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a ´i_ã@sxdZddlmZddlZddlZddlZddlZddlZdgZej j Z ej j Z e dejejB¡ZGdd„dejƒZdS)z+Fraction, infinite-precision, real numbers.é©ÚDecimalNÚFractionaC \A\s* # optional whitespace at the start, then (?P[-+]?) # an optional sign, then (?=\d|\.\d) # lookahead for digit or .digit (?P\d*) # numerator (possibly empty) (?: # followed by (?:/(?P\d+))? # an optional denominator | # or (?:\.(?P\d*))? # an optional fractional part (?:E(?P[-+]?\d+))? # and optional exponent ) \s*\Z # and optional whitespace to finish csÎeZdZdZdZdRddœ‡fdd„ Zed d „ƒZed d „ƒZd d„Z dSdd„Z e dd„ƒZ e dd„ƒZ dd„Zdd„Zdd„Zdd„Zeeejƒ\ZZdd„Zeeejƒ\ZZd d!„Zeeejƒ\ZZd"d#„Zeeejƒ\Z Z!d$d%„Z"ee"ej#ƒ\Z$Z%d&d'„Z&ee&e'ƒ\Z(Z)d(d)„Z*ee*ej+ƒ\Z,Z-d*d+„Z.d,d-„Z/d.d/„Z0d0d1„Z1d2d3„Z2d4d5„Z3d6d7„Z4d8d9„Z5dTd:d;„Z6dd?„Z8d@dA„Z9dBdC„Z:dDdE„Z;dFdG„ZdLdM„Z?dNdO„Z@dPdQ„ZA‡ZBS)Ura]This class implements rational numbers. In the two-argument form of the constructor, Fraction(8, 6) will produce a rational number equivalent to 4/3. Both arguments must be Rational. The numerator defaults to 0 and the denominator defaults to 1 so that Fraction(3) == 3 and Fraction() == 0. Fractions can also be constructed from: - numeric strings similar to those accepted by the float constructor (for example, '-2.3' or '1e10') - strings of the form '123/456' - float and Decimal instances - other Rational instances (including integers) ©Ú _numeratorÚ _denominatorrNT©Ú _normalizec s"tt|ƒ |¡}|du�rdt|ƒtur6||_d|_|St|tj ƒrV|j |_|j |_|St|t t fƒrx| ¡\|_|_|St|tƒ�rZt |¡}|dur¢td|ƒ‚t| d¡p°dƒ}| d¡}|rÌt|ƒ}nvd}| d¡}|�rdt|ƒ}||t|ƒ}||9}| d ¡} | �rBt| ƒ} | d k�r4|d| 9}n|d| 9}| d ¡d k�rb| }ntd ƒ‚nft|ƒtu�r„t|ƒu�rŠnnn@t|tj ƒ�rÂt|tj ƒ�rÂ|j |j |j |j }}ntdƒ‚|d k�ràtd|ƒ‚|�rt ||¡} |d k�r| } || }|| }||_||_|S)a£Constructs a Rational. Takes a string like '3/2' or '1.5', another Rational instance, a numerator/denominator pair, or a float. Examples -------- >>> Fraction(10, -8) Fraction(-5, 4) >>> Fraction(Fraction(1, 7), 5) Fraction(1, 35) >>> Fraction(Fraction(1, 7), Fraction(2, 3)) Fraction(3, 14) >>> Fraction('314') Fraction(314, 1) >>> Fraction('-35/4') Fraction(-35, 4) >>> Fraction('3.1415') # conversion from numeric string Fraction(6283, 2000) >>> Fraction('-47e-2') # string may include a decimal exponent Fraction(-47, 100) >>> Fraction(1.47) # direct construction from float (exact conversion) Fraction(6620291452234629, 4503599627370496) >>> Fraction(2.25) Fraction(9, 4) >>> Fraction(Decimal('1.47')) Fraction(147, 100) Néz Invalid literal for Fraction: %rÚnumÚ0ÚdenomÚdecimalé ÚexprÚsignú-z2argument should be a string or a Rational instancez+both arguments should be Rational instanceszFraction(%s, 0))ÚsuperrÚ__new__ÚtypeÚintrrÚ isinstanceÚnumbersÚRationalÚ numeratorÚ denominatorÚfloatrÚas_integer_ratioÚstrÚ_RATIONAL_FORMATÚmatchÚ ValueErrorÚgroupÚlenÚ TypeErrorÚZeroDivisionErrorÚmathÚgcd) Úclsrrr ÚselfÚmr rZscalerÚg©Ú __class__©ú!/usr/lib64/python3.9/fractions.pyr>st     ÿ       $ ÿ  þ    zFraction.__new__cCsDt|tjƒr||ƒSt|tƒs8td|j|t|ƒjfƒ‚|| ¡ŽS)z‚Converts a finite float to a rational number, exactly. Beware that Fraction.from_float(0.3) != Fraction(3, 10). z.%s.from_float() only takes floats, not %r (%s))rrÚIntegralrr$Ú__name__rr)r(Úfr.r.r/Ú from_float§s  ÿzFraction.from_floatcCsVddlm}t|tjƒr&|t|ƒƒ}n$t||ƒsJtd|j|t|ƒjfƒ‚||  ¡ŽS)zAConverts a finite Decimal instance to a rational number, exactly.rrz2%s.from_decimal() only takes Decimals, not %r (%s)) rrrrr0rr$r1rr)r(Zdecrr.r.r/Ú from_decimalµs   ÿÿzFraction.from_decimalcCs |j|jfS)z¤Return the integer ratio as a tuple. Return a tuple of two integers, whose ratio is equal to the Fraction and with a positive denominator. r©r)r.r.r/rÁszFraction.as_integer_ratioé@Bc CsÞ|dkrtdƒ‚|j|kr"t|ƒSd\}}}}|j|j}}||}|||} | |krZqŠ|||||| f\}}}}||||}}q<|||} t|| ||| |ƒ} t||ƒ} t| |ƒt| |ƒkrÖ| S| SdS)aWClosest Fraction to self with denominator at most max_denominator. >>> Fraction('3.141592653589793').limit_denominator(10) Fraction(22, 7) >>> Fraction('3.141592653589793').limit_denominator(100) Fraction(311, 99) >>> Fraction(4321, 8765).limit_denominator(10000) Fraction(4321, 8765) r z$max_denominator should be at least 1)rr r rN)r!rrrÚabs) r)Zmax_denominatorZp0Zq0Zp1Zq1ÚnÚdÚaZq2ÚkZbound1Zbound2r.r.r/Úlimit_denominatorÉs$      zFraction.limit_denominatorcCs|jS©N)r©r:r.r.r/rszFraction.numeratorcCs|jSr=)rr>r.r.r/rszFraction.denominatorcCsd|jj|j|jfS)z repr(self)z %s(%s, %s))r-r1rrr5r.r.r/Ú__repr__sÿzFraction.__repr__cCs(|jdkrt|jƒSd|j|jfSdS)z str(self)r z%s/%sN)rrrr5r.r.r/Ú__str__ s  zFraction.__str__csT‡‡fdd„}dˆjd|_ˆj|_‡‡fdd„}dˆjd|_ˆj|_||fS)aÕGenerates forward and reverse operators given a purely-rational operator and a function from the operator module. Use this like: __op__, __rop__ = _operator_fallbacks(just_rational_op, operator.op) In general, we want to implement the arithmetic operations so that mixed-mode operations either call an implementation whose author knew about the types of both arguments, or convert both to the nearest built in type and do the operation there. In Fraction, that means that we define __add__ and __radd__ as: def __add__(self, other): # Both types have numerators/denominator attributes, # so do the operation directly if isinstance(other, (int, Fraction)): return Fraction(self.numerator * other.denominator + other.numerator * self.denominator, self.denominator * other.denominator) # float and complex don't have those operations, but we # know about those types, so special case them. elif isinstance(other, float): return float(self) + other elif isinstance(other, complex): return complex(self) + other # Let the other type take over. return NotImplemented def __radd__(self, other): # radd handles more types than add because there's # nothing left to fall back to. if isinstance(other, numbers.Rational): return Fraction(self.numerator * other.denominator + other.numerator * self.denominator, self.denominator * other.denominator) elif isinstance(other, Real): return float(other) + float(self) elif isinstance(other, Complex): return complex(other) + complex(self) return NotImplemented There are 5 different cases for a mixed-type addition on Fraction. I'll refer to all of the above code that doesn't refer to Fraction, float, or complex as "boilerplate". 'r' will be an instance of Fraction, which is a subtype of Rational (r : Fraction <: Rational), and b : B <: Complex. The first three involve 'r + b': 1. If B <: Fraction, int, float, or complex, we handle that specially, and all is well. 2. If Fraction falls back to the boilerplate code, and it were to return a value from __add__, we'd miss the possibility that B defines a more intelligent __radd__, so the boilerplate should return NotImplemented from __add__. In particular, we don't handle Rational here, even though we could get an exact answer, in case the other type wants to do something special. 3. If B <: Fraction, Python tries B.__radd__ before Fraction.__add__. This is ok, because it was implemented with knowledge of Fraction, so it can handle those instances before delegating to Real or Complex. The next two situations describe 'b + r'. We assume that b didn't know about Fraction in its implementation, and that it uses similar boilerplate code: 4. If B <: Rational, then __radd_ converts both to the builtin rational type (hey look, that's us) and proceeds. 5. Otherwise, __radd__ tries to find the nearest common base ABC, and fall back to its builtin type. Since this class doesn't subclass a concrete type, there's no implementation to fall back to, so we need to try as hard as possible to return an actual value, or the user will get a TypeError. csPt|ttfƒrˆ||ƒSt|tƒr0ˆt|ƒ|ƒSt|tƒrHˆt|ƒ|ƒStSdSr=)rrrrÚcomplexÚNotImplemented©r:Úb©Úfallback_operatorÚmonomorphic_operatorr.r/Úforwardds   z-Fraction._operator_fallbacks..forwardÚ__csZt|tjƒrˆ||ƒSt|tjƒr4ˆt|ƒt|ƒƒSt|tjƒrRˆt|ƒt|ƒƒStSdSr=)rrrZRealrÚComplexrArB©rDr:rEr.r/Úreverseps    z-Fraction._operator_fallbacks..reverseZ__r)r1Ú__doc__)rGrFrHrLr.rEr/Ú_operator_fallbackssP  zFraction._operator_fallbackscCs,|j|j}}t|j||j|||ƒS)za + b©rrr©r:rDÚdaÚdbr.r.r/Ú_addsÿz Fraction._addcCs,|j|j}}t|j||j|||ƒS)za - brOrPr.r.r/Ú_sub‡sÿz Fraction._subcCst|j|j|j|jƒS)za * b©rrrrCr.r.r/Ú_mul�sz Fraction._mulcCst|j|j|j|jƒS)za / brUrCr.r.r/Ú_div•s  ÿz Fraction._divcCs|j|j|j|jS)za // b©rrrCr.r.r/Ú _floordivœszFraction._floordivcCs:|j|j}}t|j|||jƒ\}}|t|||ƒfS)z(a // b, a % b))rÚdivmodrr)r:rDrQrRZdivZn_modr.r.r/Ú_divmod¢szFraction._divmodcCs,|j|j}}t|j||j|||ƒS)za % brOrPr.r.r/Ú_modªsz Fraction._modcCs¬t|tjƒrœ|jdkrŠ|j}|dkr>t|j||j|dd�S|jdkrft|j| |j| dd�St|j | |j | dd�Sq¨t|ƒt|ƒSn t|ƒ|SdS)z¾a ** b If b is not an integer, the result will be a float or complex since roots are generally irrational. If b is an integer, the result will be rational. r rFrN) rrrrrrrrr)r:rDZpowerr.r.r/Ú__pow__±s&   þ   þ þzFraction.__pow__cCs\|jdkr|jdkr||jSt|tjƒrr.r.r/Ú__pos__ÝszFraction.__pos__cCst|j |jdd�S)z-aFrr_r>r.r.r/Ú__neg__ászFraction.__neg__cCstt|jƒ|jdd�S)zabs(a)Fr)rr7rrr>r.r.r/Ú__abs__åszFraction.__abs__cCs*|jdkr|j |j S|j|jSdS)ztrunc(a)rNrr>r.r.r/Ú __trunc__és zFraction.__trunc__cCs |j|jS)z math.floor(a)rXr>r.r.r/Ú __floor__ðszFraction.__floor__cCs|j |j S)z math.ceil(a)rXr>r.r.r/Ú__ceil__ôszFraction.__ceil__cCs˜|durZt|j|jƒ\}}|d|jkr,|S|d|jkrB|dS|ddkrR|S|dSdt|ƒ}|dkr€tt||ƒ|ƒStt||ƒ|ƒSdS)z?round(self, ndigits) Rounds half toward even. Nér rr)rZrrr7rÚround)r)ÚndigitsÚfloorÚ remainderÚshiftr.r.r/Ú __round__ùs  zFraction.__round__cCsdzt|jdtƒ}Wnty(t}Yn0ttt|jƒƒ|ƒ}|jdkrN|n| }|dkr`dS|S)z hash(self)éÿÿÿÿréþÿÿÿ)ÚpowrÚ_PyHASH_MODULUSr!Ú _PyHASH_INFÚhashr7r)r)ZdinvZhash_Úresultr.r.r/Ú__hash__s  zFraction.__hash__cCsžt|ƒtur |j|ko|jdkSt|tjƒrD|j|jkoB|j|jkSt|tj ƒr`|j dkr`|j }t|t ƒr–t  |¡s~t  |¡r†d|kS|| |¡kSntSdS)za == br rçN)rrrrrrrrrrJÚimagÚrealrr&ÚisnanÚisinfr3rBrCr.r.r/Ú__eq__3s    ÿ zFraction.__eq__cCsht|tjƒr&||j|j|j|jƒSt|tƒr`t  |¡sDt  |¡rN|d|ƒS|||  |¡ƒSnt SdS)acHelper for comparison operators, for internal use only. Implement comparison between a Rational instance `self`, and either another Rational instance or a float `other`. If `other` is not a Rational instance or a float, return NotImplemented. `op` should be one of the six standard comparison operators. ruN) rrrrrrrrr&rxryr3rB)r)ÚotherÚopr.r.r/Ú_richcmpHs   ÿ  zFraction._richcmpcCs| |tj¡S)za < b)r}ÚoperatorÚltrCr.r.r/Ú__lt__^szFraction.__lt__cCs| |tj¡S)za > b)r}r~ÚgtrCr.r.r/Ú__gt__bszFraction.__gt__cCs| |tj¡S)za <= b)r}r~ÚlerCr.r.r/Ú__le__fszFraction.__le__cCs| |tj¡S)za >= b)r}r~ÚgerCr.r.r/Ú__ge__jszFraction.__ge__cCs t|jƒS)za != 0)Úboolrr>r.r.r/Ú__bool__nszFraction.__bool__cCs|jt|ƒffSr=)r-rr5r.r.r/Ú __reduce__vszFraction.__reduce__cCs t|ƒtkr|S| |j|j¡Sr=©rrr-rrr5r.r.r/Ú__copy__ys zFraction.__copy__cCs t|ƒtkr|S| |j|j¡Sr=rŠ)r)Úmemor.r.r/Ú __deepcopy__~s zFraction.__deepcopy__)rN)r6)N)Cr1Ú __module__Ú __qualname__rMÚ __slots__rÚ classmethodr3r4rr<Úpropertyrrr?r@rNrSr~ÚaddÚ__add__Ú__radd__rTÚsubÚ__sub__Ú__rsub__rVÚmulÚ__mul__Ú__rmul__rWÚtruedivÚ __truediv__Ú __rtruediv__rYÚfloordivÚ __floordiv__Ú __rfloordiv__r[rZÚ __divmod__Ú __rdivmod__r\ÚmodÚ__mod__Ú__rmod__r]r^r`rarbrcrdrerlrtrzr}r€r‚r„r†rˆr‰r‹r�Ú __classcell__r.r.r,r/r&sdi   7  k !)rMrrr&rr~ÚreÚsysÚ__all__Ú hash_infoÚmodulusrpÚinfrqÚcompileÚVERBOSEÚ IGNORECASErrrr.r.r.r/Ús  ô