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275 lines
7.3 KiB
Python
275 lines
7.3 KiB
Python
# Complex numbers
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# ---------------
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# This module represents complex numbers as instances of the class Complex.
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# A Complex instance z has two data attribues, z.re (the real part) and z.im
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# (the imaginary part). In fact, z.re and z.im can have any value -- all
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# arithmetic operators work regardless of the type of z.re and z.im (as long
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# as they support numerical operations).
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#
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# The following functions exist (Complex is actually a class):
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# Complex([re [,im]) -> creates a complex number from a real and an imaginary part
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# IsComplex(z) -> true iff z is a complex number (== has .re and .im attributes)
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# Polar([r [,phi [,fullcircle]]]) ->
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# the complex number z for which r == z.radius() and phi == z.angle(fullcircle)
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# (r and phi default to 0)
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#
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# Complex numbers have the following methods:
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# z.abs() -> absolute value of z
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# z.radius() == z.abs()
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# z.angle([fullcircle]) -> angle from positive X axis; fullcircle gives units
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# z.phi([fullcircle]) == z.angle(fullcircle)
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#
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# These standard functions and unary operators accept complex arguments:
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# abs(z)
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# -z
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# +z
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# not z
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# repr(z) == `z`
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# str(z)
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# hash(z) -> a combination of hash(z.re) and hash(z.im) such that if z.im is zero
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# the result equals hash(z.re)
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# Note that hex(z) and oct(z) are not defined.
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#
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# These conversions accept complex arguments only if their imaginary part is zero:
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# int(z)
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# long(z)
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# float(z)
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#
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# The following operators accept two complex numbers, or one complex number
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# and one real number (int, long or float):
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# z1 + z2
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# z1 - z2
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# z1 * z2
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# z1 / z2
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# pow(z1, z2)
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# cmp(z1, z2)
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# Note that z1 % z2 and divmod(z1, z2) are not defined,
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# nor are shift and mask operations.
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#
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# The standard module math does not support complex numbers.
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# (I suppose it would be easy to implement a cmath module.)
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#
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# Idea:
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# add a class Polar(r, phi) and mixed-mode arithmetic which
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# chooses the most appropriate type for the result:
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# Complex for +,-,cmp
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# Polar for *,/,pow
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import types, math
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if not hasattr(math, 'hypot'):
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def hypot(x, y):
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# XXX I know there's a way to compute this without possibly causing
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# overflow, but I can't remember what it is right now...
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return math.sqrt(x*x + y*y)
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math.hypot = hypot
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twopi = math.pi*2.0
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halfpi = math.pi/2.0
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def IsComplex(obj):
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return hasattr(obj, 're') and hasattr(obj, 'im')
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def Polar(r = 0, phi = 0, fullcircle = twopi):
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phi = phi * (twopi / fullcircle)
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return Complex(math.cos(phi)*r, math.sin(phi)*r)
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class Complex:
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def __init__(self, re=0, im=0):
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if IsComplex(re):
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im = im + re.im
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re = re.re
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if IsComplex(im):
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re = re - im.im
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im = im.re
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self.re = re
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self.im = im
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def __setattr__(self, name, value):
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if hasattr(self, name):
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raise TypeError, "Complex numbers have set-once attributes"
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self.__dict__[name] = value
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def __repr__(self):
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if not self.im:
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return 'Complex(%s)' % `self.re`
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else:
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return 'Complex(%s, %s)' % (`self.re`, `self.im`)
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def __str__(self):
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if not self.im:
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return `self.re`
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else:
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return 'Complex(%s, %s)' % (`self.re`, `self.im`)
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def __coerce__(self, other):
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if IsComplex(other):
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return self, other
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return self, Complex(other) # May fail
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def __cmp__(self, other):
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return cmp(self.re, other.re) or cmp(self.im, other.im)
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def __hash__(self):
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if not self.im: return hash(self.re)
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mod = sys.maxint + 1L
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return int((hash(self.re) + 2L*hash(self.im) + mod) % (2L*mod) - mod)
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def __neg__(self):
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return Complex(-self.re, -self.im)
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def __pos__(self):
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return self
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def __abs__(self):
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return math.hypot(self.re, self.im)
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##return math.sqrt(self.re*self.re + self.im*self.im)
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def __int__(self):
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if self.im:
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raise ValueError, "can't convert Complex with nonzero im to int"
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return int(self.re)
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def __long__(self):
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if self.im:
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raise ValueError, "can't convert Complex with nonzero im to long"
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return long(self.re)
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def __float__(self):
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if self.im:
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raise ValueError, "can't convert Complex with nonzero im to float"
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return float(self.re)
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def __nonzero__(self):
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return not (self.re == self.im == 0)
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abs = radius = __abs__
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def angle(self, fullcircle = twopi):
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return (fullcircle/twopi) * ((halfpi - math.atan2(self.re, self.im)) % twopi)
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phi = angle
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def __add__(self, other):
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return Complex(self.re + other.re, self.im + other.im)
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__radd__ = __add__
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def __sub__(self, other):
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return Complex(self.re - other.re, self.im - other.im)
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def __rsub__(self, other):
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return Complex(other.re - self.re, other.im - self.im)
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def __mul__(self, other):
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return Complex(self.re*other.re - self.im*other.im,
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self.re*other.im + self.im*other.re)
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__rmul__ = __mul__
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def __div__(self, other):
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# Deviating from the general principle of not forcing re or im
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# to be floats, we cast to float here, otherwise division
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# of Complex numbers with integer re and im parts would use
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# the (truncating) integer division
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d = float(other.re*other.re + other.im*other.im)
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if not d: raise ZeroDivisionError, 'Complex division'
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return Complex((self.re*other.re + self.im*other.im) / d,
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(self.im*other.re - self.re*other.im) / d)
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def __rdiv__(self, other):
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return other / self
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def __pow__(self, n, z=None):
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if z is not None:
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raise TypeError, 'Complex does not support ternary pow()'
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if IsComplex(n):
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if n.im: raise TypeError, 'Complex to the Complex power'
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n = n.re
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r = pow(self.abs(), n)
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phi = n*self.angle()
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return Complex(math.cos(phi)*r, math.sin(phi)*r)
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def __rpow__(self, base):
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return pow(base, self)
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# Everything below this point is part of the test suite
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def checkop(expr, a, b, value, fuzz = 1e-6):
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import sys
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print ' ', a, 'and', b,
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try:
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result = eval(expr)
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except:
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result = sys.exc_type
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print '->', result
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if (type(result) == type('') or type(value) == type('')):
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ok = result == value
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else:
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ok = abs(result - value) <= fuzz
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if not ok:
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print '!!\t!!\t!! should be', value, 'diff', abs(result - value)
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def test():
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testsuite = {
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'a+b': [
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(1, 10, 11),
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(1, Complex(0,10), Complex(1,10)),
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(Complex(0,10), 1, Complex(1,10)),
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(Complex(0,10), Complex(1), Complex(1,10)),
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(Complex(1), Complex(0,10), Complex(1,10)),
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],
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'a-b': [
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(1, 10, -9),
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(1, Complex(0,10), Complex(1,-10)),
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(Complex(0,10), 1, Complex(-1,10)),
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(Complex(0,10), Complex(1), Complex(-1,10)),
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(Complex(1), Complex(0,10), Complex(1,-10)),
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],
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'a*b': [
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(1, 10, 10),
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(1, Complex(0,10), Complex(0, 10)),
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(Complex(0,10), 1, Complex(0,10)),
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(Complex(0,10), Complex(1), Complex(0,10)),
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(Complex(1), Complex(0,10), Complex(0,10)),
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],
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'a/b': [
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(1., 10, 0.1),
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(1, Complex(0,10), Complex(0, -0.1)),
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(Complex(0, 10), 1, Complex(0, 10)),
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(Complex(0, 10), Complex(1), Complex(0, 10)),
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(Complex(1), Complex(0,10), Complex(0, -0.1)),
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],
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'pow(a,b)': [
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(1, 10, 1),
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(1, Complex(0,10), 'TypeError'),
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(Complex(0,10), 1, Complex(0,10)),
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(Complex(0,10), Complex(1), Complex(0,10)),
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(Complex(1), Complex(0,10), 'TypeError'),
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(2, Complex(4,0), 16),
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],
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'cmp(a,b)': [
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(1, 10, -1),
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(1, Complex(0,10), 1),
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(Complex(0,10), 1, -1),
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(Complex(0,10), Complex(1), -1),
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(Complex(1), Complex(0,10), 1),
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],
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}
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exprs = testsuite.keys()
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exprs.sort()
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for expr in exprs:
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print expr + ':'
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t = (expr,)
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for item in testsuite[expr]:
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apply(checkop, t+item)
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if __name__ == '__main__':
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test()
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