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Z e j d e j e j Bƒ Z Gd d „ d e j ƒ Z d S( u+ Fraction, infinite-precision, real numbers.i ( u DecimalNu Fractionu gcdc C s" x | r | | | } } q W| S( u¶ Calculate the Greatest Common Divisor of a and b.
Unless b==0, the result will have the same sign as b (so that when
b is divided by it, the result comes out positive).
( ( u au b( ( u. /opt/alt/python33/lib64/python3.3/fractions.pyu gcd s uC
\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
c sf | Ee Z d Z d Z dP Z d dQ ‡ f d d † Z e d d „ ƒ Z e d d
„ ƒ Z d d d
„ Z
e d d „ ƒ Z e d d „ ƒ Z
d d „ Z d d „ Z d d „ Z d d „ Z e e e j ƒ \ Z Z d d „ Z e e e j ƒ \ Z Z d d „ Z e e e j ƒ \ Z Z d d „ Z e e e j ƒ \ Z Z! d d! „ Z" d" d# „ Z# d$ d% „ Z$ d& d' „ Z% d( d) „ Z& d* d+ „ Z' d, d- „ Z( d. d/ „ Z) d0 d1 „ Z* d2 d3 „ Z+ d4 d5 „ Z, d6 d7 „ Z- dQ d8 d9 „ Z. d: d; „ Z/ d< d= „ Z0 d> d? „ Z1 d@ dA „ Z2 dB dC „ Z3 dD dE „ Z4 dF dG „ Z5 dH dI „ Z6 dJ dK „ Z7 dL dM „ Z8 dN dO „ Z9 ‡ S(R u Fractionu] 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)
u
_numeratoru _denominatori c s¯ t t | ƒ j | ƒ } | d k rt | t j ƒ rR | j | _ | j | _
| St | t ƒ rŒ t j | ƒ } | j | _ | j
| _
| St | t
ƒ rÆ t j | ƒ } | j | _ | j
| _
| St | t ƒ r t j | ƒ } | d k rt d | ƒ ‚ n t | j d ƒ pd ƒ } | j d ƒ } | rBt | ƒ } nœ d } | j d ƒ } | rŽd t | ƒ } | | t | ƒ } | | 9} n | j d ƒ } | rÞt | ƒ } | d k rÌ| d | 9} qÞ| d | 9} n | j d
ƒ d k r | } q qct d ƒ ‚ nT t | t j ƒ rWt | t j ƒ rW| j | j | j | j } } n t d
ƒ ‚ | d k r‚t d | ƒ ‚ n t | | ƒ }
| |
| _ | |
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| S( u£ 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)
u Invalid literal for Fraction: %ru numu 0u denomi u decimali
u expi u signu -u2 argument should be a string or a Rational instanceu+ both arguments should be Rational instancesu Fraction(%s, 0)N( u superu Fractionu __new__u Noneu
isinstanceu numbersu Rationalu numeratoru
_numeratoru denominatoru _denominatoru floatu
from_floatu Decimalu from_decimalu stru _RATIONAL_FORMATu matchu
ValueErroru intu groupu lenu TypeErroru ZeroDivisionErroru gcd( u clsu numeratoru denominatoru selfu valueu mu denomu decimalu scaleu expu g( u __class__( u. /opt/alt/python33/lib64/python3.3/fractions.pyu __new__I sf
u Fraction.__new__c C s t | t j ƒ r | | ƒ St | t ƒ sS t d | j | t | ƒ j f ƒ ‚ n t j | ƒ sq t j | ƒ r t d | | j f ƒ ‚ n | | j
ƒ Œ S( u‚ Converts a finite float to a rational number, exactly.
Beware that Fraction.from_float(0.3) != Fraction(3, 10).
u. %s.from_float() only takes floats, not %r (%s)u Cannot convert %r to %s.( u
isinstanceu numbersu Integralu floatu TypeErroru __name__u typeu mathu isnanu isinfu as_integer_ratio( u clsu f( ( u. /opt/alt/python33/lib64/python3.3/fractions.pyu
from_float s
" u Fraction.from_floatc C s
d d l m } t | t j ƒ r7 | t | ƒ ƒ } n7 t | | ƒ sn t d | j | t | ƒ j f ƒ ‚ n | j ƒ s– t d | | j f ƒ ‚ n | j
ƒ \ } } } t d j t t
| ƒ ƒ ƒ } | rÙ | } n | d k r÷ | | d | ƒ S| | d | ƒ Sd S( uA Converts a finite Decimal instance to a rational number, exactly.i ( u Decimalu2 %s.from_decimal() only takes Decimals, not %r (%s)u Cannot convert %s to %s.u i
N( u decimalu Decimalu
isinstanceu numbersu Integralu intu TypeErroru __name__u typeu is_finiteu as_tupleu joinu mapu str( u clsu decu Decimalu signu digitsu exp( ( u. /opt/alt/python33/lib64/python3.3/fractions.pyu from_decimal½ s "
u Fraction.from_decimali@B c
C s* | d k r t d ƒ ‚ n | j | k r4 t | ƒ Sd \ } } } } | j | j } } xf | | } | | | } | | k r„ Pn | | | | | | f \ } } } } | | | | } } q\ | | | }
t | |
| | |
| ƒ } t | | ƒ } t | | ƒ t | | ƒ k r"| S| Sd S( uW Closest 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)
i u$ max_denominator should be at least 1i N( i i i i ( u
ValueErroru _denominatoru Fractionu
_numeratoru abs(
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s u Fraction.numeratorc C s | j S( N( u _denominator( u a( ( u. /opt/alt/python33/lib64/python3.3/fractions.pyu denominator s u Fraction.denominatorc C s d | j | j f S( u
repr(self)u Fraction(%s, %s)( u
_numeratoru _denominator( u self( ( u. /opt/alt/python33/lib64/python3.3/fractions.pyu __repr__ s u Fraction.__repr__c C s4 | j d k r t | j ƒ Sd | j | j f Sd S( u str(self)i u %s/%sN( u _denominatoru stru
_numerator( u self( ( u. /opt/alt/python33/lib64/python3.3/fractions.pyu __str__ s
u Fraction.__str__c st ‡ ‡ f d d † } d ˆ j d | _ ˆ j | _ ‡ ‡ f d d † } d ˆ j d | _ ˆ j | _ | | f S( uÕ Generates 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.
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isinstanceu intu Fractionu floatu complexu NotImplemented( u au b( u fallback_operatoru monomorphic_operator( u. /opt/alt/python33/lib64/python3.3/fractions.pyu forwardm s
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Fraction._subc C s! t | j | j | j | j ƒ S( u a * b( u Fractionu numeratoru denominator( u au b( ( u. /opt/alt/python33/lib64/python3.3/fractions.pyu _mul˜ s u
Fraction._mulc C s! t | j | j | j | j ƒ S( u a / b( u Fractionu numeratoru denominator( u au b( ( u. /opt/alt/python33/lib64/python3.3/fractions.pyu _divž s u
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__rfloordiv__© s u Fraction.__rfloordiv__c C s | | } | | | S( u a % b( ( u au bu div( ( u. /opt/alt/python33/lib64/python3.3/fractions.pyu __mod__ s
u Fraction.__mod__c C s | | } | | | S( u a % b( ( u bu au div( ( u. /opt/alt/python33/lib64/python3.3/fractions.pyu __rmod__² s
u Fraction.__rmod__c C sš t | t j ƒ rˆ | j d k rq | j } | d k rQ t | j | | j | ƒ St | j | | j | ƒ Sq– t | ƒ t | ƒ Sn t | ƒ | Sd S( u¾ a ** 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.
i i N( u
isinstanceu numbersu Rationalu denominatoru numeratoru Fractionu
_numeratoru _denominatoru float( u au bu power( ( u. /opt/alt/python33/lib64/python3.3/fractions.pyu __pow__· s
u Fraction.__pow__c C sz | j d k r) | j d k r) | | j St | t j ƒ rR t | j | j ƒ | S| j d k rl | | j S| t | ƒ S( u a ** bi i ( u _denominatoru
_numeratoru
isinstanceu numbersu Rationalu Fractionu numeratoru denominatoru float( u bu a( ( u. /opt/alt/python33/lib64/python3.3/fractions.pyu __rpow__Ï s u Fraction.__rpow__c C s t | j | j ƒ S( u+ +a: Coerces a subclass instance to Fraction( u Fractionu
_numeratoru _denominator( u a( ( u. /opt/alt/python33/lib64/python3.3/fractions.pyu __pos__Ý s u Fraction.__pos__c C s t | j | j ƒ S( u -a( u Fractionu
_numeratoru _denominator( u a( ( u. /opt/alt/python33/lib64/python3.3/fractions.pyu __neg__á s u Fraction.__neg__c C s t t | j ƒ | j ƒ S( u abs(a)( u Fractionu absu
_numeratoru _denominator( u a( ( u. /opt/alt/python33/lib64/python3.3/fractions.pyu __abs__å s u Fraction.__abs__c C s1 | j d k r | j | j S| j | j Sd S( u trunc(a)i N( u
_numeratoru _denominator( u a( ( u. /opt/alt/python33/lib64/python3.3/fractions.pyu __trunc__é s u Fraction.__trunc__c C s | j | j S( u Will be math.floor(a) in 3.0.( u numeratoru denominator( u a( ( u. /opt/alt/python33/lib64/python3.3/fractions.pyu __floor__ð s u Fraction.__floor__c C s | j | j S( u Will be math.ceil(a) in 3.0.( u numeratoru denominator( u a( ( u. /opt/alt/python33/lib64/python3.3/fractions.pyu __ceil__ô s u Fraction.__ceil__c C sÇ | d k rx t | j | j ƒ \ } } | d | j k r> | S| d | j k rY | d S| d d k rm | S| d Sn d t | ƒ } | d k r« t t | | ƒ | ƒ St t | | ƒ | ƒ Sd S( uO Will be round(self, ndigits) in 3.0.
Rounds half toward even.
i i i i
N( u Noneu divmodu numeratoru denominatoru absu Fractionu round( u selfu ndigitsu flooru remainderu shift( ( u. /opt/alt/python33/lib64/python3.3/fractions.pyu __round__ù s u Fraction.__round__c C sl t | j t d t ƒ } | s( t } n t | j ƒ | t } | d k rQ | n | } | d k rh d S| S( u
hash(self)i i i iÿÿÿÿiþÿÿÿ( u powu _denominatoru _PyHASH_MODULUSu _PyHASH_INFu absu
_numerator( u selfu dinvu hash_u result( ( u. /opt/alt/python33/lib64/python3.3/fractions.pyu __hash__ s u Fraction.__hash__c C s¶ t | t j ƒ r4 | j | j k o3 | j | j k St | t j ƒ ra | j d k ra | j } n t | t
ƒ r® t j | ƒ sŽ t j
| ƒ r˜ d | k S| | j | ƒ k Sn t Sd S( u a == bi g N( u
isinstanceu numbersu Rationalu
_numeratoru numeratoru _denominatoru denominatoru Complexu imagu realu floatu mathu isnanu isinfu
from_floatu NotImplemented( u au b( ( u. /opt/alt/python33/lib64/python3.3/fractions.pyu __eq__( s !
u Fraction.__eq__c C sŽ t | t j ƒ r3 | | j | j | j | j ƒ St | t ƒ r† t j | ƒ s` t j
| ƒ rm | d | ƒ S| | | j | ƒ ƒ Sn t Sd S( uc Helper 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.
g N(
u
isinstanceu numbersu Rationalu
_numeratoru denominatoru _denominatoru numeratoru floatu mathu isnanu isinfu
from_floatu NotImplemented( u selfu otheru op( ( u. /opt/alt/python33/lib64/python3.3/fractions.pyu _richcmp; s
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| j d k S( u a != 0i ( u
_numerator( u a( ( u. /opt/alt/python33/lib64/python3.3/fractions.pyu __bool__a s u Fraction.__bool__c C s | j t | ƒ f f S( N( u __class__u str( u self( ( u. /opt/alt/python33/lib64/python3.3/fractions.pyu
__reduce__g s u Fraction.__reduce__c C s, t | ƒ t k r | S| j | j | j ƒ S( N( u typeu Fractionu __class__u
_numeratoru _denominator( u self( ( u. /opt/alt/python33/lib64/python3.3/fractions.pyu __copy__j s u Fraction.__copy__c C s, t | ƒ t k r | S| j | j | j ƒ S( N( u typeu Fractionu __class__u
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_numeratoru _denominatorN(: u __name__u
__module__u __qualname__u __doc__u __slots__u Noneu __new__u classmethodu
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__rfloordiv__u __mod__u __rmod__u __pow__u __rpow__u __pos__u __neg__u __abs__u __trunc__u __floor__u __ceil__u __round__u __hash__u __eq__u _richcmpu __lt__u __gt__u __le__u __ge__u __bool__u
__reduce__u __copy__u __deepcopy__( u
__locals__( ( u __class__u. /opt/alt/python33/lib64/python3.3/fractions.pyu Fraction1 sV d7 k ( u __doc__u decimalu Decimalu mathu numbersu operatoru reu sysu __all__u gcdu hash_infou modulusu _PyHASH_MODULUSu infu _PyHASH_INFu compileu VERBOSEu
IGNORECASEu _RATIONAL_FORMATu Rationalu Fraction( ( ( u. /opt/alt/python33/lib64/python3.3/fractions.pyu s