U
e5d _ @ s d Z ddlmZ ddlZddlZddlZddlZddlZddgZdd Z dd Z
ej j Z
ej jZed ejejB ZG d
d dejZdS ) z+Fraction, infinite-precision, real numbers. DecimalNFractiongcdc C sf ddl }|dtd t| t kr2t|kr\n n&|p<| dk rPt| | S t| |S t| |S )zCalculate 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).
r Nz6fractions.gcd() is deprecated. Use math.gcd() instead. )warningswarnDeprecationWarningtypeintmathr _gcd)abr r !/usr/lib64/python3.8/fractions.pyr s c C s |r|| | } }q | S Nr r r r r r r
s r
aC
\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 s e Zd ZdZdZdRdd fdd Zed d
Zed d Zd
d Z dSddZ
e dd Z e dd Z
dd Zdd Zdd Zd d Zeeej\ZZd d Zeeej\ZZd d! Zeeej\Z Z d"d# Z ee ej \Z Z!d$d% Z"ee"ej#\Z$Z%d&d' Z&ee&e'\Z(Z)d(d) Z*ee*ej+\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 )Ur a] 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)
_numerator _denominatorr NT
_normalizec sR t t| | }|dkrdt|tkr6||_d|_|S t|tj rV|j
|_|j |_|S t|t t
frx| \|_|_|S t|trZt|}|dkrtd| t|dpd}|d}|rt|}nvd}|d}|rdt| }|| t| }||9 }|d } | rBt| } | d
kr4|d| 9 }n|d| 9 }|d d krb| }ntd
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}||_||_|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)
N z Invalid literal for Fraction: %rZnum0denomdecimal
expr Zsign-z2argument should be a string or a Rational instancez+both arguments should be Rational instanceszFraction(%s, 0))superr __new__r
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) clsr% r& r selfmr r Zscaler g __class__r r r! T sx
$
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zFraction.__new__c C sD t |tjr| |S t |ts8td| j|t|jf | | S )zConverts 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))r" r# Integralr' r/ __name__r
r( )r1 fr r r
from_float s
zFraction.from_floatc C sV ddl m} t|tjr&|t|}n$t||sJtd| j|t|jf | | S )zAConverts a finite Decimal instance to a rational number, exactly.r r z2%s.from_decimal() only takes Decimals, not %r (%s))
r r r" r# r7 r r/ r8 r
r( )r1 Zdecr r r r from_decimal s
zFraction.from_decimalc C s | j | jfS )zReturn the integer ratio as a tuple.
Return a tuple of two integers, whose ratio is equal to the
Fraction and with a positive denominator.
r r2 r r r r( s zFraction.as_integer_ratio@B c
C s |dk rt d| j|kr"t| S d\}}}}| j| j }}|| }||| } | |krZq||||| | f\}}}}|||| }}q<|| | }
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| } t||} t| | t| | kr| S | S dS )aW Closest Fraction to self with denominator at most max_denominator.
>>> Fraction('3.141592653589793').limit_denominator(10)
Fraction(22, 7)
>>> Fraction('3.141592653589793').limit_denominator(100)
Fraction(311, 99)
>>> Fraction(4321, 8765).limit_denominator(10000)
Fraction(4321, 8765)
r z$max_denominator should be at least 1)r r r r N)r, r r r abs)
r2 Zmax_denominatorZp0Zq0Zp1Zq1ndr Zq2kZbound1Zbound2r r r limit_denominator s$
zFraction.limit_denominatorc C s | j S r )r r r r r r% s zFraction.numeratorc C s | j S r )r rC r r r r& s zFraction.denominatorc C s d| j j| j| jf S )z
repr(self)z
%s(%s, %s))r6 r8 r r r< r r r __repr__" s zFraction.__repr__c C s( | j dkrt| jS d| j| j f S dS )z str(self)r z%s/%sN)r r) r r< r r r __str__' s
zFraction.__str__c sT fdd}d j d |_ j|_ fdd}d j d |_ j|_||fS )a 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.
c sP t |ttfr| |S t |tr0 t| |S t |trH t| |S tS d S r )r" r r r' complexNotImplementedr fallback_operatormonomorphic_operatorr r forward~ s
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z Fraction._operator_fallbacksc C s, | j |j }}t| j| |j| || S )za + br& r r% r r dadbr r r _add s z
Fraction._addc C s, | j |j }}t| j| |j| || S )za - brR rS r r r _sub s z
Fraction._subc C s t | j|j | j|j S )za * br r% r& r r r r _mul s z
Fraction._mulc C s t | j|j | j|j S )za / brX r r r r _div s
z
Fraction._divc C s | j |j | j|j S )za // br% r& r r r r _floordiv s zFraction._floordivc C s: | j |j }}t| j| ||j \}}|t||| fS )z(a // b, a % b))r& divmodr% r )r r rT rU ZdivZn_modr r r _divmod s zFraction._divmodc C s, | j |j }}t| j| |j| || S )za % brR rS r r r _mod s z
Fraction._modc C s t |tjr|jdkr|j}|dkr>t| j| | j| ddS | jdkrft| j| | j| ddS t| j | | j | ddS qt| t| S n t| | S dS )za ** 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 r Fr N) r" r# r$ r& r% r r r r' )r r Zpowerr r r __pow__ s&
zFraction.__pow__c C s\ | j dkr | jdkr || j S t|tjr r r rC r r r __abs__ s zFraction.__abs__c C s* | j dk r| j | j S | j | j S dS )ztrunc(a)r Nr rC r r r __trunc__ s
zFraction.__trunc__c C s | j | j S )z
math.floor(a)r[ rC r r r __floor__
s zFraction.__floor__c C s | j | j S )z math.ceil(a)r[ rC r r r __ceil__ s zFraction.__ceil__c C s |dkrZt | j| j\}}|d | jk r,|S |d | jkrB|d S |d dkrR|S |d S dt| }|dkrtt| | |S tt| | | S dS )z?round(self, ndigits)
Rounds half toward even.
Nr r r r )r] r% r& r> r round)r2 ZndigitsZfloorZ remaindershiftr r r __round__ s zFraction.__round__c C sP t | jtd t}|s t}nt| j| t }| dkr:|n| }|dkrLdS |S )z
hash(self)r r )powr _PyHASH_MODULUS _PyHASH_INFr> r )r2 ZdinvZhash_resultr r r __hash__, s zFraction.__hash__c C s t |tkr | j|ko | jdkS t|tjrD| j|jkoB| j|jkS t|tj r`|j
dkr`|j }t|t rt
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|rd|kS | | |kS ntS dS )za == br r N)r
r r r r" r# r$ r% r& rM imagrealr' r isnanisinfr: rG r r r r __eq__B s
zFraction.__eq__c C sh t |tjr&|| j|j | j|j S t |tr`t |sDt
|rN|d|S || | |S nt S dS )ac 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.
rs N)
r" r# r$ r r& r r% r' r rv rw r: rG )r2 otheropr r r _richcmpW s
zFraction._richcmpc C s | |tjS )za < b)r{ operatorltr r r r __lt__m s zFraction.__lt__c C s | |tjS )za > b)r{ r| gtr r r r __gt__q s zFraction.__gt__c C s | |tjS )za <= b)r{ r| ler r r r __le__u s zFraction.__le__c C s | |tjS )za >= b)r{ r| ger r r r __ge__y s zFraction.__ge__c C s
t | jS )za != 0)boolr rC r r r __bool__} s zFraction.__bool__c C s | j t| ffS r )r6 r) r< r r r
__reduce__ s zFraction.__reduce__c C s t | tkr| S | | j| jS r r
r r6 r r r< r r r __copy__ s zFraction.__copy__c C s t | tkr| S | | j| jS r r )r2 Zmemor r r __deepcopy__ s zFraction.__deepcopy__)r N)r= )N)Cr8
__module__ __qualname__rP __slots__r! classmethodr: r; r( rB propertyr% r& rD rE rQ rV r| add__add____radd__rW sub__sub____rsub__rY mul__mul____rmul__rZ truediv __truediv__ __rtruediv__r\ floordiv __floordiv__
__rfloordiv__r^ r]
__divmod__ __rdivmod__r_ mod__mod____rmod__r` ra rc rd re rf rg rh rk rr rx r{ r~ r r r r r r r
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