| # Originally contributed by Sjoerd Mullender. |
| # Significantly modified by Jeffrey Yasskin <jyasskin at gmail.com>. |
| |
| """Rational, infinite-precision, real numbers.""" |
| |
| from __future__ import division |
| from decimal import Decimal |
| import math |
| import numbers |
| import operator |
| import re |
| |
| __all__ = ['Fraction', 'gcd'] |
| |
| Rational = numbers.Rational |
| |
| |
| def gcd(a, b): |
| """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). |
| """ |
| while b: |
| a, b = b, a%b |
| return a |
| |
| |
| _RATIONAL_FORMAT = re.compile(r""" |
| \A\s* # optional whitespace at the start, then |
| (?P<sign>[-+]?) # an optional sign, then |
| (?=\d|\.\d) # lookahead for digit or .digit |
| (?P<num>\d*) # numerator (possibly empty) |
| (?: # followed by |
| (?:/(?P<denom>\d+))? # an optional denominator |
| | # or |
| (?:\.(?P<decimal>\d*))? # an optional fractional part |
| (?:E(?P<exp>[-+]?\d+))? # and optional exponent |
| ) |
| \s*\Z # and optional whitespace to finish |
| """, re.VERBOSE | re.IGNORECASE) |
| |
| |
| class Fraction(Rational): |
| """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) |
| |
| """ |
| |
| __slots__ = ('_numerator', '_denominator') |
| |
| # We're immutable, so use __new__ not __init__ |
| def __new__(cls, numerator=0, denominator=None): |
| """Constructs a Fraction. |
| |
| 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) |
| |
| """ |
| self = super(Fraction, cls).__new__(cls) |
| |
| if denominator is None: |
| if isinstance(numerator, Rational): |
| self._numerator = numerator.numerator |
| self._denominator = numerator.denominator |
| return self |
| |
| elif isinstance(numerator, float): |
| # Exact conversion from float |
| value = Fraction.from_float(numerator) |
| self._numerator = value._numerator |
| self._denominator = value._denominator |
| return self |
| |
| elif isinstance(numerator, Decimal): |
| value = Fraction.from_decimal(numerator) |
| self._numerator = value._numerator |
| self._denominator = value._denominator |
| return self |
| |
| elif isinstance(numerator, basestring): |
| # Handle construction from strings. |
| m = _RATIONAL_FORMAT.match(numerator) |
| if m is None: |
| raise ValueError('Invalid literal for Fraction: %r' % |
| numerator) |
| numerator = int(m.group('num') or '0') |
| denom = m.group('denom') |
| if denom: |
| denominator = int(denom) |
| else: |
| denominator = 1 |
| decimal = m.group('decimal') |
| if decimal: |
| scale = 10**len(decimal) |
| numerator = numerator * scale + int(decimal) |
| denominator *= scale |
| exp = m.group('exp') |
| if exp: |
| exp = int(exp) |
| if exp >= 0: |
| numerator *= 10**exp |
| else: |
| denominator *= 10**-exp |
| if m.group('sign') == '-': |
| numerator = -numerator |
| |
| else: |
| raise TypeError("argument should be a string " |
| "or a Rational instance") |
| |
| elif (isinstance(numerator, Rational) and |
| isinstance(denominator, Rational)): |
| numerator, denominator = ( |
| numerator.numerator * denominator.denominator, |
| denominator.numerator * numerator.denominator |
| ) |
| else: |
| raise TypeError("both arguments should be " |
| "Rational instances") |
| |
| if denominator == 0: |
| raise ZeroDivisionError('Fraction(%s, 0)' % numerator) |
| g = gcd(numerator, denominator) |
| self._numerator = numerator // g |
| self._denominator = denominator // g |
| return self |
| |
| @classmethod |
| def from_float(cls, f): |
| """Converts a finite float to a rational number, exactly. |
| |
| Beware that Fraction.from_float(0.3) != Fraction(3, 10). |
| |
| """ |
| if isinstance(f, numbers.Integral): |
| return cls(f) |
| elif not isinstance(f, float): |
| raise TypeError("%s.from_float() only takes floats, not %r (%s)" % |
| (cls.__name__, f, type(f).__name__)) |
| if math.isnan(f) or math.isinf(f): |
| raise TypeError("Cannot convert %r to %s." % (f, cls.__name__)) |
| return cls(*f.as_integer_ratio()) |
| |
| @classmethod |
| def from_decimal(cls, dec): |
| """Converts a finite Decimal instance to a rational number, exactly.""" |
| from decimal import Decimal |
| if isinstance(dec, numbers.Integral): |
| dec = Decimal(int(dec)) |
| elif not isinstance(dec, Decimal): |
| raise TypeError( |
| "%s.from_decimal() only takes Decimals, not %r (%s)" % |
| (cls.__name__, dec, type(dec).__name__)) |
| if not dec.is_finite(): |
| # Catches infinities and nans. |
| raise TypeError("Cannot convert %s to %s." % (dec, cls.__name__)) |
| sign, digits, exp = dec.as_tuple() |
| digits = int(''.join(map(str, digits))) |
| if sign: |
| digits = -digits |
| if exp >= 0: |
| return cls(digits * 10 ** exp) |
| else: |
| return cls(digits, 10 ** -exp) |
| |
| def limit_denominator(self, max_denominator=1000000): |
| """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) |
| |
| """ |
| # Algorithm notes: For any real number x, define a *best upper |
| # approximation* to x to be a rational number p/q such that: |
| # |
| # (1) p/q >= x, and |
| # (2) if p/q > r/s >= x then s > q, for any rational r/s. |
| # |
| # Define *best lower approximation* similarly. Then it can be |
| # proved that a rational number is a best upper or lower |
| # approximation to x if, and only if, it is a convergent or |
| # semiconvergent of the (unique shortest) continued fraction |
| # associated to x. |
| # |
| # To find a best rational approximation with denominator <= M, |
| # we find the best upper and lower approximations with |
| # denominator <= M and take whichever of these is closer to x. |
| # In the event of a tie, the bound with smaller denominator is |
| # chosen. If both denominators are equal (which can happen |
| # only when max_denominator == 1 and self is midway between |
| # two integers) the lower bound---i.e., the floor of self, is |
| # taken. |
| |
| if max_denominator < 1: |
| raise ValueError("max_denominator should be at least 1") |
| if self._denominator <= max_denominator: |
| return Fraction(self) |
| |
| p0, q0, p1, q1 = 0, 1, 1, 0 |
| n, d = self._numerator, self._denominator |
| while True: |
| a = n//d |
| q2 = q0+a*q1 |
| if q2 > max_denominator: |
| break |
| p0, q0, p1, q1 = p1, q1, p0+a*p1, q2 |
| n, d = d, n-a*d |
| |
| k = (max_denominator-q0)//q1 |
| bound1 = Fraction(p0+k*p1, q0+k*q1) |
| bound2 = Fraction(p1, q1) |
| if abs(bound2 - self) <= abs(bound1-self): |
| return bound2 |
| else: |
| return bound1 |
| |
| @property |
| def numerator(a): |
| return a._numerator |
| |
| @property |
| def denominator(a): |
| return a._denominator |
| |
| def __repr__(self): |
| """repr(self)""" |
| return ('Fraction(%s, %s)' % (self._numerator, self._denominator)) |
| |
| def __str__(self): |
| """str(self)""" |
| if self._denominator == 1: |
| return str(self._numerator) |
| else: |
| return '%s/%s' % (self._numerator, self._denominator) |
| |
| def _operator_fallbacks(monomorphic_operator, fallback_operator): |
| """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, long, 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, 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. |
| |
| """ |
| def forward(a, b): |
| if isinstance(b, (int, long, Fraction)): |
| return monomorphic_operator(a, b) |
| elif isinstance(b, float): |
| return fallback_operator(float(a), b) |
| elif isinstance(b, complex): |
| return fallback_operator(complex(a), b) |
| else: |
| return NotImplemented |
| forward.__name__ = '__' + fallback_operator.__name__ + '__' |
| forward.__doc__ = monomorphic_operator.__doc__ |
| |
| def reverse(b, a): |
| if isinstance(a, Rational): |
| # Includes ints. |
| return monomorphic_operator(a, b) |
| elif isinstance(a, numbers.Real): |
| return fallback_operator(float(a), float(b)) |
| elif isinstance(a, numbers.Complex): |
| return fallback_operator(complex(a), complex(b)) |
| else: |
| return NotImplemented |
| reverse.__name__ = '__r' + fallback_operator.__name__ + '__' |
| reverse.__doc__ = monomorphic_operator.__doc__ |
| |
| return forward, reverse |
| |
| def _add(a, b): |
| """a + b""" |
| return Fraction(a.numerator * b.denominator + |
| b.numerator * a.denominator, |
| a.denominator * b.denominator) |
| |
| __add__, __radd__ = _operator_fallbacks(_add, operator.add) |
| |
| def _sub(a, b): |
| """a - b""" |
| return Fraction(a.numerator * b.denominator - |
| b.numerator * a.denominator, |
| a.denominator * b.denominator) |
| |
| __sub__, __rsub__ = _operator_fallbacks(_sub, operator.sub) |
| |
| def _mul(a, b): |
| """a * b""" |
| return Fraction(a.numerator * b.numerator, a.denominator * b.denominator) |
| |
| __mul__, __rmul__ = _operator_fallbacks(_mul, operator.mul) |
| |
| def _div(a, b): |
| """a / b""" |
| return Fraction(a.numerator * b.denominator, |
| a.denominator * b.numerator) |
| |
| __truediv__, __rtruediv__ = _operator_fallbacks(_div, operator.truediv) |
| __div__, __rdiv__ = _operator_fallbacks(_div, operator.div) |
| |
| def __floordiv__(a, b): |
| """a // b""" |
| # Will be math.floor(a / b) in 3.0. |
| div = a / b |
| if isinstance(div, Rational): |
| # trunc(math.floor(div)) doesn't work if the rational is |
| # more precise than a float because the intermediate |
| # rounding may cross an integer boundary. |
| return div.numerator // div.denominator |
| else: |
| return math.floor(div) |
| |
| def __rfloordiv__(b, a): |
| """a // b""" |
| # Will be math.floor(a / b) in 3.0. |
| div = a / b |
| if isinstance(div, Rational): |
| # trunc(math.floor(div)) doesn't work if the rational is |
| # more precise than a float because the intermediate |
| # rounding may cross an integer boundary. |
| return div.numerator // div.denominator |
| else: |
| return math.floor(div) |
| |
| def __mod__(a, b): |
| """a % b""" |
| div = a // b |
| return a - b * div |
| |
| def __rmod__(b, a): |
| """a % b""" |
| div = a // b |
| return a - b * div |
| |
| def __pow__(a, b): |
| """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. |
| |
| """ |
| if isinstance(b, Rational): |
| if b.denominator == 1: |
| power = b.numerator |
| if power >= 0: |
| return Fraction(a._numerator ** power, |
| a._denominator ** power) |
| else: |
| return Fraction(a._denominator ** -power, |
| a._numerator ** -power) |
| else: |
| # A fractional power will generally produce an |
| # irrational number. |
| return float(a) ** float(b) |
| else: |
| return float(a) ** b |
| |
| def __rpow__(b, a): |
| """a ** b""" |
| if b._denominator == 1 and b._numerator >= 0: |
| # If a is an int, keep it that way if possible. |
| return a ** b._numerator |
| |
| if isinstance(a, Rational): |
| return Fraction(a.numerator, a.denominator) ** b |
| |
| if b._denominator == 1: |
| return a ** b._numerator |
| |
| return a ** float(b) |
| |
| def __pos__(a): |
| """+a: Coerces a subclass instance to Fraction""" |
| return Fraction(a._numerator, a._denominator) |
| |
| def __neg__(a): |
| """-a""" |
| return Fraction(-a._numerator, a._denominator) |
| |
| def __abs__(a): |
| """abs(a)""" |
| return Fraction(abs(a._numerator), a._denominator) |
| |
| def __trunc__(a): |
| """trunc(a)""" |
| if a._numerator < 0: |
| return -(-a._numerator // a._denominator) |
| else: |
| return a._numerator // a._denominator |
| |
| def __hash__(self): |
| """hash(self) |
| |
| Tricky because values that are exactly representable as a |
| float must have the same hash as that float. |
| |
| """ |
| # XXX since this method is expensive, consider caching the result |
| if self._denominator == 1: |
| # Get integers right. |
| return hash(self._numerator) |
| # Expensive check, but definitely correct. |
| if self == float(self): |
| return hash(float(self)) |
| else: |
| # Use tuple's hash to avoid a high collision rate on |
| # simple fractions. |
| return hash((self._numerator, self._denominator)) |
| |
| def __eq__(a, b): |
| """a == b""" |
| if isinstance(b, Rational): |
| return (a._numerator == b.numerator and |
| a._denominator == b.denominator) |
| if isinstance(b, numbers.Complex) and b.imag == 0: |
| b = b.real |
| if isinstance(b, float): |
| if math.isnan(b) or math.isinf(b): |
| # comparisons with an infinity or nan should behave in |
| # the same way for any finite a, so treat a as zero. |
| return 0.0 == b |
| else: |
| return a == a.from_float(b) |
| else: |
| # Since a doesn't know how to compare with b, let's give b |
| # a chance to compare itself with a. |
| return NotImplemented |
| |
| def _richcmp(self, other, op): |
| """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. |
| |
| """ |
| # convert other to a Rational instance where reasonable. |
| if isinstance(other, Rational): |
| return op(self._numerator * other.denominator, |
| self._denominator * other.numerator) |
| # comparisons with complex should raise a TypeError, for consistency |
| # with int<->complex, float<->complex, and complex<->complex comparisons. |
| if isinstance(other, complex): |
| raise TypeError("no ordering relation is defined for complex numbers") |
| if isinstance(other, float): |
| if math.isnan(other) or math.isinf(other): |
| return op(0.0, other) |
| else: |
| return op(self, self.from_float(other)) |
| else: |
| return NotImplemented |
| |
| def __lt__(a, b): |
| """a < b""" |
| return a._richcmp(b, operator.lt) |
| |
| def __gt__(a, b): |
| """a > b""" |
| return a._richcmp(b, operator.gt) |
| |
| def __le__(a, b): |
| """a <= b""" |
| return a._richcmp(b, operator.le) |
| |
| def __ge__(a, b): |
| """a >= b""" |
| return a._richcmp(b, operator.ge) |
| |
| def __nonzero__(a): |
| """a != 0""" |
| return a._numerator != 0 |
| |
| # support for pickling, copy, and deepcopy |
| |
| def __reduce__(self): |
| return (self.__class__, (str(self),)) |
| |
| def __copy__(self): |
| if type(self) == Fraction: |
| return self # I'm immutable; therefore I am my own clone |
| return self.__class__(self._numerator, self._denominator) |
| |
| def __deepcopy__(self, memo): |
| if type(self) == Fraction: |
| return self # My components are also immutable |
| return self.__class__(self._numerator, self._denominator) |
