calculator.py

'''
	Copyright 2015 Teemu Piippo
	All rights reserved.

	Redistribution and use in source and binary forms, with or without
	modification, are permitted provided that the following conditions
	are met:

	1. Redistributions of source code must retain the above copyright
	   notice, this list of conditions and the following disclaimer.
	2. Redistributions in binary form must reproduce the above copyright
	   notice, this list of conditions and the following disclaimer in the
	   documentation and/or other materials provided with the distribution.
	3. The name of the author may not be used to endorse or promote products
	   derived from this software without specific prior written permission.

	THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR
	IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES
	OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED.
	IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT,
	INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
	NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
	DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
	THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
	(INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF
	THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
'''

from __future__ import print_function
import re
import cmath
import math
import random
import time
import operator
import string
from fraction import Fraction
from copy import deepcopy
from math import pi as π

ε = 1e-10

class Operator (object):
	def __init__ (self, name, symbol, operands, priority, function, assign):
		self.name = name
		self.symbol = symbol
		self.operands = operands
		self.priority = priority
		self.function = function
		self.assign = assign

	def __repr__ (self):
		return '<operator %s>' % (self.name)
		"""return ('Operator(name={name}, symbol={symbol}, operands={operands}, '
			'priority={priority}, function={function})'.format(**self.__dict__))"""

class FunctionCall (object):
	def __init__ (self, funcname, args):
		assert (type(args) is list)
		self.funcname = funcname
		self.args = args

	def __repr__ (self):
		return '''FunctionCall(%r, %r)''' % (self.funcname, self.args)

class Value (object):
	def __init__ (self, value):
		if isinstance (value, Fraction) and value.denominator == 1:
			value = value.numerator
		self.value = value

	def __repr__ (self):
		return '''Value(%r)''' % (self.value)

class Name (object):
	def __init__ (self, name):
		self.name = name

	def __repr__ (self):
		return '''Name(%r)''' % (self.name)

class AssignmentResult (object):
	def __init__ (self, name, value):
		self.name, self.value = name, value

	def __repr__ (self):
		return '''AssignmentResult(%r, %r)''' % (self.name, self.value)

def do_realf (func, *args):
	for x in args:
		if x.imag:
			raise ValueError ('%s called with a complex number' % func.__name__)

	return func (*[x.real for x in args])

def do_intf (func, *args):
	for x in args:
		if x.imag:
			raise ValueError ('%s called with a complex number' % func.__name__)

		if x.real - math.floor (x.real):
			raise ValueError ('%s called with a floating point number' % func.__name__)

	return func (*[int (x.real) for x in args])

def realf (func):
	return lambda *args: do_realf (func, *args)

def intf (func):
	return lambda *args: do_intf (func, *args)

def dice (numRolls, maxValue):
	sumValue = 0

	for i in range (0, numRolls):
		sumValue += random.randint (1, maxValue)

	return sumValue

def scientific (mantissa, exp):
	return mantissa * 10 ** exp

def cbrt (x):
	return x ** (1/3)

def factorial (x):
	return math.gamma (x + 1)

def is_int (x):
	return math.fabs (x - math.floor(x)) < ε

def div(a,b):
	if is_int(a) and is_int(b):
		return Fraction(a,b)
	else:
		return a / b

def assignment (name, value, calculator):
	calculator.variables[name.name] = value.value

Operators = {}
OperatorData = {
	'not':		{ 'symbol': '!',  'operands': 1, 'priority': 5, 'function': lambda x: not x },
	'compl':	{ 'symbol': '~',  'operands': 1, 'priority': 5, 'function': intf (operator.inv) },
	'neg':		{ 'symbol': '-',  'operands': 1, 'priority': 5, 'function': lambda x: -x },
	'sqrt':		{ 'symbol': '√',  'operands': 1, 'priority': 7, 'function': cmath.sqrt },
	'cbrt':		{ 'symbol': '∛',  'operands': 1, 'priority': 7, 'function': cbrt },
	'fact':		{ 'symbol': '!',  'operands': -1, 'priority': 8, 'function': realf(factorial) },
	'pow':		{ 'symbol': '**', 'operands': 2, 'priority': 10, 'function': lambda x, y: x ** y },
	'mul':		{ 'symbol': '*',  'operands': 2, 'priority': 50, 'function': lambda x, y: x * y },
	'div':		{ 'symbol': '/',  'operands': 2, 'priority': 50, 'function': div },
	'mod':		{ 'symbol': '%',  'operands': 2, 'priority': 50, 'function': realf (math.fmod) },
	'add':		{ 'symbol': '+',  'operands': 2, 'priority': 100, 'function': lambda x, y: x + y },
	'sub':		{ 'symbol': '-',  'operands': 2, 'priority': 100, 'function': lambda x, y: x - y },
	'eq':		{ 'symbol': '==', 'operands': 2, 'priority': 500, 'function': lambda x, y: x == y },
	'neq':		{ 'symbol': '!=', 'operands': 2, 'priority': 500, 'function': lambda x, y: x != y },
	'lt':		{ 'symbol': '<',  'operands': 2, 'priority': 500, 'function': lambda x, y: x < y },
	'lteq':		{ 'symbol': '<=', 'operands': 2, 'priority': 500, 'function': lambda x, y: x <= y },
	'gt':		{ 'symbol': '>',  'operands': 2, 'priority': 500, 'function': lambda x, y: x > y },
	'gteq':		{ 'symbol': '>=', 'operands': 2, 'priority': 500, 'function': lambda x, y: x >= y },
	'btand':	{ 'symbol': '&',  'operands': 2, 'priority': 600, 'function': intf (operator.and_) },
	'btxor':	{ 'symbol': '^',  'operands': 2, 'priority': 601, 'function': intf (operator.xor) },
	'btor':		{ 'symbol': '|',  'operands': 2, 'priority': 602, 'function': intf (operator.or_) },
	'and':		{ 'symbol': '&&', 'operands': 2, 'priority': 603, 'function': lambda x, y: x and y },
	'or':		{ 'symbol': '||', 'operands': 2, 'priority': 604, 'function': lambda x, y: x or y },
	'assign':	{ 'symbol': '=',  'operands': 2, 'priority': 999, 'function': assignment, 'assign':True },
}

def powerfunction (n):
	return lambda x: x**n

superscripts='⁰¹²³⁴⁵⁶⁷⁸⁹'
for i, sup in enumerate(superscripts):
	func = powerfunction(i)
	func.__name__ = 'x' + sup
	OperatorData['sup' + str(i)] = { 'symbol': sup, 'operands': -1, 'priority': 3, 'function': func }

for name, data in OperatorData.items():
	if 'assign' not in data:
		data['assign'] = False

	Operators[name] = Operator (name=name, **data)

OperatorSymbols={}
for op in Operators.values():
	if op.symbol in OperatorSymbols:
		OperatorSymbols[op.symbol].append (op)
	else:
		OperatorSymbols[op.symbol] = [op]

def sgn (x):
	return cmp (x, 0)

def integerclamp (x, bits, signed):
	x = x % (2 ** bits)

	if signed and x >= (2 ** (bits - 1)):
		x = -(2 ** bits) + x

	return x

Constants = {
	'pi': π,
	'e': math.ℯ,
	'phi': 1.6180339887498948482,
	'epsilon': ε,
}

random.seed (time.time())

Functions = {
	'round':		{ 'function': realf (round) },
	'floor':		{ 'function': realf (math.floor) },
	'ceil':			{ 'function': realf (math.ceil) },
	'factorial':	{ 'function': intf (math.factorial) },
	'abs':			{ 'function': realf (math.fabs) },
	'degrees':		{ 'function': realf (math.degrees) },
	'radians':		{ 'function': realf (math.radians) },
	'erf':			{ 'function': realf (math.erf) },
	'erfc':			{ 'function': realf (math.erfc) },
	'gamma':		{ 'function': realf (math.gamma) },
	'lgamma':		{ 'function': realf (math.lgamma) },
	'sqrt':			{ 'function': cmath.sqrt },
	'cbrt':			{ 'function': cbrt },
	'ln':			{ 'function': cmath.log },
	'log':			{ 'function': cmath.log10 },
	'sin':			{ 'function': cmath.sin },
	'sinh':			{ 'function': cmath.sinh },
	'asin':			{ 'function': cmath.asin },
	'asinh':		{ 'function': cmath.asinh },
	'cos':			{ 'function': cmath.cos },
	'cosh':			{ 'function': cmath.cosh },
	'acos':			{ 'function': cmath.acos },
	'acosh':		{ 'function': cmath.acosh },
	'tan':			{ 'function': cmath.tan },
	'tanh':			{ 'function': cmath.tanh },
	'atan':			{ 'function': cmath.atan },
	'atanh':		{ 'function': cmath.atanh },
	'exp':			{ 'function': cmath.exp },
	'phase':		{ 'function': cmath.phase },
	'lg':			{ 'function': lambda x: cmath.log (x, 2) },
	're':			{ 'function': lambda x: x.real },
	'im':			{ 'function': lambda x: x.imag },
	'conjugate':	{ 'function': lambda x: x.conjugate() },
	'rand':			{ 'function': random.random },
	'sgn':			{ 'function': realf (sgn) },
	'byte':			{ 'function': intf (lambda x: integerclamp (x, bits=8, signed=False)) },
	'sbyte':		{ 'function': intf (lambda x: integerclamp (x, bits=8, signed=True)) },
	'word':			{ 'function': intf (lambda x: integerclamp (x, bits=16, signed=False)) },
	'sword':		{ 'function': intf (lambda x: integerclamp (x, bits=16, signed=True)) },
	'dword':		{ 'function': intf (lambda x: integerclamp (x, bits=32, signed=False)) },
	'sdword':		{ 'function': intf (lambda x: integerclamp (x, bits=32, signed=True)) },
	'qword':		{ 'function': intf (lambda x: integerclamp (x, bits=64, signed=False)) },
	'sqword':		{ 'function': intf (lambda x: integerclamp (x, bits=64, signed=True)) },
	'cdf':			{ 'function': realf (lambda x: (1 + math.erf (x / math.sqrt(2))) / 2) },
}

Tokens = {'(', ')'}

# Symbol table
class SymbolType:
	constant = 0
	function = 1
	operator = 2
	token = 3
SymbolTypes = {}
Symbols = {'ans'}
SymbolTypes['ans'] = SymbolType.constant

Aliases = {
	'π': 'pi',
	'ℯ': 'e',
	'φ': 'phi',
	'ε': 'epsilon',
	'∨': '||',
	'∧': '&&',
	'Φ': 'cdf',
}

for name, value in Constants.items():
	Symbols.add (name)
	SymbolTypes[name] = SymbolType.constant

for name, data in Functions.items():
	Symbols.add (name)
	SymbolTypes[name] = SymbolType.function

for op in Operators.values():
	if op.symbol not in Symbols:
		Symbols.add (op.symbol)
	SymbolTypes[op.symbol] = SymbolType.operator

for name in Tokens:
	SymbolTypes[name] = SymbolType.token

Symbols |= Tokens
Symbols |= set(Aliases.keys())
Symbols = sorted (Symbols, key=lambda x: len (x), reverse=True)
PreferredBase = 10

def set_preferred_base(x):
	global PreferredBase
	PreferredBase = x

def rindex (li, value):
	return (len(li) - 1) - li[::-1].index(value)

Attributes = {
	'hex': lambda self: self.set_preferred_base (0x10),
	'binary': lambda self: self.set_preferred_base (0b10),
	'decimal': lambda self: self.set_preferred_base (10),
}

Attributes['bin'] = Attributes['binary']
Attributes['dec'] = Attributes['decimal']
AttributeNames = sorted ((key for key in Attributes), key=lambda x:len(x), reverse=True)

def is_int (x):
	return math.fabs (x - math.floor(x)) < ε

identifierchars = set(string.ascii_lowercase + string.ascii_uppercase + string.digits + '_')
def is_identifier(x):
	return x and (x[0] not in string.digits) and (set(x) - identifierchars == set())

class Calculator (object):
	def __init__ (self, verbose=False):
		self.previous_result = None
		self.preferred_base = None
		self.verbose = verbose
		self.variables = {}
		self.namepattern = re.compile (r'^([A-Za-z_][A-Za-z0-9_]*)')

	def set_preferred_base (self, x):
		self.preferred_base = x

	def trim_spaces (self, expr):
		return re.sub ('\s+', '', expr.strip())

	def parse_attributes (self, expr):
		if expr[0] != '<':
			return expr

		i = 1
		while expr[i] != '>':
			if expr[i] == '|':
				i += 1

			for name in AttributeNames:
				if expr[i:i + len(name)] == name:
					Attributes[name] (self)
					i += len(name)

					if expr[i] == '>':
						break

					if expr[i] != '|':
						raise ValueError ('malformed attributes')
					break
			else:
				raise ValueError ('bad attributes: %s' % expr[i:])

		return expr[i + 1:]

	def parse_number (self, expr):
		"""Tries to parse a number from the expression. Returns (value, length) on success."""
		i = 0
		value = None
		base = 10

		# Possibly it's hexadecimal
		if expr[0:2].lower() == '0x':
			base = 0x10
			digits = string.hexdigits
			digitname = 'hexit'
			i = 2
		elif expr[0:2].lower() == '0b':
			base = 0b10
			digits = ['0', '1']
			digitname = 'bit'
			i = 2

		if base != 10:
			if not self.preferred_base:
				self.preferred_base = base

			# Skip leading zeroes
			while i < len (expr) and expr[i] == '0':
				i += 1

			startingIndex = i
			while i < len (expr) and expr[i] in digits:
				i += 1

			if i < len(expr) and expr[i] == '.':
				raise ValueError ('non-decimal floating point numbers are not supported')

			if i == startingIndex:
				if i < len (expr):
					raise ValueError ('not a valid %s "%s" in %s' % (digitname, expr[i], expr[0:i+1]))
				else:
					raise ValueError ('end of expression encountered while parsing '
						'base-%d literal' % base)

			return (Value (int (expr[startingIndex:i], base)), i)

		if expr[0] == 'i' or expr[0] == 'ι':
			# Special case, we just have 'i' -- need special handling here because otherwise this would
			# call float('') in the complex number routine, which throws an error.
			return (Value(1j), 1)

		# Try parse increasingly long substrings until we are unable to create a float or complex number
		# from it.
		try:
			while i < len (expr):
				if expr[i] == 'i' or expr[i] == 'ι':
					value = Value (float (expr[:i]) * 1j)
				else:
					value = Value (float (expr [:i + 1]))
				i += 1

			return (value, i)
		except ValueError:
			if i != 0:
				# Got a number (the error happened when we tried to parse past the number)
				return (value, i)
			else:
				# The error happened on the first character. So this is not a number.
				return None

	def parse_symbol (self, expr):
		for sym in Symbols:
			if len(sym) == 1 and ord(sym) > 256:
				if expr[0] == sym:
					return sym
			else:
				if expr[:len (sym)].lower() == sym.lower():
					return sym.lower()

		return None

	def tokenize (self, expr):
		i=0
		tokens = []

		while i < len(expr):
			sym = self.parse_symbol (expr[i:])

			if sym:
				i += len(sym)
				if sym in Aliases:
					sym = Aliases[sym]

				if sym == 'ans':
					if self.previous_result is None:
						raise ValueError ('Cannot use "ans" because there is no prior result')
					tokens.append (Value (self.previous_result))
				else:
					symtype = SymbolTypes[sym]
					if symtype == SymbolType.constant:
						tokens.append (Value(Constants[sym]))
					else:
						tokens.append (sym)
				continue

			result = self.parse_number (expr[i:])
			if result:
				num, length = result
				tokens.append (num)
				i += length
				continue

			match = self.namepattern.match (expr[i:])
			if match:
				name = match.group(1)
				i += len(name)
				tokens.append (Name(name))
				continue

			raise ValueError ("""couldn't parse: %s""" % expr[i:])

		return tokens

	def process_parens (self, expr):
		"""Processes parentheses of expr into sublists in-place.
		[1.0, '*', '(', 3.5, '+', 1j, ')']
		-> [1.0, '*', [3.5, '+', 1j]]"""
		if '(' not in expr and ')' not in expr:
			return

		try:
			parenStart = rindex (expr, '(')
			parenEnd = expr.index (')', parenStart)
		except ValueError:
			raise ValueError ("""mismatched parentheses in expression: %s""" % expr)

		subexpr = expr[parenStart + 1:parenEnd]
		del expr[parenStart + 1:parenEnd + 1]
		expr[parenStart] = subexpr
		self.process_parens (expr)

	def process_functions (self, expr):
		"""Processes functions in-place"""
		i = 0
		while i < len (expr):
			if type (expr[i]) is list:
				self.process_functions (expr[i])

			if (type(expr[i]) is not str) or (expr[i] not in Functions):
				i += 1
				continue

			# Ensure that arguments follow
			if (i + 1 >= len (expr)) or (type (expr[i + 1]) is not list):
				raise ValueError ("""function %s used without arguments""" % expr[i])

			args = expr[i + 1]
			del expr[i + 1]
			expr[i] = FunctionCall (expr[i], args)
			i += 1

	def process_variables (self, expr):
		for i, x in enumerate(expr):
			if isinstance(x, list):
				self.process_variables(x)
			elif isinstance(x, Name):
				try:
					expr[i].value = Value(self.variables[x.name])
				except KeyError as e:
					pass

	def is_operand (self, x):
		# Operands can be either lists (which also mean parens, thus can be single-expressions)
		# or complex numbers
		return type(x) in [list, FunctionCall, Value, Name]

	def find_fitting_operator (self, sym, numOperands):
		# Pass 1: exact numOperands match
		foundByName = False
		for op in Operators.values():
			if op.symbol != sym:
				continue

			foundByName = True
			if op.operands == numOperands:
				return op

		if foundByName:
			raise ValueError ('''Operator %s used incorrectly''' % sym)
		else:
			raise ValueError ('''unknown operator %s!''' % sym)

	def process_operators (self, expr):
		"""Processes operators"""
		i = 0

		# Find all operators in this expression
		while i < len (expr):
			if type (expr[i]) is list:
				self.process_functions (expr[i])
				self.process_operators (expr[i])

			if type (expr[i]) is FunctionCall:
				self.process_functions (expr[i].args)
				self.process_operators (expr[i].args)

			if (type(expr[i]) is not str) or (expr[i] not in OperatorSymbols):
				i += 1
				continue

			# Test for suffix operators
			if i > 0 and self.is_operand (expr[i - 1]):
				try:
					op = self.find_fitting_operator (expr[i], -1)
				except:
					pass
				else:
					expr[i] = op
					continue

			args = []
			argIndices = []
			if i > 0 and self.is_operand (expr[i - 1]):
				args.append (expr[i - 1])
				argIndices.append (i - 1)

			if i + 1 < len(expr) and self.is_operand (expr[i + 1]):
				args.append (expr[i + 1])
				argIndices.append (i + 1)

			# Resolve operators with the same symbol based on operand count
			numOperands = 0
			for arg in args:
				if self.is_operand (arg):
					numOperands += 1

			expr[i] = self.find_fitting_operator (expr[i], numOperands)
			i += 1

	def amend_multiplication (self, expr):
		i = 0
		while i + 1 < len(expr):
			if self.is_operand (expr[i]) and self.is_operand (expr[i + 1]):
				expr.insert(i + 1, Operators['mul'])
				i += 2
			else:
				i += 1

		# Also amend multiplication in sub-expressions
		for x in expr:
			if isinstance(x, list):
				self.amend_multiplication(x)

	def find_priority_operator (self, expr):
		"""Finds the operator with most priority in the expression"""
		bestOp = None
		bestOpIndex = -1

		for i in range (0, len (expr)):
			sym = expr[i]

			if type (sym) is not Operator:
				continue

			if not bestOp or sym.priority < bestOp.priority:
				bestOp = sym
				bestOpIndex = i

		return (bestOp, bestOpIndex)

	def process_values (self, values):
		result = []
		for x in values:
			if isinstance (x, Name) and not hasattr (x, 'value'):
				raise ValueError ('%s is unknown' % (x.name))
			value = x
			while hasattr (value, 'value'):
				value = value.value
			result.append (value)
		return result

	def evaluate (self, expr, verbose=False):
		printFunc = print if verbose else lambda x:None
		printFunc (self.tabs + 'Preprocess: %s' % expr)

		# If there are sub-expressions in here, those need to be evaluated first
		i = 0
		while i < len (expr):
			sym = expr[i]

			if type (sym) is list and sym:
				printFunc (self.tabs + 'Evaluating sub-expression: %s' % (sym))
				self.tabs += '\t'
				expr[i] = self.evaluate (list (sym), verbose)
				self.tabs = self.tabs[:-1]
				printFunc (self.tabs + '-> %s' % expr[i])

			# If there are function calls, evaluate those
			if type (sym) is FunctionCall:
				self.tabs += '\t'
				if sym.args:
					sym.args = [self.evaluate (sym.args, verbose)]
				self.tabs = self.tabs[:-1]

				printFunc (self.tabs + 'Evaluating function call: %s' % (sym))
				expr[i] = Value (Functions[sym.funcname]['function'] (*self.process_values(sym.args)))
				printFunc (self.tabs + '-> %s' % expr[i])

			i += 1

		printFunc (self.tabs + 'Evaluate: %s' % expr)
		runaway = 0
		while True:
			runaway += 1
			if runaway > 1000:
				raise ValueError ('infinite loop detected')

			op, i = self.find_priority_operator (expr)
			if not op:
				break

			if op.operands == 2:
				argIndices = [i - 1, i + 1]
			elif op.operands == -1:
				argIndices = [i - 1]
			else:
				argIndices = [i + 1]

			args = [expr[x] for x in argIndices]
			argIndices = sorted (argIndices, reverse=True)
			printFunc (self.tabs + 'Processing: (%s, %d) with args %s' % (op, i, args))
			if op.assign:
				op.function (*args, calculator=self)
				expr[i] = AssignmentResult(name=args[0], value=Value(self.variables[args[0].name]))
			else:
				expr[i] = Value (op.function (*self.process_values(args)))
			printFunc (self.tabs + '-> %s' % expr[i])

			for i2 in argIndices:
				del expr[i2]

			if op.assign and len(expr) != 1:
				raise ValueError ('expression did not evaluate into an assignment: %s' % expr)

		printFunc (self.tabs + 'Result: %s' % expr[0])

		if len(expr) != 1:
			printFunc (self.tabs + 'Bad expression detected, tokens: %s' % expr)
			raise ValueError ('malformed expression')

		return expr[0]

	def repr_number (self, x):
		"""Returns a string representation for a real number"""
		base = self.preferred_base if self.preferred_base else 10
		if is_int (x) and base != 10:
			digits='0123456789abcdef'
			assert base <= len (digits), '''base %d is too large''' % base

			divisor = base
			rep = ''
			x = int (x)
			runaway = 0

			if not x:
				return '0x0'

			if x < 0:
				return '-' + self.repr_number (abs (x))

			while x > 0:
				runaway += 1
				if runaway > 1000:
					raise ValueError('runaway triggered')

				i = (x % divisor) / (divisor / base)
				x -= i * (divisor / base)
				rep += digits[i]
				divisor *= base

			rep += 'x' if base == 16 else 'b'
			rep += '0'
			return rep[::-1]

		rep = '%.10f' % x

		if '.' in rep:
			while rep[-1] == '0':
				rep = rep[:-1]

			if rep[-1] == '.':
				rep = rep[:-1]

		return rep

	def repr_imaginary (self, x):
		rep = self.repr_number (x)

		if rep == '1':
			return 'i'
		elif rep == '-1':
			return '-i'
		else:
			return rep + 'i'

	def represent (self, x):
		"""Returns a string representation of a float or complex number"""
		if isinstance (x, Value):
			x = x.value
			if isinstance (x, Fraction):
				return '%s / %s' % (x.numerator, x.denominator)
			elif math.fabs (x.imag) > ε:
				if math.fabs (x.real) > ε:
					# Complex number
					return '%s %s %s' % (self.repr_number (x.real),
						'+' if x.imag >= 0 else '-',
						self.repr_imaginary (math.fabs (x.imag)))
				else:
					# Pure imaginary number
					return self.repr_imaginary (x.imag)
			else:
				# Real number
				return self.repr_number (x.real)
		elif isinstance (x, AssignmentResult):
			return '%s = %s' % (x.name.name, self.represent (x.value))
		elif isinstance (x, Name) and hasattr(x, 'value'):
			return self.represent(x.value)
		else:
			return '<%s %s>' % (type(x).__name__, x)

	def calc (self, expr, verbose=None):
		if verbose is None:
			verbose = self.verbose
		self.state = {}
		self.tabs = ''
		expr = self.trim_spaces (expr)
		expr = self.parse_attributes (expr)
		expr = self.tokenize (expr)
		self.process_parens (expr)
		self.process_functions (expr)
		self.process_variables (expr)
		self.amend_multiplication (expr)
		self.process_operators (expr)
		result = self.evaluate (expr, verbose)
		self.previous_result = result
		return self.represent (result)

	def __call__ (self, expr):
		return self.calc (expr)

calc = Calculator()
dcalc = Calculator(verbose=True)