def _rebuild(expr):
        if not isinstance(expr, Basic):
            return expr

        if not expr.args:
            return expr

        if iterable(expr):
            new_args = [_rebuild(arg) for arg in expr]
            return expr.func(*new_args)

        if expr in subs:
            return subs[expr]

        orig_expr = expr
        if expr in opt_subs:
            expr = opt_subs[expr]

        # If enabled, parse Muls and Adds arguments by order to ensure
        # replacement order independent from hashes
        if order != 'none':
            if isinstance(expr, (Mul, MatMul)):
                c, nc = expr.args_cnc()
                if c == [1]:
                    args = nc
                else:
                    args = list(ordered(c)) + nc
            elif isinstance(expr, (Add, MatAdd)):
                args = list(ordered(expr.args))
            else:
                args = expr.args
        else:
            args = expr.args

        new_args = list(map(_rebuild, args))
        if new_args != args:
            new_expr = expr.func(*new_args)
        else:
            new_expr = expr

        if orig_expr in to_eliminate:
            try:
                sym = next(symbols)
            except StopIteration:
                raise ValueError("Symbols iterator ran out of symbols.")

            if isinstance(orig_expr, MatrixExpr):
                sym = MatrixSymbol(sym.name, orig_expr.rows,
                    orig_expr.cols)

            subs[orig_expr] = sym
            replacements.append((sym, new_expr))
            return sym

        else:
            return new_expr

    def _match_common_args(Func, funcs):
        if order != 'none':
            funcs = list(ordered(funcs))
        else:
            funcs = sorted(funcs, key=lambda x: len(x.args))

        func_args = [set(e.args) for e in funcs]
        for i in xrange(len(func_args)):
            for j in xrange(i + 1, len(func_args)):
                com_args = func_args[i].intersection(func_args[j])
                if len(com_args) > 1:
                    com_func = Func(*com_args)

                    # for all sets, replace the common symbols by the function
                    # over them, to allow recursive matches

                    diff_i = func_args[i].difference(com_args)
                    func_args[i] = diff_i | set([com_func])
                    if diff_i:
                        opt_subs[funcs[i]] = Func(Func(*diff_i), com_func,
                                                  evaluate=False)

                    diff_j = func_args[j].difference(com_args)
                    func_args[j] = diff_j | set([com_func])
                    opt_subs[funcs[j]] = Func(Func(*diff_j), com_func,
                                              evaluate=False)

                    for k in xrange(j + 1, len(func_args)):
                        if not com_args.difference(func_args[k]):
                            diff_k = func_args[k].difference(com_args)
                            func_args[k] = diff_k | set([com_func])
                            opt_subs[funcs[k]] = Func(Func(*diff_k), com_func,
                                                      evaluate=False)

 def _eval_simplify(self, ratio, measure, rational, inverse):
     args = [a._eval_simplify(ratio, measure, rational, inverse)
         for a in self.args]
     for i, (expr, cond) in enumerate(args):
         # try to simplify conditions and the expression for
         # equalities that are part of the condition, e.g.
         # Piecewise((n, And(Eq(n,0), Eq(n + m, 0))), (1, True))
         # -> Piecewise((0, And(Eq(n, 0), Eq(m, 0))), (1, True))
         if isinstance(cond, And):
             eqs, other = sift(cond.args,
                 lambda i: isinstance(i, Equality), binary=True)
         elif isinstance(cond, Equality):
             eqs, other = [cond], []
         else:
             eqs = other = []
         if eqs:
             eqs = list(ordered(eqs))
             for j, e in enumerate(eqs):
                 # these blessed lhs objects behave like Symbols
                 # and the rhs are simple replacements for the "symbols"
                 if isinstance(e.lhs, (Symbol, UndefinedFunction)) and \
                     isinstance(e.rhs,
                         (Rational, NumberSymbol,
                         Symbol, UndefinedFunction)):
                     expr = expr.subs(*e.args)
                     eqs[j + 1:] = [ei.subs(*e.args) for ei in eqs[j + 1:]]
                     other = [ei.subs(*e.args) for ei in other]
             cond = And(*(eqs + other))
             args[i] = args[i].func(expr, cond)
     return self.func(*args)

    def _rebuild(expr):

        if expr.is_Atom:
            return expr

        if iterable(expr):
            new_args = [_rebuild(arg) for arg in expr]
            return expr.func(*new_args)

        if expr in subs:
            return subs[expr]

        orig_expr = expr
        if expr in opt_subs:
            expr = opt_subs[expr]

        # If enabled, parse Muls and Adds arguments by order to ensure
        # replacement order independent from hashes
        if order != 'none':
            if expr.is_Mul:
                c, nc = expr.args_cnc()
                args = list(ordered(c)) + nc
            elif expr.is_Add:
                args = list(ordered(expr.args))
            else:
                args = expr.args
        else:
            args = expr.args

        new_args = list(map(_rebuild, args))
        if new_args != args:
            new_expr = expr.func(*new_args)
        else:
            new_expr = expr

        if orig_expr in to_eliminate:
            try:
                sym = next(symbols)
            except StopIteration:
                raise ValueError("Symbols iterator ran out of symbols.")
            subs[orig_expr] = sym
            replacements.append((sym, new_expr))
            return sym

        else:
            return new_expr

def test_ordered():
    assert list(ordered((x, y), hash, default=False)) in [[x, y], [y, x]]
    assert list(ordered((x, y), hash, default=False)) == list(ordered((y, x), hash, default=False))
    assert list(ordered((x, y))) == [x, y]

    seq, keys = [[[1, 2, 1], [0, 3, 1], [1, 1, 3], [2], [1]], (lambda x: len(x), lambda x: sum(x))]
    assert list(ordered(seq, keys, default=False, warn=False)) == [[1], [2], [1, 2, 1], [0, 3, 1], [1, 1, 3]]
    raises(ValueError, lambda: list(ordered(seq, keys, default=False, warn=True)))

 def __repr__(self):  # Factors
     return "Factors({%s})" % ', '.join(
         ['%s: %s' % (k, v) for k, v in ordered(self.factors.items())])

 def __hash__(self):  # Factors
     keys = tuple(ordered(self.factors.keys()))
     values = [self.factors[k] for k in keys]
     return hash((keys, values))

 def _handle_irel(self, x, handler):
     """Return either None (if the conditions of self depend only on x) else
     a Piecewise expression whose expressions (handled by the handler that
     was passed) are paired with the governing x-independent relationals,
     e.g. Piecewise((A, a(x) & b(y)), (B, c(x) | c(y)) ->
     Piecewise(
         (handler(Piecewise((A, a(x) & True), (B, c(x) | True)), b(y) & c(y)),
         (handler(Piecewise((A, a(x) & True), (B, c(x) | False)), b(y)),
         (handler(Piecewise((A, a(x) & False), (B, c(x) | True)), c(y)),
         (handler(Piecewise((A, a(x) & False), (B, c(x) | False)), True))
     """
     # identify governing relationals
     rel = self.atoms(Relational)
     irel = list(ordered([r for r in rel if x not in r.free_symbols
         and r not in (S.true, S.false)]))
     if irel:
         args = {}
         exprinorder = []
         for truth in product((1, 0), repeat=len(irel)):
             reps = dict(zip(irel, truth))
             # only store the true conditions since the false are implied
             # when they appear lower in the Piecewise args
             if 1 not in truth:
                 cond = None  # flag this one so it doesn't get combined
             else:
                 andargs = Tuple(*[i for i in reps if reps[i]])
                 free = list(andargs.free_symbols)
                 if len(free) == 1:
                     from sympy.solvers.inequalities import (
                         reduce_inequalities, _solve_inequality)
                     try:
                         t = reduce_inequalities(andargs, free[0])
                         # ValueError when there are potentially
                         # nonvanishing imaginary parts
                     except (ValueError, NotImplementedError):
                         # at least isolate free symbol on left
                         t = And(*[_solve_inequality(
                             a, free[0], linear=True)
                             for a in andargs])
                 else:
                     t = And(*andargs)
                 if t is S.false:
                     continue  # an impossible combination
                 cond = t
             expr = handler(self.xreplace(reps))
             if isinstance(expr, self.func) and len(expr.args) == 1:
                 expr, econd = expr.args[0]
                 cond = And(econd, True if cond is None else cond)
             # the ec pairs are being collected since all possibilities
             # are being enumerated, but don't put the last one in since
             # its expr might match a previous expression and it
             # must appear last in the args
             if cond is not None:
                 args.setdefault(expr, []).append(cond)
                 # but since we only store the true conditions we must maintain
                 # the order so that the expression with the most true values
                 # comes first
                 exprinorder.append(expr)
         # convert collected conditions as args of Or
         for k in args:
             args[k] = Or(*args[k])
         # take them in the order obtained
         args = [(e, args[e]) for e in uniq(exprinorder)]
         # add in the last arg
         args.append((expr, True))
         # if any condition reduced to True, it needs to go last
         # and there should only be one of them or else the exprs
         # should agree
         trues = [i for i in range(len(args)) if args[i][1] is S.true]
         if not trues:
             # make the last one True since all cases were enumerated
             e, c = args[-1]
             args[-1] = (e, S.true)
         else:
             assert len(set([e for e, c in [args[i] for i in trues]])) == 1
             args.append(args.pop(trues.pop()))
             while trues:
                 args.pop(trues.pop())
         return Piecewise(*args)

 def __repr__(self):  # Factors
     return "Factors({%s})" % ", ".join(["%s: %s" % (k, v) for k, v in ordered(self.factors.items())])

    def _match_common_args(Func, funcs):
        if order != 'none':
            funcs = list(ordered(funcs))
        else:
            funcs = sorted(funcs, key=lambda x: len(x.args))

        if Func is Mul:
            F = Pow
            meth = 'as_powers_dict'
            from sympy.core.add import _addsort as inplace_sorter
        elif Func is Add:
            F = Mul
            meth = 'as_coefficients_dict'
            from sympy.core.mul import _mulsort as inplace_sorter
        else:
            assert None  # expected Mul or Add

        # ----------------- helpers ---------------------------
        def ufunc(*args):
            # return a well formed unevaluated function from the args
            # SHARES Func, inplace_sorter
            args = list(args)
            inplace_sorter(args)
            return Func(*args, evaluate=False)

        def as_dict(e):
            # creates a dictionary of the expression using either
            # as_coefficients_dict or as_powers_dict, depending on Func
            # SHARES meth
            d = getattr(e, meth, lambda: {a: S.One for a in e.args})()
            for k in list(d.keys()):
                try:
                    as_int(d[k])
                except ValueError:
                    d[F(k, d.pop(k))] = S.One
            return d

        def from_dict(d):
            # build expression from dict from
            # as_coefficients_dict or as_powers_dict
            # SHARES F
            return ufunc(*[F(k, v) for k, v in d.items()])

        def update(k):
            # updates all of the info associated with k using
            # the com_dict: func_dicts, func_args, opt_subs
            # returns True if all values were updated, else None
            # SHARES com_dict, com_func, func_dicts, func_args,
            #        opt_subs, funcs, verbose
            for di in com_dict:
                # don't allow a sign to change
                if com_dict[di] > func_dicts[k][di]:
                    return
            # remove it
            if Func is Add:
                take = min(func_dicts[k][i] for i in com_dict)
                com_func_take = Mul(take, from_dict(com_dict), evaluate=False)
            else:
                take = igcd(*[func_dicts[k][i] for i in com_dict])
                com_func_take = Pow(from_dict(com_dict), take, evaluate=False)
            for di in com_dict:
                func_dicts[k][di] -= take*com_dict[di]
            # compute the remaining expression
            rem = from_dict(func_dicts[k])
            # reject hollow change, e.g extracting x + 1 from x + 3
            if Func is Add and rem and rem.is_Integer and 1 in com_dict:
                return
            if verbose:
                print('\nfunc %s (%s) \ncontains %s \nas %s \nleaving %s' %
                    (funcs[k], func_dicts[k], com_func, com_func_take, rem))
            # recompute the dict since some keys may now
            # have corresponding values of 0; one could
            # keep track of which ones went to zero but
            # this seems cleaner
            func_dicts[k] = as_dict(rem)
            # update associated info
            func_dicts[k][com_func] = take
            func_args[k] = set(func_dicts[k])
            # keep the constant separate from the remaining
            # part of the expression, e.g. 2*(a*b) rather than 2*a*b
            opt_subs[funcs[k]] = ufunc(rem, com_func_take)
            # everything was updated
            return True

        def get_copy(i):
            return [func_dicts[i].copy(), func_args[i].copy(), funcs[i], i]

        def restore(dafi):
            i = dafi.pop()
            func_dicts[i], func_args[i], funcs[i] = dafi

        # ----------------- end helpers -----------------------

        func_dicts = [as_dict(f) for f in funcs]
        func_args = [set(d) for d in func_dicts]
        while True:
            hit = pairwise_most_common(func_args)
            if not hit or len(hit[0][0]) <= 1:
                break
            changed = False
#.........这里部分代码省略.........

def cse(exprs, symbols=None, optimizations=None, postprocess=None):
    """ Perform common subexpression elimination on an expression.

    Parameters
    ==========

    exprs : list of sympy expressions, or a single sympy expression
        The expressions to reduce.
    symbols : infinite iterator yielding unique Symbols
        The symbols used to label the common subexpressions which are pulled
        out. The ``numbered_symbols`` generator is useful. The default is a
        stream of symbols of the form "x0", "x1", etc. This must be an infinite
        iterator.
    optimizations : list of (callable, callable) pairs, optional
        The (preprocessor, postprocessor) pairs. If not provided,
        ``sympy.simplify.cse.cse_optimizations`` is used.
    postprocess : a function which accepts the two return values of cse and
        returns the desired form of output from cse, e.g. if you want the
        replacements reversed the function might be the following lambda:
        lambda r, e: return reversed(r), e

    Returns
    =======

    replacements : list of (Symbol, expression) pairs
        All of the common subexpressions that were replaced. Subexpressions
        earlier in this list might show up in subexpressions later in this list.
    reduced_exprs : list of sympy expressions
        The reduced expressions with all of the replacements above.
    """
    from sympy.matrices import Matrix

    if symbols is None:
        symbols = numbered_symbols()
    else:
        # In case we get passed an iterable with an __iter__ method instead of
        # an actual iterator.
        symbols = iter(symbols)
    tmp_symbols = numbered_symbols('_csetmp')
    subexp_iv = dict()
    muls = set()
    adds = set()

    if optimizations is None:
        # Pull out the default here just in case there are some weird
        # manipulations of the module-level list in some other thread.
        optimizations = list(cse_optimizations)

    # Handle the case if just one expression was passed.
    if isinstance(exprs, Basic):
        exprs = [exprs]

    # Preprocess the expressions to give us better optimization opportunities.
    prep_exprs = [preprocess_for_cse(e, optimizations) for e in exprs]

    # Find all subexpressions.
    def _parse(expr):
        
        if expr.is_Atom:
            # Exclude atoms, since there is no point in renaming them.
            return expr
        
        if iterable(expr):
            return expr
        
        subexpr = type(expr)(*map(_parse, expr.args))

        if subexpr in subexp_iv:
            return subexp_iv[subexpr]
        
        if subexpr.is_Mul:
            muls.add(subexpr)
        elif subexpr.is_Add:
            adds.add(subexpr)

        ivar = next(tmp_symbols)
        subexp_iv[subexpr] = ivar
        
        return ivar
    
    tmp_exprs = list()
    for expr in prep_exprs:
        if isinstance(expr, Basic):
            tmp_exprs.append(_parse(expr))
        else:
            tmp_exprs.append(expr)
    
    # process adds - any adds that weren't repeated might contain
    # subpatterns that are repeated, e.g. x+y+z and x+y have x+y in common
    adds = list(ordered(adds))
    addargs = [set(a.args) for a in adds]
    for i in xrange(len(addargs)):
        for j in xrange(i + 1, len(addargs)):
            com = addargs[i].intersection(addargs[j])
            if len(com) > 1:
                
                add_subexp = Add(*com)
                
                diff_add_i = addargs[i].difference(com)
                diff_add_j = addargs[j].difference(com)
#.........这里部分代码省略.........

    def _collapse_arguments(cls, args, **assumptions):
        """Remove redundant args.

        Examples
        ========

        >>> from sympy import Min, Max
        >>> from sympy.abc import a, b, c, d, e

        Any arg in parent that appears in any
        parent-like function in any of the flat args
        of parent can be removed from that sub-arg:

        >>> Min(a, Max(b, Min(a, c, d)))
        Min(a, Max(b, Min(c, d)))

        If the arg of parent appears in an opposite-than parent
        function in any of the flat args of parent that function
        can be replaced with the arg:

        >>> Min(a, Max(b, Min(c, d, Max(a, e))))
        Min(a, Max(b, Min(a, c, d)))

        """
        from sympy.utilities.iterables import ordered
        from sympy.simplify.simplify import walk

        if not args:
            return args
        args = list(ordered(args))
        if cls == Min:
            other = Max
        else:
            other = Min

        # find global comparable max of Max and min of Min if a new
        # value is being introduced in these args at position 0 of
        # the ordered args
        if args[0].is_number:
            sifted = mins, maxs = [], []
            for i in args:
                for v in walk(i, Min, Max):
                    if v.args[0].is_comparable:
                        sifted[isinstance(v, Max)].append(v)
            small = Min.identity
            for i in mins:
                v = i.args[0]
                if v.is_number and (v < small) == True:
                    small = v
            big = Max.identity
            for i in maxs:
                v = i.args[0]
                if v.is_number and (v > big) == True:
                    big = v
            # at the point when this function is called from __new__,
            # there may be more than one numeric arg present since
            # local zeros have not been handled yet, so look through
            # more than the first arg
            if cls == Min:
                for i in range(len(args)):
                    if not args[i].is_number:
                        break
                    if (args[i] < small) == True:
                        small = args[i]
            elif cls == Max:
                for i in range(len(args)):
                    if not args[i].is_number:
                        break
                    if (args[i] > big) == True:
                        big = args[i]
            T = None
            if cls == Min:
                if small != Min.identity:
                    other = Max
                    T = small
            elif big != Max.identity:
                other = Min
                T = big
            if T is not None:
                # remove numerical redundancy
                for i in range(len(args)):
                    a = args[i]
                    if isinstance(a, other):
                        a0 = a.args[0]
                        if ((a0 > T) if other == Max else (a0 < T)) == True:
                            args[i] = cls.identity

        # remove redundant symbolic args
        def do(ai, a):
            if not isinstance(ai, (Min, Max)):
                return ai
            cond = a in ai.args
            if not cond:
                return ai.func(*[do(i, a) for i in ai.args],
                    evaluate=False)
            if isinstance(ai, cls):
                return ai.func(*[do(i, a) for i in ai.args if i != a],
                    evaluate=False)
            return a
        for i, a in enumerate(args):
#.........这里部分代码省略.........

def tree_cse(exprs, symbols, opt_subs=None, order='canonical', ignore=()):
    """Perform raw CSE on expression tree, taking opt_subs into account.

    Parameters
    ==========

    exprs : list of sympy expressions
        The expressions to reduce.
    symbols : infinite iterator yielding unique Symbols
        The symbols used to label the common subexpressions which are pulled
        out.
    opt_subs : dictionary of expression substitutions
        The expressions to be substituted before any CSE action is performed.
    order : string, 'none' or 'canonical'
        The order by which Mul and Add arguments are processed. For large
        expressions where speed is a concern, use the setting order='none'.
    ignore : iterable of Symbols
        Substitutions containing any Symbol from ``ignore`` will be ignored.
    """
    from sympy.matrices.expressions import MatrixExpr, MatrixSymbol, MatMul, MatAdd

    if opt_subs is None:
        opt_subs = dict()

    ## Find repeated sub-expressions

    to_eliminate = set()

    seen_subexp = set()

    def _find_repeated(expr):
        if not isinstance(expr, Basic):
            return

        if expr.is_Atom or expr.is_Order:
            return

        if iterable(expr):
            args = expr

        else:
            if expr in seen_subexp:
                for ign in ignore:
                    if ign in expr.free_symbols:
                        break
                else:
                    to_eliminate.add(expr)
                    return

            seen_subexp.add(expr)

            if expr in opt_subs:
                expr = opt_subs[expr]

            args = expr.args

        list(map(_find_repeated, args))

    for e in exprs:
        if isinstance(e, Basic):
            _find_repeated(e)

    ## Rebuild tree

    replacements = []

    subs = dict()

    def _rebuild(expr):
        if not isinstance(expr, Basic):
            return expr

        if not expr.args:
            return expr

        if iterable(expr):
            new_args = [_rebuild(arg) for arg in expr]
            return expr.func(*new_args)

        if expr in subs:
            return subs[expr]

        orig_expr = expr
        if expr in opt_subs:
            expr = opt_subs[expr]

        # If enabled, parse Muls and Adds arguments by order to ensure
        # replacement order independent from hashes
        if order != 'none':
            if isinstance(expr, (Mul, MatMul)):
                c, nc = expr.args_cnc()
                if c == [1]:
                    args = nc
                else:
                    args = list(ordered(c)) + nc
            elif isinstance(expr, (Add, MatAdd)):
                args = list(ordered(expr.args))
            else:
                args = expr.args
        else:
#.........这里部分代码省略.........

 def _eval_simplify(self, ratio, measure, rational, inverse):
     args = [a._eval_simplify(ratio, measure, rational, inverse)
         for a in self.args]
     for i, (expr, cond) in enumerate(args):
         # try to simplify conditions and the expression for
         # equalities that are part of the condition, e.g.
         # Piecewise((n, And(Eq(n,0), Eq(n + m, 0))), (1, True))
         # -> Piecewise((0, And(Eq(n, 0), Eq(m, 0))), (1, True))
         if isinstance(cond, And):
             eqs, other = sift(cond.args,
                 lambda i: isinstance(i, Equality), binary=True)
         elif isinstance(cond, Equality):
             eqs, other = [cond], []
         else:
             eqs = other = []
         if eqs:
             eqs = list(ordered(eqs))
             for j, e in enumerate(eqs):
                 # these blessed lhs objects behave like Symbols
                 # and the rhs are simple replacements for the "symbols"
                 if isinstance(e.lhs, (Symbol, UndefinedFunction)) and \
                     isinstance(e.rhs,
                         (Rational, NumberSymbol,
                         Symbol, UndefinedFunction)):
                     expr = expr.subs(*e.args)
                     eqs[j + 1:] = [ei.subs(*e.args) for ei in eqs[j + 1:]]
                     other = [ei.subs(*e.args) for ei in other]
             cond = And(*(eqs + other))
             args[i] = args[i].func(expr, cond)
     # See if expressions valid for a single point happens to evaluate
     # to the same function as in the next piecewise segment, see:
     # https://github.com/sympy/sympy/issues/8458
     prevexpr = None
     for i, (expr, cond) in reversed(list(enumerate(args))):
         if prevexpr is not None:
             _prevexpr = prevexpr
             _expr = expr
             if isinstance(cond, And):
                 eqs, other = sift(cond.args,
                     lambda i: isinstance(i, Equality), binary=True)
             elif isinstance(cond, Equality):
                 eqs, other = [cond], []
             else:
                 eqs = other = []
             if eqs:
                 eqs = list(ordered(eqs))
                 for j, e in enumerate(eqs):
                     # these blessed lhs objects behave like Symbols
                     # and the rhs are simple replacements for the "symbols"
                     if isinstance(e.lhs, (Symbol, UndefinedFunction)) and \
                         isinstance(e.rhs,
                             (Rational, NumberSymbol,
                             Symbol, UndefinedFunction)):
                         _prevexpr = _prevexpr.subs(*e.args)
                         _expr = _expr.subs(*e.args)
             if _prevexpr == _expr:
                 args[i] = args[i].func(args[i+1][0], cond)
             else:
                 prevexpr = expr
         else:
             prevexpr = expr
     return self.func(*args)

def piecewise_fold(expr):
    """
    Takes an expression containing a piecewise function and returns the
    expression in piecewise form. In addition, any ITE conditions are
    rewritten in negation normal form and simplified.

    Examples
    ========

    >>> from sympy import Piecewise, piecewise_fold, sympify as S
    >>> from sympy.abc import x
    >>> p = Piecewise((x, x < 1), (1, S(1) <= x))
    >>> piecewise_fold(x*p)
    Piecewise((x**2, x < 1), (x, True))

    See Also
    ========

    Piecewise
    """
    if not isinstance(expr, Basic) or not expr.has(Piecewise):
        return expr

    new_args = []
    if isinstance(expr, (ExprCondPair, Piecewise)):
        for e, c in expr.args:
            if not isinstance(e, Piecewise):
                e = piecewise_fold(e)
            # we don't keep Piecewise in condition because
            # it has to be checked to see that it's complete
            # and we convert it to ITE at that time
            assert not c.has(Piecewise)  # pragma: no cover
            if isinstance(c, ITE):
                c = c.to_nnf()
                c = simplify_logic(c, form='cnf')
            if isinstance(e, Piecewise):
                new_args.extend([(piecewise_fold(ei), And(ci, c))
                    for ei, ci in e.args])
            else:
                new_args.append((e, c))
    else:
        from sympy.utilities.iterables import cartes, sift, common_prefix
        # Given
        #     P1 = Piecewise((e11, c1), (e12, c2), A)
        #     P2 = Piecewise((e21, c1), (e22, c2), B)
        #     ...
        # the folding of f(P1, P2) is trivially
        # Piecewise(
        #   (f(e11, e21), c1),
        #   (f(e12, e22), c2),
        #   (f(Piecewise(A), Piecewise(B)), True))
        # Certain objects end up rewriting themselves as thus, so
        # we do that grouping before the more generic folding.
        # The following applies this idea when f = Add or f = Mul
        # (and the expression is commutative).
        if expr.is_Add or expr.is_Mul and expr.is_commutative:
            p, args = sift(expr.args, lambda x: x.is_Piecewise, binary=True)
            pc = sift(p, lambda x: tuple([c for e,c in x.args]))
            for c in list(ordered(pc)):
                if len(pc[c]) > 1:
                    pargs = [list(i.args) for i in pc[c]]
                    # the first one is the same; there may be more
                    com = common_prefix(*[
                        [i.cond for i in j] for j in pargs])
                    n = len(com)
                    collected = []
                    for i in range(n):
                        collected.append((
                            expr.func(*[ai[i].expr for ai in pargs]),
                            com[i]))
                    remains = []
                    for a in pargs:
                        if n == len(a):  # no more args
                            continue
                        if a[n].cond == True:  # no longer Piecewise
                            remains.append(a[n].expr)
                        else:  # restore the remaining Piecewise
                            remains.append(
                                Piecewise(*a[n:], evaluate=False))
                    if remains:
                        collected.append((expr.func(*remains), True))
                    args.append(Piecewise(*collected, evaluate=False))
                    continue
                args.extend(pc[c])
        else:
            args = expr.args
        # fold
        folded = list(map(piecewise_fold, args))
        for ec in cartes(*[
                (i.args if isinstance(i, Piecewise) else
                 [(i, true)]) for i in folded]):
            e, c = zip(*ec)
            new_args.append((expr.func(*e), And(*c)))

    return Piecewise(*new_args)

def cse(exprs, symbols=None, optimizations=None, postprocess=None):
    """ Perform common subexpression elimination on an expression.

    Parameters
    ==========

    exprs : list of sympy expressions, or a single sympy expression
        The expressions to reduce.
    symbols : infinite iterator yielding unique Symbols
        The symbols used to label the common subexpressions which are pulled
        out. The ``numbered_symbols`` generator is useful. The default is a
        stream of symbols of the form "x0", "x1", etc. This must be an infinite
        iterator.
    optimizations : list of (callable, callable) pairs, optional
        The (preprocessor, postprocessor) pairs. If not provided,
        ``sympy.simplify.cse.cse_optimizations`` is used.
    postprocess : a function which accepts the two return values of cse and
        returns the desired form of output from cse, e.g. if you want the
        replacements reversed the function might be the following lambda:
        lambda r, e: return reversed(r), e

    Returns
    =======

    replacements : list of (Symbol, expression) pairs
        All of the common subexpressions that were replaced. Subexpressions
        earlier in this list might show up in subexpressions later in this list.
    reduced_exprs : list of sympy expressions
        The reduced expressions with all of the replacements above.
    """
    from sympy.matrices import Matrix

    if symbols is None:
        symbols = numbered_symbols()
    else:
        # In case we get passed an iterable with an __iter__ method instead of
        # an actual iterator.
        symbols = iter(symbols)
    seen_subexp = set()
    muls = set()
    adds = set()
    to_eliminate = set()

    if optimizations is None:
        # Pull out the default here just in case there are some weird
        # manipulations of the module-level list in some other thread.
        optimizations = list(cse_optimizations)

    # Handle the case if just one expression was passed.
    if isinstance(exprs, Basic):
        exprs = [exprs]

    # Preprocess the expressions to give us better optimization opportunities.
    reduced_exprs = [preprocess_for_cse(e, optimizations) for e in exprs]

    # Find all of the repeated subexpressions.
    for expr in reduced_exprs:
        if not isinstance(expr, Basic):
            continue
        pt = preorder_traversal(expr)
        for subtree in pt:

            inv = 1/subtree if subtree.is_Pow else None

            if subtree.is_Atom or iterable(subtree) or inv and inv.is_Atom:
                # Exclude atoms, since there is no point in renaming them.
                continue

            if subtree in seen_subexp:
                if inv and _coeff_isneg(subtree.exp):
                    # save the form with positive exponent
                    subtree = inv
                to_eliminate.add(subtree)
                pt.skip()
                continue

            if inv and inv in seen_subexp:
                if _coeff_isneg(subtree.exp):
                    # save the form with positive exponent
                    subtree = inv
                to_eliminate.add(subtree)
                pt.skip()
                continue
            elif subtree.is_Mul:
                muls.add(subtree)
            elif subtree.is_Add:
                adds.add(subtree)

            seen_subexp.add(subtree)

    # process adds - any adds that weren't repeated might contain
    # subpatterns that are repeated, e.g. x+y+z and x+y have x+y in common
    adds = [set(a.args) for a in ordered(adds)]
    for i in xrange(len(adds)):
        for j in xrange(i + 1, len(adds)):
            com = adds[i].intersection(adds[j])
            if len(com) > 1:
                to_eliminate.add(Add(*com))

                # remove this set of symbols so it doesn't appear again
#.........这里部分代码省略.........