftdigits - elif spec['type'] in 'gG': - if self._exp <= 0 and leftdigits > -6: - dotplace = leftdigits - else: - dotplace = 1 - - # find digits before and after decimal point, and get exponent - if dotplace < 0: - intpart = '0' - fracpart = '0'*(-dotplace) + self._int - elif dotplace > len(self._int): - intpart = self._int + '0'*(dotplace-len(self._int)) - fracpart = '' - else: - intpart = self._int[:dotplace] or '0' - fracpart = self._int[dotplace:] - exp = leftdigits-dotplace - - # done with the decimal-specific stuff; hand over the rest - # of the formatting to the _format_number function - return _format_number(self._sign, intpart, fracpart, exp, spec) - -def _dec_from_triple(sign, coefficient, exponent, special=False): - """Create a decimal instance directly, without any validation, - normalization (e.g. removal of leading zeros) or argument - conversion. - - This function is for *internal use only*. - """ - - self = object.__new__(Decimal) - self._sign = sign - self._int = coefficient - self._exp = exponent - self._is_special = special - - return self - -# Register Decimal as a kind of Number (an abstract base class). -# However, do not register it as Real (because Decimals are not -# interoperable with floats). -_numbers.Number.register(Decimal) - - -##### Context class ####################################################### - -class _ContextManager(object): - """Context manager class to support localcontext(). - - Sets a copy of the supplied context in __enter__() and restores - the previous decimal context in __exit__() - """ - def __init__(self, new_context): - self.new_context = new_context.copy() - def __enter__(self): - self.saved_context = getcontext() - setcontext(self.new_context) - return self.new_context - def __exit__(self, t, v, tb): - setcontext(self.saved_context) - -class Context(object): - """Contains the context for a Decimal instance. - - Contains: - prec - precision (for use in rounding, division, square roots..) - rounding - rounding type (how you round) - traps - If traps[exception] = 1, then the exception is - raised when it is caused. Otherwise, a value is - substituted in. - flags - When an exception is caused, flags[exception] is set. - (Whether or not the trap_enabler is set) - Should be reset by user of Decimal instance. - Emin - Minimum exponent - Emax - Maximum exponent - capitals - If 1, 1*10^1 is printed as 1E+1. - If 0, printed as 1e1 - clamp - If 1, change exponents if too high (Default 0) - """ - - def __init__(self, prec=None, rounding=None, Emin=None, Emax=None, - capitals=None, clamp=None, flags=None, traps=None, - _ignored_flags=None): - # Set defaults; for everything except flags and _ignored_flags, - # inherit from DefaultContext. - try: - dc = DefaultContext - except NameError: - pass - - self.prec = prec if prec is not None else dc.prec - self.rounding = rounding if rounding is not None else dc.rounding - self.Emin = Emin if Emin is not None else dc.Emin - self.Emax = Emax if Emax is not None else dc.Emax - self.capitals = capitals if capitals is not None else dc.capitals - self.clamp = clamp if clamp is not None else dc.clamp - - if _ignored_flags is None: - self._ignored_flags = [] - else: - self._ignored_flags = _ignored_flags - - if traps is None: - self.traps = dc.traps.copy() - elif not isinstance(traps, dict): - self.traps = dict((s, int(s in traps)) for s in _signals + traps) - else: - self.traps = traps - - if flags is None: - self.flags = dict.fromkeys(_signals, 0) - elif not isinstance(flags, dict): - self.flags = dict((s, int(s in flags)) for s in _signals + flags) - else: - self.flags = flags - - def _set_integer_check(self, name, value, vmin, vmax): - if not isinstance(value, int): - raise TypeError("%s must be an integer" % name) - if vmin == '-inf': - if value > vmax: - raise ValueError("%s must be in [%s, %d]. got: %s" % (name, vmin, vmax, value)) - elif vmax == 'inf': - if value < vmin: - raise ValueError("%s must be in [%d, %s]. got: %s" % (name, vmin, vmax, value)) - else: - if value < vmin or value > vmax: - raise ValueError("%s must be in [%d, %d]. got %s" % (name, vmin, vmax, value)) - return object.__setattr__(self, name, value) - - def _set_signal_dict(self, name, d): - if not isinstance(d, dict): - raise TypeError("%s must be a signal dict" % d) - for key in d: - if not key in _signals: - raise KeyError("%s is not a valid signal dict" % d) - for key in _signals: - if not key in d: - raise KeyError("%s is not a valid signal dict" % d) - return object.__setattr__(self, name, d) - - def __setattr__(self, name, value): - if name == 'prec': - return self._set_integer_check(name, value, 1, 'inf') - elif name == 'Emin': - return self._set_integer_check(name, value, '-inf', 0) - elif name == 'Emax': - return self._set_integer_check(name, value, 0, 'inf') - elif name == 'capitals': - return self._set_integer_check(name, value, 0, 1) - elif name == 'clamp': - return self._set_integer_check(name, value, 0, 1) - elif name == 'rounding': - if not value in _rounding_modes: - # raise TypeError even for strings to have consistency - # among various implementations. - raise TypeError("%s: invalid rounding mode" % value) - return object.__setattr__(self, name, value) - elif name == 'flags' or name == 'traps': - return self._set_signal_dict(name, value) - elif name == '_ignored_flags': - return object.__setattr__(self, name, value) - else: - raise AttributeError( - "'decimal.Context' object has no attribute '%s'" % name) - - def __delattr__(self, name): - raise AttributeError("%s cannot be deleted" % name) - - # Support for pickling, copy, and deepcopy - def __reduce__(self): - flags = [sig for sig, v in self.flags.items() if v] - traps = [sig for sig, v in self.traps.items() if v] - return (self.__class__, - (self.prec, self.rounding, self.Emin, self.Emax, - self.capitals, self.clamp, flags, traps)) - - def __repr__(self): - """Show the current context.""" - s = [] - s.append('Context(prec=%(prec)d, rounding=%(rounding)s, ' - 'Emin=%(Emin)d, Emax=%(Emax)d, capitals=%(capitals)d, ' - 'clamp=%(clamp)d' - % vars(self)) - names = [f.__name__ for f, v in self.flags.items() if v] - s.append('flags=[' + ', '.join(names) + ']') - names = [t.__name__ for t, v in self.traps.items() if v] - s.append('traps=[' + ', '.join(names) + ']') - return ', '.join(s) + ')' - - def clear_flags(self): - """Reset all flags to zero""" - for flag in self.flags: - self.flags[flag] = 0 - - def clear_traps(self): - """Reset all traps to zero""" - for flag in self.traps: - self.traps[flag] = 0 - - def _shallow_copy(self): - """Returns a shallow copy from self.""" - nc = Context(self.prec, self.rounding, self.Emin, self.Emax, - self.capitals, self.clamp, self.flags, self.traps, - self._ignored_flags) - return nc - - def copy(self): - """Returns a deep copy from self.""" - nc = Context(self.prec, self.rounding, self.Emin, self.Emax, - self.capitals, self.clamp, - self.flags.copy(), self.traps.copy(), - self._ignored_flags) - return nc - __copy__ = copy - - def _raise_error(self, condition, explanation = None, *args): - """Handles an error - - If the flag is in _ignored_flags, returns the default response. - Otherwise, it sets the flag, then, if the corresponding - trap_enabler is set, it reraises the exception. Otherwise, it returns - the default value after setting the flag. - """ - error = _condition_map.get(condition, condition) - if error in self._ignored_flags: - # Don't touch the flag - return error().handle(self, *args) - - self.flags[error] = 1 - if not self.traps[error]: - # The errors define how to handle themselves. - return condition().handle(self, *args) - - # Errors should only be risked on copies of the context - # self._ignored_flags = [] - raise error(explanation) - - def _ignore_all_flags(self): - """Ignore all flags, if they are raised""" - return self._ignore_flags(*_signals) - - def _ignore_flags(self, *flags): - """Ignore the flags, if they are raised""" - # Do not mutate-- This way, copies of a context leave the original - # alone. - self._ignored_flags = (self._ignored_flags + list(flags)) - return list(flags) - - def _regard_flags(self, *flags): - """Stop ignoring the flags, if they are raised""" - if flags and isinstance(flags[0], (tuple,list)): - flags = flags[0] - for flag in flags: - self._ignored_flags.remove(flag) - - # We inherit object.__hash__, so we must deny this explicitly - __hash__ = None - - def Etiny(self): - """Returns Etiny (= Emin - prec + 1)""" - return int(self.Emin - self.prec + 1) - - def Etop(self): - """Returns maximum exponent (= Emax - prec + 1)""" - return int(self.Emax - self.prec + 1) - - def _set_rounding(self, type): - """Sets the rounding type. - - Sets the rounding type, and returns the current (previous) - rounding type. Often used like: - - context = context.copy() - # so you don't change the calling context - # if an error occurs in the middle. - rounding = context._set_rounding(ROUND_UP) - val = self.__sub__(other, context=context) - context._set_rounding(rounding) - - This will make it round up for that operation. - """ - rounding = self.rounding - self.rounding= type - return rounding - - def create_decimal(self, num='0'): - """Creates a new Decimal instance but using self as context. - - This method implements the to-number operation of the - IBM Decimal specification.""" - - if isinstance(num, str) and num != num.strip(): - return self._raise_error(ConversionSyntax, - "no trailing or leading whitespace is " - "permitted.") - - d = Decimal(num, context=self) - if d._isnan() and len(d._int) > self.prec - self.clamp: - return self._raise_error(ConversionSyntax, - "diagnostic info too long in NaN") - return d._fix(self) - - def create_decimal_from_float(self, f): - """Creates a new Decimal instance from a float but rounding using self - as the context. - - >>> context = Context(prec=5, rounding=ROUND_DOWN) - >>> context.create_decimal_from_float(3.1415926535897932) - Decimal('3.1415') - >>> context = Context(prec=5, traps=[Inexact]) - >>> context.create_decimal_from_float(3.1415926535897932) - Traceback (most recent call last): - ... - decimal.Inexact: None - - """ - d = Decimal.from_float(f) # An exact conversion - return d._fix(self) # Apply the context rounding - - # Methods - def abs(self, a): - """Returns the absolute value of the operand. - - If the operand is negative, the result is the same as using the minus - operation on the operand. Otherwise, the result is the same as using - the plus operation on the operand. - - >>> ExtendedContext.abs(Decimal('2.1')) - Decimal('2.1') - >>> ExtendedContext.abs(Decimal('-100')) - Decimal('100') - >>> ExtendedContext.abs(Decimal('101.5')) - Decimal('101.5') - >>> ExtendedContext.abs(Decimal('-101.5')) - Decimal('101.5') - >>> ExtendedContext.abs(-1) - Decimal('1') - """ - a = _convert_other(a, raiseit=True) - return a.__abs__(context=self) - - def add(self, a, b): - """Return the sum of the two operands. - - >>> ExtendedContext.add(Decimal('12'), Decimal('7.00')) - Decimal('19.00') - >>> ExtendedContext.add(Decimal('1E+2'), Decimal('1.01E+4')) - Decimal('1.02E+4') - >>> ExtendedContext.add(1, Decimal(2)) - Decimal('3') - >>> ExtendedContext.add(Decimal(8), 5) - Decimal('13') - >>> ExtendedContext.add(5, 5) - Decimal('10') - """ - a = _convert_other(a, raiseit=True) - r = a.__add__(b, context=self) - if r is NotImplemented: - raise TypeError("Unable to convert %s to Decimal" % b) - else: - return r - - def _apply(self, a): - return str(a._fix(self)) - - def canonical(self, a): - """Returns the same Decimal object. - - As we do not have different encodings for the same number, the - received object already is in its canonical form. - - >>> ExtendedContext.canonical(Decimal('2.50')) - Decimal('2.50') - """ - if not isinstance(a, Decimal): - raise TypeError("canonical requires a Decimal as an argument.") - return a.canonical() - - def compare(self, a, b): - """Compares values numerically. - - If the signs of the operands differ, a value representing each operand - ('-1' if the operand is less than zero, '0' if the operand is zero or - negative zero, or '1' if the operand is greater than zero) is used in - place of that operand for the comparison instead of the actual - operand. - - The comparison is then effected by subtracting the second operand from - the first and then returning a value according to the result of the - subtraction: '-1' if the result is less than zero, '0' if the result is - zero or negative zero, or '1' if the result is greater than zero. - - >>> ExtendedContext.compare(Decimal('2.1'), Decimal('3')) - Decimal('-1') - >>> ExtendedContext.compare(Decimal('2.1'), Decimal('2.1')) - Decimal('0') - >>> ExtendedContext.compare(Decimal('2.1'), Decimal('2.10')) - Decimal('0') - >>> ExtendedContext.compare(Decimal('3'), Decimal('2.1')) - Decimal('1') - >>> ExtendedContext.compare(Decimal('2.1'), Decimal('-3')) - Decimal('1') - >>> ExtendedContext.compare(Decimal('-3'), Decimal('2.1')) - Decimal('-1') - >>> ExtendedContext.compare(1, 2) - Decimal('-1') - >>> ExtendedContext.compare(Decimal(1), 2) - Decimal('-1') - >>> ExtendedContext.compare(1, Decimal(2)) - Decimal('-1') - """ - a = _convert_other(a, raiseit=True) - return a.compare(b, context=self) - - def compare_signal(self, a, b): - """Compares the values of the two operands numerically. - - It's pretty much like compare(), but all NaNs signal, with signaling - NaNs taking precedence over quiet NaNs. - - >>> c = ExtendedContext - >>> c.compare_signal(Decimal('2.1'), Decimal('3')) - Decimal('-1') - >>> c.compare_signal(Decimal('2.1'), Decimal('2.1')) - Decimal('0') - >>> c.flags[InvalidOperation] = 0 - >>> print(c.flags[InvalidOperation]) - 0 - >>> c.compare_signal(Decimal('NaN'), Decimal('2.1')) - Decimal('NaN') - >>> print(c.flags[InvalidOperation]) - 1 - >>> c.flags[InvalidOperation] = 0 - >>> print(c.flags[InvalidOperation]) - 0 - >>> c.compare_signal(Decimal('sNaN'), Decimal('2.1')) - Decimal('NaN') - >>> print(c.flags[InvalidOperation]) - 1 - >>> c.compare_signal(-1, 2) - Decimal('-1') - >>> c.compare_signal(Decimal(-1), 2) - Decimal('-1') - >>> c.compare_signal(-1, Decimal(2)) - Decimal('-1') - """ - a = _convert_other(a, raiseit=True) - return a.compare_signal(b, context=self) - - def compare_total(self, a, b): - """Compares two operands using their abstract representation. - - This is not like the standard compare, which use their numerical - value. Note that a total ordering is defined for all possible abstract - representations. - - >>> ExtendedContext.compare_total(Decimal('12.73'), Decimal('127.9')) - Decimal('-1') - >>> ExtendedContext.compare_total(Decimal('-127'), Decimal('12')) - Decimal('-1') - >>> ExtendedContext.compare_total(Decimal('12.30'), Decimal('12.3')) - Decimal('-1') - >>> ExtendedContext.compare_total(Decimal('12.30'), Decimal('12.30')) - Decimal('0') - >>> ExtendedContext.compare_total(Decimal('12.3'), Decimal('12.300')) - Decimal('1') - >>> ExtendedContext.compare_total(Decimal('12.3'), Decimal('NaN')) - Decimal('-1') - >>> ExtendedContext.compare_total(1, 2) - Decimal('-1') - >>> ExtendedContext.compare_total(Decimal(1), 2) - Decimal('-1') - >>> ExtendedContext.compare_total(1, Decimal(2)) - Decimal('-1') - """ - a = _convert_other(a, raiseit=True) - return a.compare_total(b) - - def compare_total_mag(self, a, b): - """Compares two operands using their abstract representation ignoring sign. - - Like compare_total, but with operand's sign ignored and assumed to be 0. - """ - a = _convert_other(a, raiseit=True) - return a.compare_total_mag(b) - - def copy_abs(self, a): - """Returns a copy of the operand with the sign set to 0. - - >>> ExtendedContext.copy_abs(Decimal('2.1')) - Decimal('2.1') - >>> ExtendedContext.copy_abs(Decimal('-100')) - Decimal('100') - >>> ExtendedContext.copy_abs(-1) - Decimal('1') - """ - a = _convert_other(a, raiseit=True) - return a.copy_abs() - - def copy_decimal(self, a): - """Returns a copy of the decimal object. - - >>> ExtendedContext.copy_decimal(Decimal('2.1')) - Decimal('2.1') - >>> ExtendedContext.copy_decimal(Decimal('-1.00')) - Decimal('-1.00') - >>> ExtendedContext.copy_decimal(1) - Decimal('1') - """ - a = _convert_other(a, raiseit=True) - return Decimal(a) - - def copy_negate(self, a): - """Returns a copy of the operand with the sign inverted. - - >>> ExtendedContext.copy_negate(Decimal('101.5')) - Decimal('-101.5') - >>> ExtendedContext.copy_negate(Decimal('-101.5')) - Decimal('101.5') - >>> ExtendedContext.copy_negate(1) - Decimal('-1') - """ - a = _convert_other(a, raiseit=True) - return a.copy_negate() - - def copy_sign(self, a, b): - """Copies the second operand's sign to the first one. - - In detail, it returns a copy of the first operand with the sign - equal to the sign of the second operand. - - >>> ExtendedContext.copy_sign(Decimal( '1.50'), Decimal('7.33')) - Decimal('1.50') - >>> ExtendedContext.copy_sign(Decimal('-1.50'), Decimal('7.33')) - Decimal('1.50') - >>> ExtendedContext.copy_sign(Decimal( '1.50'), Decimal('-7.33')) - Decimal('-1.50') - >>> ExtendedContext.copy_sign(Decimal('-1.50'), Decimal('-7.33')) - Decimal('-1.50') - >>> ExtendedContext.copy_sign(1, -2) - Decimal('-1') - >>> ExtendedContext.copy_sign(Decimal(1), -2) - Decimal('-1') - >>> ExtendedContext.copy_sign(1, Decimal(-2)) - Decimal('-1') - """ - a = _convert_other(a, raiseit=True) - return a.copy_sign(b) - - def divide(self, a, b): - """Decimal division in a specified context. - - >>> ExtendedContext.divide(Decimal('1'), Decimal('3')) - Decimal('0.333333333') - >>> ExtendedContext.divide(Decimal('2'), Decimal('3')) - Decimal('0.666666667') - >>> ExtendedContext.divide(Decimal('5'), Decimal('2')) - Decimal('2.5') - >>> ExtendedContext.divide(Decimal('1'), Decimal('10')) - Decimal('0.1') - >>> ExtendedContext.divide(Decimal('12'), Decimal('12')) - Decimal('1') - >>> ExtendedContext.divide(Decimal('8.00'), Decimal('2')) - Decimal('4.00') - >>> ExtendedContext.divide(Decimal('2.400'), Decimal('2.0')) - Decimal('1.20') - >>> ExtendedContext.divide(Decimal('1000'), Decimal('100')) - Decimal('10') - >>> ExtendedContext.divide(Decimal('1000'), Decimal('1')) - Decimal('1000') - >>> ExtendedContext.divide(Decimal('2.40E+6'), Decimal('2')) - Decimal('1.20E+6') - >>> ExtendedContext.divide(5, 5) - Decimal('1') - >>> ExtendedContext.divide(Decimal(5), 5) - Decimal('1') - >>> ExtendedContext.divide(5, Decimal(5)) - Decimal('1') - """ - a = _convert_other(a, raiseit=True) - r = a.__truediv__(b, context=self) - if r is NotImplemented: - raise TypeError("Unable to convert %s to Decimal" % b) - else: - return r - - def divide_int(self, a, b): - """Divides two numbers and returns the integer part of the result. - - >>> ExtendedContext.divide_int(Decimal('2'), Decimal('3')) - Decimal('0') - >>> ExtendedContext.divide_int(Decimal('10'), Decimal('3')) - Decimal('3') - >>> ExtendedContext.divide_int(Decimal('1'), Decimal('0.3')) - Decimal('3') - >>> ExtendedContext.divide_int(10, 3) - Decimal('3') - >>> ExtendedContext.divide_int(Decimal(10), 3) - Decimal('3') - >>> ExtendedContext.divide_int(10, Decimal(3)) - Decimal('3') - """ - a = _convert_other(a, raiseit=True) - r = a.__floordiv__(b, context=self) - if r is NotImplemented: - raise TypeError("Unable to convert %s to Decimal" % b) - else: - return r - - def divmod(self, a, b): - """Return (a // b, a % b). - - >>> ExtendedContext.divmod(Decimal(8), Decimal(3)) - (Decimal('2'), Decimal('2')) - >>> ExtendedContext.divmod(Decimal(8), Decimal(4)) - (Decimal('2'), Decimal('0')) - >>> ExtendedContext.divmod(8, 4) - (Decimal('2'), Decimal('0')) - >>> ExtendedContext.divmod(Decimal(8), 4) - (Decimal('2'), Decimal('0')) - >>> ExtendedContext.divmod(8, Decimal(4)) - (Decimal('2'), Decimal('0')) - """ - a = _convert_other(a, raiseit=True) - r = a.__divmod__(b, context=self) - if r is NotImplemented: - raise TypeError("Unable to convert %s to Decimal" % b) - else: - return r - - def exp(self, a): - """Returns e ** a. - - >>> c = ExtendedContext.copy() - >>> c.Emin = -999 - >>> c.Emax = 999 - >>> c.exp(Decimal('-Infinity')) - Decimal('0') - >>> c.exp(Decimal('-1')) - Decimal('0.367879441') - >>> c.exp(Decimal('0')) - Decimal('1') - >>> c.exp(Decimal('1')) - Decimal('2.71828183') - >>> c.exp(Decimal('0.693147181')) - Decimal('2.00000000') - >>> c.exp(Decimal('+Infinity')) - Decimal('Infinity') - >>> c.exp(10) - Decimal('22026.4658') - """ - a =_convert_other(a, raiseit=True) - return a.exp(context=self) - - def fma(self, a, b, c): - """Returns a multiplied by b, plus c. - - The first two operands are multiplied together, using multiply, - the third operand is then added to the result of that - multiplication, using add, all with only one final rounding. - - >>> ExtendedContext.fma(Decimal('3'), Decimal('5'), Decimal('7')) - Decimal('22') - >>> ExtendedContext.fma(Decimal('3'), Decimal('-5'), Decimal('7')) - Decimal('-8') - >>> ExtendedContext.fma(Decimal('888565290'), Decimal('1557.96930'), Decimal('-86087.7578')) - Decimal('1.38435736E+12') - >>> ExtendedContext.fma(1, 3, 4) - Decimal('7') - >>> ExtendedContext.fma(1, Decimal(3), 4) - Decimal('7') - >>> ExtendedContext.fma(1, 3, Decimal(4)) - Decimal('7') - """ - a = _convert_other(a, raiseit=True) - return a.fma(b, c, context=self) - - def is_canonical(self, a): - """Return True if the operand is canonical; otherwise return False. - - Currently, the encoding of a Decimal instance is always - canonical, so this method returns True for any Decimal. - - >>> ExtendedContext.is_canonical(Decimal('2.50')) - True - """ - if not isinstance(a, Decimal): - raise TypeError("is_canonical requires a Decimal as an argument.") - return a.is_canonical() - - def is_finite(self, a): - """Return True if the operand is finite; otherwise return False. - - A Decimal instance is considered finite if it is neither - infinite nor a NaN. - - >>> ExtendedContext.is_finite(Decimal('2.50')) - True - >>> ExtendedContext.is_finite(Decimal('-0.3')) - True - >>> ExtendedContext.is_finite(Decimal('0')) - True - >>> ExtendedContext.is_finite(Decimal('Inf')) - False - >>> ExtendedContext.is_finite(Decimal('NaN')) - False - >>> ExtendedContext.is_finite(1) - True - """ - a = _convert_other(a, raiseit=True) - return a.is_finite() - - def is_infinite(self, a): - """Return True if the operand is infinite; otherwise return False. - - >>> ExtendedContext.is_infinite(Decimal('2.50')) - False - >>> ExtendedContext.is_infinite(Decimal('-Inf')) - True - >>> ExtendedContext.is_infinite(Decimal('NaN')) - False - >>> ExtendedContext.is_infinite(1) - False - """ - a = _convert_other(a, raiseit=True) - return a.is_infinite() - - def is_nan(self, a): - """Return True if the operand is a qNaN or sNaN; - otherwise return False. - - >>> ExtendedContext.is_nan(Decimal('2.50')) - False - >>> ExtendedContext.is_nan(Decimal('NaN')) - True - >>> ExtendedContext.is_nan(Decimal('-sNaN')) - True - >>> ExtendedContext.is_nan(1) - False - """ - a = _convert_other(a, raiseit=True) - return a.is_nan() - - def is_normal(self, a): - """Return True if the operand is a normal number; - otherwise return False. - - >>> c = ExtendedContext.copy() - >>> c.Emin = -999 - >>> c.Emax = 999 - >>> c.is_normal(Decimal('2.50')) - True - >>> c.is_normal(Decimal('0.1E-999')) - False - >>> c.is_normal(Decimal('0.00')) - False - >>> c.is_normal(Decimal('-Inf')) - False - >>> c.is_normal(Decimal('NaN')) - False - >>> c.is_normal(1) - True - """ - a = _convert_other(a, raiseit=True) - return a.is_normal(context=self) - - def is_qnan(self, a): - """Return True if the operand is a quiet NaN; otherwise return False. - - >>> ExtendedContext.is_qnan(Decimal('2.50')) - False - >>> ExtendedContext.is_qnan(Decimal('NaN')) - True - >>> ExtendedContext.is_qnan(Decimal('sNaN')) - False - >>> ExtendedContext.is_qnan(1) - False - """ - a = _convert_other(a, raiseit=True) - return a.is_qnan() - - def is_signed(self, a): - """Return True if the operand is negative; otherwise return False. - - >>> ExtendedContext.is_signed(Decimal('2.50')) - False - >>> ExtendedContext.is_signed(Decimal('-12')) - True - >>> ExtendedContext.is_signed(Decimal('-0')) - True - >>> ExtendedContext.is_signed(8) - False - >>> ExtendedContext.is_signed(-8) - True - """ - a = _convert_other(a, raiseit=True) - return a.is_signed() - - def is_snan(self, a): - """Return True if the operand is a signaling NaN; - otherwise return False. - - >>> ExtendedContext.is_snan(Decimal('2.50')) - False - >>> ExtendedContext.is_snan(Decimal('NaN')) - False - >>> ExtendedContext.is_snan(Decimal('sNaN')) - True - >>> ExtendedContext.is_snan(1) - False - """ - a = _convert_other(a, raiseit=True) - return a.is_snan() - - def is_subnormal(self, a): - """Return True if the operand is subnormal; otherwise return False. - - >>> c = ExtendedContext.copy() - >>> c.Emin = -999 - >>> c.Emax = 999 - >>> c.is_subnormal(Decimal('2.50')) - False - >>> c.is_subnormal(Decimal('0.1E-999')) - True - >>> c.is_subnormal(Decimal('0.00')) - False - >>> c.is_subnormal(Decimal('-Inf')) - False - >>> c.is_subnormal(Decimal('NaN')) - False - >>> c.is_subnormal(1) - False - """ - a = _convert_other(a, raiseit=True) - return a.is_subnormal(context=self) - - def is_zero(self, a): - """Return True if the operand is a zero; otherwise return False. - - >>> ExtendedContext.is_zero(Decimal('0')) - True - >>> ExtendedContext.is_zero(Decimal('2.50')) - False - >>> ExtendedContext.is_zero(Decimal('-0E+2')) - True - >>> ExtendedContext.is_zero(1) - False - >>> ExtendedContext.is_zero(0) - True - """ - a = _convert_other(a, raiseit=True) - return a.is_zero() - - def ln(self, a): - """Returns the natural (base e) logarithm of the operand. - - >>> c = ExtendedContext.copy() - >>> c.Emin = -999 - >>> c.Emax = 999 - >>> c.ln(Decimal('0')) - Decimal('-Infinity') - >>> c.ln(Decimal('1.000')) - Decimal('0') - >>> c.ln(Decimal('2.71828183')) - Decimal('1.00000000') - >>> c.ln(Decimal('10')) - Decimal('2.30258509') - >>> c.ln(Decimal('+Infinity')) - Decimal('Infinity') - >>> c.ln(1) - Decimal('0') - """ - a = _convert_other(a, raiseit=True) - return a.ln(context=self) - - def log10(self, a): - """Returns the base 10 logarithm of the operand. - - >>> c = ExtendedContext.copy() - >>> c.Emin = -999 - >>> c.Emax = 999 - >>> c.log10(Decimal('0')) - Decimal('-Infinity') - >>> c.log10(Decimal('0.001')) - Decimal('-3') - >>> c.log10(Decimal('1.000')) - Decimal('0') - >>> c.log10(Decimal('2')) - Decimal('0.301029996') - >>> c.log10(Decimal('10')) - Decimal('1') - >>> c.log10(Decimal('70')) - Decimal('1.84509804') - >>> c.log10(Decimal('+Infinity')) - Decimal('Infinity') - >>> c.log10(0) - Decimal('-Infinity') - >>> c.log10(1) - Decimal('0') - """ - a = _convert_other(a, raiseit=True) - return a.log10(context=self) - - def logb(self, a): - """ Returns the exponent of the magnitude of the operand's MSD. - - The result is the integer which is the exponent of the magnitude - of the most significant digit of the operand (as though the - operand were truncated to a single digit while maintaining the - value of that digit and without limiting the resulting exponent). - - >>> ExtendedContext.logb(Decimal('250')) - Decimal('2') - >>> ExtendedContext.logb(Decimal('2.50')) - Decimal('0') - >>> ExtendedContext.logb(Decimal('0.03')) - Decimal('-2') - >>> ExtendedContext.logb(Decimal('0')) - Decimal('-Infinity') - >>> ExtendedContext.logb(1) - Decimal('0') - >>> ExtendedContext.logb(10) - Decimal('1') - >>> ExtendedContext.logb(100) - Decimal('2') - """ - a = _convert_other(a, raiseit=True) - return a.logb(context=self) - - def logical_and(self, a, b): - """Applies the logical operation 'and' between each operand's digits. - - The operands must be both logical numbers. - - >>> ExtendedContext.logical_and(Decimal('0'), Decimal('0')) - Decimal('0') - >>> ExtendedContext.logical_and(Decimal('0'), Decimal('1')) - Decimal('0') - >>> ExtendedContext.logical_and(Decimal('1'), Decimal('0')) - Decimal('0') - >>> ExtendedContext.logical_and(Decimal('1'), Decimal('1')) - Decimal('1') - >>> ExtendedContext.logical_and(Decimal('1100'), Decimal('1010')) - Decimal('1000') - >>> ExtendedContext.logical_and(Decimal('1111'), Decimal('10')) - Decimal('10') - >>> ExtendedContext.logical_and(110, 1101) - Decimal('100') - >>> ExtendedContext.logical_and(Decimal(110), 1101) - Decimal('100') - >>> ExtendedContext.logical_and(110, Decimal(1101)) - Decimal('100') - """ - a = _convert_other(a, raiseit=True) - return a.logical_and(b, context=self) - - def logical_invert(self, a): - """Invert all the digits in the operand. - - The operand must be a logical number. - - >>> ExtendedContext.logical_invert(Decimal('0')) - Decimal('111111111') - >>> ExtendedContext.logical_invert(Decimal('1')) - Decimal('111111110') - >>> ExtendedContext.logical_invert(Decimal('111111111')) - Decimal('0') - >>> ExtendedContext.logical_invert(Decimal('101010101')) - Decimal('10101010') - >>> ExtendedContext.logical_invert(1101) - Decimal('111110010') - """ - a = _convert_other(a, raiseit=True) - return a.logical_invert(context=self) - - def logical_or(self, a, b): - """Applies the logical operation 'or' between each operand's digits. - - The operands must be both logical numbers. - - >>> ExtendedContext.logical_or(Decimal('0'), Decimal('0')) - Decimal('0') - >>> ExtendedContext.logical_or(Decimal('0'), Decimal('1')) - Decimal('1') - >>> ExtendedContext.logical_or(Decimal('1'), Decimal('0')) - Decimal('1') - >>> ExtendedContext.logical_or(Decimal('1'), Decimal('1')) - Decimal('1') - >>> ExtendedContext.logical_or(Decimal('1100'), Decimal('1010')) - Decimal('1110') - >>> ExtendedContext.logical_or(Decimal('1110'), Decimal('10')) - Decimal('1110') - >>> ExtendedContext.logical_or(110, 1101) - Decimal('1111') - >>> ExtendedContext.logical_or(Decimal(110), 1101) - Decimal('1111') - >>> ExtendedContext.logical_or(110, Decimal(1101)) - Decimal('1111') - """ - a = _convert_other(a, raiseit=True) - return a.logical_or(b, context=self) - - def logical_xor(self, a, b): - """Applies the logical operation 'xor' between each operand's digits. - - The operands must be both logical numbers. - - >>> ExtendedContext.logical_xor(Decimal('0'), Decimal('0')) - Decimal('0') - >>> ExtendedContext.logical_xor(Decimal('0'), Decimal('1')) - Decimal('1') - >>> ExtendedContext.logical_xor(Decimal('1'), Decimal('0')) - Decimal('1') - >>> ExtendedContext.logical_xor(Decimal('1'), Decimal('1')) - Decimal('0') - >>> ExtendedContext.logical_xor(Decimal('1100'), Decimal('1010')) - Decimal('110') - >>> ExtendedContext.logical_xor(Decimal('1111'), Decimal('10')) - Decimal('1101') - >>> ExtendedContext.logical_xor(110, 1101) - Decimal('1011') - >>> ExtendedContext.logical_xor(Decimal(110), 1101) - Decimal('1011') - >>> ExtendedContext.logical_xor(110, Decimal(1101)) - Decimal('1011') - """ - a = _convert_other(a, raiseit=True) - return a.logical_xor(b, context=self) - - def max(self, a, b): - """max compares two values numerically and returns the maximum. - - If either operand is a NaN then the general rules apply. - Otherwise, the operands are compared as though by the compare - operation. If they are numerically equal then the left-hand operand - is chosen as the result. Otherwise the maximum (closer to positive - infinity) of the two operands is chosen as the result. - - >>> ExtendedContext.max(Decimal('3'), Decimal('2')) - Decimal('3') - >>> ExtendedContext.max(Decimal('-10'), Decimal('3')) - Decimal('3') - >>> ExtendedContext.max(Decimal('1.0'), Decimal('1')) - Decimal('1') - >>> ExtendedContext.max(Decimal('7'), Decimal('NaN')) - Decimal('7') - >>> ExtendedContext.max(1, 2) - Decimal('2') - >>> ExtendedContext.max(Decimal(1), 2) - Decimal('2') - >>> ExtendedContext.max(1, Decimal(2)) - Decimal('2') - """ - a = _convert_other(a, raiseit=True) - return a.max(b, context=self) - - def max_mag(self, a, b): - """Compares the values numerically with their sign ignored. - - >>> ExtendedContext.max_mag(Decimal('7'), Decimal('NaN')) - Decimal('7') - >>> ExtendedContext.max_mag(Decimal('7'), Decimal('-10')) - Decimal('-10') - >>> ExtendedContext.max_mag(1, -2) - Decimal('-2') - >>> ExtendedContext.max_mag(Decimal(1), -2) - Decimal('-2') - >>> ExtendedContext.max_mag(1, Decimal(-2)) - Decimal('-2') - """ - a = _convert_other(a, raiseit=True) - return a.max_mag(b, context=self) - - def min(self, a, b): - """min compares two values numerically and returns the minimum. - - If either operand is a NaN then the general rules apply. - Otherwise, the operands are compared as though by the compare - operation. If they are numerically equal then the left-hand operand - is chosen as the result. Otherwise the minimum (closer to negative - infinity) of the two operands is chosen as the result. - - >>> ExtendedContext.min(Decimal('3'), Decimal('2')) - Decimal('2') - >>> ExtendedContext.min(Decimal('-10'), Decimal('3')) - Decimal('-10') - >>> ExtendedContext.min(Decimal('1.0'), Decimal('1')) - Decimal('1.0') - >>> ExtendedContext.min(Decimal('7'), Decimal('NaN')) - Decimal('7') - >>> ExtendedContext.min(1, 2) - Decimal('1') - >>> ExtendedContext.min(Decimal(1), 2) - Decimal('1') - >>> ExtendedContext.min(1, Decimal(29)) - Decimal('1') - """ - a = _convert_other(a, raiseit=True) - return a.min(b, context=self) - - def min_mag(self, a, b): - """Compares the values numerically with their sign ignored. - - >>> ExtendedContext.min_mag(Decimal('3'), Decimal('-2')) - Decimal('-2') - >>> ExtendedContext.min_mag(Decimal('-3'), Decimal('NaN')) - Decimal('-3') - >>> ExtendedContext.min_mag(1, -2) - Decimal('1') - >>> ExtendedContext.min_mag(Decimal(1), -2) - Decimal('1') - >>> ExtendedContext.min_mag(1, Decimal(-2)) - Decimal('1') - """ - a = _convert_other(a, raiseit=True) - return a.min_mag(b, context=self) - - def minus(self, a): - """Minus corresponds to unary prefix minus in Python. - - The operation is evaluated using the same rules as subtract; the - operation minus(a) is calculated as subtract('0', a) where the '0' - has the same exponent as the operand. - - >>> ExtendedContext.minus(Decimal('1.3')) - Decimal('-1.3') - >>> ExtendedContext.minus(Decimal('-1.3')) - Decimal('1.3') - >>> ExtendedContext.minus(1) - Decimal('-1') - """ - a = _convert_other(a, raiseit=True) - return a.__neg__(context=self) - - def multiply(self, a, b): - """multiply multiplies two operands. - - If either operand is a special value then the general rules apply. - Otherwise, the operands are multiplied together - ('long multiplication'), resulting in a number which may be as long as - the sum of the lengths of the two operands. - - >>> ExtendedContext.multiply(Decimal('1.20'), Decimal('3')) - Decimal('3.60') - >>> ExtendedContext.multiply(Decimal('7'), Decimal('3')) - Decimal('21') - >>> ExtendedContext.multiply(Decimal('0.9'), Decimal('0.8')) - Decimal('0.72') - >>> ExtendedContext.multiply(Decimal('0.9'), Decimal('-0')) - Decimal('-0.0') - >>> ExtendedContext.multiply(Decimal('654321'), Decimal('654321')) - Decimal('4.28135971E+11') - >>> ExtendedContext.multiply(7, 7) - Decimal('49') - >>> ExtendedContext.multiply(Decimal(7), 7) - Decimal('49') - >>> ExtendedContext.multiply(7, Decimal(7)) - Decimal('49') - """ - a = _convert_other(a, raiseit=True) - r = a.__mul__(b, context=self) - if r is NotImplemented: - raise TypeError("Unable to convert %s to Decimal" % b) - else: - return r - - def next_minus(self, a): - """Returns the largest representable number smaller than a. - - >>> c = ExtendedContext.copy() - >>> c.Emin = -999 - >>> c.Emax = 999 - >>> ExtendedContext.next_minus(Decimal('1')) - Decimal('0.999999999') - >>> c.next_minus(Decimal('1E-1007')) - Decimal('0E-1007') - >>> ExtendedContext.next_minus(Decimal('-1.00000003')) - Decimal('-1.00000004') - >>> c.next_minus(Decimal('Infinity')) - Decimal('9.99999999E+999') - >>> c.next_minus(1) - Decimal('0.999999999') - """ - a = _convert_other(a, raiseit=True) - return a.next_minus(context=self) - - def next_plus(self, a): - """Returns the smallest representable number larger than a. - - >>> c = ExtendedContext.copy() - >>> c.Emin = -999 - >>> c.Emax = 999 - >>> ExtendedContext.next_plus(Decimal('1')) - Decimal('1.00000001') - >>> c.next_plus(Decimal('-1E-1007')) - Decimal('-0E-1007') - >>> ExtendedContext.next_plus(Decimal('-1.00000003')) - Decimal('-1.00000002') - >>> c.next_plus(Decimal('-Infinity')) - Decimal('-9.99999999E+999') - >>> c.next_plus(1) - Decimal('1.00000001') - """ - a = _convert_other(a, raiseit=True) - return a.next_plus(context=self) - - def next_toward(self, a, b): - """Returns the number closest to a, in direction towards b. - - The result is the closest representable number from the first - operand (but not the first operand) that is in the direction - towards the second operand, unless the operands have the same - value. - - >>> c = ExtendedContext.copy() - >>> c.Emin = -999 - >>> c.Emax = 999 - >>> c.next_toward(Decimal('1'), Decimal('2')) - Decimal('1.00000001') - >>> c.next_toward(Decimal('-1E-1007'), Decimal('1')) - Decimal('-0E-1007') - >>> c.next_toward(Decimal('-1.00000003'), Decimal('0')) - Decimal('-1.00000002') - >>> c.next_toward(Decimal('1'), Decimal('0')) - Decimal('0.999999999') - >>> c.next_toward(Decimal('1E-1007'), Decimal('-100')) - Decimal('0E-1007') - >>> c.next_toward(Decimal('-1.00000003'), Decimal('-10')) - Decimal('-1.00000004') - >>> c.next_toward(Decimal('0.00'), Decimal('-0.0000')) - Decimal('-0.00') - >>> c.next_toward(0, 1) - Decimal('1E-1007') - >>> c.next_toward(Decimal(0), 1) - Decimal('1E-1007') - >>> c.next_toward(0, Decimal(1)) - Decimal('1E-1007') - """ - a = _convert_other(a, raiseit=True) - return a.next_toward(b, context=self) - - def normalize(self, a): - """normalize reduces an operand to its simplest form. - - Essentially a plus operation with all trailing zeros removed from the - result. - - >>> ExtendedContext.normalize(Decimal('2.1')) - Decimal('2.1') - >>> ExtendedContext.normalize(Decimal('-2.0')) - Decimal('-2') - >>> ExtendedContext.normalize(Decimal('1.200')) - Decimal('1.2') - >>> ExtendedContext.normalize(Decimal('-120')) - Decimal('-1.2E+2') - >>> ExtendedContext.normalize(Decimal('120.00')) - Decimal('1.2E+2') - >>> ExtendedContext.normalize(Decimal('0.00')) - Decimal('0') - >>> ExtendedContext.normalize(6) - Decimal('6') - """ - a = _convert_other(a, raiseit=True) - return a.normalize(context=self) - - def number_class(self, a): - """Returns an indication of the class of the operand. - - The class is one of the following strings: - -sNaN - -NaN - -Infinity - -Normal - -Subnormal - -Zero - +Zero - +Subnormal - +Normal - +Infinity - - >>> c = ExtendedContext.copy() - >>> c.Emin = -999 - >>> c.Emax = 999 - >>> c.number_class(Decimal('Infinity')) - '+Infinity' - >>> c.number_class(Decimal('1E-10')) - '+Normal' - >>> c.number_class(Decimal('2.50')) - '+Normal' - >>> c.number_class(Decimal('0.1E-999')) - '+Subnormal' - >>> c.number_class(Decimal('0')) - '+Zero' - >>> c.number_class(Decimal('-0')) - '-Zero' - >>> c.number_class(Decimal('-0.1E-999')) - '-Subnormal' - >>> c.number_class(Decimal('-1E-10')) - '-Normal' - >>> c.number_class(Decimal('-2.50')) - '-Normal' - >>> c.number_class(Decimal('-Infinity')) - '-Infinity' - >>> c.number_class(Decimal('NaN')) - 'NaN' - >>> c.number_class(Decimal('-NaN')) - 'NaN' - >>> c.number_class(Decimal('sNaN')) - 'sNaN' - >>> c.number_class(123) - '+Normal' - """ - a = _convert_other(a, raiseit=True) - return a.number_class(context=self) - - def plus(self, a): - """Plus corresponds to unary prefix plus in Python. - - The operation is evaluated using the same rules as add; the - operation plus(a) is calculated as add('0', a) where the '0' - has the same exponent as the operand. - - >>> ExtendedContext.plus(Decimal('1.3')) - Decimal('1.3') - >>> ExtendedContext.plus(Decimal('-1.3')) - Decimal('-1.3') - >>> ExtendedContext.plus(-1) - Decimal('-1') - """ - a = _convert_other(a, raiseit=True) - return a.__pos__(context=self) - - def power(self, a, b, modulo=None): - """Raises a to the power of b, to modulo if given. - - With two arguments, compute a**b. If a is negative then b - must be integral. The result will be inexact unless b is - integral and the result is finite and can be expressed exactly - in 'precision' digits. - - With three arguments, compute (a**b) % modulo. For the - three argument form, the following restrictions on the - arguments hold: - - - all three arguments must be integral - - b must be nonnegative - - at least one of a or b must be nonzero - - modulo must be nonzero and have at most 'precision' digits - - The result of pow(a, b, modulo) is identical to the result - that would be obtained by computing (a**b) % modulo with - unbounded precision, but is computed more efficiently. It is - always exact. - - >>> c = ExtendedContext.copy() - >>> c.Emin = -999 - >>> c.Emax = 999 - >>> c.power(Decimal('2'), Decimal('3')) - Decimal('8') - >>> c.power(Decimal('-2'), Decimal('3')) - Decimal('-8') - >>> c.power(Decimal('2'), Decimal('-3')) - Decimal('0.125') - >>> c.power(Decimal('1.7'), Decimal('8')) - Decimal('69.7575744') - >>> c.power(Decimal('10'), Decimal('0.301029996')) - Decimal('2.00000000') - >>> c.power(Decimal('Infinity'), Decimal('-1')) - Decimal('0') - >>> c.power(Decimal('Infinity'), Decimal('0')) - Decimal('1') - >>> c.power(Decimal('Infinity'), Decimal('1')) - Decimal('Infinity') - >>> c.power(Decimal('-Infinity'), Decimal('-1')) - Decimal('-0') - >>> c.power(Decimal('-Infinity'), Decimal('0')) - Decimal('1') - >>> c.power(Decimal('-Infinity'), Decimal('1')) - Decimal('-Infinity') - >>> c.power(Decimal('-Infinity'), Decimal('2')) - Decimal('Infinity') - >>> c.power(Decimal('0'), Decimal('0')) - Decimal('NaN') - - >>> c.power(Decimal('3'), Decimal('7'), Decimal('16')) - Decimal('11') - >>> c.power(Decimal('-3'), Decimal('7'), Decimal('16')) - Decimal('-11') - >>> c.power(Decimal('-3'), Decimal('8'), Decimal('16')) - Decimal('1') - >>> c.power(Decimal('3'), Decimal('7'), Decimal('-16')) - Decimal('11') - >>> c.power(Decimal('23E12345'), Decimal('67E189'), Decimal('123456789')) - Decimal('11729830') - >>> c.power(Decimal('-0'), Decimal('17'), Decimal('1729')) - Decimal('-0') - >>> c.power(Decimal('-23'), Decimal('0'), Decimal('65537')) - Decimal('1') - >>> ExtendedContext.power(7, 7) - Decimal('823543') - >>> ExtendedContext.power(Decimal(7), 7) - Decimal('823543') - >>> ExtendedContext.power(7, Decimal(7), 2) - Decimal('1') - """ - a = _convert_other(a, raiseit=True) - r = a.__pow__(b, modulo, context=self) - if r is NotImplemented: - raise TypeError("Unable to convert %s to Decimal" % b) - else: - return r - - def quantize(self, a, b): - """Returns a value equal to 'a' (rounded), having the exponent of 'b'. - - The coefficient of the result is derived from that of the left-hand - operand. It may be rounded using the current rounding setting (if the - exponent is being increased), multiplied by a positive power of ten (if - the exponent is being decreased), or is unchanged (if the exponent is - already equal to that of the right-hand operand). - - Unlike other operations, if the length of the coefficient after the - quantize operation would be greater than precision then an Invalid - operation condition is raised. This guarantees that, unless there is - an error condition, the exponent of the result of a quantize is always - equal to that of the right-hand operand. - - Also unlike other operations, quantize will never raise Underflow, even - if the result is subnormal and inexact. - - >>> ExtendedContext.quantize(Decimal('2.17'), Decimal('0.001')) - Decimal('2.170') - >>> ExtendedContext.quantize(Decimal('2.17'), Decimal('0.01')) - Decimal('2.17') - >>> ExtendedContext.quantize(Decimal('2.17'), Decimal('0.1')) - Decimal('2.2') - >>> ExtendedContext.quantize(Decimal('2.17'), Decimal('1e+0')) - Decimal('2') - >>> ExtendedContext.quantize(Decimal('2.17'), Decimal('1e+1')) - Decimal('0E+1') - >>> ExtendedContext.quantize(Decimal('-Inf'), Decimal('Infinity')) - Decimal('-Infinity') - >>> ExtendedContext.quantize(Decimal('2'), Decimal('Infinity')) - Decimal('NaN') - >>> ExtendedContext.quantize(Decimal('-0.1'), Decimal('1')) - Decimal('-0') - >>> ExtendedContext.quantize(Decimal('-0'), Decimal('1e+5')) - Decimal('-0E+5') - >>> ExtendedContext.quantize(Decimal('+35236450.6'), Decimal('1e-2')) - Decimal('NaN') - >>> ExtendedContext.quantize(Decimal('-35236450.6'), Decimal('1e-2')) - Decimal('NaN') - >>> ExtendedContext.quantize(Decimal('217'), Decimal('1e-1')) - Decimal('217.0') - >>> ExtendedContext.quantize(Decimal('217'), Decimal('1e-0')) - Decimal('217') - >>> ExtendedContext.quantize(Decimal('217'), Decimal('1e+1')) - Decimal('2.2E+2') - >>> ExtendedContext.quantize(Decimal('217'), Decimal('1e+2')) - Decimal('2E+2') - >>> ExtendedContext.quantize(1, 2) - Decimal('1') - >>> ExtendedContext.quantize(Decimal(1), 2) - Decimal('1') - >>> ExtendedContext.quantize(1, Decimal(2)) - Decimal('1') - """ - a = _convert_other(a, raiseit=True) - return a.quantize(b, context=self) - - def radix(self): - """Just returns 10, as this is Decimal, :) - - >>> ExtendedContext.radix() - Decimal('10') - """ - return Decimal(10) - - def remainder(self, a, b): - """Returns the remainder from integer division. - - The result is the residue of the dividend after the operation of - calculating integer division as described for divide-integer, rounded - to precision digits if necessary. The sign of the result, if - non-zero, is the same as that of the original dividend. - - This operation will fail under the same conditions as integer division - (that is, if integer division on the same two operands would fail, the - remainder cannot be calculated). - - >>> ExtendedContext.remainder(Decimal('2.1'), Decimal('3')) - Decimal('2.1') - >>> ExtendedContext.remainder(Decimal('10'), Decimal('3')) - Decimal('1') - >>> ExtendedContext.remainder(Decimal('-10'), Decimal('3')) - Decimal('-1') - >>> ExtendedContext.remainder(Decimal('10.2'), Decimal('1')) - Decimal('0.2') - >>> ExtendedContext.remainder(Decimal('10'), Decimal('0.3')) - Decimal('0.1') - >>> ExtendedContext.remainder(Decimal('3.6'), Decimal('1.3')) - Decimal('1.0') - >>> ExtendedContext.remainder(22, 6) - Decimal('4') - >>> ExtendedContext.remainder(Decimal(22), 6) - Decimal('4') - >>> ExtendedContext.remainder(22, Decimal(6)) - Decimal('4') - """ - a = _convert_other(a, raiseit=True) - r = a.__mod__(b, context=self) - if r is NotImplemented: - raise TypeError("Unable to convert %s to Decimal" % b) - else: - return r - - def remainder_near(self, a, b): - """Returns to be "a - b * n", where n is the integer nearest the exact - value of "x / b" (if two integers are equally near then the even one - is chosen). If the result is equal to 0 then its sign will be the - sign of a. - - This operation will fail under the same conditions as integer division - (that is, if integer division on the same two operands would fail, the - remainder cannot be calculated). - - >>> ExtendedContext.remainder_near(Decimal('2.1'), Decimal('3')) - Decimal('-0.9') - >>> ExtendedContext.remainder_near(Decimal('10'), Decimal('6')) - Decimal('-2') - >>> ExtendedContext.remainder_near(Decimal('10'), Decimal('3')) - Decimal('1') - >>> ExtendedContext.remainder_near(Decimal('-10'), Decimal('3')) - Decimal('-1') - >>> ExtendedContext.remainder_near(Decimal('10.2'), Decimal('1')) - Decimal('0.2') - >>> ExtendedContext.remainder_near(Decimal('10'), Decimal('0.3')) - Decimal('0.1') - >>> ExtendedContext.remainder_near(Decimal('3.6'), Decimal('1.3')) - Decimal('-0.3') - >>> ExtendedContext.remainder_near(3, 11) - Decimal('3') - >>> ExtendedContext.remainder_near(Decimal(3), 11) - Decimal('3') - >>> ExtendedContext.remainder_near(3, Decimal(11)) - Decimal('3') - """ - a = _convert_other(a, raiseit=True) - return a.remainder_near(b, context=self) - - def rotate(self, a, b): - """Returns a rotated copy of a, b times. - - The coefficient of the result is a rotated copy of the digits in - the coefficient of the first operand. The number of places of - rotation is taken from the absolute value of the second operand, - with the rotation being to the left if the second operand is - positive or to the right otherwise. - - >>> ExtendedContext.rotate(Decimal('34'), Decimal('8')) - Decimal('400000003') - >>> ExtendedContext.rotate(Decimal('12'), Decimal('9')) - Decimal('12') - >>> ExtendedContext.rotate(Decimal('123456789'), Decimal('-2')) - Decimal('891234567') - >>> ExtendedContext.rotate(Decimal('123456789'), Decimal('0')) - Decimal('123456789') - >>> ExtendedContext.rotate(Decimal('123456789'), Decimal('+2')) - Decimal('345678912') - >>> ExtendedContext.rotate(1333333, 1) - Decimal('13333330') - >>> ExtendedContext.rotate(Decimal(1333333), 1) - Decimal('13333330') - >>> ExtendedContext.rotate(1333333, Decimal(1)) - Decimal('13333330') - """ - a = _convert_other(a, raiseit=True) - return a.rotate(b, context=self) - - def same_quantum(self, a, b): - """Returns True if the two operands have the same exponent. - - The result is never affected by either the sign or the coefficient of - either operand. - - >>> ExtendedContext.same_quantum(Decimal('2.17'), Decimal('0.001')) - False - >>> ExtendedContext.same_quantum(Decimal('2.17'), Decimal('0.01')) - True - >>> ExtendedContext.same_quantum(Decimal('2.17'), Decimal('1')) - False - >>> ExtendedContext.same_quantum(Decimal('Inf'), Decimal('-Inf')) - True - >>> ExtendedContext.same_quantum(10000, -1) - True - >>> ExtendedContext.same_quantum(Decimal(10000), -1) - True - >>> ExtendedContext.same_quantum(10000, Decimal(-1)) - True - """ - a = _convert_other(a, raiseit=True) - return a.same_quantum(b) - - def scaleb (self, a, b): - """Returns the first operand after adding the second value its exp. - - >>> ExtendedContext.scaleb(Decimal('7.50'), Decimal('-2')) - Decimal('0.0750') - >>> ExtendedContext.scaleb(Decimal('7.50'), Decimal('0')) - Decimal('7.50') - >>> ExtendedContext.scaleb(Decimal('7.50'), Decimal('3')) - Decimal('7.50E+3') - >>> ExtendedContext.scaleb(1, 4) - Decimal('1E+4') - >>> ExtendedContext.scaleb(Decimal(1), 4) - Decimal('1E+4') - >>> ExtendedContext.scaleb(1, Decimal(4)) - Decimal('1E+4') - """ - a = _convert_other(a, raiseit=True) - return a.scaleb(b, context=self) - - def shift(self, a, b): - """Returns a shifted copy of a, b times. - - The coefficient of the result is a shifted copy of the digits - in the coefficient of the first operand. The number of places - to shift is taken from the absolute value of the second operand, - with the shift being to the left if the second operand is - positive or to the right otherwise. Digits shifted into the - coefficient are zeros. - - >>> ExtendedContext.shift(Decimal('34'), Decimal('8')) - Decimal('400000000') - >>> ExtendedContext.shift(Decimal('12'), Decimal('9')) - Decimal('0') - >>> ExtendedContext.shift(Decimal('123456789'), Decimal('-2')) - Decimal('1234567') - >>> ExtendedContext.shift(Decimal('123456789'), Decimal('0')) - Decimal('123456789') - >>> ExtendedContext.shift(Decimal('123456789'), Decimal('+2')) - Decimal('345678900') - >>> ExtendedContext.shift(88888888, 2) - Decimal('888888800') - >>> ExtendedContext.shift(Decimal(88888888), 2) - Decimal('888888800') - >>> ExtendedContext.shift(88888888, Decimal(2)) - Decimal('888888800') - """ - a = _convert_other(a, raiseit=True) - return a.shift(b, context=self) - - def sqrt(self, a): - """Square root of a non-negative number to context precision. - - If the result must be inexact, it is rounded using the round-half-even - algorithm. - - >>> ExtendedContext.sqrt(Decimal('0')) - Decimal('0') - >>> ExtendedContext.sqrt(Decimal('-0')) - Decimal('-0') - >>> ExtendedContext.sqrt(Decimal('0.39')) - Decimal('0.624499800') - >>> ExtendedContext.sqrt(Decimal('100')) - Decimal('10') - >>> ExtendedContext.sqrt(Decimal('1')) - Decimal('1') - >>> ExtendedContext.sqrt(Decimal('1.0')) - Decimal('1.0') - >>> ExtendedContext.sqrt(Decimal('1.00')) - Decimal('1.0') - >>> ExtendedContext.sqrt(Decimal('7')) - Decimal('2.64575131') - >>> ExtendedContext.sqrt(Decimal('10')) - Decimal('3.16227766') - >>> ExtendedContext.sqrt(2) - Decimal('1.41421356') - >>> ExtendedContext.prec - 9 - """ - a = _convert_other(a, raiseit=True) - return a.sqrt(context=self) - - def subtract(self, a, b): - """Return the difference between the two operands. - - >>> ExtendedContext.subtract(Decimal('1.3'), Decimal('1.07')) - Decimal('0.23') - >>> ExtendedContext.subtract(Decimal('1.3'), Decimal('1.30')) - Decimal('0.00') - >>> ExtendedContext.subtract(Decimal('1.3'), Decimal('2.07')) - Decimal('-0.77') - >>> ExtendedContext.subtract(8, 5) - Decimal('3') - >>> ExtendedContext.subtract(Decimal(8), 5) - Decimal('3') - >>> ExtendedContext.subtract(8, Decimal(5)) - Decimal('3') - """ - a = _convert_other(a, raiseit=True) - r = a.__sub__(b, context=self) - if r is NotImplemented: - raise TypeError("Unable to convert %s to Decimal" % b) - else: - return r - - def to_eng_string(self, a): - """Converts a number to a string, using scientific notation. - - The operation is not affected by the context. - """ - a = _convert_other(a, raiseit=True) - return a.to_eng_string(context=self) - - def to_sci_string(self, a): - """Converts a number to a string, using scientific notation. - - The operation is not affected by the context. - """ - a = _convert_other(a, raiseit=True) - return a.__str__(context=self) - - def to_integral_exact(self, a): - """Rounds to an integer. - - When the operand has a negative exponent, the result is the same - as using the quantize() operation using the given operand as the - left-hand-operand, 1E+0 as the right-hand-operand, and the precision - of the operand as the precision setting; Inexact and Rounded flags - are allowed in this operation. The rounding mode is taken from the - context. - - >>> ExtendedContext.to_integral_exact(Decimal('2.1')) - Decimal('2') - >>> ExtendedContext.to_integral_exact(Decimal('100')) - Decimal('100') - >>> ExtendedContext.to_integral_exact(Decimal('100.0')) - Decimal('100') - >>> ExtendedContext.to_integral_exact(Decimal('101.5')) - Decimal('102') - >>> ExtendedContext.to_integral_exact(Decimal('-101.5')) - Decimal('-102') - >>> ExtendedContext.to_integral_exact(Decimal('10E+5')) - Decimal('1.0E+6') - >>> ExtendedContext.to_integral_exact(Decimal('7.89E+77')) - Decimal('7.89E+77') - >>> ExtendedContext.to_integral_exact(Decimal('-Inf')) - Decimal('-Infinity') - """ - a = _convert_other(a, raiseit=True) - return a.to_integral_exact(context=self) - - def to_integral_value(self, a): - """Rounds to an integer. - - When the operand has a negative exponent, the result is the same - as using the quantize() operation using the given operand as the - left-hand-operand, 1E+0 as the right-hand-operand, and the precision - of the operand as the precision setting, except that no flags will - be set. The rounding mode is taken from the context. - - >>> ExtendedContext.to_integral_value(Decimal('2.1')) - Decimal('2') - >>> ExtendedContext.to_integral_value(Decimal('100')) - Decimal('100') - >>> ExtendedContext.to_integral_value(Decimal('100.0')) - Decimal('100') - >>> ExtendedContext.to_integral_value(Decimal('101.5')) - Decimal('102') - >>> ExtendedContext.to_integral_value(Decimal('-101.5')) - Decimal('-102') - >>> ExtendedContext.to_integral_value(Decimal('10E+5')) - Decimal('1.0E+6') - >>> ExtendedContext.to_integral_value(Decimal('7.89E+77')) - Decimal('7.89E+77') - >>> ExtendedContext.to_integral_value(Decimal('-Inf')) - Decimal('-Infinity') - """ - a = _convert_other(a, raiseit=True) - return a.to_integral_value(context=self) - - # the method name changed, but we provide also the old one, for compatibility - to_integral = to_integral_value - -class _WorkRep(object): - __slots__ = ('sign','int','exp') - # sign: 0 or 1 - # int: int - # exp: None, int, or string - - def __init__(self, value=None): - if value is None: - self.sign = None - self.int = 0 - self.exp = None - elif isinstance(value, Decimal): - self.sign = value._sign - self.int = int(value._int) - self.exp = value._exp - else: - # assert isinstance(value, tuple) - self.sign = value[0] - self.int = value[1] - self.exp = value[2] - - def __repr__(self): - return "(%r, %r, %r)" % (self.sign, self.int, self.exp) - - __str__ = __repr__ - - - -def _normalize(op1, op2, prec = 0): - """Normalizes op1, op2 to have the same exp and length of coefficient. - - Done during addition. - """ - if op1.exp < op2.exp: - tmp = op2 - other = op1 - else: - tmp = op1 - other = op2 - - # Let exp = min(tmp.exp - 1, tmp.adjusted() - precision - 1). - # Then adding 10**exp to tmp has the same effect (after rounding) - # as adding any positive quantity smaller than 10**exp; similarly - # for subtraction. So if other is smaller than 10**exp we replace - # it with 10**exp. This avoids tmp.exp - other.exp getting too large. - tmp_len = len(str(tmp.int)) - other_len = len(str(other.int)) - exp = tmp.exp + min(-1, tmp_len - prec - 2) - if other_len + other.exp - 1 < exp: - other.int = 1 - other.exp = exp - - tmp.int *= 10 ** (tmp.exp - other.exp) - tmp.exp = other.exp - return op1, op2 - -##### Integer arithmetic functions used by ln, log10, exp and __pow__ ##### - -_nbits = int.bit_length - -def _decimal_lshift_exact(n, e): - """ Given integers n and e, return n * 10**e if it's an integer, else None. - - The computation is designed to avoid computing large powers of 10 - unnecessarily. - - >>> _decimal_lshift_exact(3, 4) - 30000 - >>> _decimal_lshift_exact(300, -999999999) # returns None - - """ - if n == 0: - return 0 - elif e >= 0: - return n * 10**e - else: - # val_n = largest power of 10 dividing n. - str_n = str(abs(n)) - val_n = len(str_n) - len(str_n.rstrip('0')) - return None if val_n < -e else n // 10**-e - -def _sqrt_nearest(n, a): - """Closest integer to the square root of the positive integer n. a is - an initial approximation to the square root. Any positive integer - will do for a, but the closer a is to the square root of n the - faster convergence will be. - - """ - if n <= 0 or a <= 0: - raise ValueError("Both arguments to _sqrt_nearest should be positive.") - - b=0 - while a != b: - b, a = a, a--n//a>>1 - return a - -def _rshift_nearest(x, shift): - """Given an integer x and a nonnegative integer shift, return closest - integer to x / 2**shift; use round-to-even in case of a tie. - - """ - b, q = 1 << shift, x >> shift - return q + (2*(x & (b-1)) + (q&1) > b) - -def _div_nearest(a, b): - """Closest integer to a/b, a and b positive integers; rounds to even - in the case of a tie. - - """ - q, r = divmod(a, b) - return q + (2*r + (q&1) > b) - -def _ilog(x, M, L = 8): - """Integer approximation to M*log(x/M), with absolute error boundable - in terms only of x/M. - - Given positive integers x and M, return an integer approximation to - M * log(x/M). For L = 8 and 0.1 <= x/M <= 10 the difference - between the approximation and the exact result is at most 22. For - L = 8 and 1.0 <= x/M <= 10.0 the difference is at most 15. In - both cases these are upper bounds on the error; it will usually be - much smaller.""" - - # The basic algorithm is the following: let log1p be the function - # log1p(x) = log(1+x). Then log(x/M) = log1p((x-M)/M). We use - # the reduction - # - # log1p(y) = 2*log1p(y/(1+sqrt(1+y))) - # - # repeatedly until the argument to log1p is small (< 2**-L in - # absolute value). For small y we can use the Taylor series - # expansion - # - # log1p(y) ~ y - y**2/2 + y**3/3 - ... - (-y)**T/T - # - # truncating at T such that y**T is small enough. The whole - # computation is carried out in a form of fixed-point arithmetic, - # with a real number z being represented by an integer - # approximation to z*M. To avoid loss of precision, the y below - # is actually an integer approximation to 2**R*y*M, where R is the - # number of reductions performed so far. - - y = x-M - # argument reduction; R = number of reductions performed - R = 0 - while (R <= L and abs(y) << L-R >= M or - R > L and abs(y) >> R-L >= M): - y = _div_nearest((M*y) << 1, - M + _sqrt_nearest(M*(M+_rshift_nearest(y, R)), M)) - R += 1 - - # Taylor series with T terms - T = -int(-10*len(str(M))//(3*L)) - yshift = _rshift_nearest(y, R) - w = _div_nearest(M, T) - for k in range(T-1, 0, -1): - w = _div_nearest(M, k) - _div_nearest(yshift*w, M) - - return _div_nearest(w*y, M) - -def _dlog10(c, e, p): - """Given integers c, e and p with c > 0, p >= 0, compute an integer - approximation to 10**p * log10(c*10**e), with an absolute error of - at most 1. Assumes that c*10**e is not exactly 1.""" - - # increase precision by 2; compensate for this by dividing - # final result by 100 - p += 2 - - # write c*10**e as d*10**f with either: - # f >= 0 and 1 <= d <= 10, or - # f <= 0 and 0.1 <= d <= 1. - # Thus for c*10**e close to 1, f = 0 - l = len(str(c)) - f = e+l - (e+l >= 1) - - if p > 0: - M = 10**p - k = e+p-f - if k >= 0: - c *= 10**k - else: - c = _div_nearest(c, 10**-k) - - log_d = _ilog(c, M) # error < 5 + 22 = 27 - log_10 = _log10_digits(p) # error < 1 - log_d = _div_nearest(log_d*M, log_10) - log_tenpower = f*M # exact - else: - log_d = 0 # error < 2.31 - log_tenpower = _div_nearest(f, 10**-p) # error < 0.5 - - return _div_nearest(log_tenpower+log_d, 100) - -def _dlog(c, e, p): - """Given integers c, e and p with c > 0, compute an integer - approximation to 10**p * log(c*10**e), with an absolute error of - at most 1. Assumes that c*10**e is not exactly 1.""" - - # Increase precision by 2. The precision increase is compensated - # for at the end with a division by 100. - p += 2 - - # rewrite c*10**e as d*10**f with either f >= 0 and 1 <= d <= 10, - # or f <= 0 and 0.1 <= d <= 1. Then we can compute 10**p * log(c*10**e) - # as 10**p * log(d) + 10**p*f * log(10). - l = len(str(c)) - f = e+l - (e+l >= 1) - - # compute approximation to 10**p*log(d), with error < 27 - if p > 0: - k = e+p-f - if k >= 0: - c *= 10**k - else: - c = _div_nearest(c, 10**-k) # error of <= 0.5 in c - - # _ilog magnifies existing error in c by a factor of at most 10 - log_d = _ilog(c, 10**p) # error < 5 + 22 = 27 - else: - # p <= 0: just approximate the whole thing by 0; error < 2.31 - log_d = 0 - - # compute approximation to f*10**p*log(10), with error < 11. - if f: - extra = len(str(abs(f)))-1 - if p + extra >= 0: - # error in f * _log10_digits(p+extra) < |f| * 1 = |f| - # after division, error < |f|/10**extra + 0.5 < 10 + 0.5 < 11 - f_log_ten = _div_nearest(f*_log10_digits(p+extra), 10**extra) - else: - f_log_ten = 0 - else: - f_log_ten = 0 - - # error in sum < 11+27 = 38; error after division < 0.38 + 0.5 < 1 - return _div_nearest(f_log_ten + log_d, 100) - -class _Log10Memoize(object): - """Class to compute, store, and allow retrieval of, digits of the - constant log(10) = 2.302585.... This constant is needed by - Decimal.ln, Decimal.log10, Decimal.exp and Decimal.__pow__.""" - def __init__(self): - self.digits = "23025850929940456840179914546843642076011014886" - - def getdigits(self, p): - """Given an integer p >= 0, return floor(10**p)*log(10). - - For example, self.getdigits(3) returns 2302. - """ - # digits are stored as a string, for quick conversion to - # integer in the case that we've already computed enough - # digits; the stored digits should always be correct - # (truncated, not rounded to nearest). - if p < 0: - raise ValueError("p should be nonnegative") - - if p >= len(self.digits): - # compute p+3, p+6, p+9, ... digits; continue until at - # least one of the extra digits is nonzero - extra = 3 - while True: - # compute p+extra digits, correct to within 1ulp - M = 10**(p+extra+2) - digits = str(_div_nearest(_ilog(10*M, M), 100)) - if digits[-extra:] != '0'*extra: - break - extra += 3 - # keep all reliable digits so far; remove trailing zeros - # and next nonzero digit - self.digits = digits.rstrip('0')[:-1] - return int(self.digits[:p+1]) - -_log10_digits = _Log10Memoize().getdigits - -def _iexp(x, M, L=8): - """Given integers x and M, M > 0, such that x/M is small in absolute - value, compute an integer approximation to M*exp(x/M). For 0 <= - x/M <= 2.4, the absolute error in the result is bounded by 60 (and - is usually much smaller).""" - - # Algorithm: to compute exp(z) for a real number z, first divide z - # by a suitable power R of 2 so that |z/2**R| < 2**-L. Then - # compute expm1(z/2**R) = exp(z/2**R) - 1 using the usual Taylor - # series - # - # expm1(x) = x + x**2/2! + x**3/3! + ... - # - # Now use the identity - # - # expm1(2x) = expm1(x)*(expm1(x)+2) - # - # R times to compute the sequence expm1(z/2**R), - # expm1(z/2**(R-1)), ... , exp(z/2), exp(z). - - # Find R such that x/2**R/M <= 2**-L - R = _nbits((x<<L)//M) - - # Taylor series. (2**L)**T > M - T = -int(-10*len(str(M))//(3*L)) - y = _div_nearest(x, T) - Mshift = M<<R - for i in range(T-1, 0, -1): - y = _div_nearest(x*(Mshift + y), Mshift * i) - - # Expansion - for k in range(R-1, -1, -1): - Mshift = M<<(k+2) - y = _div_nearest(y*(y+Mshift), Mshift) - - return M+y - -def _dexp(c, e, p): - """Compute an approximation to exp(c*10**e), with p decimal places of - precision. - - Returns integers d, f such that: - - 10**(p-1) <= d <= 10**p, and - (d-1)*10**f < exp(c*10**e) < (d+1)*10**f - - In other words, d*10**f is an approximation to exp(c*10**e) with p - digits of precision, and with an error in d of at most 1. This is - almost, but not quite, the same as the error being < 1ulp: when d - = 10**(p-1) the error could be up to 10 ulp.""" - - # we'll call iexp with M = 10**(p+2), giving p+3 digits of precision - p += 2 - - # compute log(10) with extra precision = adjusted exponent of c*10**e - extra = max(0, e + len(str(c)) - 1) - q = p + extra - - # compute quotient c*10**e/(log(10)) = c*10**(e+q)/(log(10)*10**q), - # rounding down - shift = e+q - if shift >= 0: - cshift = c*10**shift - else: - cshift = c//10**-shift - quot, rem = divmod(cshift, _log10_digits(q)) - - # reduce remainder back to original precision - rem = _div_nearest(rem, 10**extra) - - # error in result of _iexp < 120; error after division < 0.62 - return _div_nearest(_iexp(rem, 10**p), 1000), quot - p + 3 - -def _dpower(xc, xe, yc, ye, p): - """Given integers xc, xe, yc and ye representing Decimals x = xc*10**xe and - y = yc*10**ye, compute x**y. Returns a pair of integers (c, e) such that: - - 10**(p-1) <= c <= 10**p, and - (c-1)*10**e < x**y < (c+1)*10**e - - in other words, c*10**e is an approximation to x**y with p digits - of precision, and with an error in c of at most 1. (This is - almost, but not quite, the same as the error being < 1ulp: when c - == 10**(p-1) we can only guarantee error < 10ulp.) - - We assume that: x is positive and not equal to 1, and y is nonzero. - """ - - # Find b such that 10**(b-1) <= |y| <= 10**b - b = len(str(abs(yc))) + ye - - # log(x) = lxc*10**(-p-b-1), to p+b+1 places after the decimal point - lxc = _dlog(xc, xe, p+b+1) - - # compute product y*log(x) = yc*lxc*10**(-p-b-1+ye) = pc*10**(-p-1) - shift = ye-b - if shift >= 0: - pc = lxc*yc*10**shift - else: - pc = _div_nearest(lxc*yc, 10**-shift) - - if pc == 0: - # we prefer a result that isn't exactly 1; this makes it - # easier to compute a correctly rounded result in __pow__ - if ((len(str(xc)) + xe >= 1) == (yc > 0)): # if x**y > 1: - coeff, exp = 10**(p-1)+1, 1-p - else: - coeff, exp = 10**p-1, -p - else: - coeff, exp = _dexp(pc, -(p+1), p+1) - coeff = _div_nearest(coeff, 10) - exp += 1 - - return coeff, exp - -def _log10_lb(c, correction = { - '1': 100, '2': 70, '3': 53, '4': 40, '5': 31, - '6': 23, '7': 16, '8': 10, '9': 5}): - """Compute a lower bound for 100*log10(c) for a positive integer c.""" - if c <= 0: - raise ValueError("The argument to _log10_lb should be nonnegative.") - str_c = str(c) - return 100*len(str_c) - correction[str_c[0]] - -##### Helper Functions #################################################### - -def _convert_other(other, raiseit=False, allow_float=False): - """Convert other to Decimal. - - Verifies that it's ok to use in an implicit construction. - If allow_float is true, allow conversion from float; this - is used in the comparison methods (__eq__ and friends). - - """ - if isinstance(other, Decimal): - return other - if isinstance(other, int): - return Decimal(other) - if allow_float and isinstance(other, float): - return Decimal.from_float(other) - - if raiseit: - raise TypeError("Unable to convert %s to Decimal" % other) - return NotImplemented - -def _convert_for_comparison(self, other, equality_op=False): - """Given a Decimal instance self and a Python object other, return - a pair (s, o) of Decimal instances such that "s op o" is - equivalent to "self op other" for any of the 6 comparison - operators "op". - - """ - if isinstance(other, Decimal): - return self, other - - # Comparison with a Rational instance (also includes integers): - # self op n/d <=> self*d op n (for n and d integers, d positive). - # A NaN or infinity can be left unchanged without affecting the - # comparison result. - if isinstance(other, _numbers.Rational): - if not self._is_special: - self = _dec_from_triple(self._sign, - str(int(self._int) * other.denominator), - self._exp) - return self, Decimal(other.numerator) - - # Comparisons with float and complex types. == and != comparisons - # with complex numbers should succeed, returning either True or False - # as appropriate. Other comparisons return NotImplemented. - if equality_op and isinstance(other, _numbers.Complex) and other.imag == 0: - other = other.real - if isinstance(other, float): - context = getcontext() - if equality_op: - context.flags[FloatOperation] = 1 - else: - context._raise_error(FloatOperation, - "strict semantics for mixing floats and Decimals are enabled") - return self, Decimal.from_float(other) - return NotImplemented, NotImplemented - - -##### Setup Specific Contexts ############################################ - -# The default context prototype used by Context() -# Is mutable, so that new contexts can have different default values - -DefaultContext = Context( - prec=28, rounding=ROUND_HALF_EVEN, - traps=[DivisionByZero, Overflow, InvalidOperation], - flags=[], - Emax=999999, - Emin=-999999, - capitals=1, - clamp=0 -) - -# Pre-made alternate contexts offered by the specification -# Don't change these; the user should be able to select these -# contexts and be able to reproduce results from other implementations -# of the spec. - -BasicContext = Context( - prec=9, rounding=ROUND_HALF_UP, - traps=[DivisionByZero, Overflow, InvalidOperation, Clamped, Underflow], - flags=[], -) - -ExtendedContext = Context( - prec=9, rounding=ROUND_HALF_EVEN, - traps=[], - flags=[], -) - - -##### crud for parsing strings ############################################# -# -# Regular expression used for parsing numeric strings. Additional -# comments: -# -# 1. Uncomment the two '\s*' lines to allow leading and/or trailing -# whitespace. But note that the specification disallows whitespace in -# a numeric string. -# -# 2. For finite numbers (not infinities and NaNs) the body of the -# number between the optional sign and the optional exponent must have -# at least one decimal digit, possibly after the decimal point. The -# lookahead expression '(?=\d|\.\d)' checks this. - -import re -_parser = re.compile(r""" # A numeric string consists of: -# \s* - (?P<sign>[-+])? # an optional sign, followed by either... - ( - (?=\d|\.\d) # ...a number (with at least one digit) - (?P<int>\d*) # having a (possibly empty) integer part - (\.(?P<frac>\d*))? # followed by an optional fractional part - (E(?P<exp>[-+]?\d+))? # followed by an optional exponent, or... - | - Inf(inity)? # ...an infinity, or... - | - (?P<signal>s)? # ...an (optionally signaling) - NaN # NaN - (?P<diag>\d*) # with (possibly empty) diagnostic info. - ) -# \s* - \Z -""", re.VERBOSE | re.IGNORECASE).match - -_all_zeros = re.compile('0*$').match -_exact_half = re.compile('50*$').match - -##### PEP3101 support functions ############################################## -# The functions in this section have little to do with the Decimal -# class, and could potentially be reused or adapted for other pure -# Python numeric classes that want to implement __format__ -# -# A format specifier for Decimal looks like: -# -# [[fill]align][sign][#][0][minimumwidth][,][.precision][type] - -_parse_format_specifier_regex = re.compile(r"""\A -(?: - (?P<fill>.)? - (?P<align>[<>=^]) -)? -(?P<sign>[-+ ])? -(?P<alt>\#)? -(?P<zeropad>0)? -(?P<minimumwidth>(?!0)\d+)? -(?P<thousands_sep>,)? -(?:\.(?P<precision>0|(?!0)\d+))? -(?P<type>[eEfFgGn%])? -\Z -""", re.VERBOSE|re.DOTALL) - -del re - -# The locale module is only needed for the 'n' format specifier. The -# rest of the PEP 3101 code functions quite happily without it, so we -# don't care too much if locale isn't present. -try: - import locale as _locale + from _decimal import * + from _decimal import __doc__ + from _decimal import __version__ + from _decimal import __libmpdec_version__ except ImportError: - pass - -def _parse_format_specifier(format_spec, _localeconv=None): - """Parse and validate a format specifier. - - Turns a standard numeric format specifier into a dict, with the - following entries: - - fill: fill character to pad field to minimum width - align: alignment type, either '<', '>', '=' or '^' - sign: either '+', '-' or ' ' - minimumwidth: nonnegative integer giving minimum width - zeropad: boolean, indicating whether to pad with zeros - thousands_sep: string to use as thousands separator, or '' - grouping: grouping for thousands separators, in format - used by localeconv - decimal_point: string to use for decimal point - precision: nonnegative integer giving precision, or None - type: one of the characters 'eEfFgG%', or None - - """ - m = _parse_format_specifier_regex.match(format_spec) - if m is None: - raise ValueError("Invalid format specifier: " + format_spec) - - # get the dictionary - format_dict = m.groupdict() - - # zeropad; defaults for fill and alignment. If zero padding - # is requested, the fill and align fields should be absent. - fill = format_dict['fill'] - align = format_dict['align'] - format_dict['zeropad'] = (format_dict['zeropad'] is not None) - if format_dict['zeropad']: - if fill is not None: - raise ValueError("Fill character conflicts with '0'" - " in format specifier: " + format_spec) - if align is not None: - raise ValueError("Alignment conflicts with '0' in " - "format specifier: " + format_spec) - format_dict['fill'] = fill or ' ' - # PEP 3101 originally specified that the default alignment should - # be left; it was later agreed that right-aligned makes more sense - # for numeric types. See http://bugs.python.org/issue6857. - format_dict['align'] = align or '>' - - # default sign handling: '-' for negative, '' for positive - if format_dict['sign'] is None: - format_dict['sign'] = '-' - - # minimumwidth defaults to 0; precision remains None if not given - format_dict['minimumwidth'] = int(format_dict['minimumwidth'] or '0') - if format_dict['precision'] is not None: - format_dict['precision'] = int(format_dict['precision']) - - # if format type is 'g' or 'G' then a precision of 0 makes little - # sense; convert it to 1. Same if format type is unspecified. - if format_dict['precision'] == 0: - if format_dict['type'] is None or format_dict['type'] in 'gGn': - format_dict['precision'] = 1 - - # determine thousands separator, grouping, and decimal separator, and - # add appropriate entries to format_dict - if format_dict['type'] == 'n': - # apart from separators, 'n' behaves just like 'g' - format_dict['type'] = 'g' - if _localeconv is None: - _localeconv = _locale.localeconv() - if format_dict['thousands_sep'] is not None: - raise ValueError("Explicit thousands separator conflicts with " - "'n' type in format specifier: " + format_spec) - format_dict['thousands_sep'] = _localeconv['thousands_sep'] - format_dict['grouping'] = _localeconv['grouping'] - format_dict['decimal_point'] = _localeconv['decimal_point'] - else: - if format_dict['thousands_sep'] is None: - format_dict['thousands_sep'] = '' - format_dict['grouping'] = [3, 0] - format_dict['decimal_point'] = '.' - - return format_dict - -def _format_align(sign, body, spec): - """Given an unpadded, non-aligned numeric string 'body' and sign - string 'sign', add padding and alignment conforming to the given - format specifier dictionary 'spec' (as produced by - parse_format_specifier). - - """ - # how much extra space do we have to play with? - minimumwidth = spec['minimumwidth'] - fill = spec['fill'] - padding = fill*(minimumwidth - len(sign) - len(body)) - - align = spec['align'] - if align == '<': - result = sign + body + padding - elif align == '>': - result = padding + sign + body - elif align == '=': - result = sign + padding + body - elif align == '^': - half = len(padding)//2 - result = padding[:half] + sign + body + padding[half:] - else: - raise ValueError('Unrecognised alignment field') + from _pydecimal import * + from _pydecimal import __doc__ + from _pydecimal import __version__ + from _pydecimal import __libmpdec_version__ - return result -def _group_lengths(grouping): - """Convert a localeconv-style grouping into a (possibly infinite) - iterable of integers representing group lengths. - - """ - # The result from localeconv()['grouping'], and the input to this - # function, should be a list of integers in one of the - # following three forms: - # - # (1) an empty list, or - # (2) nonempty list of positive integers + [0] - # (3) list of positive integers + [locale.CHAR_MAX], or - - from itertools import chain, repeat - if not grouping: - return [] - elif grouping[-1] == 0 and len(grouping) >= 2: - return chain(grouping[:-1], repeat(grouping[-2])) - elif grouping[-1] == _locale.CHAR_MAX: - return grouping[:-1] - else: - raise ValueError('unrecognised format for grouping') - -def _insert_thousands_sep(digits, spec, min_width=1): - """Insert thousands separators into a digit string. - - spec is a dictionary whose keys should include 'thousands_sep' and - 'grouping'; typically it's the result of parsing the format - specifier using _parse_format_specifier. - - The min_width keyword argument gives the minimum length of the - result, which will be padded on the left with zeros if necessary. - - If necessary, the zero padding adds an extra '0' on the left to - avoid a leading thousands separator. For example, inserting - commas every three digits in '123456', with min_width=8, gives - '0,123,456', even though that has length 9. - - """ - - sep = spec['thousands_sep'] - grouping = spec['grouping'] - - groups = [] - for l in _group_lengths(grouping): - if l <= 0: - raise ValueError("group length should be positive") - # max(..., 1) forces at least 1 digit to the left of a separator - l = min(max(len(digits), min_width, 1), l) - groups.append('0'*(l - len(digits)) + digits[-l:]) - digits = digits[:-l] - min_width -= l - if not digits and min_width <= 0: - break - min_width -= len(sep) - else: - l = max(len(digits), min_width, 1) - groups.append('0'*(l - len(digits)) + digits[-l:]) - return sep.join(reversed(groups)) - -def _format_sign(is_negative, spec): - """Determine sign character.""" - - if is_negative: - return '-' - elif spec['sign'] in ' +': - return spec['sign'] - else: - return '' - -def _format_number(is_negative, intpart, fracpart, exp, spec): - """Format a number, given the following data: - - is_negative: true if the number is negative, else false - intpart: string of digits that must appear before the decimal point - fracpart: string of digits that must come after the point - exp: exponent, as an integer - spec: dictionary resulting from parsing the format specifier - - This function uses the information in spec to: - insert separators (decimal separator and thousands separators) - format the sign - format the exponent - add trailing '%' for the '%' type - zero-pad if necessary - fill and align if necessary - """ - - sign = _format_sign(is_negative, spec) - - if fracpart or spec['alt']: - fracpart = spec['decimal_point'] + fracpart - - if exp != 0 or spec['type'] in 'eE': - echar = {'E': 'E', 'e': 'e', 'G': 'E', 'g': 'e'}[spec['type']] - fracpart += "{0}{1:+}".format(echar, exp) - if spec['type'] == '%': - fracpart += '%' - - if spec['zeropad']: - min_width = spec['minimumwidth'] - len(fracpart) - len(sign) - else: - min_width = 0 - intpart = _insert_thousands_sep(intpart, spec, min_width) - - return _format_align(sign, intpart+fracpart, spec) - - -##### Useful Constants (internal use only) ################################ - -# Reusable defaults -_Infinity = Decimal('Inf') -_NegativeInfinity = Decimal('-Inf') -_NaN = Decimal('NaN') -_Zero = Decimal(0) -_One = Decimal(1) -_NegativeOne = Decimal(-1) - -# _SignedInfinity[sign] is infinity w/ that sign -_SignedInfinity = (_Infinity, _NegativeInfinity) - -# Constants related to the hash implementation; hash(x) is based -# on the reduction of x modulo _PyHASH_MODULUS -_PyHASH_MODULUS = sys.hash_info.modulus -# hash values to use for positive and negative infinities, and nans -_PyHASH_INF = sys.hash_info.inf -_PyHASH_NAN = sys.hash_info.nan - -# _PyHASH_10INV is the inverse of 10 modulo the prime _PyHASH_MODULUS -_PyHASH_10INV = pow(10, _PyHASH_MODULUS - 2, _PyHASH_MODULUS) -del sys - -try: - import _decimal -except ImportError: - pass -else: - s1 = set(dir()) - s2 = set(dir(_decimal)) - for name in s1 - s2: - del globals()[name] - del s1, s2, name - from _decimal import * -if __name__ == '__main__': - import doctest, decimal - doctest.testmod(decimal) |