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| author | Facundo Batista <facundobatista@gmail.com> | 2007-09-13 18:13:15 (GMT) |
|---|---|---|
| committer | Facundo Batista <facundobatista@gmail.com> | 2007-09-13 18:13:15 (GMT) |
| commit | 353750c405c9099d0be69c6af1d17037b38c4ddf (patch) | |
| tree | 8b3a0deb454d83c12b3d06365fa3f9c86d75d169 /Lib/decimal.py | |
| parent | ddca9f0823e44dd9c35d38de65417c09521ab5aa (diff) | |
| download | cpython-353750c405c9099d0be69c6af1d17037b38c4ddf.zip cpython-353750c405c9099d0be69c6af1d17037b38c4ddf.tar.gz cpython-353750c405c9099d0be69c6af1d17037b38c4ddf.tar.bz2 | |
Merged the decimal-branch (revisions 54886 to 58140). Decimal is now
fully updated to the latests Decimal Specification (v1.66) and the
latests test cases (v2.56).
Thanks to Mark Dickinson for all his help during this process.
Diffstat (limited to 'Lib/decimal.py')
| -rw-r--r-- | Lib/decimal.py | 3548 |
1 files changed, 2846 insertions, 702 deletions
diff --git a/Lib/decimal.py b/Lib/decimal.py index 7c67895..8543e10 100644 --- a/Lib/decimal.py +++ b/Lib/decimal.py @@ -128,7 +128,7 @@ __all__ = [ # 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_FLOOR', 'ROUND_UP', 'ROUND_HALF_DOWN', 'ROUND_05UP', # Functions for manipulating contexts 'setcontext', 'getcontext', 'localcontext' @@ -144,6 +144,7 @@ ROUND_CEILING = 'ROUND_CEILING' ROUND_FLOOR = 'ROUND_FLOOR' ROUND_UP = 'ROUND_UP' ROUND_HALF_DOWN = 'ROUND_HALF_DOWN' +ROUND_05UP = 'ROUND_05UP' # Rounding decision (not part of the public API) NEVER_ROUND = 'NEVER_ROUND' # Round in division (non-divmod), sqrt ONLY @@ -204,13 +205,22 @@ class InvalidOperation(DecimalException): 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: if args[0] == 1: # sNaN, must drop 's' but keep diagnostics - return Decimal( (args[1]._sign, args[1]._int, 'n') ) + ans = Decimal((args[1]._sign, args[1]._int, 'n')) + return ans._fix_nan(context) + elif args[0] == 2: + return Decimal( (args[1], args[2], 'n') ) return NaN + class ConversionSyntax(InvalidOperation): """Trying to convert badly formed string. @@ -218,9 +228,8 @@ class ConversionSyntax(InvalidOperation): 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 (0, (0,), 'n') # Passed to something which uses a tuple. + return NaN class DivisionByZero(DecimalException, ZeroDivisionError): """Division by 0. @@ -340,7 +349,7 @@ class Overflow(Inexact, Rounded): def handle(self, context, sign, *args): if context.rounding in (ROUND_HALF_UP, ROUND_HALF_EVEN, - ROUND_HALF_DOWN, ROUND_UP): + ROUND_HALF_DOWN, ROUND_UP): return Infsign[sign] if sign == 0: if context.rounding == ROUND_CEILING: @@ -562,7 +571,7 @@ class Decimal(object): raise ValueError('Invalid sign') for digit in value[1]: if not isinstance(digit, (int,long)) or digit < 0: - raise ValueError("The second value in the tuple must be" + raise ValueError("The second value in the tuple must be " "composed of non negative integer elements.") self._sign = value[0] self._int = tuple(value[1]) @@ -596,10 +605,6 @@ class Decimal(object): if _isnan(value): sig, sign, diag = _isnan(value) self._is_special = True - if len(diag) > context.prec: # Diagnostic info too long - self._sign, self._int, self._exp = \ - context._raise_error(ConversionSyntax) - return self if sig == 1: self._exp = 'n' # qNaN else: # sig == 2 @@ -611,9 +616,8 @@ class Decimal(object): self._sign, self._int, self._exp = _string2exact(value) except ValueError: self._is_special = True - self._sign, self._int, self._exp = \ - context._raise_error(ConversionSyntax, - "Invalid literal for Decimal: %r" % value) + return context._raise_error(ConversionSyntax, + "Invalid literal for Decimal: %r" % value) return self raise TypeError("Cannot convert %r to Decimal" % value) @@ -622,7 +626,7 @@ class Decimal(object): """Returns whether the number is not actually one. 0 if a number - 1 if NaN + 1 if NaN (it could be a normal quiet NaN or a phantom one) 2 if sNaN """ if self._is_special: @@ -646,7 +650,7 @@ class Decimal(object): return 1 return 0 - def _check_nans(self, other = None, context=None): + def _check_nans(self, other=None, context=None): """Returns whether the number is not actually one. if self, other are sNaN, signal @@ -673,9 +677,9 @@ class Decimal(object): return context._raise_error(InvalidOperation, 'sNaN', 1, other) if self_is_nan: - return self + return self._fix_nan(context) - return other + return other._fix_nan(context) return 0 def __nonzero__(self): @@ -685,24 +689,31 @@ class Decimal(object): 1 if self != 0 """ if self._is_special: - return 1 + return True return sum(self._int) != 0 - def __cmp__(self, other, context=None): + def __cmp__(self, other): other = _convert_other(other) if other is NotImplemented: - return other + # Never return NotImplemented + return 1 if self._is_special or other._is_special: - ans = self._check_nans(other, context) - if ans: + # check for nans, without raising on a signaling nan + if self._isnan() or other._isnan(): return 1 # Comparison involving NaN's always reports self > other # INF = INF return cmp(self._isinfinity(), other._isinfinity()) - if not self and not other: - return 0 # If both 0, sign comparison isn't certain. + # check for zeros; note that cmp(0, -0) should return 0 + if not self: + if not other: + return 0 + else: + return -((-1)**other._sign) + if not other: + return (-1)**self._sign # If different signs, neg one is less if other._sign < self._sign: @@ -712,35 +723,15 @@ class Decimal(object): self_adjusted = self.adjusted() other_adjusted = other.adjusted() - if self_adjusted == other_adjusted and \ - self._int + (0,)*(self._exp - other._exp) == \ - other._int + (0,)*(other._exp - self._exp): - return 0 # equal, except in precision. ([0]*(-x) = []) - elif self_adjusted > other_adjusted and self._int[0] != 0: + if self_adjusted == other_adjusted: + self_padded = self._int + (0,)*(self._exp - other._exp) + other_padded = other._int + (0,)*(other._exp - self._exp) + return cmp(self_padded, other_padded) * (-1)**self._sign + elif self_adjusted > other_adjusted: return (-1)**self._sign - elif self_adjusted < other_adjusted and other._int[0] != 0: + else: # self_adjusted < other_adjusted return -((-1)**self._sign) - # Need to round, so make sure we have a valid context - if context is None: - context = getcontext() - - context = context._shallow_copy() - rounding = context._set_rounding(ROUND_UP) # round away from 0 - - flags = context._ignore_all_flags() - res = self.__sub__(other, context=context) - - context._regard_flags(*flags) - - context.rounding = rounding - - if not res: - return 0 - elif res._sign: - return -1 - return 1 - def __eq__(self, other): if not isinstance(other, (Decimal, int, long)): return NotImplemented @@ -760,9 +751,7 @@ class Decimal(object): NaN => one is NaN Like __cmp__, but returns Decimal instances. """ - other = _convert_other(other) - if other is NotImplemented: - return other + other = _convert_other(other, raiseit=True) # Compare(NaN, NaN) = NaN if (self._is_special or other and other._is_special): @@ -770,7 +759,7 @@ class Decimal(object): if ans: return ans - return Decimal(self.__cmp__(other, context)) + return Decimal(self.__cmp__(other)) def __hash__(self): """x.__hash__() <==> hash(x)""" @@ -799,7 +788,7 @@ class Decimal(object): # Invariant: eval(repr(d)) == d return 'Decimal("%s")' % str(self) - def __str__(self, eng = 0, context=None): + 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. @@ -889,7 +878,7 @@ class Decimal(object): Same rules for when in exponential and when as a value as in __str__. """ - return self.__str__(eng=1, context=context) + return self.__str__(eng=True, context=context) def __neg__(self, context=None): """Returns a copy with the sign switched. @@ -903,17 +892,15 @@ class Decimal(object): if not self: # -Decimal('0') is Decimal('0'), not Decimal('-0') - sign = 0 - elif self._sign: - sign = 0 + ans = self.copy_sign(Dec_0) else: - sign = 1 + ans = self.copy_negate() if context is None: context = getcontext() if context._rounding_decision == ALWAYS_ROUND: - return Decimal((sign, self._int, self._exp))._fix(context) - return Decimal( (sign, self._int, self._exp)) + return ans._fix(context) + return ans def __pos__(self, context=None): """Returns a copy, unless it is a sNaN. @@ -925,19 +912,16 @@ class Decimal(object): if ans: return ans - sign = self._sign if not self: # + (-0) = 0 - sign = 0 + ans = self.copy_sign(Dec_0) + else: + ans = Decimal(self) if context is None: context = getcontext() - if context._rounding_decision == ALWAYS_ROUND: - ans = self._fix(context) - else: - ans = Decimal(self) - ans._sign = sign + return ans._fix(context) return ans def __abs__(self, round=1, context=None): @@ -1000,16 +984,19 @@ class Decimal(object): sign = min(self._sign, other._sign) if negativezero: sign = 1 - return Decimal( (sign, (0,), exp)) + ans = Decimal( (sign, (0,), exp)) + if shouldround: + ans = ans._fix(context) + return ans if not self: exp = max(exp, other._exp - context.prec-1) - ans = other._rescale(exp, watchexp=0, context=context) + ans = other._rescale(exp, context.rounding) if shouldround: ans = ans._fix(context) return ans if not other: exp = max(exp, self._exp - context.prec-1) - ans = self._rescale(exp, watchexp=0, context=context) + ans = self._rescale(exp, context.rounding) if shouldround: ans = ans._fix(context) return ans @@ -1022,10 +1009,10 @@ class Decimal(object): if op1.sign != op2.sign: # Equal and opposite if op1.int == op2.int: - if exp < context.Etiny(): - exp = context.Etiny() - context._raise_error(Clamped) - return Decimal((negativezero, (0,), exp)) + ans = Decimal((negativezero, (0,), exp)) + if shouldround: + ans = ans._fix(context) + return ans if op1.int < op2.int: op1, op2 = op2, op1 # OK, now abs(op1) > abs(op2) @@ -1056,7 +1043,7 @@ class Decimal(object): __radd__ = __add__ def __sub__(self, other, context=None): - """Return self + (-other)""" + """Return self - other""" other = _convert_other(other) if other is NotImplemented: return other @@ -1066,41 +1053,28 @@ class Decimal(object): if ans: return ans - # -Decimal(0) = Decimal(0), which we don't want since - # (-0 - 0 = -0 + (-0) = -0, but -0 + 0 = 0.) - # so we change the sign directly to a copy - tmp = Decimal(other) - tmp._sign = 1-tmp._sign - - return self.__add__(tmp, context=context) + # 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)""" + """Return other - self""" other = _convert_other(other) if other is NotImplemented: return other - tmp = Decimal(self) - tmp._sign = 1 - tmp._sign - return other.__add__(tmp, context=context) + return other.__sub__(self, context=context) - def _increment(self, round=1, context=None): + def _increment(self): """Special case of add, adding 1eExponent Since it is common, (rounding, for example) this adds (sign)*one E self._exp to the number more efficiently than add. + Assumes that self is nonspecial. + For example: Decimal('5.624e10')._increment() == Decimal('5.625e10') """ - if self._is_special: - ans = self._check_nans(context=context) - if ans: - return ans - - # Must be infinite, and incrementing makes no difference - return Decimal(self) - L = list(self._int) L[-1] += 1 spot = len(L)-1 @@ -1111,13 +1085,7 @@ class Decimal(object): break L[spot-1] += 1 spot -= 1 - ans = Decimal((self._sign, L, self._exp)) - - if context is None: - context = getcontext() - if round and context._rounding_decision == ALWAYS_ROUND: - ans = ans._fix(context) - return ans + return Decimal((self._sign, L, self._exp)) def __mul__(self, other, context=None): """Return self * other. @@ -1207,6 +1175,8 @@ class Decimal(object): if context is None: context = getcontext() + shouldround = context._rounding_decision == ALWAYS_ROUND + sign = self._sign ^ other._sign if self._is_special or other._is_special: @@ -1218,10 +1188,11 @@ class Decimal(object): if self._isinfinity() and other._isinfinity(): if divmod: - return (context._raise_error(InvalidOperation, + reloco = (context._raise_error(InvalidOperation, '(+-)INF // (+-)INF'), context._raise_error(InvalidOperation, '(+-)INF % (+-)INF')) + return reloco return context._raise_error(InvalidOperation, '(+-)INF/(+-)INF') if self._isinfinity(): @@ -1237,7 +1208,10 @@ class Decimal(object): if other._isinfinity(): if divmod: - return (Decimal((sign, (0,), 0)), Decimal(self)) + otherside = Decimal(self) + if shouldround and (divmod == 1 or divmod == 3): + otherside = otherside._fix(context) + return (Decimal((sign, (0,), 0)), otherside) context._raise_error(Clamped, 'Division by infinity') return Decimal((sign, (0,), context.Etiny())) @@ -1249,17 +1223,14 @@ class Decimal(object): if not self: if divmod: - otherside = Decimal(self) - otherside._exp = min(self._exp, other._exp) + otherside = Decimal((self._sign, (0,), min(self._exp, other._exp))) + if shouldround and (divmod == 1 or divmod == 3): + otherside = otherside._fix(context) return (Decimal((sign, (0,), 0)), otherside) exp = self._exp - other._exp - if exp < context.Etiny(): - exp = context.Etiny() - context._raise_error(Clamped, '0e-x / y') - if exp > context.Emax: - exp = context.Emax - context._raise_error(Clamped, '0e+x / y') - return Decimal( (sign, (0,), exp) ) + ans = Decimal((sign, (0,), exp)) + ans = ans._fix(context) + return ans if not other: if divmod: @@ -1268,7 +1239,6 @@ class Decimal(object): return context._raise_error(DivisionByZero, 'x / 0', sign) # OK, so neither = 0, INF or NaN - shouldround = context._rounding_decision == ALWAYS_ROUND # If we're dividing into ints, and self < other, stop. # self.__abs__(0) does not round. @@ -1276,7 +1246,7 @@ class Decimal(object): if divmod == 1 or divmod == 3: exp = min(self._exp, other._exp) - ans2 = self._rescale(exp, context=context, watchexp=0) + ans2 = self._rescale(exp, context.rounding) if shouldround: ans2 = ans2._fix(context) return (Decimal( (sign, (0,), 0) ), @@ -1302,12 +1272,9 @@ class Decimal(object): if res.int >= prec_limit and shouldround: return context._raise_error(DivisionImpossible) otherside = Decimal(op1) - frozen = context._ignore_all_flags() - exp = min(self._exp, other._exp) - otherside = otherside._rescale(exp, context=context, watchexp=0) - context._regard_flags(*frozen) - if shouldround: + otherside = otherside._rescale(exp, context.rounding) + if shouldround and (divmod == 1 or divmod == 3): otherside = otherside._fix(context) return (Decimal(res), otherside) @@ -1331,21 +1298,6 @@ class Decimal(object): op1.int *= 10 op1.exp -= 1 - if res.exp == 0 and divmod and op2.int > op1.int: - # Solves an error in precision. Same as a previous block. - - if res.int >= prec_limit and shouldround: - return context._raise_error(DivisionImpossible) - otherside = Decimal(op1) - frozen = context._ignore_all_flags() - - exp = min(self._exp, other._exp) - otherside = otherside._rescale(exp, context=context) - - context._regard_flags(*frozen) - - return (Decimal(res), otherside) - ans = Decimal(res) if shouldround: ans = ans._fix(context) @@ -1401,81 +1353,76 @@ class Decimal(object): """ Remainder nearest to 0- abs(remainder-near) <= other/2 """ - other = _convert_other(other) - if other is NotImplemented: - return other - - if self._is_special or other._is_special: - ans = self._check_nans(other, context) - if ans: - return ans - if self and not other: - return context._raise_error(InvalidOperation, 'x % 0') - if context is None: context = getcontext() - # If DivisionImpossible causes an error, do not leave Rounded/Inexact - # ignored in the calling function. - context = context._shallow_copy() - flags = context._ignore_flags(Rounded, Inexact) - # Keep DivisionImpossible flags - (side, r) = self.__divmod__(other, context=context) - if r._isnan(): - context._regard_flags(*flags) - return r + other = _convert_other(other, raiseit=True) - context = context._shallow_copy() - rounding = context._set_rounding_decision(NEVER_ROUND) - - if other._sign: - comparison = other.__div__(Decimal(-2), context=context) - else: - comparison = other.__div__(Decimal(2), context=context) - - context._set_rounding_decision(rounding) - context._regard_flags(*flags) + ans = self._check_nans(other, context) + if ans: + return ans - s1, s2 = r._sign, comparison._sign - r._sign, comparison._sign = 0, 0 + # self == +/-infinity -> InvalidOperation + if self._isinfinity(): + return context._raise_error(InvalidOperation, + 'remainder_near(infinity, x)') - if r < comparison: - r._sign, comparison._sign = s1, s2 - # Get flags now - self.__divmod__(other, context=context) - return r._fix(context) - r._sign, comparison._sign = s1, s2 + # 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)') - rounding = context._set_rounding_decision(NEVER_ROUND) + # other = +/-infinity -> remainder = self + if other._isinfinity(): + ans = Decimal(self) + return ans._fix(context) - (side, r) = self.__divmod__(other, context=context) - context._set_rounding_decision(rounding) - if r._isnan(): - return r + # self = 0 -> remainder = self, with ideal exponent + ideal_exponent = min(self._exp, other._exp) + if not self: + ans = Decimal((self._sign, (0,), ideal_exponent)) + return ans._fix(context) - decrease = not side._iseven() - rounding = context._set_rounding_decision(NEVER_ROUND) - side = side.__abs__(context=context) - context._set_rounding_decision(rounding) + # 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)[0] + if expdiff <= -2: + # expdiff <= -2 => abs(self/other) < 0.1 + ans = self._rescale(ideal_exponent, context.rounding) + return ans._fix(context) - s1, s2 = r._sign, comparison._sign - r._sign, comparison._sign = 0, 0 - if r > comparison or decrease and r == comparison: - r._sign, comparison._sign = s1, s2 - context.prec += 1 - numbsquant = len(side.__add__(Decimal(1), context=context)._int) - if numbsquant >= context.prec: - context.prec -= 1 - return context._raise_error(DivisionImpossible)[1] - context.prec -= 1 - if self._sign == other._sign: - r = r.__sub__(other, context=context) - else: - r = r.__add__(other, context=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: - r._sign, comparison._sign = s1, s2 + 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)[0] + + # result has same sign as self unless r is negative + sign = self._sign + if r < 0: + sign = 1-sign + r = -r - return r._fix(context) + ans = Decimal((sign, map(int, str(r)), ideal_exponent)) + return ans._fix(context) def __floordiv__(self, other, context=None): """self // other""" @@ -1500,14 +1447,11 @@ class Decimal(object): return context._raise_error(InvalidContext) elif self._isinfinity(): raise OverflowError("Cannot convert infinity to long") + s = (-1)**self._sign if self._exp >= 0: - s = ''.join(map(str, self._int)) + '0'*self._exp + return s*int(''.join(map(str, self._int)))*10**self._exp else: - s = ''.join(map(str, self._int))[:self._exp] - if s == '': - s = '0' - sign = '-'*self._sign - return int(sign + s) + return s*int(''.join(map(str, self._int))[:self._exp] or '0') def __long__(self): """Converts to a long. @@ -1516,6 +1460,21 @@ class Decimal(object): """ return long(self.__int__()) + def _fix_nan(self, context): + """Decapitate the payload of a NaN to fit the context""" + payload = self._int + + # maximum length of payload is precision if _clamp=0, + # precision-1 if _clamp=1. + max_payload_len = context.prec - context._clamp + if len(payload) > max_payload_len: + pos = len(payload)-max_payload_len + while pos < len(payload) and payload[pos] == 0: + pos += 1 + payload = payload[pos:] + return Decimal((self._sign, payload, self._exp)) + return self + def _fix(self, context): """Round if it is necessary to keep self within prec precision. @@ -1525,303 +1484,649 @@ class Decimal(object): self - Decimal instance context - context used. """ - if self._is_special: - return self - if context is None: - context = getcontext() - prec = context.prec - ans = self._fixexponents(context) - if len(ans._int) > prec: - ans = ans._round(prec, context=context) - ans = ans._fixexponents(context) - return ans - - def _fixexponents(self, context): - """Fix the exponents and return a copy with the exponent in bounds. - Only call if known to not be a special value. - """ - folddown = context._clamp - Emin = context.Emin - ans = self - ans_adjusted = ans.adjusted() - if ans_adjusted < Emin: - Etiny = context.Etiny() - if ans._exp < Etiny: - if not ans: - ans = Decimal(self) - ans._exp = Etiny - context._raise_error(Clamped) - return ans - ans = ans._rescale(Etiny, context=context) - # It isn't zero, and exp < Emin => subnormal - context._raise_error(Subnormal) - if context.flags[Inexact]: - context._raise_error(Underflow) - else: - if ans: - # Only raise subnormal if non-zero. - context._raise_error(Subnormal) - else: - Etop = context.Etop() - if folddown and ans._exp > Etop: - context._raise_error(Clamped) - ans = ans._rescale(Etop, context=context) - else: - Emax = context.Emax - if ans_adjusted > Emax: - if not ans: - ans = Decimal(self) - ans._exp = Emax - context._raise_error(Clamped) - return ans - context._raise_error(Inexact) - context._raise_error(Rounded) - c = context._raise_error(Overflow, 'above Emax', ans._sign) - return c - return ans - - def _round(self, prec=None, rounding=None, context=None): - """Returns a rounded version of self. - - You can specify the precision or rounding method. Otherwise, the - context determines it. - """ - - if self._is_special: - ans = self._check_nans(context=context) - if ans: - return ans - - if self._isinfinity(): - return Decimal(self) if context is None: context = getcontext() - if rounding is None: - rounding = context.rounding - if prec is None: - prec = context.prec + if self._is_special: + if self._isnan(): + # decapitate payload if necessary + return self._fix_nan(context) + else: + # self is +/-Infinity; return unaltered + return 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: - if prec <= 0: - dig = (0,) - exp = len(self._int) - prec + self._exp + 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 Decimal((self._sign, (0,), new_exp)) else: - dig = (0,) * prec - exp = len(self._int) + self._exp - prec - ans = Decimal((self._sign, dig, exp)) - context._raise_error(Rounded) - return ans + return self - if prec == 0: - temp = Decimal(self) - temp._int = (0,)+temp._int - prec = 1 - elif prec < 0: - exp = self._exp + len(self._int) - prec - 1 - temp = Decimal( (self._sign, (0, 1), exp)) - prec = 1 - else: - temp = Decimal(self) - - numdigits = len(temp._int) - if prec == numdigits: - return temp - - # See if we need to extend precision - expdiff = prec - numdigits - if expdiff > 0: - tmp = list(temp._int) - tmp.extend([0] * expdiff) - ans = Decimal( (temp._sign, tmp, temp._exp - expdiff)) - return ans + # exp_min is the smallest allowable exponent of the result, + # equal to max(self.adjusted()-context.prec+1, Etiny) + exp_min = len(self._int) + self._exp - context.prec + if exp_min > Etop: + # overflow: exp_min > Etop iff self.adjusted() > Emax + context._raise_error(Inexact) + context._raise_error(Rounded) + return context._raise_error(Overflow, 'above Emax', self._sign) + self_is_subnormal = exp_min < Etiny + if self_is_subnormal: + context._raise_error(Subnormal) + exp_min = Etiny - # OK, but maybe all the lost digits are 0. - lostdigits = self._int[expdiff:] - if lostdigits == (0,) * len(lostdigits): - ans = Decimal( (temp._sign, temp._int[:prec], temp._exp - expdiff)) - # Rounded, but not Inexact + # round if self has too many digits + if self._exp < exp_min: context._raise_error(Rounded) + ans = self._rescale(exp_min, context.rounding) + if ans != self: + context._raise_error(Inexact) + if self_is_subnormal: + context._raise_error(Underflow) + if not ans: + # raise Clamped on underflow to 0 + context._raise_error(Clamped) + elif len(ans._int) == context.prec+1: + # we get here only if rescaling rounds the + # cofficient up to exactly 10**context.prec + if ans._exp < Etop: + ans = Decimal((ans._sign, ans._int[:-1], ans._exp+1)) + else: + # Inexact and Rounded have already been raised + ans = context._raise_error(Overflow, 'above Emax', + self._sign) return ans - # Okay, let's round and lose data - - this_function = getattr(temp, self._pick_rounding_function[rounding]) - # Now we've got the rounding function - - if prec != context.prec: - context = context._shallow_copy() - context.prec = prec - ans = this_function(prec, expdiff, context) - context._raise_error(Rounded) - context._raise_error(Inexact, 'Changed in rounding') + # 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 Decimal((self._sign, self_padded, Etop)) - return ans + # here self was representable to begin with; return unchanged + return self _pick_rounding_function = {} - def _round_down(self, prec, expdiff, context): - """Also known as round-towards-0, truncate.""" - return Decimal( (self._sign, self._int[:prec], self._exp - expdiff) ) + # for each of the rounding functions below: + # self is a finite, nonzero Decimal + # prec is an integer satisfying 0 <= prec < len(self._int) + # the rounded result will have exponent self._exp + len(self._int) - prec; - def _round_half_up(self, prec, expdiff, context, tmp = None): - """Rounds 5 up (away from 0)""" + def _round_down(self, prec): + """Also known as round-towards-0, truncate.""" + newexp = self._exp + len(self._int) - prec + return Decimal((self._sign, self._int[:prec] or (0,), newexp)) - if tmp is None: - tmp = Decimal( (self._sign,self._int[:prec], self._exp - expdiff)) - if self._int[prec] >= 5: - tmp = tmp._increment(round=0, context=context) - if len(tmp._int) > prec: - return Decimal( (tmp._sign, tmp._int[:-1], tmp._exp + 1)) + def _round_up(self, prec): + """Rounds away from 0.""" + newexp = self._exp + len(self._int) - prec + tmp = Decimal((self._sign, self._int[:prec] or (0,), newexp)) + for digit in self._int[prec:]: + if digit != 0: + return tmp._increment() return tmp - def _round_half_even(self, prec, expdiff, context): - """Round 5 to even, rest to nearest.""" - - tmp = Decimal( (self._sign, self._int[:prec], self._exp - expdiff)) - half = (self._int[prec] == 5) - if half: - for digit in self._int[prec+1:]: - if digit != 0: - half = 0 - break - if half: - if self._int[prec-1] & 1 == 0: - return tmp - return self._round_half_up(prec, expdiff, context, tmp) + def _round_half_up(self, prec): + """Rounds 5 up (away from 0)""" + if self._int[prec] >= 5: + return self._round_up(prec) + else: + return self._round_down(prec) - def _round_half_down(self, prec, expdiff, context): + def _round_half_down(self, prec): """Round 5 down""" - - tmp = Decimal( (self._sign, self._int[:prec], self._exp - expdiff)) - half = (self._int[prec] == 5) - if half: + if self._int[prec] == 5: for digit in self._int[prec+1:]: if digit != 0: - half = 0 break - if half: - return tmp - return self._round_half_up(prec, expdiff, context, tmp) + else: + return self._round_down(prec) + return self._round_half_up(prec) - def _round_up(self, prec, expdiff, context): - """Rounds away from 0.""" - tmp = Decimal( (self._sign, self._int[:prec], self._exp - expdiff) ) - for digit in self._int[prec:]: - if digit != 0: - tmp = tmp._increment(round=1, context=context) - if len(tmp._int) > prec: - return Decimal( (tmp._sign, tmp._int[:-1], tmp._exp + 1)) - else: - return tmp - return tmp + def _round_half_even(self, prec): + """Round 5 to even, rest to nearest.""" + if prec and self._int[prec-1] & 1: + return self._round_half_up(prec) + else: + return self._round_half_down(prec) - def _round_ceiling(self, prec, expdiff, context): + def _round_ceiling(self, prec): """Rounds up (not away from 0 if negative.)""" if self._sign: - return self._round_down(prec, expdiff, context) + return self._round_down(prec) else: - return self._round_up(prec, expdiff, context) + return self._round_up(prec) - def _round_floor(self, prec, expdiff, context): + def _round_floor(self, prec): """Rounds down (not towards 0 if negative)""" if not self._sign: - return self._round_down(prec, expdiff, context) + return self._round_down(prec) + else: + return self._round_up(prec) + + def _round_05up(self, prec): + """Round down unless digit prec-1 is 0 or 5.""" + if prec == 0 or self._int[prec-1] in (0, 5): + return self._round_up(prec) else: - return self._round_up(prec, expdiff, context) + return self._round_down(prec) - def __pow__(self, n, modulo = None, context=None): - """Return self ** n (mod modulo) + def fma(self, other, third, context=None): + """Fused multiply-add. - If modulo is None (default), don't take it mod modulo. + 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. """ - n = _convert_other(n) - if n is NotImplemented: - return n + + other = _convert_other(other, raiseit=True) + third = _convert_other(third, raiseit=True) if context is None: context = getcontext() - if self._is_special or n._is_special or n.adjusted() > 8: - # Because the spot << doesn't work with really big exponents - if n._isinfinity() or n.adjusted() > 8: - return context._raise_error(InvalidOperation, 'x ** INF') + # do self*other in fresh context with no traps and no rounding + mul_context = Context(traps=[], flags=[], + _rounding_decision=NEVER_ROUND) + product = self.__mul__(other, mul_context) - ans = self._check_nans(n, context) - if ans: - return ans + if mul_context.flags[InvalidOperation]: + # reraise in current context + return context._raise_error(InvalidOperation, + 'invalid multiplication in fma', + 1, product) + + ans = product.__add__(third, context) + return ans + + def _power_modulo(self, other, modulo, context=None): + """Three argument version of __pow__""" + + # if can't convert other and modulo to Decimal, raise + # TypeError; there's no point returning NotImplemented (no + # equivalent of __rpow__ for three argument pow) + other = _convert_other(other, raiseit=True) + modulo = _convert_other(modulo, raiseit=True) + + if context is None: + context = getcontext() + + # deal with NaNs: if there are any sNaNs then first one wins, + # (i.e. behaviour for NaNs is identical to that of fma) + self_is_nan = self._isnan() + other_is_nan = other._isnan() + modulo_is_nan = modulo._isnan() + if self_is_nan or other_is_nan or modulo_is_nan: + if self_is_nan == 2: + return context._raise_error(InvalidOperation, 'sNaN', + 1, self) + if other_is_nan == 2: + return context._raise_error(InvalidOperation, 'sNaN', + 1, other) + if modulo_is_nan == 2: + return context._raise_error(InvalidOperation, 'sNaN', + 1, modulo) + if self_is_nan: + return self + if other_is_nan: + return other + return modulo + + # check inputs: we apply same restrictions as Python's pow() + if not (self._isinteger() and + other._isinteger() and + modulo._isinteger()): + return context._raise_error(InvalidOperation, + 'pow() 3rd argument not allowed ' + 'unless all arguments are integers') + if other < 0: + return context._raise_error(InvalidOperation, + 'pow() 2nd argument cannot be ' + 'negative when 3rd argument specified') + if not modulo: + return context._raise_error(InvalidOperation, + 'pow() 3rd argument cannot be 0') + + # additional restriction for decimal: the modulus must be less + # than 10**prec in absolute value + if modulo.adjusted() >= context.prec: + return context._raise_error(InvalidOperation, + 'insufficient precision: pow() 3rd ' + 'argument must not have more than ' + 'precision digits') + + # define 0**0 == NaN, for consistency with two-argument pow + # (even though it hurts!) + if not other and not self: + return context._raise_error(InvalidOperation, + 'at least one of pow() 1st argument ' + 'and 2nd argument must be nonzero ;' + '0**0 is not defined') + + # compute sign of result + if other._iseven(): + sign = 0 + else: + sign = self._sign + + # convert modulo to a Python integer, and self and other to + # Decimal integers (i.e. force their exponents to be >= 0) + modulo = abs(int(modulo)) + base = _WorkRep(self.to_integral_value()) + exponent = _WorkRep(other.to_integral_value()) + + # compute result using integer pow() + base = (base.int % modulo * pow(10, base.exp, modulo)) % modulo + for i in xrange(exponent.exp): + base = pow(base, 10, modulo) + base = pow(base, exponent.int, modulo) + + return Decimal((sign, map(int, str(base)), 0)) + + def _power_exact(self, other, p): + """Attempt to compute self**other exactly. + + Given Decimals self and other and an integer p, attempt to + compute an exact result for the power self**other, with p + digits of precision. Return None if self**other is not + exactly representable in p digits. + + Assumes that elimination of special cases has already been + performed: self and other must both be nonspecial; self must + be positive and not numerically equal to 1; other must be + nonzero. For efficiency, other._exp should not be too large, + so that 10**abs(other._exp) is a feasible calculation.""" + + # In the comments below, we write x for the value of self and + # y for the value of other. Write x = xc*10**xe and y = + # yc*10**ye. + + # The main purpose of this method is to identify the *failure* + # of x**y to be exactly representable with as little effort as + # possible. So we look for cheap and easy tests that + # eliminate the possibility of x**y being exact. Only if all + # these tests are passed do we go on to actually compute x**y. + + # Here's the main idea. First normalize both x and y. We + # express y as a rational m/n, with m and n relatively prime + # and n>0. Then for x**y to be exactly representable (at + # *any* precision), xc must be the nth power of a positive + # integer and xe must be divisible by n. If m is negative + # then additionally xc must be a power of either 2 or 5, hence + # a power of 2**n or 5**n. + # + # There's a limit to how small |y| can be: if y=m/n as above + # then: + # + # (1) if xc != 1 then for the result to be representable we + # need xc**(1/n) >= 2, and hence also xc**|y| >= 2. So + # if |y| <= 1/nbits(xc) then xc < 2**nbits(xc) <= + # 2**(1/|y|), hence xc**|y| < 2 and the result is not + # representable. + # + # (2) if xe != 0, |xe|*(1/n) >= 1, so |xe|*|y| >= 1. Hence if + # |y| < 1/|xe| then the result is not representable. + # + # Note that since x is not equal to 1, at least one of (1) and + # (2) must apply. Now |y| < 1/nbits(xc) iff |yc|*nbits(xc) < + # 10**-ye iff len(str(|yc|*nbits(xc)) <= -ye. + # + # There's also a limit to how large y can be, at least if it's + # positive: the normalized result will have coefficient xc**y, + # so if it's representable then xc**y < 10**p, and y < + # p/log10(xc). Hence if y*log10(xc) >= p then the result is + # not exactly representable. + + # if len(str(abs(yc*xe)) <= -ye then abs(yc*xe) < 10**-ye, + # so |y| < 1/xe and the result is not representable. + # Similarly, len(str(abs(yc)*xc_bits)) <= -ye implies |y| + # < 1/nbits(xc). + + x = _WorkRep(self) + xc, xe = x.int, x.exp + while xc % 10 == 0: + xc //= 10 + xe += 1 + + y = _WorkRep(other) + yc, ye = y.int, y.exp + while yc % 10 == 0: + yc //= 10 + ye += 1 + + # case where xc == 1: result is 10**(xe*y), with xe*y + # required to be an integer + if xc == 1: + if ye >= 0: + exponent = xe*yc*10**ye + else: + exponent, remainder = divmod(xe*yc, 10**-ye) + if remainder: + return None + if y.sign == 1: + exponent = -exponent + # if other is a nonnegative integer, use ideal exponent + if other._isinteger() and other._sign == 0: + ideal_exponent = self._exp*int(other) + zeros = min(exponent-ideal_exponent, p-1) + else: + zeros = 0 + return Decimal((0, (1,) + (0,)*zeros, exponent-zeros)) + + # case where y is negative: xc must be either a power + # of 2 or a power of 5. + if y.sign == 1: + last_digit = xc % 10 + if last_digit in (2,4,6,8): + # quick test for power of 2 + if xc & -xc != xc: + return None + # now xc is a power of 2; e is its exponent + e = _nbits(xc)-1 + # find e*y and xe*y; both must be integers + if ye >= 0: + y_as_int = yc*10**ye + e = e*y_as_int + xe = xe*y_as_int + else: + ten_pow = 10**-ye + e, remainder = divmod(e*yc, ten_pow) + if remainder: + return None + xe, remainder = divmod(xe*yc, ten_pow) + if remainder: + return None + + if e*65 >= p*93: # 93/65 > log(10)/log(5) + return None + xc = 5**e + + elif last_digit == 5: + # e >= log_5(xc) if xc is a power of 5; we have + # equality all the way up to xc=5**2658 + e = _nbits(xc)*28//65 + xc, remainder = divmod(5**e, xc) + if remainder: + return None + while xc % 5 == 0: + xc //= 5 + e -= 1 + if ye >= 0: + y_as_integer = yc*10**ye + e = e*y_as_integer + xe = xe*y_as_integer + else: + ten_pow = 10**-ye + e, remainder = divmod(e*yc, ten_pow) + if remainder: + return None + xe, remainder = divmod(xe*yc, ten_pow) + if remainder: + return None + if e*3 >= p*10: # 10/3 > log(10)/log(2) + return None + xc = 2**e + else: + return None + + if xc >= 10**p: + return None + xe = -e-xe + return Decimal((0, map(int, str(xc)), xe)) + + # now y is positive; find m and n such that y = m/n + if ye >= 0: + m, n = yc*10**ye, 1 + else: + if xe != 0 and len(str(abs(yc*xe))) <= -ye: + return None + xc_bits = _nbits(xc) + if xc != 1 and len(str(abs(yc)*xc_bits)) <= -ye: + return None + m, n = yc, 10**(-ye) + while m % 2 == n % 2 == 0: + m //= 2 + n //= 2 + while m % 5 == n % 5 == 0: + m //= 5 + n //= 5 + + # compute nth root of xc*10**xe + if n > 1: + # if 1 < xc < 2**n then xc isn't an nth power + if xc != 1 and xc_bits <= n: + return None + + xe, rem = divmod(xe, n) + if rem != 0: + return None + + # compute nth root of xc using Newton's method + a = 1L << -(-_nbits(xc)//n) # initial estimate + while True: + q, r = divmod(xc, a**(n-1)) + if a <= q: + break + else: + a = (a*(n-1) + q)//n + if not (a == q and r == 0): + return None + xc = a + + # now xc*10**xe is the nth root of the original xc*10**xe + # compute mth power of xc*10**xe + + # if m > p*100//_log10_lb(xc) then m > p/log10(xc), hence xc**m > + # 10**p and the result is not representable. + if xc > 1 and m > p*100//_log10_lb(xc): + return None + xc = xc**m + xe *= m + if xc > 10**p: + return None + + # by this point the result *is* exactly representable + # adjust the exponent to get as close as possible to the ideal + # exponent, if necessary + str_xc = str(xc) + if other._isinteger() and other._sign == 0: + ideal_exponent = self._exp*int(other) + zeros = min(xe-ideal_exponent, p-len(str_xc)) + else: + zeros = 0 + return Decimal((0, map(int, str_xc)+[0,]*zeros, xe-zeros)) + + def __pow__(self, other, modulo=None, context=None): + """Return self ** other [ % modulo]. - if not n._isinteger(): - return context._raise_error(InvalidOperation, 'x ** (non-integer)') + With two arguments, compute self**other. - if not self and not n: - return context._raise_error(InvalidOperation, '0 ** 0') + With three arguments, compute (self**other) % modulo. For the + three argument form, the following restrictions on the + arguments hold: - if not n: - return Decimal(1) + - 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 self == Decimal(1): - return Decimal(1) + If any of these restrictions is violated the InvalidOperation + flag is raised. - sign = self._sign and not n._iseven() - n = int(n) + The result of pow(self, other, modulo) is identical to the + result that would be obtained by computing (self**other) % + modulo with unbounded precision, but is computed more + efficiently. It is always exact. + """ + + if modulo is not None: + return self._power_modulo(other, modulo, context) + + other = _convert_other(other) + if other is NotImplemented: + return other + + if context is None: + context = getcontext() + + # either argument is a NaN => result is NaN + ans = self._check_nans(other, context) + if ans: + return ans + + # 0**0 = NaN (!), x**0 = 1 for nonzero x (including +/-Infinity) + if not other: + if not self: + return context._raise_error(InvalidOperation, '0 ** 0') + else: + return Dec_p1 + + # result has sign 1 iff self._sign is 1 and other is an odd integer + result_sign = 0 + if self._sign == 1: + if other._isinteger(): + if not other._iseven(): + result_sign = 1 + else: + # -ve**noninteger = NaN + # (-0)**noninteger = 0**noninteger + if self: + return context._raise_error(InvalidOperation, + 'x ** y with x negative and y not an integer') + # negate self, without doing any unwanted rounding + self = Decimal((0, self._int, self._exp)) + + # 0**(+ve or Inf)= 0; 0**(-ve or -Inf) = Infinity + if not self: + if other._sign == 0: + return Decimal((result_sign, (0,), 0)) + else: + return Infsign[result_sign] + # Inf**(+ve or Inf) = Inf; Inf**(-ve or -Inf) = 0 if self._isinfinity(): - if modulo: - return context._raise_error(InvalidOperation, 'INF % x') - if n > 0: - return Infsign[sign] - return Decimal( (sign, (0,), 0) ) + if other._sign == 0: + return Infsign[result_sign] + else: + return Decimal((result_sign, (0,), 0)) + + # 1**other = 1, but the choice of exponent and the flags + # depend on the exponent of self, and on whether other is a + # positive integer, a negative integer, or neither + if self == Dec_p1: + if other._isinteger(): + # exp = max(self._exp*max(int(other), 0), + # 1-context.prec) but evaluating int(other) directly + # is dangerous until we know other is small (other + # could be 1e999999999) + if other._sign == 1: + multiplier = 0 + elif other > context.prec: + multiplier = context.prec + else: + multiplier = int(other) - # With ludicrously large exponent, just raise an overflow - # and return inf. - if not modulo and n > 0 and \ - (self._exp + len(self._int) - 1) * n > context.Emax and self: + 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 - tmp = Decimal('inf') - tmp._sign = sign - context._raise_error(Rounded) - context._raise_error(Inexact) - context._raise_error(Overflow, 'Big power', sign) - return tmp + return Decimal((result_sign, (1,)+(0,)*-exp, exp)) - elength = len(str(abs(n))) - firstprec = context.prec + # compute adjusted exponent of self + self_adj = self.adjusted() - if not modulo and firstprec + elength + 1 > DefaultContext.Emax: - return context._raise_error(Overflow, 'Too much precision.', sign) + # 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 Decimal((result_sign, (0,), 0)) + else: + return Infsign[result_sign] + + # from here on, the result always goes through the call + # to _fix at the end of this function. + ans = None + + # crude test to catch cases of extreme overflow/underflow. If + # log10(self)*other >= 10**bound and bound >= len(str(Emax)) + # then 10**bound >= 10**len(str(Emax)) >= Emax+1 and hence + # self**other >= 10**(Emax+1), so overflow occurs. The test + # for underflow is similar. + bound = self._log10_exp_bound() + other.adjusted() + if (self_adj >= 0) == (other._sign == 0): + # self > 1 and other +ve, or self < 1 and other -ve + # possibility of overflow + if bound >= len(str(context.Emax)): + ans = Decimal((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 = Decimal((result_sign, (1,), Etiny-1)) + + # try for an exact result with precision +1 + if ans is None: + ans = self._power_exact(other, context.prec + 1) + if ans is not None and result_sign == 1: + ans = Decimal((1, ans._int, ans._exp)) + + # usual case: inexact result, x**y computed directly as exp(y*log(x)) + if ans is None: + p = context.prec + x = _WorkRep(self) + xc, xe = x.int, x.exp + y = _WorkRep(other) + yc, ye = y.int, y.exp + if y.sign == 1: + yc = -yc + + # compute correctly rounded result: start with precision +3, + # then increase precision until result is unambiguously roundable + extra = 3 + while True: + coeff, exp = _dpower(xc, xe, yc, ye, p+extra) + if coeff % (5*10**(len(str(coeff))-p-1)): + break + extra += 3 - mul = Decimal(self) - val = Decimal(1) - context = context._shallow_copy() - context.prec = firstprec + elength + 1 - if n < 0: - # n is a long now, not Decimal instance - n = -n - mul = Decimal(1).__div__(mul, context=context) - - spot = 1 - while spot <= n: - spot <<= 1 - - spot >>= 1 - # spot is the highest power of 2 less than n - while spot: - val = val.__mul__(val, context=context) - if val._isinfinity(): - val = Infsign[sign] - break - if spot & n: - val = val.__mul__(mul, context=context) - if modulo is not None: - val = val.__mod__(modulo, context=context) - spot >>= 1 - context.prec = firstprec + ans = Decimal((result_sign, map(int, str(coeff)), exp)) - if context._rounding_decision == ALWAYS_ROUND: - return val._fix(context) - return val + # the specification says that for non-integer other we need to + # raise Inexact, even when the result is actually exact. In + # the same way, we need to raise Underflow here if the result + # is subnormal. (The call to _fix will take care of raising + # Rounded and Subnormal, as usual.) + if not other._isinteger(): + context._raise_error(Inexact) + # pad with zeros up to length context.prec+1 if necessary + if len(ans._int) <= context.prec: + expdiff = context.prec+1 - len(ans._int) + ans = Decimal((ans._sign, ans._int+(0,)*expdiff, ans._exp-expdiff)) + if ans.adjusted() < context.Emin: + context._raise_error(Underflow) + + # unlike exp, ln and log10, the power function respects the + # rounding mode; no need to use ROUND_HALF_EVEN here + ans = ans._fix(context) + return ans def __rpow__(self, other, context=None): """Swaps self/other and returns __pow__.""" @@ -1833,6 +2138,9 @@ class Decimal(object): 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: @@ -1844,19 +2152,25 @@ class Decimal(object): if not dup: return Decimal( (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: + while dup._int[end-1] == 0 and exp < exp_max: exp += 1 end -= 1 return Decimal( (dup._sign, dup._int[:end], exp) ) - def quantize(self, exp, rounding=None, context=None, watchexp=1): + 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. """ + 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: @@ -1865,11 +2179,45 @@ class Decimal(object): if exp._isinfinity() or self._isinfinity(): if exp._isinfinity() and self._isinfinity(): return self # if both are inf, it is OK - if context is None: - context = getcontext() return context._raise_error(InvalidOperation, 'quantize with one INF') - return self._rescale(exp._exp, rounding, context, watchexp) + + # 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 = Decimal((self._sign, (0,), exp._exp)) + return ans._fix(context) + + self_adjusted = self.adjusted() + if self_adjusted > context.Emax: + return context._raise_error(InvalidOperation, + 'exponent of quantize result too large for current context') + if self_adjusted - exp._exp + 1 > context.prec: + return context._raise_error(InvalidOperation, + 'quantize result has too many digits for current context') + + ans = self._rescale(exp._exp, rounding) + if ans.adjusted() > context.Emax: + return context._raise_error(InvalidOperation, + 'exponent of quantize result too large for current context') + if len(ans._int) > context.prec: + return context._raise_error(InvalidOperation, + 'quantize result has too many digits for current context') + + # raise appropriate flags + if ans._exp > self._exp: + context._raise_error(Rounded) + if ans != self: + context._raise_error(Inexact) + if ans and ans.adjusted() < context.Emin: + context._raise_error(Subnormal) + + # call to fix takes care of any necessary folddown + ans = ans._fix(context) + return ans def same_quantum(self, other): """Test whether self and other have the same exponent. @@ -1883,82 +2231,85 @@ class Decimal(object): return self._isinfinity() and other._isinfinity() and True return self._exp == other._exp - def _rescale(self, exp, rounding=None, context=None, watchexp=1): - """Rescales so that the exponent is 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 version - watchexp: if set (default) an error is returned if exp is greater - than Emax or less than Etiny. + rounding = rounding mode """ - if context is None: - context = getcontext() - if self._is_special: - if self._isinfinity(): - return context._raise_error(InvalidOperation, 'rescale with an INF') - - ans = self._check_nans(context=context) - if ans: - return ans - - if watchexp and (context.Emax < exp or context.Etiny() > exp): - return context._raise_error(InvalidOperation, 'rescale(a, INF)') - + return self if not self: - ans = Decimal(self) - ans._int = (0,) - ans._exp = exp - return ans - - diff = self._exp - exp - digits = len(self._int) + diff - - if watchexp and digits > context.prec: - return context._raise_error(InvalidOperation, 'Rescale > prec') + return Decimal((self._sign, (0,), exp)) - tmp = Decimal(self) - tmp._int = (0,) + tmp._int - digits += 1 + if self._exp >= exp: + # pad answer with zeros if necessary + return Decimal((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: - tmp._exp = -digits + tmp._exp - tmp._int = (0,1) - digits = 1 - tmp = tmp._round(digits, rounding, context=context) + self = Decimal((self._sign, (1,), exp-1)) + digits = 0 + this_function = getattr(self, self._pick_rounding_function[rounding]) + return this_function(digits) - if tmp._int[0] == 0 and len(tmp._int) > 1: - tmp._int = tmp._int[1:] - tmp._exp = exp + def to_integral_exact(self, rounding=None, context=None): + """Rounds to a nearby integer. - tmp_adjusted = tmp.adjusted() - if tmp and tmp_adjusted < context.Emin: - context._raise_error(Subnormal) - elif tmp and tmp_adjusted > context.Emax: - return context._raise_error(InvalidOperation, 'rescale(a, INF)') - return tmp + If no rounding mode is specified, take the rounding mode from + the context. This method raises the Rounded and Inexact flags + when appropriate. - def to_integral(self, rounding=None, context=None): - """Rounds to the nearest integer, without raising inexact, rounded.""" + 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 self if self._exp >= 0: return self + if not self: + return Decimal((self._sign, (0,), 0)) if context is None: context = getcontext() - flags = context._ignore_flags(Rounded, Inexact) - ans = self._rescale(0, rounding, context=context) - context._regard_flags(flags) + if rounding is None: + rounding = context.rounding + context._raise_error(Rounded) + ans = self._rescale(0, rounding) + if ans != self: + context._raise_error(Inexact) return ans - def sqrt(self, context=None): - """Return the square root of self. + 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 self + if self._exp >= 0: + return self + else: + return self._rescale(0, rounding) - Uses a converging algorithm (Xn+1 = 0.5*(Xn + self / Xn)) - Should quadratically approach the right answer. - """ + # the method name changed, but we provide also the old one, for compatibility + to_integral = to_integral_value + + def sqrt(self, context=None): + """Return the square root of self.""" if self._is_special: ans = self._check_nans(context=context) if ans: @@ -1968,16 +2319,9 @@ class Decimal(object): return Decimal(self) if not self: - # exponent = self._exp / 2, using round_down. - # if self._exp < 0: - # exp = (self._exp+1) // 2 - # else: - exp = (self._exp) // 2 - if self._sign == 1: - # sqrt(-0) = -0 - return Decimal( (1, (0,), exp)) - else: - return Decimal( (0, (0,), exp)) + # exponent = self._exp // 2. sqrt(-0) = -0 + ans = Decimal((self._sign, (0,), self._exp // 2)) + return ans._fix(context) if context is None: context = getcontext() @@ -1985,104 +2329,91 @@ class Decimal(object): if self._sign == 1: return context._raise_error(InvalidOperation, 'sqrt(-x), x > 0') - tmp = Decimal(self) - - expadd = tmp._exp // 2 - if tmp._exp & 1: - tmp._int += (0,) - tmp._exp = 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: - tmp._exp = 0 - - context = context._shallow_copy() - flags = context._ignore_all_flags() - firstprec = context.prec - context.prec = 3 - if tmp.adjusted() & 1 == 0: - ans = Decimal( (0, (8,1,9), tmp.adjusted() - 2) ) - ans = ans.__add__(tmp.__mul__(Decimal((0, (2,5,9), -2)), - context=context), context=context) - ans._exp -= 1 + tmp.adjusted() // 2 + 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: - ans = Decimal( (0, (2,5,9), tmp._exp + len(tmp._int)- 3) ) - ans = ans.__add__(tmp.__mul__(Decimal((0, (8,1,9), -3)), - context=context), context=context) - ans._exp -= 1 + tmp.adjusted() // 2 - - # ans is now a linear approximation. - Emax, Emin = context.Emax, context.Emin - context.Emax, context.Emin = DefaultContext.Emax, DefaultContext.Emin - - half = Decimal('0.5') - - maxp = firstprec + 2 - rounding = context._set_rounding(ROUND_HALF_EVEN) - while 1: - context.prec = min(2*context.prec - 2, maxp) - ans = half.__mul__(ans.__add__(tmp.__div__(ans, context=context), - context=context), context=context) - if context.prec == maxp: + 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 - - # Round to the answer's precision-- the only error can be 1 ulp. - context.prec = firstprec - prevexp = ans.adjusted() - ans = ans._round(context=context) - - # Now, check if the other last digits are better. - context.prec = firstprec + 1 - # In case we rounded up another digit and we should actually go lower. - if prevexp != ans.adjusted(): - ans._int += (0,) - ans._exp -= 1 - - - lower = ans.__sub__(Decimal((0, (5,), ans._exp-1)), context=context) - context._set_rounding(ROUND_UP) - if lower.__mul__(lower, context=context) > (tmp): - ans = ans.__sub__(Decimal((0, (1,), ans._exp)), context=context) - + 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: - upper = ans.__add__(Decimal((0, (5,), ans._exp-1)),context=context) - context._set_rounding(ROUND_DOWN) - if upper.__mul__(upper, context=context) < tmp: - ans = ans.__add__(Decimal((0, (1,), ans._exp)),context=context) + # result is not exact; fix last digit as described above + if n % 5 == 0: + n += 1 - ans._exp += expadd + ans = Decimal((0, map(int, str(n)), e)) - context.prec = firstprec - context.rounding = rounding + # round, and fit to current context + context = context._shallow_copy() + rounding = context._set_rounding(ROUND_HALF_EVEN) ans = ans._fix(context) + context.rounding = rounding - rounding = context._set_rounding_decision(NEVER_ROUND) - if not ans.__mul__(ans, context=context) == self: - # Only rounded/inexact if here. - context._regard_flags(flags) - context._raise_error(Rounded) - context._raise_error(Inexact) - else: - # Exact answer, so let's set the exponent right. - # if self._exp < 0: - # exp = (self._exp +1)// 2 - # else: - exp = self._exp // 2 - context.prec += ans._exp - exp - ans = ans._rescale(exp, context=context) - context.prec = firstprec - context._regard_flags(flags) - context.Emax, context.Emin = Emax, Emin - - return ans._fix(context) + return ans def max(self, other, context=None): """Returns the larger value. - like max(self, other) except if one is not a number, returns + 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) - if other is NotImplemented: - return other + other = _convert_other(other, raiseit=True) if self._is_special or other._is_special: # If one operand is a quiet NaN and the other is number, then the @@ -2096,7 +2427,6 @@ class Decimal(object): return other return self._check_nans(other, context) - ans = self c = self.__cmp__(other) if c == 0: # If both operands are finite and equal in numerical value @@ -2106,16 +2436,13 @@ class Decimal(object): # 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. - if self._sign != other._sign: - if self._sign: - ans = other - elif self._exp < other._exp and not self._sign: - ans = other - elif self._exp > other._exp and self._sign: - ans = other - elif c == -1: + # the result. This is exactly the ordering used in compare_total. + c = self.compare_total(other) + + if c == -1: ans = other + else: + ans = self if context is None: context = getcontext() @@ -2129,9 +2456,7 @@ class Decimal(object): 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) - if other is NotImplemented: - return other + other = _convert_other(other, raiseit=True) if self._is_special or other._is_special: # If one operand is a quiet NaN and the other is number, then the @@ -2145,25 +2470,13 @@ class Decimal(object): return other return self._check_nans(other, context) - ans = self 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. - if self._sign != other._sign: - if other._sign: - ans = other - elif self._exp > other._exp and not self._sign: - ans = other - elif self._exp < other._exp and self._sign: - ans = other - elif c == 1: + c = self.compare_total(other) + + if c == -1: + ans = self + else: ans = other if context is None: @@ -2174,15 +2487,17 @@ class Decimal(object): 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 1 if self is even. Assumes self is an integer.""" - if self._exp > 0: - return 1 + """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] & 1 == 0 def adjusted(self): @@ -2193,6 +2508,872 @@ class Decimal(object): except TypeError: return 0 + def canonical(self, context=None): + """Returns the same Decimal object. + + As we do not have different encodings for the same number, the + received object already is in its canonical form. + """ + return self + + def compare_signal(self, other, context=None): + """Compares self to the other operand numerically. + + It's pretty much like compare(), but all NaNs signal, with signaling + NaNs taking precedence over quiet NaNs. + """ + if context is None: + context = getcontext() + + self_is_nan = self._isnan() + other_is_nan = other._isnan() + if self_is_nan == 2: + return context._raise_error(InvalidOperation, 'sNaN', + 1, self) + if other_is_nan == 2: + return context._raise_error(InvalidOperation, 'sNaN', + 1, other) + if self_is_nan: + return context._raise_error(InvalidOperation, 'NaN in compare_signal', + 1, self) + if other_is_nan: + return context._raise_error(InvalidOperation, 'NaN in compare_signal', + 1, other) + return self.compare(other, context=context) + + def compare_total(self, other): + """Compares self to other using the abstract representations. + + This is not like the standard compare, which use their numerical + value. Note that a total ordering is defined for all possible abstract + representations. + """ + # if one is negative and the other is positive, it's easy + if self._sign and not other._sign: + return Dec_n1 + if not self._sign and other._sign: + return Dec_p1 + sign = self._sign + + # let's handle both NaN types + self_nan = self._isnan() + other_nan = other._isnan() + if self_nan or other_nan: + if self_nan == other_nan: + if self._int < other._int: + if sign: + return Dec_p1 + else: + return Dec_n1 + if self._int > other._int: + if sign: + return Dec_n1 + else: + return Dec_p1 + return Dec_0 + + if sign: + if self_nan == 1: + return Dec_n1 + if other_nan == 1: + return Dec_p1 + if self_nan == 2: + return Dec_n1 + if other_nan == 2: + return Dec_p1 + else: + if self_nan == 1: + return Dec_p1 + if other_nan == 1: + return Dec_n1 + if self_nan == 2: + return Dec_p1 + if other_nan == 2: + return Dec_n1 + + if self < other: + return Dec_n1 + if self > other: + return Dec_p1 + + if self._exp < other._exp: + if sign: + return Dec_p1 + else: + return Dec_n1 + if self._exp > other._exp: + if sign: + return Dec_n1 + else: + return Dec_p1 + return Dec_0 + + + def compare_total_mag(self, other): + """Compares self to other using abstract repr., ignoring sign. + + Like compare_total, but with operand's sign ignored and assumed to be 0. + """ + s = self.copy_abs() + o = other.copy_abs() + return s.compare_total(o) + + def copy_abs(self): + """Returns a copy with the sign set to 0. """ + return Decimal((0, self._int, self._exp)) + + def copy_negate(self): + """Returns a copy with the sign inverted.""" + if self._sign: + return Decimal((0, self._int, self._exp)) + else: + return Decimal((1, self._int, self._exp)) + + def copy_sign(self, other): + """Returns self with the sign of other.""" + return Decimal((other._sign, self._int, self._exp)) + + def exp(self, context=None): + """Returns e ** self.""" + + if context is None: + context = getcontext() + + # exp(NaN) = NaN + ans = self._check_nans(context=context) + if ans: + return ans + + # exp(-Infinity) = 0 + if self._isinfinity() == -1: + return Dec_0 + + # exp(0) = 1 + if not self: + return Dec_p1 + + # exp(Infinity) = Infinity + if self._isinfinity() == 1: + return Decimal(self) + + # the result is now guaranteed to be inexact (the true + # mathematical result is transcendental). There's no need to + # raise Rounded and Inexact here---they'll always be raised as + # a result of the call to _fix. + p = context.prec + adj = self.adjusted() + + # we only need to do any computation for quite a small range + # of adjusted exponents---for example, -29 <= adj <= 10 for + # the default context. For smaller exponent the result is + # indistinguishable from 1 at the given precision, while for + # larger exponent the result either overflows or underflows. + if self._sign == 0 and adj > len(str((context.Emax+1)*3)): + # overflow + ans = Decimal((0, (1,), context.Emax+1)) + elif self._sign == 1 and adj > len(str((-context.Etiny()+1)*3)): + # underflow to 0 + ans = Decimal((0, (1,), context.Etiny()-1)) + elif self._sign == 0 and adj < -p: + # p+1 digits; final round will raise correct flags + ans = Decimal((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 = Decimal((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 = Decimal((0, map(int, 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): + """Returns 1 if self is canonical; otherwise returns 0.""" + return Dec_p1 + + def is_finite(self): + """Returns 1 if self is finite, otherwise returns 0. + + For it to be finite, it must be neither infinite nor a NaN. + """ + if self._is_special: + return Dec_0 + else: + return Dec_p1 + + def is_infinite(self): + """Returns 1 if self is an Infinite, otherwise returns 0.""" + if self._isinfinity(): + return Dec_p1 + else: + return Dec_0 + + def is_nan(self): + """Returns 1 if self is qNaN or sNaN, otherwise returns 0.""" + if self._isnan(): + return Dec_p1 + else: + return Dec_0 + + def is_normal(self, context=None): + """Returns 1 if self is a normal number, otherwise returns 0.""" + if self._is_special: + return Dec_0 + if not self: + return Dec_0 + if context is None: + context = getcontext() + if context.Emin <= self.adjusted() <= context.Emax: + return Dec_p1 + else: + return Dec_0 + + def is_qnan(self): + """Returns 1 if self is a quiet NaN, otherwise returns 0.""" + if self._isnan() == 1: + return Dec_p1 + else: + return Dec_0 + + def is_signed(self): + """Returns 1 if self is negative, otherwise returns 0.""" + return Decimal(self._sign) + + def is_snan(self): + """Returns 1 if self is a signaling NaN, otherwise returns 0.""" + if self._isnan() == 2: + return Dec_p1 + else: + return Dec_0 + + def is_subnormal(self, context=None): + """Returns 1 if self is subnormal, otherwise returns 0.""" + if self._is_special: + return Dec_0 + if not self: + return Dec_0 + if context is None: + context = getcontext() + + r = self._exp + len(self._int) + if r <= context.Emin: + return Dec_p1 + return Dec_0 + + def is_zero(self): + """Returns 1 if self is a zero, otherwise returns 0.""" + if self: + return Dec_0 + else: + return Dec_p1 + + def _ln_exp_bound(self): + """Compute a lower bound for the adjusted exponent of self.ln(). + In other words, compute r such that self.ln() >= 10**r. Assumes + that self is finite and positive and that self != 1. + """ + + # for 0.1 <= x <= 10 we use the inequalities 1-1/x <= ln(x) <= x-1 + adj = self._exp + len(self._int) - 1 + if adj >= 1: + # argument >= 10; we use 23/10 = 2.3 as a lower bound for ln(10) + return len(str(adj*23//10)) - 1 + if adj <= -2: + # argument <= 0.1 + return len(str((-1-adj)*23//10)) - 1 + op = _WorkRep(self) + c, e = op.int, op.exp + if adj == 0: + # 1 < self < 10 + num = str(c-10**-e) + den = str(c) + return len(num) - len(den) - (num < den) + # adj == -1, 0.1 <= self < 1 + return e + len(str(10**-e - c)) - 1 + + + def ln(self, context=None): + """Returns the natural (base e) logarithm of self.""" + + if context is None: + context = getcontext() + + # ln(NaN) = NaN + ans = self._check_nans(context=context) + if ans: + return ans + + # ln(0.0) == -Infinity + if not self: + return negInf + + # ln(Infinity) = Infinity + if self._isinfinity() == 1: + return Inf + + # ln(1.0) == 0.0 + if self == Dec_p1: + return Dec_0 + + # ln(negative) raises InvalidOperation + if self._sign == 1: + return context._raise_error(InvalidOperation, + 'ln of a negative value') + + # result is irrational, so necessarily inexact + op = _WorkRep(self) + c, e = op.int, op.exp + p = context.prec + + # correctly rounded result: repeatedly increase precision by 3 + # until we get an unambiguously roundable result + places = p - self._ln_exp_bound() + 2 # at least p+3 places + while True: + coeff = _dlog(c, e, places) + # assert len(str(abs(coeff)))-p >= 1 + if coeff % (5*10**(len(str(abs(coeff)))-p-1)): + break + places += 3 + ans = Decimal((int(coeff<0), map(int, str(abs(coeff))), -places)) + + context = context._shallow_copy() + rounding = context._set_rounding(ROUND_HALF_EVEN) + ans = ans._fix(context) + context.rounding = rounding + return ans + + def _log10_exp_bound(self): + """Compute a lower bound for the adjusted exponent of self.log10(). + In other words, find r such that self.log10() >= 10**r. + Assumes that self is finite and positive and that self != 1. + """ + + # For x >= 10 or x < 0.1 we only need a bound on the integer + # part of log10(self), and this comes directly from the + # exponent of x. For 0.1 <= x <= 10 we use the inequalities + # 1-1/x <= log(x) <= x-1. If x > 1 we have |log10(x)| > + # (1-1/x)/2.31 > 0. If x < 1 then |log10(x)| > (1-x)/2.31 > 0 + + adj = self._exp + len(self._int) - 1 + if adj >= 1: + # self >= 10 + return len(str(adj))-1 + if adj <= -2: + # self < 0.1 + return len(str(-1-adj))-1 + op = _WorkRep(self) + c, e = op.int, op.exp + if adj == 0: + # 1 < self < 10 + num = str(c-10**-e) + den = str(231*c) + return len(num) - len(den) - (num < den) + 2 + # adj == -1, 0.1 <= self < 1 + num = str(10**-e-c) + return len(num) + e - (num < "231") - 1 + + def log10(self, context=None): + """Returns the base 10 logarithm of self.""" + + if context is None: + context = getcontext() + + # log10(NaN) = NaN + ans = self._check_nans(context=context) + if ans: + return ans + + # log10(0.0) == -Infinity + if not self: + return negInf + + # log10(Infinity) = Infinity + if self._isinfinity() == 1: + return Inf + + # log10(negative or -Infinity) raises InvalidOperation + if self._sign == 1: + return context._raise_error(InvalidOperation, + 'log10 of a negative value') + + # log10(10**n) = n + if self._int[0] == 1 and self._int[1:] == (0,)*(len(self._int) - 1): + # answer may need rounding + ans = Decimal(self._exp + len(self._int) - 1) + else: + # result is irrational, so necessarily inexact + op = _WorkRep(self) + c, e = op.int, op.exp + p = context.prec + + # correctly rounded result: repeatedly increase precision + # until result is unambiguously roundable + places = p-self._log10_exp_bound()+2 + while True: + coeff = _dlog10(c, e, places) + # assert len(str(abs(coeff)))-p >= 1 + if coeff % (5*10**(len(str(abs(coeff)))-p-1)): + break + places += 3 + ans = Decimal((int(coeff<0), map(int, str(abs(coeff))), -places)) + + context = context._shallow_copy() + rounding = context._set_rounding(ROUND_HALF_EVEN) + ans = ans._fix(context) + context.rounding = rounding + return ans + + def logb(self, context=None): + """ Returns the exponent of the magnitude of self's MSD. + + The result is the integer which is the exponent of the magnitude + of the most significant digit of self (as though it were truncated + to a single digit while maintaining the value of that digit and + without limiting the resulting exponent). + """ + # logb(NaN) = NaN + ans = self._check_nans(context=context) + if ans: + return ans + + if context is None: + context = getcontext() + + # logb(+/-Inf) = +Inf + if self._isinfinity(): + return Inf + + # logb(0) = -Inf, DivisionByZero + if not self: + return context._raise_error(DivisionByZero, 'logb(0)', -1) + + # otherwise, simply return the adjusted exponent of self, as a + # Decimal. Note that no attempt is made to fit the result + # into the current context. + return Decimal(self.adjusted()) + + def _islogical(self): + """Return True if self is a logical operand. + + For being logical, it must be a finite numbers with a sign of 0, + an exponent of 0, and a coefficient whose digits must all be + either 0 or 1. + """ + if self._sign != 0 or self._exp != 0: + return False + for dig in self._int: + if dig not in (0, 1): + return False + return True + + def _fill_logical(self, context, opa, opb): + dif = context.prec - len(opa) + if dif > 0: + opa = (0,)*dif + opa + elif dif < 0: + opa = opa[-context.prec:] + dif = context.prec - len(opb) + if dif > 0: + opb = (0,)*dif + opb + elif dif < 0: + opb = opb[-context.prec:] + return opa, opb + + def logical_and(self, other, context=None): + """Applies an 'and' operation between self and other's digits.""" + if context is None: + context = getcontext() + if not self._islogical() or not other._islogical(): + return context._raise_error(InvalidOperation) + + # fill to context.prec + (opa, opb) = self._fill_logical(context, self._int, other._int) + + # make the operation, and clean starting zeroes + result = [a&b for a,b in zip(opa,opb)] + for i,d in enumerate(result): + if d == 1: + break + result = tuple(result[i:]) + + # if empty, we must have at least a zero + if not result: + result = (0,) + return Decimal((0, result, 0)) + + def logical_invert(self, context=None): + """Invert all its digits.""" + if context is None: + context = getcontext() + return self.logical_xor(Decimal((0,(1,)*context.prec,0)), context) + + def logical_or(self, other, context=None): + """Applies an 'or' operation between self and other's digits.""" + if context is None: + context = getcontext() + if not self._islogical() or not other._islogical(): + return context._raise_error(InvalidOperation) + + # fill to context.prec + (opa, opb) = self._fill_logical(context, self._int, other._int) + + # make the operation, and clean starting zeroes + result = [a|b for a,b in zip(opa,opb)] + for i,d in enumerate(result): + if d == 1: + break + result = tuple(result[i:]) + + # if empty, we must have at least a zero + if not result: + result = (0,) + return Decimal((0, result, 0)) + + def logical_xor(self, other, context=None): + """Applies an 'xor' operation between self and other's digits.""" + if context is None: + context = getcontext() + if not self._islogical() or not other._islogical(): + return context._raise_error(InvalidOperation) + + # fill to context.prec + (opa, opb) = self._fill_logical(context, self._int, other._int) + + # make the operation, and clean starting zeroes + result = [a^b for a,b in zip(opa,opb)] + for i,d in enumerate(result): + if d == 1: + break + result = tuple(result[i:]) + + # if empty, we must have at least a zero + if not result: + result = (0,) + return Decimal((0, result, 0)) + + def max_mag(self, other, context=None): + """Compares the values numerically with their sign ignored.""" + other = _convert_other(other, raiseit=True) + + if self._is_special or other._is_special: + # If one operand is a quiet NaN and the other is number, then the + # number is always returned + sn = self._isnan() + on = other._isnan() + if sn or on: + if on == 1 and sn != 2: + return self + if sn == 1 and on != 2: + return other + 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 + + if context is None: + context = getcontext() + if context._rounding_decision == ALWAYS_ROUND: + return ans._fix(context) + return ans + + def min_mag(self, other, context=None): + """Compares the values numerically with their sign ignored.""" + other = _convert_other(other, raiseit=True) + + if self._is_special or other._is_special: + # If one operand is a quiet NaN and the other is number, then the + # number is always returned + sn = self._isnan() + on = other._isnan() + if sn or on: + if on == 1 and sn != 2: + return self + if sn == 1 and on != 2: + return other + 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 + + if context is None: + context = getcontext() + if context._rounding_decision == ALWAYS_ROUND: + return ans._fix(context) + return ans + + def next_minus(self, context=None): + """Returns the largest representable number smaller than itself.""" + if context is None: + context = getcontext() + + ans = self._check_nans(context=context) + if ans: + return ans + + if self._isinfinity() == -1: + return negInf + if self._isinfinity() == 1: + return Decimal((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__(Decimal((0, (1,), context.Etiny()-1)), context) + + def next_plus(self, context=None): + """Returns the smallest representable number larger than itself.""" + if context is None: + context = getcontext() + + ans = self._check_nans(context=context) + if ans: + return ans + + if self._isinfinity() == 1: + return Inf + if self._isinfinity() == -1: + return Decimal((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__(Decimal((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 Decimal((other._sign, self._int, self._exp)) + + if comparison == -1: + ans = self.next_plus(context) + else: # comparison == 1 + ans = self.next_minus(context) + + # decide which flags to raise using value of ans + if ans._isinfinity(): + context._raise_error(Overflow, + 'Infinite result from next_toward', + ans._sign) + context._raise_error(Rounded) + context._raise_error(Inexact) + elif ans.adjusted() < context.Emin: + context._raise_error(Underflow) + context._raise_error(Subnormal) + context._raise_error(Rounded) + context._raise_error(Inexact) + # if precision == 1 then we don't raise Clamped for a + # result 0E-Etiny. + if not ans: + context._raise_error(Clamped) + + return ans + + def number_class(self, context=None): + """Returns an indication of the class of self. + + The class is one of the following strings: + -sNaN + -NaN + -Infinity + -Normal + -Subnormal + -Zero + +Zero + +Subnormal + +Normal + +Infinity + """ + if self.is_snan(): + return "sNaN" + if self.is_qnan(): + return "NaN" + inf = self._isinfinity() + if inf == 1: + return "+Infinity" + if inf == -1: + return "-Infinity" + if self.is_zero(): + if self._sign: + return "-Zero" + else: + return "+Zero" + if context is None: + context = getcontext() + if self.is_subnormal(context=context): + if self._sign: + return "-Subnormal" + else: + return "+Subnormal" + # just a normal, regular, boring number, :) + if self._sign: + return "-Normal" + else: + return "+Normal" + + def radix(self): + """Just returns 10, as this is Decimal, :)""" + return Decimal(10) + + def rotate(self, other, context=None): + """Returns a rotated copy of self, value-of-other times.""" + if context is None: + context = getcontext() + + ans = self._check_nans(other, context) + if ans: + return ans + + if other._exp != 0: + return context._raise_error(InvalidOperation) + if not (-context.prec <= int(other) <= context.prec): + return context._raise_error(InvalidOperation) + + if self._isinfinity(): + return self + + # get values, pad if necessary + torot = int(other) + rotdig = self._int + topad = context.prec - len(rotdig) + if topad: + rotdig = ((0,)*topad) + rotdig + + # let's rotate! + rotated = rotdig[torot:] + rotdig[:torot] + + # clean starting zeroes + for i,d in enumerate(rotated): + if d != 0: + break + rotated = rotated[i:] + + return Decimal((self._sign, rotated, self._exp)) + + + def scaleb (self, other, context=None): + """Returns self operand after adding the second value to its exp.""" + if context is None: + context = getcontext() + + ans = self._check_nans(other, context) + if ans: + return ans + + if other._exp != 0: + return context._raise_error(InvalidOperation) + liminf = -2 * (context.Emax + context.prec) + limsup = 2 * (context.Emax + context.prec) + if not (liminf <= int(other) <= limsup): + return context._raise_error(InvalidOperation) + + if self._isinfinity(): + return self + + d = Decimal((self._sign, self._int, self._exp + int(other))) + d = d._fix(context) + return d + + def shift(self, other, context=None): + """Returns a shifted copy of self, value-of-other times.""" + if context is None: + context = getcontext() + + ans = self._check_nans(other, context) + if ans: + return ans + + if other._exp != 0: + return context._raise_error(InvalidOperation) + if not (-context.prec <= int(other) <= context.prec): + return context._raise_error(InvalidOperation) + + if self._isinfinity(): + return self + + # get values, pad if necessary + torot = int(other) + if not torot: + return self + rotdig = self._int + topad = context.prec - len(rotdig) + if topad: + rotdig = ((0,)*topad) + rotdig + + # let's shift! + if torot < 0: + rotated = rotdig[:torot] + else: + rotated = (rotdig + ((0,) * torot)) + rotated = rotated[-context.prec:] + + # clean starting zeroes + if rotated: + for i,d in enumerate(rotated): + if d != 0: + break + rotated = rotated[i:] + else: + rotated = (0,) + + return Decimal((self._sign, rotated, self._exp)) + + # Support for pickling, copy, and deepcopy def __reduce__(self): return (self.__class__, (str(self),)) @@ -2407,6 +3588,9 @@ class Context(object): def create_decimal(self, num='0'): """Creates a new Decimal instance but using self as context.""" d = Decimal(num, context=self) + if d._isnan() and len(d._int) > self.prec - self._clamp: + return self._raise_error(ConversionSyntax, + "diagnostic info too long in NaN") return d._fix(self) # Methods @@ -2441,6 +3625,17 @@ class Context(object): def _apply(self, a): return str(a._fix(self)) + def canonical(self, a): + """Returns the same Decimal object. + + As we do not have different encodings for the same number, the + received object already is in its canonical form. + + >>> ExtendedContext.canonical(Decimal('2.50')) + Decimal("2.50") + """ + return a.canonical(context=self) + def compare(self, a, b): """Compares values numerically. @@ -2470,6 +3665,110 @@ class Context(object): """ return a.compare(b, context=self) + def compare_signal(self, a, b): + """Compares the values of the two operands numerically. + + It's pretty much like compare(), but all NaNs signal, with signaling + NaNs taking precedence over quiet NaNs. + + >>> c = ExtendedContext + >>> c.compare_signal(Decimal('2.1'), Decimal('3')) + Decimal("-1") + >>> c.compare_signal(Decimal('2.1'), Decimal('2.1')) + Decimal("0") + >>> c.flags[InvalidOperation] = 0 + >>> print c.flags[InvalidOperation] + 0 + >>> c.compare_signal(Decimal('NaN'), Decimal('2.1')) + Decimal("NaN") + >>> print c.flags[InvalidOperation] + 1 + >>> c.flags[InvalidOperation] = 0 + >>> print c.flags[InvalidOperation] + 0 + >>> c.compare_signal(Decimal('sNaN'), Decimal('2.1')) + Decimal("NaN") + >>> print c.flags[InvalidOperation] + 1 + """ + return a.compare_signal(b, context=self) + + def compare_total(self, a, b): + """Compares two operands using their abstract representation. + + This is not like the standard compare, which use their numerical + value. Note that a total ordering is defined for all possible abstract + representations. + + >>> ExtendedContext.compare_total(Decimal('12.73'), Decimal('127.9')) + Decimal("-1") + >>> ExtendedContext.compare_total(Decimal('-127'), Decimal('12')) + Decimal("-1") + >>> ExtendedContext.compare_total(Decimal('12.30'), Decimal('12.3')) + Decimal("-1") + >>> ExtendedContext.compare_total(Decimal('12.30'), Decimal('12.30')) + Decimal("0") + >>> ExtendedContext.compare_total(Decimal('12.3'), Decimal('12.300')) + Decimal("1") + >>> ExtendedContext.compare_total(Decimal('12.3'), Decimal('NaN')) + Decimal("-1") + """ + return a.compare_total(b) + + def compare_total_mag(self, a, b): + """Compares two operands using their abstract representation ignoring sign. + + Like compare_total, but with operand's sign ignored and assumed to be 0. + """ + return a.compare_total_mag(b) + + def copy_abs(self, a): + """Returns a copy of the operand with the sign set to 0. + + >>> ExtendedContext.copy_abs(Decimal('2.1')) + Decimal("2.1") + >>> ExtendedContext.copy_abs(Decimal('-100')) + Decimal("100") + """ + return a.copy_abs() + + def copy_decimal(self, a): + """Returns a copy of the decimal objet. + + >>> ExtendedContext.copy_decimal(Decimal('2.1')) + Decimal("2.1") + >>> ExtendedContext.copy_decimal(Decimal('-1.00')) + Decimal("-1.00") + """ + return a + + def copy_negate(self, a): + """Returns a copy of the operand with the sign inverted. + + >>> ExtendedContext.copy_negate(Decimal('101.5')) + Decimal("-101.5") + >>> ExtendedContext.copy_negate(Decimal('-101.5')) + Decimal("101.5") + """ + return a.copy_negate() + + def copy_sign(self, a, b): + """Copies the second operand's sign to the first one. + + In detail, it returns a copy of the first operand with the sign + equal to the sign of the second operand. + + >>> ExtendedContext.copy_sign(Decimal( '1.50'), Decimal('7.33')) + Decimal("1.50") + >>> ExtendedContext.copy_sign(Decimal('-1.50'), Decimal('7.33')) + Decimal("1.50") + >>> ExtendedContext.copy_sign(Decimal( '1.50'), Decimal('-7.33')) + Decimal("-1.50") + >>> ExtendedContext.copy_sign(Decimal('-1.50'), Decimal('-7.33')) + Decimal("-1.50") + """ + return a.copy_sign(b) + def divide(self, a, b): """Decimal division in a specified context. @@ -2511,6 +3810,316 @@ class Context(object): def divmod(self, a, b): return a.__divmod__(b, context=self) + def exp(self, a): + """Returns e ** a. + + >>> c = ExtendedContext.copy() + >>> c.Emin = -999 + >>> c.Emax = 999 + >>> c.exp(Decimal('-Infinity')) + Decimal("0") + >>> c.exp(Decimal('-1')) + Decimal("0.367879441") + >>> c.exp(Decimal('0')) + Decimal("1") + >>> c.exp(Decimal('1')) + Decimal("2.71828183") + >>> c.exp(Decimal('0.693147181')) + Decimal("2.00000000") + >>> c.exp(Decimal('+Infinity')) + Decimal("Infinity") + """ + return a.exp(context=self) + + def fma(self, a, b, c): + """Returns a multiplied by b, plus c. + + The first two operands are multiplied together, using multiply, + the third operand is then added to the result of that + multiplication, using add, all with only one final rounding. + + >>> ExtendedContext.fma(Decimal('3'), Decimal('5'), Decimal('7')) + Decimal("22") + >>> ExtendedContext.fma(Decimal('3'), Decimal('-5'), Decimal('7')) + Decimal("-8") + >>> ExtendedContext.fma(Decimal('888565290'), Decimal('1557.96930'), Decimal('-86087.7578')) + Decimal("1.38435736E+12") + """ + return a.fma(b, c, context=self) + + def is_canonical(self, a): + """Returns 1 if the operand is canonical; otherwise returns 0. + + >>> ExtendedContext.is_canonical(Decimal('2.50')) + Decimal("1") + """ + return Dec_p1 + + def is_finite(self, a): + """Returns 1 if the operand is finite, otherwise returns 0. + + For it to be finite, it must be neither infinite nor a NaN. + + >>> ExtendedContext.is_finite(Decimal('2.50')) + Decimal("1") + >>> ExtendedContext.is_finite(Decimal('-0.3')) + Decimal("1") + >>> ExtendedContext.is_finite(Decimal('0')) + Decimal("1") + >>> ExtendedContext.is_finite(Decimal('Inf')) + Decimal("0") + >>> ExtendedContext.is_finite(Decimal('NaN')) + Decimal("0") + """ + return a.is_finite() + + def is_infinite(self, a): + """Returns 1 if the operand is an Infinite, otherwise returns 0. + + >>> ExtendedContext.is_infinite(Decimal('2.50')) + Decimal("0") + >>> ExtendedContext.is_infinite(Decimal('-Inf')) + Decimal("1") + >>> ExtendedContext.is_infinite(Decimal('NaN')) + Decimal("0") + """ + return a.is_infinite() + + def is_nan(self, a): + """Returns 1 if the operand is qNaN or sNaN, otherwise returns 0. + + >>> ExtendedContext.is_nan(Decimal('2.50')) + Decimal("0") + >>> ExtendedContext.is_nan(Decimal('NaN')) + Decimal("1") + >>> ExtendedContext.is_nan(Decimal('-sNaN')) + Decimal("1") + """ + return a.is_nan() + + def is_normal(self, a): + """Returns 1 if the operand is a normal number, otherwise returns 0. + + >>> c = ExtendedContext.copy() + >>> c.Emin = -999 + >>> c.Emax = 999 + >>> c.is_normal(Decimal('2.50')) + Decimal("1") + >>> c.is_normal(Decimal('0.1E-999')) + Decimal("0") + >>> c.is_normal(Decimal('0.00')) + Decimal("0") + >>> c.is_normal(Decimal('-Inf')) + Decimal("0") + >>> c.is_normal(Decimal('NaN')) + Decimal("0") + """ + return a.is_normal(context=self) + + def is_qnan(self, a): + """Returns 1 if the operand is a quiet NaN, otherwise returns 0. + + >>> ExtendedContext.is_qnan(Decimal('2.50')) + Decimal("0") + >>> ExtendedContext.is_qnan(Decimal('NaN')) + Decimal("1") + >>> ExtendedContext.is_qnan(Decimal('sNaN')) + Decimal("0") + """ + return a.is_qnan() + + def is_signed(self, a): + """Returns 1 if the operand is negative, otherwise returns 0. + + >>> ExtendedContext.is_signed(Decimal('2.50')) + Decimal("0") + >>> ExtendedContext.is_signed(Decimal('-12')) + Decimal("1") + >>> ExtendedContext.is_signed(Decimal('-0')) + Decimal("1") + """ + return a.is_signed() + + def is_snan(self, a): + """Returns 1 if the operand is a signaling NaN, otherwise returns 0. + + >>> ExtendedContext.is_snan(Decimal('2.50')) + Decimal("0") + >>> ExtendedContext.is_snan(Decimal('NaN')) + Decimal("0") + >>> ExtendedContext.is_snan(Decimal('sNaN')) + Decimal("1") + """ + return a.is_snan() + + def is_subnormal(self, a): + """Returns 1 if the operand is subnormal, otherwise returns 0. + + >>> c = ExtendedContext.copy() + >>> c.Emin = -999 + >>> c.Emax = 999 + >>> c.is_subnormal(Decimal('2.50')) + Decimal("0") + >>> c.is_subnormal(Decimal('0.1E-999')) + Decimal("1") + >>> c.is_subnormal(Decimal('0.00')) + Decimal("0") + >>> c.is_subnormal(Decimal('-Inf')) + Decimal("0") + >>> c.is_subnormal(Decimal('NaN')) + Decimal("0") + """ + return a.is_subnormal(context=self) + + def is_zero(self, a): + """Returns 1 if the operand is a zero, otherwise returns 0. + + >>> ExtendedContext.is_zero(Decimal('0')) + Decimal("1") + >>> ExtendedContext.is_zero(Decimal('2.50')) + Decimal("0") + >>> ExtendedContext.is_zero(Decimal('-0E+2')) + Decimal("1") + """ + return a.is_zero() + + def ln(self, a): + """Returns the natural (base e) logarithm of the operand. + + >>> c = ExtendedContext.copy() + >>> c.Emin = -999 + >>> c.Emax = 999 + >>> c.ln(Decimal('0')) + Decimal("-Infinity") + >>> c.ln(Decimal('1.000')) + Decimal("0") + >>> c.ln(Decimal('2.71828183')) + Decimal("1.00000000") + >>> c.ln(Decimal('10')) + Decimal("2.30258509") + >>> c.ln(Decimal('+Infinity')) + Decimal("Infinity") + """ + return a.ln(context=self) + + def log10(self, a): + """Returns the base 10 logarithm of the operand. + + >>> c = ExtendedContext.copy() + >>> c.Emin = -999 + >>> c.Emax = 999 + >>> c.log10(Decimal('0')) + Decimal("-Infinity") + >>> c.log10(Decimal('0.001')) + Decimal("-3") + >>> c.log10(Decimal('1.000')) + Decimal("0") + >>> c.log10(Decimal('2')) + Decimal("0.301029996") + >>> c.log10(Decimal('10')) + Decimal("1") + >>> c.log10(Decimal('70')) + Decimal("1.84509804") + >>> c.log10(Decimal('+Infinity')) + Decimal("Infinity") + """ + return a.log10(context=self) + + def logb(self, a): + """ Returns the exponent of the magnitude of the operand's MSD. + + The result is the integer which is the exponent of the magnitude + of the most significant digit of the operand (as though the + operand were truncated to a single digit while maintaining the + value of that digit and without limiting the resulting exponent). + + >>> ExtendedContext.logb(Decimal('250')) + Decimal("2") + >>> ExtendedContext.logb(Decimal('2.50')) + Decimal("0") + >>> ExtendedContext.logb(Decimal('0.03')) + Decimal("-2") + >>> ExtendedContext.logb(Decimal('0')) + Decimal("-Infinity") + """ + return a.logb(context=self) + + def logical_and(self, a, b): + """Applies the logical operation 'and' between each operand's digits. + + The operands must be both logical numbers. + + >>> ExtendedContext.logical_and(Decimal('0'), Decimal('0')) + Decimal("0") + >>> ExtendedContext.logical_and(Decimal('0'), Decimal('1')) + Decimal("0") + >>> ExtendedContext.logical_and(Decimal('1'), Decimal('0')) + Decimal("0") + >>> ExtendedContext.logical_and(Decimal('1'), Decimal('1')) + Decimal("1") + >>> ExtendedContext.logical_and(Decimal('1100'), Decimal('1010')) + Decimal("1000") + >>> ExtendedContext.logical_and(Decimal('1111'), Decimal('10')) + Decimal("10") + """ + return a.logical_and(b, context=self) + + def logical_invert(self, a): + """Invert all the digits in the operand. + + The operand must be a logical number. + + >>> ExtendedContext.logical_invert(Decimal('0')) + Decimal("111111111") + >>> ExtendedContext.logical_invert(Decimal('1')) + Decimal("111111110") + >>> ExtendedContext.logical_invert(Decimal('111111111')) + Decimal("0") + >>> ExtendedContext.logical_invert(Decimal('101010101')) + Decimal("10101010") + """ + return a.logical_invert(context=self) + + def logical_or(self, a, b): + """Applies the logical operation 'or' between each operand's digits. + + The operands must be both logical numbers. + + >>> ExtendedContext.logical_or(Decimal('0'), Decimal('0')) + Decimal("0") + >>> ExtendedContext.logical_or(Decimal('0'), Decimal('1')) + Decimal("1") + >>> ExtendedContext.logical_or(Decimal('1'), Decimal('0')) + Decimal("1") + >>> ExtendedContext.logical_or(Decimal('1'), Decimal('1')) + Decimal("1") + >>> ExtendedContext.logical_or(Decimal('1100'), Decimal('1010')) + Decimal("1110") + >>> ExtendedContext.logical_or(Decimal('1110'), Decimal('10')) + Decimal("1110") + """ + return a.logical_or(b, context=self) + + def logical_xor(self, a, b): + """Applies the logical operation 'xor' between each operand's digits. + + The operands must be both logical numbers. + + >>> ExtendedContext.logical_xor(Decimal('0'), Decimal('0')) + Decimal("0") + >>> ExtendedContext.logical_xor(Decimal('0'), Decimal('1')) + Decimal("1") + >>> ExtendedContext.logical_xor(Decimal('1'), Decimal('0')) + Decimal("1") + >>> ExtendedContext.logical_xor(Decimal('1'), Decimal('1')) + Decimal("0") + >>> ExtendedContext.logical_xor(Decimal('1100'), Decimal('1010')) + Decimal("110") + >>> ExtendedContext.logical_xor(Decimal('1111'), Decimal('10')) + Decimal("1101") + """ + return a.logical_xor(b, context=self) + def max(self, a,b): """max compares two values numerically and returns the maximum. @@ -2531,6 +4140,10 @@ class Context(object): """ return a.max(b, context=self) + def max_mag(self, a, b): + """Compares the values numerically with their sign ignored.""" + return a.max_mag(b, context=self) + def min(self, a,b): """min compares two values numerically and returns the minimum. @@ -2551,6 +4164,10 @@ class Context(object): """ return a.min(b, context=self) + def min_mag(self, a, b): + """Compares the values numerically with their sign ignored.""" + return a.min_mag(b, context=self) + def minus(self, a): """Minus corresponds to unary prefix minus in Python. @@ -2586,6 +4203,68 @@ class Context(object): """ return a.__mul__(b, context=self) + def next_minus(self, a): + """Returns the largest representable number smaller than a. + + >>> c = ExtendedContext.copy() + >>> c.Emin = -999 + >>> c.Emax = 999 + >>> ExtendedContext.next_minus(Decimal('1')) + Decimal("0.999999999") + >>> c.next_minus(Decimal('1E-1007')) + Decimal("0E-1007") + >>> ExtendedContext.next_minus(Decimal('-1.00000003')) + Decimal("-1.00000004") + >>> c.next_minus(Decimal('Infinity')) + Decimal("9.99999999E+999") + """ + return a.next_minus(context=self) + + def next_plus(self, a): + """Returns the smallest representable number larger than a. + + >>> c = ExtendedContext.copy() + >>> c.Emin = -999 + >>> c.Emax = 999 + >>> ExtendedContext.next_plus(Decimal('1')) + Decimal("1.00000001") + >>> c.next_plus(Decimal('-1E-1007')) + Decimal("-0E-1007") + >>> ExtendedContext.next_plus(Decimal('-1.00000003')) + Decimal("-1.00000002") + >>> c.next_plus(Decimal('-Infinity')) + Decimal("-9.99999999E+999") + """ + return a.next_plus(context=self) + + def next_toward(self, a, b): + """Returns the number closest to a, in direction towards b. + + The result is the closest representable number from the first + operand (but not the first operand) that is in the direction + towards the second operand, unless the operands have the same + value. + + >>> c = ExtendedContext.copy() + >>> c.Emin = -999 + >>> c.Emax = 999 + >>> c.next_toward(Decimal('1'), Decimal('2')) + Decimal("1.00000001") + >>> c.next_toward(Decimal('-1E-1007'), Decimal('1')) + Decimal("-0E-1007") + >>> c.next_toward(Decimal('-1.00000003'), Decimal('0')) + Decimal("-1.00000002") + >>> c.next_toward(Decimal('1'), Decimal('0')) + Decimal("0.999999999") + >>> c.next_toward(Decimal('1E-1007'), Decimal('-100')) + Decimal("0E-1007") + >>> c.next_toward(Decimal('-1.00000003'), Decimal('-10')) + Decimal("-1.00000004") + >>> c.next_toward(Decimal('0.00'), Decimal('-0.0000')) + Decimal("-0.00") + """ + return a.next_toward(b, context=self) + def normalize(self, a): """normalize reduces an operand to its simplest form. @@ -2607,6 +4286,53 @@ class Context(object): """ return a.normalize(context=self) + def number_class(self, a): + """Returns an indication of the class of the operand. + + The class is one of the following strings: + -sNaN + -NaN + -Infinity + -Normal + -Subnormal + -Zero + +Zero + +Subnormal + +Normal + +Infinity + + >>> c = Context(ExtendedContext) + >>> c.Emin = -999 + >>> c.Emax = 999 + >>> c.number_class(Decimal('Infinity')) + '+Infinity' + >>> c.number_class(Decimal('1E-10')) + '+Normal' + >>> c.number_class(Decimal('2.50')) + '+Normal' + >>> c.number_class(Decimal('0.1E-999')) + '+Subnormal' + >>> c.number_class(Decimal('0')) + '+Zero' + >>> c.number_class(Decimal('-0')) + '-Zero' + >>> c.number_class(Decimal('-0.1E-999')) + '-Subnormal' + >>> c.number_class(Decimal('-1E-10')) + '-Normal' + >>> c.number_class(Decimal('-2.50')) + '-Normal' + >>> c.number_class(Decimal('-Infinity')) + '-Infinity' + >>> c.number_class(Decimal('NaN')) + 'NaN' + >>> c.number_class(Decimal('-NaN')) + 'NaN' + >>> c.number_class(Decimal('sNaN')) + 'sNaN' + """ + return a.number_class(context=self) + def plus(self, a): """Plus corresponds to unary prefix plus in Python. @@ -2624,49 +4350,69 @@ class Context(object): def power(self, a, b, modulo=None): """Raises a to the power of b, to modulo if given. - The right-hand operand must be a whole number whose integer part (after - any exponent has been applied) has no more than 9 digits and whose - fractional part (if any) is all zeros before any rounding. The operand - may be positive, negative, or zero; if negative, the absolute value of - the power is used, and the left-hand operand is inverted (divided into - 1) before use. - - If the increased precision needed for the intermediate calculations - exceeds the capabilities of the implementation then an Invalid - operation condition is raised. - - If, when raising to a negative power, an underflow occurs during the - division into 1, the operation is not halted at that point but - continues. - - >>> ExtendedContext.power(Decimal('2'), Decimal('3')) + 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") - >>> ExtendedContext.power(Decimal('2'), Decimal('-3')) + >>> c.power(Decimal('-2'), Decimal('3')) + Decimal("-8") + >>> c.power(Decimal('2'), Decimal('-3')) Decimal("0.125") - >>> ExtendedContext.power(Decimal('1.7'), Decimal('8')) + >>> c.power(Decimal('1.7'), Decimal('8')) Decimal("69.7575744") - >>> ExtendedContext.power(Decimal('Infinity'), Decimal('-2')) - Decimal("0") - >>> ExtendedContext.power(Decimal('Infinity'), Decimal('-1')) + >>> c.power(Decimal('10'), Decimal('0.301029996')) + Decimal("2.00000000") + >>> c.power(Decimal('Infinity'), Decimal('-1')) Decimal("0") - >>> ExtendedContext.power(Decimal('Infinity'), Decimal('0')) + >>> c.power(Decimal('Infinity'), Decimal('0')) Decimal("1") - >>> ExtendedContext.power(Decimal('Infinity'), Decimal('1')) + >>> c.power(Decimal('Infinity'), Decimal('1')) Decimal("Infinity") - >>> ExtendedContext.power(Decimal('Infinity'), Decimal('2')) - Decimal("Infinity") - >>> ExtendedContext.power(Decimal('-Infinity'), Decimal('-2')) - Decimal("0") - >>> ExtendedContext.power(Decimal('-Infinity'), Decimal('-1')) + >>> c.power(Decimal('-Infinity'), Decimal('-1')) Decimal("-0") - >>> ExtendedContext.power(Decimal('-Infinity'), Decimal('0')) + >>> c.power(Decimal('-Infinity'), Decimal('0')) Decimal("1") - >>> ExtendedContext.power(Decimal('-Infinity'), Decimal('1')) + >>> c.power(Decimal('-Infinity'), Decimal('1')) Decimal("-Infinity") - >>> ExtendedContext.power(Decimal('-Infinity'), Decimal('2')) + >>> c.power(Decimal('-Infinity'), Decimal('2')) Decimal("Infinity") - >>> ExtendedContext.power(Decimal('0'), Decimal('0')) + >>> c.power(Decimal('0'), Decimal('0')) Decimal("NaN") + + >>> c.power(Decimal('3'), Decimal('7'), Decimal('16')) + Decimal("11") + >>> c.power(Decimal('-3'), Decimal('7'), Decimal('16')) + Decimal("-11") + >>> c.power(Decimal('-3'), Decimal('8'), Decimal('16')) + Decimal("1") + >>> c.power(Decimal('3'), Decimal('7'), Decimal('-16')) + Decimal("11") + >>> c.power(Decimal('23E12345'), Decimal('67E189'), Decimal('123456789')) + Decimal("11729830") + >>> c.power(Decimal('-0'), Decimal('17'), Decimal('1729')) + Decimal("-0") + >>> c.power(Decimal('-23'), Decimal('0'), Decimal('65537')) + Decimal("1") """ return a.__pow__(b, modulo, context=self) @@ -2721,6 +4467,14 @@ class Context(object): """ 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. @@ -2775,6 +4529,28 @@ class Context(object): """ return a.remainder_near(b, context=self) + def rotate(self, a, b): + """Returns a rotated copy of a, b times. + + The coefficient of the result is a rotated copy of the digits in + the coefficient of the first operand. The number of places of + rotation is taken from the absolute value of the second operand, + with the rotation being to the left if the second operand is + positive or to the right otherwise. + + >>> ExtendedContext.rotate(Decimal('34'), Decimal('8')) + Decimal("400000003") + >>> ExtendedContext.rotate(Decimal('12'), Decimal('9')) + Decimal("12") + >>> ExtendedContext.rotate(Decimal('123456789'), Decimal('-2')) + Decimal("891234567") + >>> ExtendedContext.rotate(Decimal('123456789'), Decimal('0')) + Decimal("123456789") + >>> ExtendedContext.rotate(Decimal('123456789'), Decimal('+2')) + Decimal("345678912") + """ + return a.rotate(b, context=self) + def same_quantum(self, a, b): """Returns True if the two operands have the same exponent. @@ -2792,6 +4568,41 @@ class Context(object): """ return a.same_quantum(b) + def scaleb (self, a, b): + """Returns the first operand after adding the second value its exp. + + >>> ExtendedContext.scaleb(Decimal('7.50'), Decimal('-2')) + Decimal("0.0750") + >>> ExtendedContext.scaleb(Decimal('7.50'), Decimal('0')) + Decimal("7.50") + >>> ExtendedContext.scaleb(Decimal('7.50'), Decimal('3')) + Decimal("7.50E+3") + """ + return a.scaleb (b, context=self) + + def shift(self, a, b): + """Returns a shifted copy of a, b times. + + The coefficient of the result is a shifted copy of the digits + in the coefficient of the first operand. The number of places + to shift is taken from the absolute value of the second operand, + with the shift being to the left if the second operand is + positive or to the right otherwise. Digits shifted into the + coefficient are zeros. + + >>> ExtendedContext.shift(Decimal('34'), Decimal('8')) + Decimal("400000000") + >>> ExtendedContext.shift(Decimal('12'), Decimal('9')) + Decimal("0") + >>> ExtendedContext.shift(Decimal('123456789'), Decimal('-2')) + Decimal("1234567") + >>> ExtendedContext.shift(Decimal('123456789'), Decimal('0')) + Decimal("123456789") + >>> ExtendedContext.shift(Decimal('123456789'), Decimal('+2')) + Decimal("345678900") + """ + return a.shift(b, context=self) + def sqrt(self, a): """Square root of a non-negative number to context precision. @@ -2847,7 +4658,36 @@ class Context(object): """ return a.__str__(context=self) - def to_integral(self, a): + def to_integral_exact(self, a): + """Rounds to an integer. + + When the operand has a negative exponent, the result is the same + as using the quantize() operation using the given operand as the + left-hand-operand, 1E+0 as the right-hand-operand, and the precision + of the operand as the precision setting; Inexact and Rounded flags + are allowed in this operation. The rounding mode is taken from the + context. + + >>> ExtendedContext.to_integral_exact(Decimal('2.1')) + Decimal("2") + >>> ExtendedContext.to_integral_exact(Decimal('100')) + Decimal("100") + >>> ExtendedContext.to_integral_exact(Decimal('100.0')) + Decimal("100") + >>> ExtendedContext.to_integral_exact(Decimal('101.5')) + Decimal("102") + >>> ExtendedContext.to_integral_exact(Decimal('-101.5')) + Decimal("-102") + >>> ExtendedContext.to_integral_exact(Decimal('10E+5')) + Decimal("1.0E+6") + >>> ExtendedContext.to_integral_exact(Decimal('7.89E+77')) + Decimal("7.89E+77") + >>> ExtendedContext.to_integral_exact(Decimal('-Inf')) + Decimal("-Infinity") + """ + return a.to_integral_exact(context=self) + + def to_integral_value(self, a): """Rounds to an integer. When the operand has a negative exponent, the result is the same @@ -2856,24 +4696,27 @@ class Context(object): 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(Decimal('2.1')) + >>> ExtendedContext.to_integral_value(Decimal('2.1')) Decimal("2") - >>> ExtendedContext.to_integral(Decimal('100')) + >>> ExtendedContext.to_integral_value(Decimal('100')) Decimal("100") - >>> ExtendedContext.to_integral(Decimal('100.0')) + >>> ExtendedContext.to_integral_value(Decimal('100.0')) Decimal("100") - >>> ExtendedContext.to_integral(Decimal('101.5')) + >>> ExtendedContext.to_integral_value(Decimal('101.5')) Decimal("102") - >>> ExtendedContext.to_integral(Decimal('-101.5')) + >>> ExtendedContext.to_integral_value(Decimal('-101.5')) Decimal("-102") - >>> ExtendedContext.to_integral(Decimal('10E+5')) + >>> ExtendedContext.to_integral_value(Decimal('10E+5')) Decimal("1.0E+6") - >>> ExtendedContext.to_integral(Decimal('7.89E+77')) + >>> ExtendedContext.to_integral_value(Decimal('7.89E+77')) Decimal("7.89E+77") - >>> ExtendedContext.to_integral(Decimal('-Inf')) + >>> ExtendedContext.to_integral_value(Decimal('-Inf')) Decimal("-Infinity") """ - return a.to_integral(context=self) + 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') @@ -2911,39 +4754,28 @@ def _normalize(op1, op2, shouldround = 0, prec = 0): Done during addition. """ - # Yes, the exponent is a long, but the difference between exponents - # must be an int-- otherwise you'd get a big memory problem. - numdigits = int(op1.exp - op2.exp) - if numdigits < 0: - numdigits = -numdigits + if op1.exp < op2.exp: tmp = op2 other = op1 else: tmp = op1 other = op2 - - if shouldround and numdigits > prec + 1: - # Big difference in exponents - check the adjusted exponents + # 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. + if shouldround: tmp_len = len(str(tmp.int)) other_len = len(str(other.int)) - if numdigits > (other_len + prec + 1 - tmp_len): - # If the difference in adjusted exps is > prec+1, we know - # other is insignificant, so might as well put a 1 after the - # precision (since this is only for addition). Also stops - # use of massive longs. - - extend = prec + 2 - tmp_len - if extend <= 0: - extend = 1 - tmp.int *= 10 ** extend - tmp.exp -= extend + exp = tmp.exp + min(-1, tmp_len - prec - 2) + if other_len + other.exp - 1 < exp: other.int = 1 - other.exp = tmp.exp - return op1, op2 + other.exp = exp - tmp.int *= 10 ** numdigits - tmp.exp -= numdigits + tmp.int *= 10 ** (tmp.exp - other.exp) + tmp.exp = other.exp return op1, op2 def _adjust_coefficients(op1, op2): @@ -2968,9 +4800,315 @@ def _adjust_coefficients(op1, op2): return op1, op2, adjust + +##### Integer arithmetic functions used by ln, log10, exp and __pow__ ##### + +# This function from Tim Peters was taken from here: +# http://mail.python.org/pipermail/python-list/1999-July/007758.html +# The correction being in the function definition is for speed, and +# the whole function is not resolved with math.log because of avoiding +# the use of floats. +def _nbits(n, correction = { + '0': 4, '1': 3, '2': 2, '3': 2, + '4': 1, '5': 1, '6': 1, '7': 1, + '8': 0, '9': 0, 'a': 0, 'b': 0, + 'c': 0, 'd': 0, 'e': 0, 'f': 0}): + """Number of bits in binary representation of the positive integer n, + or 0 if n == 0. + """ + if n < 0: + raise ValueError("The argument to _nbits should be nonnegative.") + hex_n = "%x" % n + return 4*len(hex_n) - correction[hex_n[0]] + +def _sqrt_nearest(n, a): + """Closest integer to the square root of the positive integer n. a is + an initial approximation to the square root. Any positive integer + will do for a, but the closer a is to the square root of n the + faster convergence will be. + + """ + if n <= 0 or a <= 0: + raise ValueError("Both arguments to _sqrt_nearest should be positive.") + + b=0 + while a != b: + b, a = a, a--n//a>>1 + return a + +def _rshift_nearest(x, shift): + """Given an integer x and a nonnegative integer shift, return closest + integer to x / 2**shift; use round-to-even in case of a tie. + + """ + b, q = 1L << shift, x >> shift + return q + (2*(x & (b-1)) + (q&1) > b) + +def _div_nearest(a, b): + """Closest integer to a/b, a and b positive integers; rounds to even + in the case of a tie. + + """ + q, r = divmod(a, b) + return q + (2*r + (q&1) > b) + +def _ilog(x, M, L = 8): + """Integer approximation to M*log(x/M), with absolute error boundable + in terms only of x/M. + + Given positive integers x and M, return an integer approximation to + M * log(x/M). For L = 8 and 0.1 <= x/M <= 10 the difference + between the approximation and the exact result is at most 22. For + L = 8 and 1.0 <= x/M <= 10.0 the difference is at most 15. In + both cases these are upper bounds on the error; it will usually be + much smaller.""" + + # The basic algorithm is the following: let log1p be the function + # log1p(x) = log(1+x). Then log(x/M) = log1p((x-M)/M). We use + # the reduction + # + # log1p(y) = 2*log1p(y/(1+sqrt(1+y))) + # + # repeatedly until the argument to log1p is small (< 2**-L in + # absolute value). For small y we can use the Taylor series + # expansion + # + # log1p(y) ~ y - y**2/2 + y**3/3 - ... - (-y)**T/T + # + # truncating at T such that y**T is small enough. The whole + # computation is carried out in a form of fixed-point arithmetic, + # with a real number z being represented by an integer + # approximation to z*M. To avoid loss of precision, the y below + # is actually an integer approximation to 2**R*y*M, where R is the + # number of reductions performed so far. + + y = x-M + # argument reduction; R = number of reductions performed + R = 0 + while (R <= L and long(abs(y)) << L-R >= M or + R > L and abs(y) >> R-L >= M): + y = _div_nearest(long(M*y) << 1, + M + _sqrt_nearest(M*(M+_rshift_nearest(y, R)), M)) + R += 1 + + # Taylor series with T terms + T = -int(-10*len(str(M))//(3*L)) + yshift = _rshift_nearest(y, R) + w = _div_nearest(M, T) + for k in xrange(T-1, 0, -1): + w = _div_nearest(M, k) - _div_nearest(yshift*w, M) + + return _div_nearest(w*y, M) + +def _dlog10(c, e, p): + """Given integers c, e and p with c > 0, p >= 0, compute an integer + approximation to 10**p * log10(c*10**e), with an absolute error of + at most 1. Assumes that c*10**e is not exactly 1.""" + + # increase precision by 2; compensate for this by dividing + # final result by 100 + p += 2 + + # write c*10**e as d*10**f with either: + # f >= 0 and 1 <= d <= 10, or + # f <= 0 and 0.1 <= d <= 1. + # Thus for c*10**e close to 1, f = 0 + l = len(str(c)) + f = e+l - (e+l >= 1) + + if p > 0: + M = 10**p + k = e+p-f + if k >= 0: + c *= 10**k + else: + c = _div_nearest(c, 10**-k) + + log_d = _ilog(c, M) # error < 5 + 22 = 27 + log_10 = _ilog(10*M, M) # error < 15 + 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 10**p*f*log(10), with error < 17 + if f: + sign_f = [-1, 1][f > 0] + if p >= 0: + M = 10**p * abs(f) + else: + M = _div_nearest(abs(f), 10**-p) # M = 10**p*|f|, error <= 0.5 + + if M: + f_log_ten = sign_f*_ilog(10*M, M) # M*log(10), error <= 1.2 + 15 < 17 + else: + f_log_ten = 0 + else: + f_log_ten = 0 + + # error in sum < 17+27 = 44; error after division < 0.44 + 0.5 < 1 + return _div_nearest(f_log_ten + log_d, 100) + +def _iexp(x, M, L=8): + """Given integers x and M, M > 0, such that x/M is small in absolute + value, compute an integer approximation to M*exp(x/M). For 0 <= + x/M <= 2.4, the absolute error in the result is bounded by 60 (and + is usually much smaller).""" + + # Algorithm: to compute exp(z) for a real number z, first divide z + # by a suitable power R of 2 so that |z/2**R| < 2**-L. Then + # compute expm1(z/2**R) = exp(z/2**R) - 1 using the usual Taylor + # series + # + # expm1(x) = x + x**2/2! + x**3/3! + ... + # + # Now use the identity + # + # expm1(2x) = expm1(x)*(expm1(x)+2) + # + # R times to compute the sequence expm1(z/2**R), + # expm1(z/2**(R-1)), ... , exp(z/2), exp(z). + + # Find R such that x/2**R/M <= 2**-L + R = _nbits((long(x)<<L)//M) + + # Taylor series. (2**L)**T > M + T = -int(-10*len(str(M))//(3*L)) + y = _div_nearest(x, T) + Mshift = long(M)<<R + for i in xrange(T-1, 0, -1): + y = _div_nearest(x*(Mshift + y), Mshift * i) + + # Expansion + for k in xrange(R-1, -1, -1): + Mshift = long(M)<<(k+2) + y = _div_nearest(y*(y+Mshift), Mshift) + + return M+y + +def _dexp(c, e, p): + """Compute an approximation to exp(c*10**e), with p decimal places of + precision. + + Returns 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 log10 with extra precision = adjusted exponent of c*10**e + extra = max(0, e + len(str(c)) - 1) + q = p + extra + log10 = _dlog(10, 0, q) # error <= 1 + + # compute quotient c*10**e/(log10/10**q) = c*10**(e+q)/log10, + # rounding down + shift = e+q + if shift >= 0: + cshift = c*10**shift + else: + cshift = c//10**-shift + quot, rem = divmod(cshift, log10) + + # 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): +def _convert_other(other, raiseit=False): """Convert other to Decimal. Verifies that it's ok to use in an implicit construction. @@ -2979,6 +5117,8 @@ def _convert_other(other): return other if isinstance(other, (int, long)): return Decimal(other) + if raiseit: + raise TypeError("Unable to convert %s to Decimal" % other) return NotImplemented _infinity_map = { @@ -3066,12 +5206,16 @@ ExtendedContext = Context( # Reusable defaults Inf = Decimal('Inf') negInf = Decimal('-Inf') +NaN = Decimal('NaN') +Dec_0 = Decimal(0) +Dec_p1 = Decimal(1) +Dec_n1 = Decimal(-1) +Dec_p2 = Decimal(2) +Dec_n2 = Decimal(-2) # Infsign[sign] is infinity w/ that sign Infsign = (Inf, negInf) -NaN = Decimal('NaN') - ##### crud for parsing strings ############################################# import re |