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# Source Generated with Decompyle++ # File: in.pyc (Python 2.6) '''Rational, infinite-precision, real numbers.''' from __future__ import division 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('\n \\A\\s* # optional whitespace at the start, then\n (?P<sign>[-+]?) # an optional sign, then\n (?=\\d|\\.\\d) # lookahead for digit or .digit\n (?P<num>\\d*) # numerator (possibly empty)\n (?: # followed by an optional\n /(?P<denom>\\d+) # / and denominator\n | # or\n \\.(?P<decimal>\\d*) # decimal point and fractional part\n )?\n \\s*\\Z # and optional whitespace to finish\n', re.VERBOSE) class Fraction(Rational): """This class implements rational numbers. Fraction(8, 6) will produce a rational number equivalent to 4/3. Both arguments must be Integral. 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 strings of the form '[-+]?[0-9]+((/|.)[0-9]+)?', optionally surrounded by spaces. """ __slots__ = ('_numerator', '_denominator') def __new__(cls, numerator = 0, denominator = 1): """Constructs a Fraction. Takes a string like '3/2' or '1.5', another Fraction, or a numerator/denominator pair. """ self = super(Fraction, cls).__new__(cls) if type(numerator) not in (int, long) and denominator == 1: if isinstance(numerator, basestring): input = numerator m = _RATIONAL_FORMAT.match(input) if m is None: raise ValueError('Invalid literal for Fraction: %r' % input) m is None numerator = m.group('num') decimal = m.group('decimal') if decimal: numerator = int(numerator + decimal) denominator = 10 ** len(decimal) else: numerator = int(numerator) if not m.group('denom'): pass denominator = int(1) if m.group('sign') == '-': numerator = -numerator elif isinstance(numerator, Rational): other_rational = numerator numerator = other_rational.numerator denominator = other_rational.denominator if denominator == 0: raise ZeroDivisionError('Fraction(%s, 0)' % numerator) denominator == 0 numerator = operator.index(numerator) denominator = operator.index(denominator) g = gcd(numerator, denominator) self._numerator = numerator // g self._denominator = denominator // g return self 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) if not isinstance(f, float): raise TypeError('%s.from_float() only takes floats, not %r (%s)' % (cls.__name__, f, type(f).__name__)) isinstance(f, float) if math.isnan(f) or math.isinf(f): raise TypeError('Cannot convert %r to %s.' % (f, cls.__name__)) math.isinf(f) return cls(*f.as_integer_ratio()) from_float = classmethod(from_float) def from_decimal(cls, dec): '''Converts a finite Decimal instance to a rational number, exactly.''' Decimal = 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(): raise TypeError('Cannot convert %s to %s.' % (dec, cls.__name__)) dec.is_finite() (sign, digits, exp) = dec.as_tuple() digits = int(''.join(map(str, digits))) if sign: digits = -digits if exp >= 0: return cls(digits * 10 ** exp) return cls(digits, 10 ** (-exp)) from_decimal = classmethod(from_decimal) 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(1234, 5678).limit_denominator(10000) Fraction(1234, 5678) """ if max_denominator < 1: raise ValueError('max_denominator should be at least 1') max_denominator < 1 if self._denominator <= max_denominator: return Fraction(self) (p0, q0, p1, q1) = (0, 1, 1, 0) n = self._numerator d = 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 return bound1 def numerator(a): return a._numerator numerator = property(numerator) def denominator(a): return a._denominator denominator = property(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) 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) if isinstance(b, float): return fallback_operator(float(a), b) if isinstance(b, complex): return fallback_operator(complex(a), b) return NotImplemented forward.__name__ = '__' + fallback_operator.__name__ + '__' forward.__doc__ = monomorphic_operator.__doc__ def reverse(b, a): if isinstance(a, Rational): return monomorphic_operator(a, b) if isinstance(a, numbers.Real): return fallback_operator(float(a), float(b)) if isinstance(a, numbers.Complex): return fallback_operator(complex(a), complex(b)) 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''' div = a / b if isinstance(div, Rational): return div.numerator // div.denominator return math.floor(div) def __rfloordiv__(b, a): '''a // b''' div = a / b if isinstance(div, Rational): return div.numerator // div.denominator 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) return Fraction(a._denominator ** (-power), a._numerator ** (-power)) b.denominator == 1 return float(a) ** float(b) isinstance(b, Rational) return float(a) ** b def __rpow__(b, a): '''a ** b''' if b._denominator == 1 and b._numerator >= 0: 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) 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. ''' if self._denominator == 1: return hash(self._numerator) if self == float(self): return hash(float(self)) return hash((self._numerator, self._denominator)) def __eq__(a, b): '''a == b''' if isinstance(b, Rational): if a._numerator == b.numerator: pass return a._denominator == b.denominator if isinstance(b, numbers.Complex) and b.imag == 0: b = b.real if isinstance(b, float): return a == a.from_float(b) return float(a) == b def _subtractAndCompareToZero(a, b, op): """Helper function for comparison operators. Subtracts b from a, exactly if possible, and compares the result with 0 using op, in such a way that the comparison won't recurse. If the difference raises a TypeError, returns NotImplemented instead. """ if isinstance(b, numbers.Complex) and b.imag == 0: b = b.real if isinstance(b, float): b = a.from_float(b) try: diff = a - b except TypeError: return NotImplemented if isinstance(diff, Rational): return op(diff.numerator, 0) return op(diff, 0) def __lt__(a, b): '''a < b''' return a._subtractAndCompareToZero(b, operator.lt) def __gt__(a, b): '''a > b''' return a._subtractAndCompareToZero(b, operator.gt) def __le__(a, b): '''a <= b''' return a._subtractAndCompareToZero(b, operator.le) def __ge__(a, b): '''a >= b''' return a._subtractAndCompareToZero(b, operator.ge) def __nonzero__(a): '''a != 0''' return a._numerator != 0 def __reduce__(self): return (self.__class__, (str(self),)) def __copy__(self): if type(self) == Fraction: return self return self.__class__(self._numerator, self._denominator) def __deepcopy__(self, memo): if type(self) == Fraction: return self return self.__class__(self._numerator, self._denominator) 
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