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diff --git a/Lib/_pydecimal.py b/Lib/_pydecimal.py new file mode 100644 index 0000000..05ba4ee --- /dev/null +++ b/Lib/_pydecimal.py @@ -0,0 +1,6380 @@ +# Copyright (c) 2004 Python Software Foundation. +# All rights reserved. + +# Written by Eric Price <eprice at tjhsst.edu> +# and Facundo Batista <facundo at taniquetil.com.ar> +# and Raymond Hettinger <python at rcn.com> +# and Aahz <aahz at pobox.com> +# and Tim Peters + +# This module should be kept in sync with the latest updates of the +# IBM specification as it evolves. Those updates will be treated +# as bug fixes (deviation from the spec is a compatibility, usability +# bug) and will be backported. At this point the spec is stabilizing +# and the updates are becoming fewer, smaller, and less significant. + +""" +This is an implementation of decimal floating point arithmetic based on +the General Decimal Arithmetic Specification: + + http://speleotrove.com/decimal/decarith.html + +and IEEE standard 854-1987: + + http://en.wikipedia.org/wiki/IEEE_854-1987 + +Decimal floating point has finite precision with arbitrarily large bounds. + +The purpose of this module is to support arithmetic using familiar +"schoolhouse" rules and to avoid some of the tricky representation +issues associated with binary floating point. The package is especially +useful for financial applications or for contexts where users have +expectations that are at odds with binary floating point (for instance, +in binary floating point, 1.00 % 0.1 gives 0.09999999999999995 instead +of 0.0; Decimal('1.00') % Decimal('0.1') returns the expected +Decimal('0.00')). + +Here are some examples of using the decimal module: + +>>> from decimal import * +>>> setcontext(ExtendedContext) +>>> Decimal(0) +Decimal('0') +>>> Decimal('1') +Decimal('1') +>>> Decimal('-.0123') +Decimal('-0.0123') +>>> Decimal(123456) +Decimal('123456') +>>> Decimal('123.45e12345678') +Decimal('1.2345E+12345680') +>>> Decimal('1.33') + Decimal('1.27') +Decimal('2.60') +>>> Decimal('12.34') + Decimal('3.87') - Decimal('18.41') +Decimal('-2.20') +>>> dig = Decimal(1) +>>> print(dig / Decimal(3)) +0.333333333 +>>> getcontext().prec = 18 +>>> print(dig / Decimal(3)) +0.333333333333333333 +>>> print(dig.sqrt()) +1 +>>> print(Decimal(3).sqrt()) +1.73205080756887729 +>>> print(Decimal(3) ** 123) +4.85192780976896427E+58 +>>> inf = Decimal(1) / Decimal(0) +>>> print(inf) +Infinity +>>> neginf = Decimal(-1) / Decimal(0) +>>> print(neginf) +-Infinity +>>> print(neginf + inf) +NaN +>>> print(neginf * inf) +-Infinity +>>> print(dig / 0) +Infinity +>>> getcontext().traps[DivisionByZero] = 1 +>>> print(dig / 0) +Traceback (most recent call last): + ... + ... + ... +decimal.DivisionByZero: x / 0 +>>> c = Context() +>>> c.traps[InvalidOperation] = 0 +>>> print(c.flags[InvalidOperation]) +0 +>>> c.divide(Decimal(0), Decimal(0)) +Decimal('NaN') +>>> c.traps[InvalidOperation] = 1 +>>> print(c.flags[InvalidOperation]) +1 +>>> c.flags[InvalidOperation] = 0 +>>> print(c.flags[InvalidOperation]) +0 +>>> print(c.divide(Decimal(0), Decimal(0))) +Traceback (most recent call last): + ... + ... + ... +decimal.InvalidOperation: 0 / 0 +>>> print(c.flags[InvalidOperation]) +1 +>>> c.flags[InvalidOperation] = 0 +>>> c.traps[InvalidOperation] = 0 +>>> print(c.divide(Decimal(0), Decimal(0))) +NaN +>>> print(c.flags[InvalidOperation]) +1 +>>> +""" + +__all__ = [ + # Two major classes + 'Decimal', 'Context', + + # Named tuple representation + 'DecimalTuple', + + # Contexts + 'DefaultContext', 'BasicContext', 'ExtendedContext', + + # Exceptions + 'DecimalException', 'Clamped', 'InvalidOperation', 'DivisionByZero', + 'Inexact', 'Rounded', 'Subnormal', 'Overflow', 'Underflow', + 'FloatOperation', + + # Exceptional conditions that trigger InvalidOperation + 'DivisionImpossible', 'InvalidContext', 'ConversionSyntax', 'DivisionUndefined', + + # Constants for use in setting up contexts + 'ROUND_DOWN', 'ROUND_HALF_UP', 'ROUND_HALF_EVEN', 'ROUND_CEILING', + 'ROUND_FLOOR', 'ROUND_UP', 'ROUND_HALF_DOWN', 'ROUND_05UP', + + # Functions for manipulating contexts + 'setcontext', 'getcontext', 'localcontext', + + # Limits for the C version for compatibility + 'MAX_PREC', 'MAX_EMAX', 'MIN_EMIN', 'MIN_ETINY', + + # C version: compile time choice that enables the thread local context + 'HAVE_THREADS' +] + +__xname__ = __name__ # sys.modules lookup (--without-threads) +__name__ = 'decimal' # For pickling +__version__ = '1.70' # Highest version of the spec this complies with + # See http://speleotrove.com/decimal/ +__libmpdec_version__ = "2.4.1" # 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[__xname__] + 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 __lt__(self, other, context=None): + self, other = _convert_for_comparison(self, other) + if other is NotImplemented: + return other + ans = self._compare_check_nans(other, context) + if ans: + return False + return self._cmp(other) < 0 + + def __le__(self, other, context=None): + self, other = _convert_for_comparison(self, other) + if other is NotImplemented: + return other + ans = self._compare_check_nans(other, context) + if ans: + return False + return self._cmp(other) <= 0 + + def __gt__(self, other, context=None): + self, other = _convert_for_comparison(self, other) + if other is NotImplemented: + return other + ans = self._compare_check_nans(other, context) + if ans: + return False + return self._cmp(other) > 0 + + def __ge__(self, other, context=None): + self, other = _convert_for_comparison(self, other) + if other is NotImplemented: + return other + ans = self._compare_check_nans(other, context) + if ans: + return False + return self._cmp(other) >= 0 + + def compare(self, other, context=None): + """Compare self to other. Return a decimal value: + + a or b is a NaN ==> Decimal('NaN') + a < b ==> Decimal('-1') + a == b ==> Decimal('0') + a > b ==> Decimal('1') + """ + other = _convert_other(other, raiseit=True) + + # Compare(NaN, NaN) = NaN + if (self._is_special or other and other._is_special): + ans = self._check_nans(other, context) + if ans: + return ans + + return Decimal(self._cmp(other)) + + def __hash__(self): + """x.__hash__() <==> hash(x)""" + + # In order to make sure that the hash of a Decimal instance + # agrees with the hash of a numerically equal integer, float + # or Fraction, we follow the rules for numeric hashes outlined + # in the documentation. (See library docs, 'Built-in Types'). + if self._is_special: + if self.is_snan(): + raise TypeError('Cannot hash a signaling NaN value.') + elif self.is_nan(): + return _PyHASH_NAN + else: + if self._sign: + return -_PyHASH_INF + else: + return _PyHASH_INF + + if self._exp >= 0: + exp_hash = pow(10, self._exp, _PyHASH_MODULUS) + else: + exp_hash = pow(_PyHASH_10INV, -self._exp, _PyHASH_MODULUS) + hash_ = int(self._int) * exp_hash % _PyHASH_MODULUS + ans = hash_ if self >= 0 else -hash_ + return -2 if ans == -1 else ans + + def as_tuple(self): + """Represents the number as a triple tuple. + + To show the internals exactly as they are. + """ + return DecimalTuple(self._sign, tuple(map(int, self._int)), self._exp) + + def __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 + + """ + 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 +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') + + 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 |