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Ë â{|j•ãóÈ—dZddlmZddlZddlZddlZddlZddlZddlZdgZ ejjZ ejjZejd¬«d„«Zej"dej$ej&z«Zd d „Zd „Zej"d ej.ej$z«j0ZGd „dej4«Zy)z/Fraction, infinite-precision, rational numbers.é©ÚDecimalNÚFractioni@)Úmaxsizecó¶— t|dt«}ttt|««|z«}|dk\r|n| }|dk(rdS|S#t$r t }YŒ$wxYw)Néÿÿÿÿréþÿÿÿ)ÚpowÚ_PyHASH_MODULUSÚhashÚabsÚ ValueErrorÚ _PyHASH_INF)Ú numeratorÚ denominatorÚdinvÚhash_Úresults ú"/usr/lib64/python3.12/fractions.pyÚ_hash_algorithmrse€ð2Ü�; ¤OÓ4ˆô(”Tœ#˜i›.Ó)¨DÑ0Ó1ˆØ 1’n‰U¨5¨&€Fؘ2’ˆ2Ð) 6Ð)øô+ òäŠðús‚AÁAÁAa¶ \A\s* # optional whitespace at the start, (?P[-+]?) # an optional sign, then (?=\d|\.\d) # lookahead for digit or .digit (?P\d*|\d+(_\d+)*) # numerator (possibly empty) (?: # followed by (?:\s*/\s*(?P\d+(_\d+)*))? # an optional denominator | # or (?:\.(?P\d*|\d+(_\d+)*))? # an optional fractional part (?:E(?P[-+]?\d+(_\d+)*))? # and optional exponent ) \s*\Z # and optional whitespace to finish có°—|dk\r |d|zz}n |d| zz}t||dz z|«\}}|dk(r |dzdk(r|dz}|r|dkn|dk}|t|«fS)aMRound a rational number to the nearest multiple of a given power of 10. Rounds the rational number n/d to the nearest integer multiple of 10**exponent, rounding to the nearest even integer multiple in the case of a tie. Returns a pair (sign: bool, significand: int) representing the rounded value (-1)**sign * significand * 10**exponent. If no_neg_zero is true, then the returned sign will always be False when the significand is zero. Otherwise, the sign reflects the sign of the input. d must be positive, but n and d need not be relatively prime. ré ér )Údivmodr )ÚnÚdÚexponentÚ no_neg_zeroÚqÚrÚsigns rÚ_round_to_exponentr"Js|€ð�1‚}Ø ˆR�‰\Ñ‰à ˆR�(�‰]шô �!�q˜A‘v‘, Ó "�D€A€q؈A‚v�!�a‘%˜1’*Ø ˆR‰ˆáˆ1ˆqŠ5 Q¨¡U€DØ ”�Q“ˆ<Ðócó—|dk(rddd|z fStt|««t|«}}t|«t|«z ||kz}||z }t|||«\}}tt|««|dzk(r |dz}|dz }|||fS)a¡Round a rational number to a given number of significant figures. Rounds the rational number n/d to the given number of significant figures using the round-ties-to-even rule, and returns a triple (sign: bool, significand: int, exponent: int) representing the rounded value (-1)**sign * significand * 10**exponent. In the special case where n = 0, returns a significand of zero and an exponent of 1 - figures, for compatibility with formatting. Otherwise, the returned significand satisfies 10**(figures - 1) <= significand < 10**figures. d must be positive, but n and d need not be relatively prime. figures must be positive. rFrr)Ústrr Úlenr") rrÚfiguresÚstr_nÚstr_dÚmrr!Ú significands rÚ_round_to_figuresr,gs¤€ð" ˆA‚vØ�a˜˜W™Ð$Ð$ô”s˜1“v“;¤ A£ˆ5€EÜ ˆE‹ ”S˜“ZÑ 5¨E¡>Ñ2€Að�7‰{€HÜ*¨1¨a°Ó:Ñ€Dˆ+ô Œ3ˆ{Ó Ó ¨!¡ Ò+ؘш Ø�A‰ ˆà �˜hÐ &Ð&r#a› (?: (?P.)? (?P[<>=^]) )? (?P[-+ ]?) (?Pz)? (?P\#)? # A '0' that's *not* followed by another digit is parsed as a minimum width # rather than a zeropad flag. (?P0(?=[0-9]))? (?P0|[1-9][0-9]*)? (?P[,_])? (?:\.(?P0|[1-9][0-9]*))? (?P[eEfFgG%]) c󊇗eZdZdZdZd,ˆfd„ Zed„«Zed„«Zeˆfd„«Z d„Z d„Z d-d „Z e d „«Ze d „«Zd „Zd „Zd„Zd„Zd„Zeeej,«\ZZd„Zeeej4«\ZZd„Zeeej<«\ZZ d„Z!ee!ejD«\Z#Z$d„Z%ee%ejL«\Z'Z(d„Z)ee)e*«\Z+Z,d„Z-ee-ej\«\Z/Z0d„Z1d„Z2d„Z3d„Z4d„Z5ejlfd„Z7d„Z8d„Z9d„Z:d.d „Z;d!„Zd$„Z?d%„Z@d&„ZAd'„ZBd(„ZCd)„ZDd*„ZEd+„ZFˆxZGS)/ra]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 _denominatorcóB•—tt|� |«}|�€©t|«tur||_d|_|St|tj«r$|j|_|j|_|St|ttf«r|j«\|_|_|St|t«rút j#|«}|€t%d|z«‚t |j'd«xsd«}|j'd«}|r t |«}n€d}|j'd«}|r6|j)dd«}d t+|«z}||zt |«z}||z}|j'd «}|r"t |«}|d k\r |d |zz}n |d | zz}|j'd «d k(r¤| }n t-d«‚t|«tcxur t|«urnnnrt|tj«rMt|tj«r3|j|jz|j|jz}}n t-d«‚|d k(rt/d|z«‚t1j2||«} |d kr| } || z}|| z}||_||_|S)a£Constructs a Rational. Takes a string like '3/2' or '1.5', another Rational instance, a numerator/denominator pair, or a float. Examples -------- >>> Fraction(10, -8) Fraction(-5, 4) >>> Fraction(Fraction(1, 7), 5) Fraction(1, 35) >>> Fraction(Fraction(1, 7), Fraction(2, 3)) Fraction(3, 14) >>> Fraction('314') Fraction(314, 1) >>> Fraction('-35/4') Fraction(-35, 4) >>> Fraction('3.1415') # conversion from numeric string Fraction(6283, 2000) >>> Fraction('-47e-2') # string may include a decimal exponent Fraction(-47, 100) >>> Fraction(1.47) # direct construction from float (exact conversion) Fraction(6620291452234629, 4503599627370496) >>> Fraction(2.25) Fraction(9, 4) >>> Fraction(Decimal('1.47')) Fraction(147, 100) rz Invalid literal for Fraction: %rÚnumÚ0ÚdenomÚdecimalÚ_ÚrÚexprr!Ú-z2argument should be a string or a Rational instancez+both arguments should be Rational instancesúFraction(%s, 0))ÚsuperrÚ__new__ÚtypeÚintr/r0Ú isinstanceÚnumbersÚRationalrrÚfloatrÚas_integer_ratior%Ú_RATIONAL_FORMATÚmatchrÚgroupÚreplacer&Ú TypeErrorÚZeroDivisionErrorÚmathÚgcd) ÚclsrrÚselfr*r4r5Úscaler8ÚgÚ __class__s €rr<zFraction.__new__ºs˜ø€ô>”X˜sÑ+¨CÓ0ˆà Ñ Ü�I‹¤#Ñ%Ø"+�”Ø$%�Ô!Ø� ä˜I¤w×'7Ñ'7Ô8Ø"+×"5Ñ"5�”Ø$-×$9Ñ$9�Ô!Ø� ä˜I¬¬wÐ'7Ô8à5>×5OÑ5OÓ5QÑ2�” Ô!2Ø� ä˜I¤sÔ+ä$×*Ñ*¨9Ó5�Ø�9Ü$Ð%GØ%.ñ&/ó0ð0ä §¡¨£Ò 5°#Ó6� ØŸ™ Ó(�ÙÜ"% e£*‘Kà"#�KØŸg™g iÓ0�GÙØ")§/¡/°#°rÓ":˜Ø "¤C¨£LÑ 0˜Ø$-°Ñ$5¼¸G» Ñ$D˜ Ø# uÑ,˜ ØŸ'™' %›.�CÙÜ! #›h˜Ø !š8Ø%¨¨S©Ñ0™Ià'¨2°¨t©8Ñ3˜KØ—7‘7˜6“? cÒ)Ø!*  ‘Iô ð!9ó:ð:ô�)‹_¤Ó 8¤t¨KÓ'8Ô 8Ø ä˜¤G×$4Ñ$4Ô5Ü �{¤G×$4Ñ$4Ô 5à×#Ñ# k×&=Ñ&=Ñ=Ø×%Ñ%¨ ×(=Ñ(=Ñ=ð#‰Iô ð1ó2ð 2ð ˜!Ò Ü#Ð$5¸ Ñ$AÓBÐ BÜ �H‰H�Y  Ó ,ˆØ ˜Š?Ø�ˆAØ�a‰ˆ ؘш Ø#ˆŒØ'ˆÔ؈ r#c ó—t|tj«r||«St|t«s1t |j ›d|›dt |«j ›d�«‚|j|j«ŽS)z‚Converts a finite float to a rational number, exactly. Beware that Fraction.from_float(0.3) != Fraction(3, 10). z%.from_float() only takes floats, not ú (Ú)) r?r@ÚIntegralrBrHÚ__name__r=Ú_from_coprime_intsrC)rLÚfs rÚ from_floatzFraction.from_float#sm€ô �aœ×)Ñ)Ô *Ù�q“6ˆMܘAœuÔ%ÜØ Ÿ\›\ª1¬d°1«g×.>Ó.>ð@óAð Aà%ˆs×%Ñ% q×'9Ñ'9Ó';Ð<ÐrHrUr=rVrC)rLÚdecrs rÚ from_decimalzFraction.from_decimal1st€õ $Ü �cœ7×+Ñ+Ô ,Ùœ#˜c›(Ó#‰CܘC Ô)Üà—“šs¤D¨£I×$6Ó$6ð8ó9ð 9ð&ˆs×%Ñ% s×';Ñ';Ó'=Ð>Ð>r#cóJ•—tt|� |«}||_||_|S)z°Convert a pair of ints to a rational number, for internal use. The ratio of integers should be in lowest terms and the denominator should be positive. )r;rr<r/r0)rLrrÚobjrPs €rrVzFraction._from_coprime_ints=s*ø€ô”H˜cÑ*¨3Ó/ˆØ"ˆŒØ&ˆÔ؈ r#có —|jdk(S)z*Return True if the Fraction is an integer.r©r0©rMs rÚ is_integerzFraction.is_integerIs€à× Ñ  AÑ%Ð%r#có2—|j|jfS)z˜Return a pair of integers, whose ratio is equal to the original Fraction. The ratio is in lowest terms and has a positive denominator. r.r`s rrCzFraction.as_integer_ratioMs€ð —‘ ×!2Ñ!2Ð3Ð3r#cóª—|dkr td«‚|j|kr t|«Sd\}}}}|j|j}} ||z}|||zz} | |kDrn|||||zz| f\}}}}||||zz }}Œ/||z |z} d|z|| |zzz|jkrtj ||«Stj || |zz|| |zz«S)aWClosest Fraction to self with denominator at most max_denominator. >>> Fraction('3.141592653589793').limit_denominator(10) Fraction(22, 7) >>> Fraction('3.141592653589793').limit_denominator(100) Fraction(311, 99) >>> Fraction(4321, 8765).limit_denominator(10000) Fraction(4321, 8765) rz$max_denominator should be at least 1)rrrré)rr0rr/rV) rMÚmax_denominatorÚp0Úq0Úp1Úq1rrÚaÚq2Úks rÚlimit_denominatorzFraction.limit_denominatorTs€ð@ ˜QÒ ÜÐCÓDÐ DØ × Ñ  Ò /ܘD“>Ð !à#‰ˆˆB��BØ�‰ × 1Ñ 1ˆ1ˆØØ�1‘ˆAØ�A�b‘D‘ˆBØ�OÒ#ØØ  R¨¨"©¡W¨bÐ0‰NˆB��B˜Ø�a˜˜!™‘eˆqˆAð ð˜RÑ  "Ñ $ˆð ˆQ‰3��1�R‘4‘‰=˜D×-Ñ-Ò -Ü×.Ñ.¨r°2Ó6Ð 6ä×.Ñ.¨r°!°B±$©w¸¸1¸R¹4¹Ó@Ð @r#có—|jS©N)r/©rjs rrzFraction.numerator�s €à�|‰|Ðr#có—|jSror_rps rrzFraction.denominator‘s €à�~‰~Ðr#cóh—|jj›d|j›d|j›d�S)z repr(self)Ú(z, rS)rPrUr/r0r`s rÚ__repr__zFraction.__repr__•s*€à#Ÿ~™~×6Ó6Ø#Ÿ›°×0AÓ0AðCð Cr#có€—|jdk(rt|j«S|j›d|j›�S)z str(self)rÚ/)r0r%r/r`s rÚ__str__zFraction.__str__šs4€à × Ñ  Ò !Ü�t—‘Ó'Ð 'à"Ÿo›o¨t×/@Ò/@ÐAÐ Ar#c óÖ‡ ‡!—|s t|«St|«}|€$td|›dt|«j›�«‚|d�*|d�%td|›dt|«j›d�«‚|dxsd}|dxsd }|d d k(rd n|d }t |d «}t |d«}t |d«}t |dxsd«} |dŠ!t |dxsd«} |d} | dvxr| } | } | dvrdnd}| dvr7| }| dk(r|dz}t|j|j||«\}}d}| }nY| dvr t| d«n| dz}t|j|j|«\}}}| dvxs|dkDxs||zd k}|r|dz n| }| dk(rd}n|r |›||zd!›�}nd }|d|dz›d"�›}|rd n|}|dt|«|z Š |t|«|z d}| r|jd«}| r|sd nd#}||z|z}|r8| t|«z t|«z }‰ j‰!r d$|zd%zdzn|«Š ‰!rIdt‰ «dz d$zz}‰ d|d jˆ ˆ!fd&„t!|t‰ «d$«D««zŠ ‰ |z}|| t|«z t|«z z}|d k(r||z|zS|d'k(r||z|zS|d(k(rt|«dz}|d||z|z||dzS||z|zS))zAFormat this fraction according to the given format specification.NzInvalid format specifier z for object of type ÚalignÚzeropadz0; can't use explicit alignment when zero-paddingÚfillÚ Ú>r!r9r7rÚaltÚ minimumwidthr3Ú thousands_sepÚ precisionÚ6Úpresentation_typeÚgGÚEFGÚEÚezfF%Ú%rdFrÚeEréüÿÿÿz+03drÚ.ééc3ó4•K—|]}‰‰||dzz–—Œy­w)rŒN©)Ú.0ÚposÚleadingr€s €€rÚ z&Fraction.__format__..s)øèø€ð4á<�C𠨨c°A©gÐ 6Õ6Ù<ùsƒÚ<Ú^)r%Ú#_FLOAT_FORMAT_SPECIFICATION_MATCHERrr=rUÚboolr>r"r/r0Úmaxr,r&ÚrstripÚzfillÚjoinÚrange)"rMÚ format_specrEr{ryÚpos_signrÚalternate_formrzrr�rƒÚ trim_zerosÚ trim_pointÚexponent_indicatorrÚnegativer+Ú scientificÚ point_posr'ÚsuffixÚdigitsr!Ú frac_partÚ separatorÚtrailingÚ min_leadingÚ first_posÚbodyÚpaddingÚhalfr’r€s" @@rÚ __format__zFraction.__format__¡sù€ñÜ�t“9Ð ô4°KÓ@ˆØ ˆ=ÜØ+¨K¨?ð;&Ü&*¨4£j×&9Ñ&9Ð%<ð>óð ð�7‰^Ð '¨E°)Ñ,<Ð,HäØ+¨K¨?ð;&Ü&*¨4£j×&9Ñ&9Ð%<ð=AðAóð ð �V‰}Ò# ˆØ�g‘Ò% #ˆØ˜v™¨#Ò-‘2°5¸±=ˆÜ˜5 Ñ/Ó0ˆ ܘe E™lÓ+ˆÜ�u˜YÑ'Ó(ˆÜ˜5 Ñ0Ò7°CÓ8ˆ ؘoÑ.ˆ ܘ˜kÑ*Ò1¨cÓ2ˆ Ø!Ð"5Ñ6ÐØ&¨$Ð.ÒE°~Ð3Eˆ Ø'Ð'ˆ Ø$5¸Ñ$>™SÀCÐð  Ñ %Ø!�zˆHØ  CÒ'ؘA‘ �Ü$6Ø—‘ ×!2Ñ!2°H¸kó%KÑ !ˆH�kàˆJØ!‰Ið%¨Ñ,ô�I˜qÔ!à ‘]ð ô /@Ø—‘ ×!2Ñ!2°Gó/=Ñ +ˆH�k 8ð" TÐ)ò,ؘa‘<ò,à˜gÑ%¨Ñ+ð ñ (2˜ !š ¸°yˆIð  Ò #؉FÙ Ø*Ð+¨H°yÑ,@ÀÐ+FÐG‰FàˆFð   )¨a¡- °Ð1Ð2ˆñ ‰s HˆØÐ2œ3˜v›;¨Ñ2Ð3ˆØœ3˜v›;¨Ñ2Ð4Ð5ˆ Ù Ø!×(Ñ(¨Ó-ˆIÙ$©Y‘B¸Cˆ ؘyÑ(¨6Ñ1ˆñ Ø&¬¨T«Ñ2´S¸³]ÑBˆKð—m‘mÙ,9��K‘ 1Ñ$ qÒ(¸{óˆGñ ØœS ›\¨AÑ-°Ñ2Ñ2ˆIؘj˜yÐ)¨B¯G©Gô4ä  ¬C°«L¸!Ô<ó4ó-ñˆGð˜Ñ!ˆØ˜,¬¨T«Ñ2´S¸³YÑ>Ñ?ˆØ �CŠ<ؘT‘> DÑ(Ð (Ø �cŠ\ؘ$‘; Ñ(Ð (Ø �cŠ\Ü�w“< 1Ñ$ˆDؘ5˜D�> DÑ(¨4Ñ/°'¸$¸%°.Ñ@Ð @à˜'‘> DÑ(Ð (r#c󯇇—ˆˆfd„}d‰jzdz|_‰j|_ˆˆfd„}d‰jzdz|_‰j|_||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. có•—t|t«r ‰||«St|t«r‰|t|««St|t«r‰t|«|«St|t«r‰t |«|«St Sro)r?rr>rBÚcomplexÚNotImplemented)rjÚbÚfallback_operatorÚmonomorphic_operators €€rÚforwardz-Fraction._operator_fallbacks..forwardesqø€Ü˜!œXÔ&Ù+¨A¨qÓ1Ð1ܘAœsÔ#Ù+¨A¬x¸«{Ó;Ð;ܘAœuÔ%Ù(¬¨q«°1Ó5Ð5ܘAœwÔ'Ù(¬°«°QÓ7Ð7ä%Ð%r#Ú__có<•—t|tj«r‰t|«|«St|tj«r‰t |«t |««St|tj «r‰t|«t|««StSro) r?r@rArÚRealrBÚComplexr³r´)rµrjr¶r·s €€rÚreversez-Fraction._operator_fallbacks..reversesspø€Ü˜!œW×-Ñ-Ô.á+¬H°Q«K¸Ó;Ð;ܘAœwŸ|™|Ô,Ù(¬¨q«´5¸³8Ó<Ð<ܘAœwŸ™Ô/Ù(¬°«´W¸Q³ZÓ@Ð@ä%Ð%r#Ú__r)rUÚ__doc__)r·r¶r¸r½s`` rÚ_operator_fallbackszFraction._operator_fallbackssiù€õ` &ð Ð"3×"<Ñ"<Ñ<¸tÑCˆÔØ.×6Ñ6ˆŒõ &ð!Ð#4×#=Ñ#=Ñ=ÀÑDˆÔØ.×6Ñ6ˆŒà˜ÐÐr#có¨—|j|j}}|j|j}}tj||«}|dk(r"tj ||z||zz||z«S||z}|||zz||zz}tj||«} | dk(rtj |||z«Stj || z||| zz«S)za + br©r/r0rJrKrrV© rjrµÚnaÚdaÚnbÚdbrOÚsÚtÚg2s rÚ_addz Fraction._addÆóÊ€à—‘˜qŸ~™~ˆBˆØ—‘˜qŸ~™~ˆBˆÜ �H‰H�R˜Ó ˆØ �Š6Ü×.Ñ.¨r°B©w¸¸b¹Ñ/@À"ÀrÁ'ÓJÐ JØ �!‰GˆØ �"˜‘'‰N˜R !™VÑ #ˆÜ �X‰X�a˜‹^ˆØ �Š7Ü×.Ñ.¨q°!°b±&Ó9Ð 9Ü×*Ñ*¨1°©7°A¸¸r¹±NÓCÐCr#có¨—|j|j}}|j|j}}tj||«}|dk(r"tj ||z||zz ||z«S||z}|||zz||zz }tj||«} | dk(rtj |||z«Stj || z||| zz«S)za - brrÂrÃs rÚ_subz Fraction._subÖrÌr#có.—|j|j}}|j|j}}tj||«}|dkDr ||z}||z}tj||«}|dkDr ||z}||z}tj ||z||z«S)za * brrÂ)rjrµrÄrÅrÆrÇÚg1rÊs rÚ_mulz Fraction._mulæs‘€à—‘˜qŸ~™~ˆBˆØ—‘˜qŸ~™~ˆBˆÜ �X‰X�b˜"Ó ˆØ �Š6Ø �2‰IˆBØ �2‰IˆBÜ �X‰X�b˜"Ó ˆØ �Š6Ø �2‰IˆBØ �2‰IˆBÜ×*Ñ*¨2°©7°B¸±GÓ<ÐÓ?‰ ˆˆUØ”H˜U B¨¡GÓ,Ð,Ð,r#cóŠ—|j|j}}t|j|z|j|zz||z«S)za % b)rrr)rjrµrÅrÇs rÚ_modz Fraction._mods;€à—‘ § ¡ ˆBˆÜ˜Ÿ™ rÑ)¨a¯k©k¸BÑ.>Ñ?ÀÀbÁÓIÐIr#cóš—t|tj«�r|jdk(rá|j}|dk\r0t j |j|z|j|z«S|jdkDr2t j |j| z|j| z«S|jdk(rtd|j| zz«‚t j |j | z|j | z«St|«t|«zSt|ttf«rt|«|zStS)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. rrr:) r?r@rArrrrVr/r0rIrBr³r´)rjrµÚpowers rÚ__pow__zFraction.__pow__!s8€ô �aœ×)Ñ)Õ *Ø�}‰} Ò!ØŸ ™ �ؘA’:Ü#×6Ñ6°q·|±|ÀuÑ7LØ78·~±~ÈÑ7NóPðPà—\‘\ AÒ%Ü#×6Ñ6°q·~±~È%ÈÑ7OØ78·|±|ÈÀvÑ7MóOðOà—\‘\ QÒ&Ü+Ð,=Ø,-¯N©N¸u¸fÑ,Dñ-EóFðFô$×6Ñ6¸¿¹¸ÈUÈFÑ7RØ9:¿¹¸ È5È&Ñ7PóRðRô ˜Q“x¤5¨£8Ñ+Ð+Ü ˜œE¤7Ð+Ô ,ܘ“8˜q‘=Ð ä!Ð !r#có.—|jdk(r|jdk\r||jzSt|tj«r#t |j |j«|zS|jdk(r||jzS|t|«zS)za ** brr) r0r/r?r@rArrrrB)rµrjs rÚ__rpow__zFraction.__rpow__As{€à �>‰>˜QÒ  1§<¡<°1Ò#4à˜Ÿ ™ Ñ$Ð $ä �aœ×)Ñ)Ô *ܘAŸK™K¨¯©Ó7¸1Ñ<Ð <à �>‰>˜QÒ Ø˜Ÿ ™ Ñ$Ð $à”E˜!“H‰}Ðr#cóV—tj|j|j«S)z++a: Coerces a subclass instance to Fraction©rrVr/r0rps rÚ__pos__zFraction.__pos__Os€ä×*Ñ*¨1¯<©<¸¿¹ÓHÐHr#cóX—tj|j |j«S)z-arãrps rÚ__neg__zFraction.__neg__Ss€ä×*Ñ*¨A¯L©L¨=¸!¿.¹.ÓIÐIr#cóh—tjt|j«|j«S)zabs(a))rrVr r/r0rps rÚ__abs__zFraction.__abs__Ws"€ä×*Ñ*¬3¨q¯|©|Ó+<¸a¿n¹nÓMÐMr#có —|jdkr!||j |jz «S||j|jz«S)zint(a)rr.)rjÚ_indexs rÚ__int__zFraction.__int__[sC€à �<‰<˜!Ò Ù˜QŸ\™\˜M¨Q¯^©^Ñ;Ð<Ó=Ð =á˜!Ÿ,™,¨!¯.©.Ñ8Ó9Ð 9r#cóˆ—|jdkr|j |jz S|j|jzS)z math.trunc(a)rr.rps rÚ __trunc__zFraction.__trunc__bs9€à �<‰<˜!Ò Ø—l‘l�] a§n¡nÑ4Ð5Ð 5à—<‘< 1§>¡>Ñ1Ð 1r#có4—|j|jzS)z math.floor(a)r.rps rÚ __floor__zFraction.__floor__is€à�|‰|˜qŸ~™~Ñ-Ð-r#có8—|j |jz S)z math.ceil(a)r.rps rÚ__ceil__zFraction.__ceil__ms€ð—,‘,� !§.¡.Ñ0Ð1Ð1r#có&—|€K|j}t|j|«\}}|dz|kr|S|dz|kDr|dzS|dzdk(r|S|dzSdt|«z}|dkDrt t ||z«|«St t ||z «|z«S)z?round(self, ndigits) Rounds half toward even. rdrrr)r0rr/r rÚround)rMÚndigitsrÚfloorÚ remainderÚshifts rÚ __round__zFraction.__round__rs°€ð ˆ?Ø×!Ñ!ˆAÜ% d§o¡o°qÓ9Ñ ˆE�9ؘ1‰}˜qÒ Ø� ؘQ‘ Ò"ؘq‘yÐ à˜‘˜a’Ø� à˜q‘yÐ Ø”C˜“LÑ ˆð �QŠ;ÜœE $¨¡,Ó/°Ó7Ð 7äœE $¨¡,Ó/°%Ñ7Ó8Ð 8r#cóB—t|j|j«S)z hash(self))rr/r0r`s rÚ__hash__zFraction.__hash__Œs€ä˜tŸ™°×0AÑ0AÓBÐBr#có—t|«tur |j|k(xr|jdk(St |t j «r4|j|jk(xr|j|jk(St |t j«r|jdk(r |j}t |t«rCtj|«stj|«rd|k(S||j!|«k(St"S)za == brrç)r=r>r/r0r?r@rArrr¼ÚimagÚrealrBrJÚisnanÚisinfrXr´rÕs rÚ__eq__zFraction.__eq__�sÊ€ä �‹7”c‰>Ø—<‘< 1Ñ$Ò<¨¯©¸1Ñ)<Ð <Ü �aœ×)Ñ)Ô *Ø—L‘L A§K¡KÑ/ò4Ø—N‘N a§m¡mÑ3ð 5ä �aœŸ™Ô )¨a¯f©f¸ªkØ—‘ˆAÜ �aœÔ Ü�z‰z˜!Œ}¤§ ¡ ¨1¤ ð˜a‘x�à˜AŸL™L¨›OÑ+Ð+ô"Ð !r#cóf—t|tj«r7||j|jz|j |j z«St|t«rKtj|«stj|«r |d|«S|||j|««StS)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. rü) r?r@rAr/rr0rrBrJrÿrrXr´)rMÚotherÚops rÚ_richcmpzFraction._richcmp¥s�€ô �eœW×-Ñ-Ô .Ù�d—o‘o¨×(9Ñ(9Ñ9Ø×'Ñ'¨%¯/©/Ñ9ó;ð ;ä �eœUÔ #Ü�z‰z˜%Ô ¤D§J¡J¨uÔ$5Ù˜#˜u“~Ð%á˜$ §¡°Ó 6Ó7Ð7ä!Ð !r#cóB—|j|tj«S)za < b)rÚoperatorÚltrÕs rÚ__lt__zFraction.__lt__»ó€à�z‰z˜!œXŸ[™[Ó)Ð)r#cóB—|j|tj«S)za > b)rrÚgtrÕs rÚ__gt__zFraction.__gt__¿r r#cóB—|j|tj«S)za <= b)rrÚlerÕs rÚ__le__zFraction.__le__Ãr r#cóB—|j|tj«S)za >= b)rrÚgerÕs rÚ__ge__zFraction.__ge__Çr r#có,—t|j«S)za != 0)r—r/rps rÚ__bool__zFraction.__bool__Ës€ô�A—L‘LÓ!Ð!r#cóJ—|j|j|jffSro)rPr/r0r`s rÚ __reduce__zFraction.__reduce__Ós €Ø—‘ §¡°$×2CÑ2CÐ DÐEÐEr#cóv—t|«tk(r|S|j|j|j«Sro©r=rrPr/r0r`s rÚ__copy__zFraction.__copy__Öó.€Ü �‹:œÒ !؈KØ�~‰~˜dŸo™o¨t×/@Ñ/@ÓAÐAr#cóv—t|«tk(r|S|j|j|j«Sror)rMÚmemos rÚ __deepcopy__zFraction.__deepcopy__Ûrr#)rN)i@Bro)HrUÚ __module__Ú __qualname__r¿Ú __slots__r<Ú classmethodrXr[rVrarCrmÚpropertyrrrtrwr°rÀrËrÚaddÚ__add__Ú__radd__rÎÚsubÚ__sub__Ú__rsub__rÑÚmulÚ__mul__Ú__rmul__rÓÚtruedivÚ __truediv__Ú __rtruediv__rÖÚfloordivÚ __floordiv__Ú __rfloordiv__rÚrÚ __divmod__Ú __rdivmod__rÜÚmodÚ__mod__Ú__rmod__rßrárärærèÚindexrërírïrñrørúrrr r rrrrrrÚ __classcell__)rPs@rrr¢sçø„ñð(/€IõgðRñ =óð =ðñ ?óð ?ðó óð ò&ò4ó7AðrñóððñóðòCò Bòr)òhk òb Dñ,¨D°(·,±,Ó?Ñ€GˆXò Dñ,¨D°(·,±,Ó?Ñ€GˆXò =ñ,¨D°(·,±,Ó?Ñ€GˆXò1ñ(!4°D¸(×:JÑ:JÓ KÑ€K�òNñ#6°iÀ×ARÑARÓ"SÑ€L�-ò-ñ 2°'¸6ÓBÑ€J� òJñ ,¨D°(·,±,Ó?Ñ€GˆXò"ò@ òIòJòNð#Ÿ.™.ó:ò2ò.ò2ó 9ò4Cò"ò*"ò,*ò*ò*ò*ò"òFòBö Br#)F)r¿r5rÚ functoolsrJr@rÚreÚsysÚ__all__Ú hash_infoÚmodulusr ÚinfrÚ lru_cacherÚcompileÚVERBOSEÚ IGNORECASErDr"r,ÚDOTALLÚ fullmatchr–rArr�r#rÚrGsæðñ6åÛÛ ÛÛÛ Û à ˆ,€ð —-‘-×'Ñ'€ð�m‰m×Ñ€ à€×јwÔ'ñ*ó(ð*ð@�2—:‘:ð ð‡Z�Z�"—-‘-Ñó !Ðó"ò:$'ðR'1 b§j¡jð2ð‡Y�Y�—‘Ñó'÷'™Yð$ô$| Bˆw×Ñõ| Br#