venv/lib/python3.11/site-packages/sympy/simplify/simplify.py

from __future__ import annotations

from typing import overload

from collections import defaultdict

from sympy.concrete.products import Product
from sympy.concrete.summations import Sum
from sympy.core import (Basic, S, Add, Mul, Pow, Symbol, sympify,
                        expand_func, Function, Dummy, Expr, factor_terms,
                        expand_power_exp, Eq)
from sympy.core.exprtools import factor_nc
from sympy.core.parameters import global_parameters
from sympy.core.function import (expand_log, count_ops, _mexpand,
    nfloat, expand_mul, expand)
from sympy.core.numbers import Float, I, pi, Rational, equal_valued
from sympy.core.relational import Relational
from sympy.core.rules import Transform
from sympy.core.sorting import ordered
from sympy.core.sympify import _sympify
from sympy.core.traversal import bottom_up as _bottom_up, walk as _walk
from sympy.functions import gamma, exp, sqrt, log, exp_polar, re
from sympy.functions.combinatorial.factorials import CombinatorialFunction
from sympy.functions.elementary.complexes import unpolarify, Abs, sign
from sympy.functions.elementary.exponential import ExpBase
from sympy.functions.elementary.hyperbolic import HyperbolicFunction
from sympy.functions.elementary.integers import ceiling
from sympy.functions.elementary.piecewise import (Piecewise, piecewise_fold,
                                                  piecewise_simplify)
from sympy.functions.elementary.trigonometric import TrigonometricFunction
from sympy.functions.special.bessel import (BesselBase, besselj, besseli,
                                            besselk, bessely, jn)
from sympy.functions.special.tensor_functions import KroneckerDelta
from sympy.integrals.integrals import Integral
from sympy.logic.boolalg import Boolean
from sympy.matrices.expressions import (MatrixExpr, MatAdd, MatMul,
                                            MatPow, MatrixSymbol)
from sympy.polys import together, cancel, factor
from sympy.polys.numberfields.minpoly import _is_sum_surds, _minimal_polynomial_sq
from sympy.sets.sets import Set
from sympy.simplify.combsimp import combsimp
from sympy.simplify.cse_opts import sub_pre, sub_post
from sympy.simplify.hyperexpand import hyperexpand
from sympy.simplify.powsimp import powsimp
from sympy.simplify.radsimp import radsimp, fraction, collect_abs
from sympy.simplify.sqrtdenest import sqrtdenest
from sympy.simplify.trigsimp import trigsimp, exptrigsimp
from sympy.utilities.decorator import deprecated
from sympy.utilities.iterables import has_variety, sift, subsets, iterable
from sympy.utilities.misc import as_int

import mpmath


def separatevars(expr, symbols=[], dict=False, force=False):
    """
    Separates variables in an expression, if possible.  By
    default, it separates with respect to all symbols in an
    expression and collects constant coefficients that are
    independent of symbols.

    Explanation
    ===========

    If ``dict=True`` then the separated terms will be returned
    in a dictionary keyed to their corresponding symbols.
    By default, all symbols in the expression will appear as
    keys; if symbols are provided, then all those symbols will
    be used as keys, and any terms in the expression containing
    other symbols or non-symbols will be returned keyed to the
    string 'coeff'. (Passing None for symbols will return the
    expression in a dictionary keyed to 'coeff'.)

    If ``force=True``, then bases of powers will be separated regardless
    of assumptions on the symbols involved.

    Notes
    =====

    The order of the factors is determined by Mul, so that the
    separated expressions may not necessarily be grouped together.

    Although factoring is necessary to separate variables in some
    expressions, it is not necessary in all cases, so one should not
    count on the returned factors being factored.

    Examples
    ========

    >>> from sympy.abc import x, y, z, alpha
    >>> from sympy import separatevars, sin
    >>> separatevars((x*y)**y)
    (x*y)**y
    >>> separatevars((x*y)**y, force=True)
    x**y*y**y

    >>> e = 2*x**2*z*sin(y)+2*z*x**2
    >>> separatevars(e)
    2*x**2*z*(sin(y) + 1)
    >>> separatevars(e, symbols=(x, y), dict=True)
    {'coeff': 2*z, x: x**2, y: sin(y) + 1}
    >>> separatevars(e, [x, y, alpha], dict=True)
    {'coeff': 2*z, alpha: 1, x: x**2, y: sin(y) + 1}

    If the expression is not really separable, or is only partially
    separable, separatevars will do the best it can to separate it
    by using factoring.

    >>> separatevars(x + x*y - 3*x**2)
    -x*(3*x - y - 1)

    If the expression is not separable then expr is returned unchanged
    or (if dict=True) then None is returned.

    >>> eq = 2*x + y*sin(x)
    >>> separatevars(eq) == eq
    True
    >>> separatevars(2*x + y*sin(x), symbols=(x, y), dict=True) is None
    True

    """
    expr = sympify(expr)
    if dict:
        return _separatevars_dict(_separatevars(expr, force), symbols)
    else:
        return _separatevars(expr, force)


def _separatevars(expr, force):
    if isinstance(expr, Abs):
        arg = expr.args[0]
        if arg.is_Mul and not arg.is_number:
            s = separatevars(arg, dict=True, force=force)
            if s is not None:
                return Mul(*map(expr.func, s.values()))
            else:
                return expr

    if len(expr.free_symbols) < 2:
        return expr

    # don't destroy a Mul since much of the work may already be done
    if expr.is_Mul:
        args = list(expr.args)
        changed = False
        for i, a in enumerate(args):
            args[i] = separatevars(a, force)
            changed = changed or args[i] != a
        if changed:
            expr = expr.func(*args)
        return expr

    # get a Pow ready for expansion
    if expr.is_Pow and expr.base != S.Exp1:
        expr = Pow(separatevars(expr.base, force=force), expr.exp)

    # First try other expansion methods
    expr = expr.expand(mul=False, multinomial=False, force=force)

    _expr, reps = posify(expr) if force else (expr, {})
    expr = factor(_expr).subs(reps)

    if not expr.is_Add:
        return expr

    # Find any common coefficients to pull out
    args = list(expr.args)
    commonc = args[0].args_cnc(cset=True, warn=False)[0]
    for i in args[1:]:
        commonc &= i.args_cnc(cset=True, warn=False)[0]
    commonc = Mul(*commonc)
    commonc = commonc.as_coeff_Mul()[1]  # ignore constants
    commonc_set = commonc.args_cnc(cset=True, warn=False)[0]

    # remove them
    for i, a in enumerate(args):
        c, nc = a.args_cnc(cset=True, warn=False)
        c = c - commonc_set
        args[i] = Mul(*c)*Mul(*nc)
    nonsepar = Add(*args)

    if len(nonsepar.free_symbols) > 1:
        _expr = nonsepar
        _expr, reps = posify(_expr) if force else (_expr, {})
        _expr = (factor(_expr)).subs(reps)

        if not _expr.is_Add:
            nonsepar = _expr

    return commonc*nonsepar


def _separatevars_dict(expr, symbols):
    if symbols:
        if not all(t.is_Atom for t in symbols):
            raise ValueError("symbols must be Atoms.")
        symbols = list(symbols)
    elif symbols is None:
        return {'coeff': expr}
    else:
        symbols = list(expr.free_symbols)
        if not symbols:
            return None

    ret = {i: [] for i in symbols + ['coeff']}

    for i in Mul.make_args(expr):
        expsym = i.free_symbols
        intersection = set(symbols).intersection(expsym)
        if len(intersection) > 1:
            return None
        if len(intersection) == 0:
            # There are no symbols, so it is part of the coefficient
            ret['coeff'].append(i)
        else:
            ret[intersection.pop()].append(i)

    # rebuild
    for k, v in ret.items():
        ret[k] = Mul(*v)

    return ret


def posify(eq):
    """Return ``eq`` (with generic symbols made positive) and a
    dictionary containing the mapping between the old and new
    symbols.

    Explanation
    ===========

    Any symbol that has positive=None will be replaced with a positive dummy
    symbol having the same name. This replacement will allow more symbolic
    processing of expressions, especially those involving powers and
    logarithms.

    A dictionary that can be sent to subs to restore ``eq`` to its original
    symbols is also returned.

    >>> from sympy import posify, Symbol, log, solve
    >>> from sympy.abc import x
    >>> posify(x + Symbol('p', positive=True) + Symbol('n', negative=True))
    (_x + n + p, {_x: x})

    >>> eq = 1/x
    >>> log(eq).expand()
    log(1/x)
    >>> log(posify(eq)[0]).expand()
    -log(_x)
    >>> p, rep = posify(eq)
    >>> log(p).expand().subs(rep)
    -log(x)

    It is possible to apply the same transformations to an iterable
    of expressions:

    >>> eq = x**2 - 4
    >>> solve(eq, x)
    [-2, 2]
    >>> eq_x, reps = posify([eq, x]); eq_x
    [_x**2 - 4, _x]
    >>> solve(*eq_x)
    [2]
    """
    eq = sympify(eq)
    if not isinstance(eq, Basic) and iterable(eq):
        f = type(eq)
        eq = list(eq)
        syms = set()
        for e in eq:
            syms = syms.union(e.atoms(Symbol))
        reps = {}
        for s in syms:
            reps.update({v: k for k, v in posify(s)[1].items()})
        for i, e in enumerate(eq):
            eq[i] = e.subs(reps)
        return f(eq), {r: s for s, r in reps.items()}

    reps = {s: Dummy(s.name, positive=True, **s.assumptions0)
                 for s in eq.free_symbols if s.is_positive is None}
    eq = eq.subs(reps)
    return eq, {r: s for s, r in reps.items()}


def hypersimp(f, k):
    """Given combinatorial term f(k) simplify its consecutive term ratio
       i.e. f(k+1)/f(k).  The input term can be composed of functions and
       integer sequences which have equivalent representation in terms
       of gamma special function.

       Explanation
       ===========

       The algorithm performs three basic steps:

       1. Rewrite all functions in terms of gamma, if possible.

       2. Rewrite all occurrences of gamma in terms of products
          of gamma and rising factorial with integer,  absolute
          constant exponent.

       3. Perform simplification of nested fractions, powers
          and if the resulting expression is a quotient of
          polynomials, reduce their total degree.

       If f(k) is hypergeometric then as result we arrive with a
       quotient of polynomials of minimal degree. Otherwise None
       is returned.

       For more information on the implemented algorithm refer to:

       1. W. Koepf, Algorithms for m-fold Hypergeometric Summation,
          Journal of Symbolic Computation (1995) 20, 399-417
    """
    f = sympify(f)

    g = f.subs(k, k + 1) / f

    g = g.rewrite(gamma)
    if g.has(Piecewise):
        g = piecewise_fold(g)
        g = g.args[-1][0]
    g = expand_func(g)
    g = powsimp(g, deep=True, combine='exp')

    if g.is_rational_function(k):
        return simplify(g, ratio=S.Infinity)
    else:
        return None


def hypersimilar(f, g, k):
    """
    Returns True if ``f`` and ``g`` are hyper-similar.

    Explanation
    ===========

    Similarity in hypergeometric sense means that a quotient of
    f(k) and g(k) is a rational function in ``k``. This procedure
    is useful in solving recurrence relations.

    For more information see hypersimp().

    """
    f, g = list(map(sympify, (f, g)))

    h = (f/g).rewrite(gamma)
    h = h.expand(func=True, basic=False)

    return h.is_rational_function(k)


def signsimp(expr, evaluate=None):
    """Make all Add sub-expressions canonical wrt sign.

    Explanation
    ===========

    If an Add subexpression, ``a``, can have a sign extracted,
    as determined by could_extract_minus_sign, it is replaced
    with Mul(-1, a, evaluate=False). This allows signs to be
    extracted from powers and products.

    Examples
    ========

    >>> from sympy import signsimp, exp, symbols
    >>> from sympy.abc import x, y
    >>> i = symbols('i', odd=True)
    >>> n = -1 + 1/x
    >>> n/x/(-n)**2 - 1/n/x
    (-1 + 1/x)/(x*(1 - 1/x)**2) - 1/(x*(-1 + 1/x))
    >>> signsimp(_)
    0
    >>> x*n + x*-n
    x*(-1 + 1/x) + x*(1 - 1/x)
    >>> signsimp(_)
    0

    Since powers automatically handle leading signs

    >>> (-2)**i
    -2**i

    signsimp can be used to put the base of a power with an integer
    exponent into canonical form:

    >>> n**i
    (-1 + 1/x)**i

    By default, signsimp does not leave behind any hollow simplification:
    if making an Add canonical wrt sign didn't change the expression, the
    original Add is restored. If this is not desired then the keyword
    ``evaluate`` can be set to False:

    >>> e = exp(y - x)
    >>> signsimp(e) == e
    True
    >>> signsimp(e, evaluate=False)
    exp(-(x - y))

    """
    if evaluate is None:
        evaluate = global_parameters.evaluate
    expr = sympify(expr)
    if not isinstance(expr, (Expr, Relational)) or expr.is_Atom:
        return expr
    # get rid of an pre-existing unevaluation regarding sign
    e = expr.replace(lambda x: x.is_Mul and -(-x) != x, lambda x: -(-x))
    e = sub_post(sub_pre(e))
    if not isinstance(e, (Expr, Relational)) or e.is_Atom:
        return e
    if e.is_Add:
        rv = e.func(*[signsimp(a) for a in e.args])
        if not evaluate and isinstance(rv, Add
                ) and rv.could_extract_minus_sign():
            return Mul(S.NegativeOne, -rv, evaluate=False)
        return rv
    if evaluate:
        e = e.replace(lambda x: x.is_Mul and -(-x) != x, lambda x: -(-x))
    return e


@overload
def simplify(expr: Expr, **kwargs) -> Expr: ...
@overload
def simplify(expr: Boolean, **kwargs) -> Boolean: ...
@overload
def simplify(expr: Set, **kwargs) -> Set: ...
@overload
def simplify(expr: Basic, **kwargs) -> Basic: ...

def simplify(expr, ratio=1.7, measure=count_ops, rational=False, inverse=False, doit=True, **kwargs):
    """Simplifies the given expression.

    Explanation
    ===========

    Simplification is not a well defined term and the exact strategies
    this function tries can change in the future versions of SymPy. If
    your algorithm relies on "simplification" (whatever it is), try to
    determine what you need exactly  -  is it powsimp()?, radsimp()?,
    together()?, logcombine()?, or something else? And use this particular
    function directly, because those are well defined and thus your algorithm
    will be robust.

    Nonetheless, especially for interactive use, or when you do not know
    anything about the structure of the expression, simplify() tries to apply
    intelligent heuristics to make the input expression "simpler".  For
    example:

    >>> from sympy import simplify, cos, sin
    >>> from sympy.abc import x, y
    >>> a = (x + x**2)/(x*sin(y)**2 + x*cos(y)**2)
    >>> a
    (x**2 + x)/(x*sin(y)**2 + x*cos(y)**2)
    >>> simplify(a)
    x + 1

    Note that we could have obtained the same result by using specific
    simplification functions:

    >>> from sympy import trigsimp, cancel
    >>> trigsimp(a)
    (x**2 + x)/x
    >>> cancel(_)
    x + 1

    In some cases, applying :func:`simplify` may actually result in some more
    complicated expression. The default ``ratio=1.7`` prevents more extreme
    cases: if (result length)/(input length) > ratio, then input is returned
    unmodified.  The ``measure`` parameter lets you specify the function used
    to determine how complex an expression is.  The function should take a
    single argument as an expression and return a number such that if
    expression ``a`` is more complex than expression ``b``, then
    ``measure(a) > measure(b)``.  The default measure function is
    :func:`~.count_ops`, which returns the total number of operations in the
    expression.

    For example, if ``ratio=1``, ``simplify`` output cannot be longer
    than input.

    ::

        >>> from sympy import sqrt, simplify, count_ops, oo
        >>> root = 1/(sqrt(2)+3)

    Since ``simplify(root)`` would result in a slightly longer expression,
    root is returned unchanged instead::

       >>> simplify(root, ratio=1) == root
       True

    If ``ratio=oo``, simplify will be applied anyway::

        >>> count_ops(simplify(root, ratio=oo)) > count_ops(root)
        True

    Note that the shortest expression is not necessary the simplest, so
    setting ``ratio`` to 1 may not be a good idea.
    Heuristically, the default value ``ratio=1.7`` seems like a reasonable
    choice.

    You can easily define your own measure function based on what you feel
    should represent the "size" or "complexity" of the input expression.  Note
    that some choices, such as ``lambda expr: len(str(expr))`` may appear to be
    good metrics, but have other problems (in this case, the measure function
    may slow down simplify too much for very large expressions).  If you do not
    know what a good metric would be, the default, ``count_ops``, is a good
    one.

    For example:

    >>> from sympy import symbols, log
    >>> a, b = symbols('a b', positive=True)
    >>> g = log(a) + log(b) + log(a)*log(1/b)
    >>> h = simplify(g)
    >>> h
    log(a*b**(1 - log(a)))
    >>> count_ops(g)
    8
    >>> count_ops(h)
    5

    So you can see that ``h`` is simpler than ``g`` using the count_ops metric.
    However, we may not like how ``simplify`` (in this case, using
    ``logcombine``) has created the ``b**(log(1/a) + 1)`` term.  A simple way
    to reduce this would be to give more weight to powers as operations in
    ``count_ops``.  We can do this by using the ``visual=True`` option:

    >>> print(count_ops(g, visual=True))
    2*ADD + DIV + 4*LOG + MUL
    >>> print(count_ops(h, visual=True))
    2*LOG + MUL + POW + SUB

    >>> from sympy import Symbol, S
    >>> def my_measure(expr):
    ...     POW = Symbol('POW')
    ...     # Discourage powers by giving POW a weight of 10
    ...     count = count_ops(expr, visual=True).subs(POW, 10)
    ...     # Every other operation gets a weight of 1 (the default)
    ...     count = count.replace(Symbol, type(S.One))
    ...     return count
    >>> my_measure(g)
    8
    >>> my_measure(h)
    14
    >>> 15./8 > 1.7 # 1.7 is the default ratio
    True
    >>> simplify(g, measure=my_measure)
    -log(a)*log(b) + log(a) + log(b)

    Note that because ``simplify()`` internally tries many different
    simplification strategies and then compares them using the measure
    function, we get a completely different result that is still different
    from the input expression by doing this.

    If ``rational=True``, Floats will be recast as Rationals before simplification.
    If ``rational=None``, Floats will be recast as Rationals but the result will
    be recast as Floats. If rational=False(default) then nothing will be done
    to the Floats.

    If ``inverse=True``, it will be assumed that a composition of inverse
    functions, such as sin and asin, can be cancelled in any order.
    For example, ``asin(sin(x))`` will yield ``x`` without checking whether
    x belongs to the set where this relation is true. The default is
    False.

    Note that ``simplify()`` automatically calls ``doit()`` on the final
    expression. You can avoid this behavior by passing ``doit=False`` as
    an argument.

    Also, it should be noted that simplifying a boolean expression is not
    well defined. If the expression prefers automatic evaluation (such as
    :obj:`~.Eq()` or :obj:`~.Or()`), simplification will return ``True`` or
    ``False`` if truth value can be determined. If the expression is not
    evaluated by default (such as :obj:`~.Predicate()`), simplification will
    not reduce it and you should use :func:`~.refine` or :func:`~.ask`
    function. This inconsistency will be resolved in future version.

    See Also
    ========

    sympy.assumptions.refine.refine : Simplification using assumptions.
    sympy.assumptions.ask.ask : Query for boolean expressions using assumptions.
    """

    def shorter(*choices):
        """
        Return the choice that has the fewest ops. In case of a tie,
        the expression listed first is selected.
        """
        if not has_variety(choices):
            return choices[0]
        return min(choices, key=measure)

    def done(e):
        rv = e.doit() if doit else e
        return shorter(rv, collect_abs(rv))

    expr = sympify(expr, rational=rational)
    kwargs = {
        "ratio": kwargs.get('ratio', ratio),
        "measure": kwargs.get('measure', measure),
        "rational": kwargs.get('rational', rational),
        "inverse": kwargs.get('inverse', inverse),
        "doit": kwargs.get('doit', doit)}
    # no routine for Expr needs to check for is_zero
    if isinstance(expr, Expr) and expr.is_zero:
        return S.Zero if not expr.is_Number else expr

    _eval_simplify = getattr(expr, '_eval_simplify', None)
    if _eval_simplify is not None:
        return _eval_simplify(**kwargs)

    original_expr = expr = collect_abs(signsimp(expr))

    if not isinstance(expr, Basic) or not expr.args:  # XXX: temporary hack
        return expr

    if inverse and expr.has(Function):
        expr = inversecombine(expr)
        if not expr.args:  # simplified to atomic
            return expr

    # do deep simplification
    handled = Add, Mul, Pow, ExpBase
    expr = expr.replace(
        # here, checking for x.args is not enough because Basic has
        # args but Basic does not always play well with replace, e.g.
        # when simultaneous is True found expressions will be masked
        # off with a Dummy but not all Basic objects in an expression
        # can be replaced with a Dummy
        lambda x: isinstance(x, Expr) and x.args and not isinstance(
            x, handled),
        lambda x: x.func(*[simplify(i, **kwargs) for i in x.args]),
        simultaneous=False)
    if not isinstance(expr, handled):
        return done(expr)

    if not expr.is_commutative:
        expr = nc_simplify(expr)

    # TODO: Apply different strategies, considering expression pattern:
    # is it a purely rational function? Is there any trigonometric function?...
    # See also https://github.com/sympy/sympy/pull/185.

    # rationalize Floats
    floats = False
    if rational is not False and expr.has(Float):
        floats = True
        expr = nsimplify(expr, rational=True)

    expr = _bottom_up(expr, lambda w: getattr(w, 'normal', lambda: w)())
    expr = Mul(*powsimp(expr).as_content_primitive())
    _e = cancel(expr)
    expr1 = shorter(_e, _mexpand(_e).cancel())  # issue 6829
    expr2 = shorter(together(expr, deep=True), together(expr1, deep=True))

    if ratio is S.Infinity:
        expr = expr2
    else:
        expr = shorter(expr2, expr1, expr)
    if not isinstance(expr, Basic):  # XXX: temporary hack
        return expr

    expr = factor_terms(expr, sign=False)

    # must come before `Piecewise` since this introduces more `Piecewise` terms
    if expr.has(sign):
        expr = expr.rewrite(Abs)

    # Deal with Piecewise separately to avoid recursive growth of expressions
    if expr.has(Piecewise):
        # Fold into a single Piecewise
        expr = piecewise_fold(expr)
        # Apply doit, if doit=True
        expr = done(expr)
        # Still a Piecewise?
        if expr.has(Piecewise):
            # Fold into a single Piecewise, in case doit lead to some
            # expressions being Piecewise
            expr = piecewise_fold(expr)
            # kroneckersimp also affects Piecewise
            if expr.has(KroneckerDelta):
                expr = kroneckersimp(expr)
            # Still a Piecewise?
            if expr.has(Piecewise):
                # Do not apply doit on the segments as it has already
                # been done above, but simplify
                expr = piecewise_simplify(expr, deep=True, doit=False)
                # Still a Piecewise?
                if expr.has(Piecewise):
                    # Try factor common terms
                    expr = shorter(expr, factor_terms(expr))
                    # As all expressions have been simplified above with the
                    # complete simplify, nothing more needs to be done here
                    return expr

    # hyperexpand automatically only works on hypergeometric terms
    # Do this after the Piecewise part to avoid recursive expansion
    expr = hyperexpand(expr)

    if expr.has(KroneckerDelta):
        expr = kroneckersimp(expr)

    if expr.has(BesselBase):
        expr = besselsimp(expr)

    if expr.has(TrigonometricFunction, HyperbolicFunction):
        expr = trigsimp(expr, deep=True)

    if expr.has(log):
        expr = shorter(expand_log(expr, deep=True), logcombine(expr))

    if expr.has(CombinatorialFunction, gamma):
        # expression with gamma functions or non-integer arguments is
        # automatically passed to gammasimp
        expr = combsimp(expr)

    if expr.has(Sum):
        expr = sum_simplify(expr, **kwargs)

    if expr.has(Integral):
        expr = expr.xreplace({
            i: factor_terms(i) for i in expr.atoms(Integral)})

    if expr.has(Product):
        expr = product_simplify(expr, **kwargs)

    from sympy.physics.units import Quantity

    if expr.has(Quantity):
        from sympy.physics.units.util import quantity_simplify
        expr = quantity_simplify(expr)

    short = shorter(powsimp(expr, combine='exp', deep=True), powsimp(expr), expr)
    short = shorter(short, cancel(short))
    short = shorter(short, factor_terms(short), expand_power_exp(expand_mul(short)))
    if short.has(TrigonometricFunction, HyperbolicFunction, ExpBase, exp):
        short = exptrigsimp(short)

    # get rid of hollow 2-arg Mul factorization
    hollow_mul = Transform(
        lambda x: Mul(*x.args),
        lambda x:
        x.is_Mul and
        len(x.args) == 2 and
        x.args[0].is_Number and
        x.args[1].is_Add and
        x.is_commutative)
    expr = short.xreplace(hollow_mul)

    numer, denom = expr.as_numer_denom()
    if denom.is_Add:
        n, d = fraction(radsimp(1/denom, symbolic=False, max_terms=1))
        if n is not S.One:
            expr = (numer*n).expand()/d

    if expr.could_extract_minus_sign():
        n, d = fraction(expr)
        if d != 0:
            expr = signsimp(-n/(-d))

    if measure(expr) > ratio*measure(original_expr):
        expr = original_expr

    # restore floats
    if floats and rational is None:
        expr = nfloat(expr, exponent=False)

    return done(expr)


def sum_simplify(s, **kwargs):
    """Main function for Sum simplification"""
    if not isinstance(s, Add):
        s = s.xreplace({a: sum_simplify(a, **kwargs)
            for a in s.atoms(Add) if a.has(Sum)})
    s = expand(s)
    if not isinstance(s, Add):
        return s

    terms = s.args
    s_t = [] # Sum Terms
    o_t = [] # Other Terms

    for term in terms:
        sum_terms, other = sift(Mul.make_args(term),
            lambda i: isinstance(i, Sum), binary=True)
        if not sum_terms:
            o_t.append(term)
            continue
        other = [Mul(*other)]
        s_t.append(Mul(*(other + [s._eval_simplify(**kwargs) for s in sum_terms])))

    result = Add(sum_combine(s_t), *o_t)

    return result


def sum_combine(s_t):
    """Helper function for Sum simplification

       Attempts to simplify a list of sums, by combining limits / sum function's
       returns the simplified sum
    """
    used = [False] * len(s_t)

    for method in range(2):
        for i, s_term1 in enumerate(s_t):
            if not used[i]:
                for j, s_term2 in enumerate(s_t):
                    if not used[j] and i != j:
                        temp = sum_add(s_term1, s_term2, method)
                        if isinstance(temp, (Sum, Mul)):
                            s_t[i] = temp
                            s_term1 = s_t[i]
                            used[j] = True

    result = S.Zero
    for i, s_term in enumerate(s_t):
        if not used[i]:
            result = Add(result, s_term)

    return result

def factor_sum(self, limits=None, radical=False, clear=False, fraction=False, sign=True):
    """Return Sum with constant factors extracted.

    If ``limits`` is specified then ``self`` is the summand; the other
    keywords are passed to ``factor_terms``.

    Examples
    ========

    >>> from sympy import Sum
    >>> from sympy.abc import x, y
    >>> from sympy.simplify.simplify import factor_sum
    >>> s = Sum(x*y, (x, 1, 3))
    >>> factor_sum(s)
    y*Sum(x, (x, 1, 3))
    >>> factor_sum(s.function, s.limits)
    y*Sum(x, (x, 1, 3))
    """

    # XXX deprecate in favor of direct call to factor_terms
    kwargs = {"radical": radical, "clear": clear,
        "fraction": fraction, "sign": sign}
    expr = Sum(self, *limits) if limits else self
    return factor_terms(expr, **kwargs)


def sum_add(self, other, method=0):
    """Helper function for Sum simplification"""
    #we know this is something in terms of a constant * a sum
    #so we temporarily put the constants inside for simplification
    #then simplify the result
    def __refactor(val):
        args = Mul.make_args(val)
        sumv = next(x for x in args if isinstance(x, Sum))
        constant = Mul(*[x for x in args if x != sumv])
        return Sum(constant * sumv.function, *sumv.limits)

    if isinstance(self, Mul):
        rself = __refactor(self)
    else:
        rself = self

    if isinstance(other, Mul):
        rother = __refactor(other)
    else:
        rother = other

    if type(rself) is type(rother):
        if method == 0:
            if rself.limits == rother.limits:
                return factor_sum(Sum(rself.function + rother.function, *rself.limits))
        elif method == 1:
            if simplify(rself.function - rother.function) == 0:
                if len(rself.limits) == len(rother.limits) == 1:
                    i = rself.limits[0][0]
                    x1 = rself.limits[0][1]
                    y1 = rself.limits[0][2]
                    j = rother.limits[0][0]
                    x2 = rother.limits[0][1]
                    y2 = rother.limits[0][2]

                    if i == j:
                        if x2 == y1 + 1:
                            return factor_sum(Sum(rself.function, (i, x1, y2)))
                        elif x1 == y2 + 1:
                            return factor_sum(Sum(rself.function, (i, x2, y1)))

    return Add(self, other)


def product_simplify(s, **kwargs):
    """Main function for Product simplification"""
    terms = Mul.make_args(s)
    p_t = [] # Product Terms
    o_t = [] # Other Terms

    deep = kwargs.get('deep', True)
    for term in terms:
        if isinstance(term, Product):
            if deep:
                p_t.append(Product(term.function.simplify(**kwargs),
                                   *term.limits))
            else:
                p_t.append(term)
        else:
            o_t.append(term)

    used = [False] * len(p_t)

    for method in range(2):
        for i, p_term1 in enumerate(p_t):
            if not used[i]:
                for j, p_term2 in enumerate(p_t):
                    if not used[j] and i != j:
                        tmp_prod = product_mul(p_term1, p_term2, method)
                        if isinstance(tmp_prod, Product):
                            p_t[i] = tmp_prod
                            used[j] = True

    result = Mul(*o_t)

    for i, p_term in enumerate(p_t):
        if not used[i]:
            result = Mul(result, p_term)

    return result


def product_mul(self, other, method=0):
    """Helper function for Product simplification"""
    if type(self) is type(other):
        if method == 0:
            if self.limits == other.limits:
                return Product(self.function * other.function, *self.limits)
        elif method == 1:
            if simplify(self.function - other.function) == 0:
                if len(self.limits) == len(other.limits) == 1:
                    i = self.limits[0][0]
                    x1 = self.limits[0][1]
                    y1 = self.limits[0][2]
                    j = other.limits[0][0]
                    x2 = other.limits[0][1]
                    y2 = other.limits[0][2]

                    if i == j:
                        if x2 == y1 + 1:
                            return Product(self.function, (i, x1, y2))
                        elif x1 == y2 + 1:
                            return Product(self.function, (i, x2, y1))

    return Mul(self, other)


def _nthroot_solve(p, n, prec):
    """
     helper function for ``nthroot``
     It denests ``p**Rational(1, n)`` using its minimal polynomial
    """
    from sympy.solvers import solve
    while n % 2 == 0:
        p = sqrtdenest(sqrt(p))
        n = n // 2
    if n == 1:
        return p
    pn = p**Rational(1, n)
    x = Symbol('x')
    f = _minimal_polynomial_sq(p, n, x)
    if f is None:
        return None
    sols = solve(f, x)
    for sol in sols:
        if abs(sol - pn).n() < 1./10**prec:
            sol = sqrtdenest(sol)
            if _mexpand(sol**n) == p:
                return sol


def logcombine(expr, force=False):
    """
    Takes logarithms and combines them using the following rules:

    - log(x) + log(y) == log(x*y) if both are positive
    - a*log(x) == log(x**a) if x is positive and a is real

    If ``force`` is ``True`` then the assumptions above will be assumed to hold if
    there is no assumption already in place on a quantity. For example, if
    ``a`` is imaginary or the argument negative, force will not perform a
    combination but if ``a`` is a symbol with no assumptions the change will
    take place.

    Examples
    ========

    >>> from sympy import Symbol, symbols, log, logcombine, I
    >>> from sympy.abc import a, x, y, z
    >>> logcombine(a*log(x) + log(y) - log(z))
    a*log(x) + log(y) - log(z)
    >>> logcombine(a*log(x) + log(y) - log(z), force=True)
    log(x**a*y/z)
    >>> x,y,z = symbols('x,y,z', positive=True)
    >>> a = Symbol('a', real=True)
    >>> logcombine(a*log(x) + log(y) - log(z))
    log(x**a*y/z)

    The transformation is limited to factors and/or terms that
    contain logs, so the result depends on the initial state of
    expansion:

    >>> eq = (2 + 3*I)*log(x)
    >>> logcombine(eq, force=True) == eq
    True
    >>> logcombine(eq.expand(), force=True)
    log(x**2) + I*log(x**3)

    See Also
    ========

    posify: replace all symbols with symbols having positive assumptions
    sympy.core.function.expand_log: expand the logarithms of products
        and powers; the opposite of logcombine

    """

    def f(rv):
        if not (rv.is_Add or rv.is_Mul):
            return rv

        def gooda(a):
            # bool to tell whether the leading ``a`` in ``a*log(x)``
            # could appear as log(x**a)
            return (a is not S.NegativeOne and  # -1 *could* go, but we disallow
                (a.is_extended_real or force and a.is_extended_real is not False))

        def goodlog(l):
            # bool to tell whether log ``l``'s argument can combine with others
            a = l.args[0]
            return a.is_positive or force and a.is_nonpositive is not False

        other = []
        logs = []
        log1 = defaultdict(list)
        for a in Add.make_args(rv):
            if isinstance(a, log) and goodlog(a):
                log1[()].append(([], a))
            elif not a.is_Mul:
                other.append(a)
            else:
                ot = []
                co = []
                lo = []
                for ai in a.args:
                    if ai.is_Rational and ai < 0:
                        ot.append(S.NegativeOne)
                        co.append(-ai)
                    elif isinstance(ai, log) and goodlog(ai):
                        lo.append(ai)
                    elif gooda(ai):
                        co.append(ai)
                    else:
                        ot.append(ai)
                if len(lo) > 1:
                    logs.append((ot, co, lo))
                elif lo:
                    log1[tuple(ot)].append((co, lo[0]))
                else:
                    other.append(a)

        # if there is only one log in other, put it with the
        # good logs
        if len(other) == 1 and isinstance(other[0], log):
            log1[()].append(([], other.pop()))
        # if there is only one log at each coefficient and none have
        # an exponent to place inside the log then there is nothing to do
        if not logs and all(len(log1[k]) == 1 and log1[k][0] == [] for k in log1):
            return rv

        # collapse multi-logs as far as possible in a canonical way
        # TODO: see if x*log(a)+x*log(a)*log(b) -> x*log(a)*(1+log(b))?
        # -- in this case, it's unambiguous, but if it were were a log(c) in
        # each term then it's arbitrary whether they are grouped by log(a) or
        # by log(c). So for now, just leave this alone; it's probably better to
        # let the user decide
        for o, e, l in logs:
            l = list(ordered(l))
            e = log(l.pop(0).args[0]**Mul(*e))
            while l:
                li = l.pop(0)
                e = log(li.args[0]**e)
            c, l = Mul(*o), e
            if isinstance(l, log):  # it should be, but check to be sure
                log1[(c,)].append(([], l))
            else:
                other.append(c*l)

        # logs that have the same coefficient can multiply
        for k in list(log1.keys()):
            log1[Mul(*k)] = log(logcombine(Mul(*[
                l.args[0]**Mul(*c) for c, l in log1.pop(k)]),
                force=force), evaluate=False)

        # logs that have oppositely signed coefficients can divide
        for k in ordered(list(log1.keys())):
            if k not in log1:  # already popped as -k
                continue
            if -k in log1:
                # figure out which has the minus sign; the one with
                # more op counts should be the one
                num, den = k, -k
                if num.count_ops() > den.count_ops():
                    num, den = den, num
                other.append(
                    num*log(log1.pop(num).args[0]/log1.pop(den).args[0],
                            evaluate=False))
            else:
                other.append(k*log1.pop(k))

        return Add(*other)

    return _bottom_up(expr, f)


def inversecombine(expr):
    """Simplify the composition of a function and its inverse.

    Explanation
    ===========

    No attention is paid to whether the inverse is a left inverse or a
    right inverse; thus, the result will in general not be equivalent
    to the original expression.

    Examples
    ========

    >>> from sympy.simplify.simplify import inversecombine
    >>> from sympy import asin, sin, log, exp
    >>> from sympy.abc import x
    >>> inversecombine(asin(sin(x)))
    x
    >>> inversecombine(2*log(exp(3*x)))
    6*x
    """

    def f(rv):
        if isinstance(rv, log):
            if isinstance(rv.args[0], exp) or (rv.args[0].is_Pow and rv.args[0].base == S.Exp1):
                rv = rv.args[0].exp
        elif rv.is_Function and hasattr(rv, "inverse"):
            if (len(rv.args) == 1 and len(rv.args[0].args) == 1 and
               isinstance(rv.args[0], rv.inverse(argindex=1))):
                rv = rv.args[0].args[0]
        if rv.is_Pow and rv.base == S.Exp1:
            if isinstance(rv.exp, log):
                rv = rv.exp.args[0]
        return rv

    return _bottom_up(expr, f)


def kroneckersimp(expr):
    """
    Simplify expressions with KroneckerDelta.

    The only simplification currently attempted is to identify multiplicative cancellation:

    Examples
    ========

    >>> from sympy import KroneckerDelta, kroneckersimp
    >>> from sympy.abc import i
    >>> kroneckersimp(1 + KroneckerDelta(0, i) * KroneckerDelta(1, i))
    1
    """
    def args_cancel(args1, args2):
        for i1 in range(2):
            for i2 in range(2):
                a1 = args1[i1]
                a2 = args2[i2]
                a3 = args1[(i1 + 1) % 2]
                a4 = args2[(i2 + 1) % 2]
                if Eq(a1, a2) is S.true and Eq(a3, a4) is S.false:
                    return True
        return False

    def cancel_kronecker_mul(m):
        args = m.args
        deltas = [a for a in args if isinstance(a, KroneckerDelta)]
        for delta1, delta2 in subsets(deltas, 2):
            args1 = delta1.args
            args2 = delta2.args
            if args_cancel(args1, args2):
                return S.Zero * m # In case of oo etc
        return m

    if not expr.has(KroneckerDelta):
        return expr

    if expr.has(Piecewise):
        expr = expr.rewrite(KroneckerDelta)

    newexpr = expr
    expr = None

    while newexpr != expr:
        expr = newexpr
        newexpr = expr.replace(lambda e: isinstance(e, Mul), cancel_kronecker_mul)

    return expr


def besselsimp(expr):
    """
    Simplify bessel-type functions.

    Explanation
    ===========

    This routine tries to simplify bessel-type functions. Currently it only
    works on the Bessel J and I functions, however. It works by looking at all
    such functions in turn, and eliminating factors of "I" and "-1" (actually
    their polar equivalents) in front of the argument. Then, functions of
    half-integer order are rewritten using trigonometric functions and
    functions of integer order (> 1) are rewritten using functions
    of low order.  Finally, if the expression was changed, compute
    factorization of the result with factor().

    >>> from sympy import besselj, besseli, besselsimp, polar_lift, I, S
    >>> from sympy.abc import z, nu
    >>> besselsimp(besselj(nu, z*polar_lift(-1)))
    exp(I*pi*nu)*besselj(nu, z)
    >>> besselsimp(besseli(nu, z*polar_lift(-I)))
    exp(-I*pi*nu/2)*besselj(nu, z)
    >>> besselsimp(besseli(S(-1)/2, z))
    sqrt(2)*cosh(z)/(sqrt(pi)*sqrt(z))
    >>> besselsimp(z*besseli(0, z) + z*(besseli(2, z))/2 + besseli(1, z))
    3*z*besseli(0, z)/2
    """
    # TODO
    # - better algorithm?
    # - simplify (cos(pi*b)*besselj(b,z) - besselj(-b,z))/sin(pi*b) ...
    # - use contiguity relations?

    def replacer(fro, to, factors):
        factors = set(factors)

        def repl(nu, z):
            if factors.intersection(Mul.make_args(z)):
                return to(nu, z)
            return fro(nu, z)
        return repl

    def torewrite(fro, to):
        def tofunc(nu, z):
            return fro(nu, z).rewrite(to)
        return tofunc

    def tominus(fro):
        def tofunc(nu, z):
            return exp(I*pi*nu)*fro(nu, exp_polar(-I*pi)*z)
        return tofunc

    orig_expr = expr

    ifactors = [I, exp_polar(I*pi/2), exp_polar(-I*pi/2)]
    expr = expr.replace(
        besselj, replacer(besselj,
        torewrite(besselj, besseli), ifactors))
    expr = expr.replace(
        besseli, replacer(besseli,
        torewrite(besseli, besselj), ifactors))

    minusfactors = [-1, exp_polar(I*pi)]
    expr = expr.replace(
        besselj, replacer(besselj, tominus(besselj), minusfactors))
    expr = expr.replace(
        besseli, replacer(besseli, tominus(besseli), minusfactors))

    z0 = Dummy('z')

    def expander(fro):
        def repl(nu, z):
            if (nu % 1) == S.Half:
                return simplify(trigsimp(unpolarify(
                        fro(nu, z0).rewrite(besselj).rewrite(jn).expand(
                            func=True)).subs(z0, z)))
            elif nu.is_Integer and nu > 1:
                return fro(nu, z).expand(func=True)
            return fro(nu, z)
        return repl

    expr = expr.replace(besselj, expander(besselj))
    expr = expr.replace(bessely, expander(bessely))
    expr = expr.replace(besseli, expander(besseli))
    expr = expr.replace(besselk, expander(besselk))

    def _bessel_simp_recursion(expr):

        def _use_recursion(bessel, expr):
            while True:
                bessels = expr.find(lambda x: isinstance(x, bessel))
                try:
                    for ba in sorted(bessels, key=lambda x: re(x.args[0])):
                        a, x = ba.args
                        bap1 = bessel(a+1, x)
                        bap2 = bessel(a+2, x)
                        if expr.has(bap1) and expr.has(bap2):
                            expr = expr.subs(ba, 2*(a+1)/x*bap1 - bap2)
                            break
                    else:
                        return expr
                except (ValueError, TypeError):
                    return expr
        if expr.has(besselj):
            expr = _use_recursion(besselj, expr)
        if expr.has(bessely):
            expr = _use_recursion(bessely, expr)
        return expr

    expr = _bessel_simp_recursion(expr)
    if expr != orig_expr:
        expr = expr.factor()

    return expr


def nthroot(expr, n, max_len=4, prec=15):
    """
    Compute a real nth-root of a sum of surds.

    Parameters
    ==========

    expr : sum of surds
    n : integer
    max_len : maximum number of surds passed as constants to ``nsimplify``

    Algorithm
    =========

    First ``nsimplify`` is used to get a candidate root; if it is not a
    root the minimal polynomial is computed; the answer is one of its
    roots.

    Examples
    ========

    >>> from sympy.simplify.simplify import nthroot
    >>> from sympy import sqrt
    >>> nthroot(90 + 34*sqrt(7), 3)
    sqrt(7) + 3

    """
    expr = sympify(expr)
    n = sympify(n)
    p = expr**Rational(1, n)
    if not n.is_integer:
        return p
    if not _is_sum_surds(expr):
        return p
    surds = []
    coeff_muls = [x.as_coeff_Mul() for x in expr.args]
    for x, y in coeff_muls:
        if not x.is_rational:
            return p
        if y is S.One:
            continue
        if not (y.is_Pow and y.exp == S.Half and y.base.is_integer):
            return p
        surds.append(y)
    surds.sort()
    surds = surds[:max_len]
    if expr < 0 and n % 2 == 1:
        p = (-expr)**Rational(1, n)
        a = nsimplify(p, constants=surds)
        res = a if _mexpand(a**n) == _mexpand(-expr) else p
        return -res
    a = nsimplify(p, constants=surds)
    if _mexpand(a) is not _mexpand(p) and _mexpand(a**n) == _mexpand(expr):
        return _mexpand(a)
    expr = _nthroot_solve(expr, n, prec)
    if expr is None:
        return p
    return expr


def nsimplify(expr, constants=(), tolerance=None, full=False, rational=None,
    rational_conversion='base10'):
    """
    Find a simple representation for a number or, if there are free symbols or
    if ``rational=True``, then replace Floats with their Rational equivalents. If
    no change is made and rational is not False then Floats will at least be
    converted to Rationals.

    Explanation
    ===========

    For numerical expressions, a simple formula that numerically matches the
    given numerical expression is sought (and the input should be possible
    to evalf to a precision of at least 30 digits).

    Optionally, a list of (rationally independent) constants to
    include in the formula may be given.

    A lower tolerance may be set to find less exact matches. If no tolerance
    is given then the least precise value will set the tolerance (e.g. Floats
    default to 15 digits of precision, so would be tolerance=10**-15).

    With ``full=True``, a more extensive search is performed
    (this is useful to find simpler numbers when the tolerance
    is set low).

    When converting to rational, if rational_conversion='base10' (the default), then
    convert floats to rationals using their base-10 (string) representation.
    When rational_conversion='exact' it uses the exact, base-2 representation.

    Examples
    ========

    >>> from sympy import nsimplify, sqrt, GoldenRatio, exp, I, pi
    >>> nsimplify(4/(1+sqrt(5)), [GoldenRatio])
    -2 + 2*GoldenRatio
    >>> nsimplify((1/(exp(3*pi*I/5)+1)))
    1/2 - I*sqrt(sqrt(5)/10 + 1/4)
    >>> nsimplify(I**I, [pi])
    exp(-pi/2)
    >>> nsimplify(pi, tolerance=0.01)
    22/7

    >>> nsimplify(0.333333333333333, rational=True, rational_conversion='exact')
    6004799503160655/18014398509481984
    >>> nsimplify(0.333333333333333, rational=True)
    1/3

    See Also
    ========

    sympy.core.function.nfloat

    """
    try:
        return sympify(as_int(expr))
    except (TypeError, ValueError):
        pass
    expr = sympify(expr).xreplace({
        Float('inf'): S.Infinity,
        Float('-inf'): S.NegativeInfinity,
        })
    if expr is S.Infinity or expr is S.NegativeInfinity:
        return expr
    if rational or expr.free_symbols:
        return _real_to_rational(expr, tolerance, rational_conversion)

    # SymPy's default tolerance for Rationals is 15; other numbers may have
    # lower tolerances set, so use them to pick the largest tolerance if None
    # was given
    if tolerance is None:
        tolerance = 10**-min([15] +
             [mpmath.libmp.libmpf.prec_to_dps(n._prec)
             for n in expr.atoms(Float)])
    # XXX should prec be set independent of tolerance or should it be computed
    # from tolerance?
    prec = 30
    bprec = int(prec*3.33)

    constants_dict = {}
    for constant in constants:
        constant = sympify(constant)
        v = constant.evalf(prec)
        if not v.is_Float:
            raise ValueError("constants must be real-valued")
        constants_dict[str(constant)] = v._to_mpmath(bprec)

    exprval = expr.evalf(prec, chop=True)
    re, im = exprval.as_real_imag()

    # safety check to make sure that this evaluated to a number
    if not (re.is_Number and im.is_Number):
        return expr

    def nsimplify_real(x):
        orig = mpmath.mp.dps
        xv = x._to_mpmath(bprec)
        try:
            # We'll be happy with low precision if a simple fraction
            if not (tolerance or full):
                mpmath.mp.dps = 15
                rat = mpmath.pslq([xv, 1])
                if rat is not None:
                    return Rational(-int(rat[1]), int(rat[0]))
            mpmath.mp.dps = prec
            newexpr = mpmath.identify(xv, constants=constants_dict,
                tol=tolerance, full=full)
            if not newexpr:
                raise ValueError
            if full:
                newexpr = newexpr[0]
            expr = sympify(newexpr)
            if x and not expr:  # don't let x become 0
                raise ValueError
            if expr.is_finite is False and xv not in [mpmath.inf, mpmath.ninf]:
                raise ValueError
            return expr
        finally:
            # even though there are returns above, this is executed
            # before leaving
            mpmath.mp.dps = orig
    try:
        if re:
            re = nsimplify_real(re)
        if im:
            im = nsimplify_real(im)
    except ValueError:
        if rational is None:
            return _real_to_rational(expr, rational_conversion=rational_conversion)
        return expr

    rv = re + im*S.ImaginaryUnit
    # if there was a change or rational is explicitly not wanted
    # return the value, else return the Rational representation
    if rv != expr or rational is False:
        return rv
    return _real_to_rational(expr, rational_conversion=rational_conversion)


def _real_to_rational(expr, tolerance=None, rational_conversion='base10'):
    """
    Replace all reals in expr with rationals.

    Examples
    ========

    >>> from sympy.simplify.simplify import _real_to_rational
    >>> from sympy.abc import x

    >>> _real_to_rational(.76 + .1*x**.5)
    sqrt(x)/10 + 19/25

    If rational_conversion='base10', this uses the base-10 string. If
    rational_conversion='exact', the exact, base-2 representation is used.

    >>> _real_to_rational(0.333333333333333, rational_conversion='exact')
    6004799503160655/18014398509481984
    >>> _real_to_rational(0.333333333333333)
    1/3

    """
    expr = _sympify(expr)
    inf = Float('inf')
    p = expr
    reps = {}
    reduce_num = None
    if tolerance is not None and tolerance < 1:
        reduce_num = ceiling(1/tolerance)
    for fl in p.atoms(Float):
        key = fl
        if reduce_num is not None:
            r = Rational(fl).limit_denominator(reduce_num)
        elif (tolerance is not None and tolerance >= 1 and
                fl.is_Integer is False):
            r = Rational(tolerance*round(fl/tolerance)
                ).limit_denominator(int(tolerance))
        else:
            if rational_conversion == 'exact':
                r = Rational(fl)
                reps[key] = r
                continue
            elif rational_conversion != 'base10':
                raise ValueError("rational_conversion must be 'base10' or 'exact'")

            r = nsimplify(fl, rational=False)
            # e.g. log(3).n() -> log(3) instead of a Rational
            if fl and not r:
                r = Rational(fl)
            elif not r.is_Rational:
                if fl in (inf, -inf):
                    r = S.ComplexInfinity
                elif fl < 0:
                    fl = -fl
                    d = Pow(10, int(mpmath.log(fl)/mpmath.log(10)))
                    r = -Rational(str(fl/d))*d
                elif fl > 0:
                    d = Pow(10, int(mpmath.log(fl)/mpmath.log(10)))
                    r = Rational(str(fl/d))*d
                else:
                    r = S.Zero
        reps[key] = r
    return p.subs(reps, simultaneous=True)


def clear_coefficients(expr, rhs=S.Zero):
    """Return `p, r` where `p` is the expression obtained when Rational
    additive and multiplicative coefficients of `expr` have been stripped
    away in a naive fashion (i.e. without simplification). The operations
    needed to remove the coefficients will be applied to `rhs` and returned
    as `r`.

    Examples
    ========

    >>> from sympy.simplify.simplify import clear_coefficients
    >>> from sympy.abc import x, y
    >>> from sympy import Dummy
    >>> expr = 4*y*(6*x + 3)
    >>> clear_coefficients(expr - 2)
    (y*(2*x + 1), 1/6)

    When solving 2 or more expressions like `expr = a`,
    `expr = b`, etc..., it is advantageous to provide a Dummy symbol
    for `rhs` and  simply replace it with `a`, `b`, etc... in `r`.

    >>> rhs = Dummy('rhs')
    >>> clear_coefficients(expr, rhs)
    (y*(2*x + 1), _rhs/12)
    >>> _[1].subs(rhs, 2)
    1/6
    """
    was = None
    free = expr.free_symbols
    if expr.is_Rational:
        return (S.Zero, rhs - expr)
    while expr and was != expr:
        was = expr
        m, expr = (
            expr.as_content_primitive()
            if free else
            factor_terms(expr).as_coeff_Mul(rational=True))
        rhs /= m
        c, expr = expr.as_coeff_Add(rational=True)
        rhs -= c
    expr = signsimp(expr, evaluate = False)
    if expr.could_extract_minus_sign():
        expr = -expr
        rhs = -rhs
    return expr, rhs

def nc_simplify(expr, deep=True):
    '''
    Simplify a non-commutative expression composed of multiplication
    and raising to a power by grouping repeated subterms into one power.
    Priority is given to simplifications that give the fewest number
    of arguments in the end (for example, in a*b*a*b*c*a*b*c simplifying
    to (a*b)**2*c*a*b*c gives 5 arguments while a*b*(a*b*c)**2 has 3).
    If ``expr`` is a sum of such terms, the sum of the simplified terms
    is returned.

    Keyword argument ``deep`` controls whether or not subexpressions
    nested deeper inside the main expression are simplified. See examples
    below. Setting `deep` to `False` can save time on nested expressions
    that do not need simplifying on all levels.

    Examples
    ========

    >>> from sympy import symbols
    >>> from sympy.simplify.simplify import nc_simplify
    >>> a, b, c = symbols("a b c", commutative=False)
    >>> nc_simplify(a*b*a*b*c*a*b*c)
    a*b*(a*b*c)**2
    >>> expr = a**2*b*a**4*b*a**4
    >>> nc_simplify(expr)
    a**2*(b*a**4)**2
    >>> nc_simplify(a*b*a*b*c**2*(a*b)**2*c**2)
    ((a*b)**2*c**2)**2
    >>> nc_simplify(a*b*a*b + 2*a*c*a**2*c*a**2*c*a)
    (a*b)**2 + 2*(a*c*a)**3
    >>> nc_simplify(b**-1*a**-1*(a*b)**2)
    a*b
    >>> nc_simplify(a**-1*b**-1*c*a)
    (b*a)**(-1)*c*a
    >>> expr = (a*b*a*b)**2*a*c*a*c
    >>> nc_simplify(expr)
    (a*b)**4*(a*c)**2
    >>> nc_simplify(expr, deep=False)
    (a*b*a*b)**2*(a*c)**2

    '''
    if isinstance(expr, MatrixExpr):
        expr = expr.doit(inv_expand=False)
        _Add, _Mul, _Pow, _Symbol = MatAdd, MatMul, MatPow, MatrixSymbol
    else:
        _Add, _Mul, _Pow, _Symbol = Add, Mul, Pow, Symbol

    # =========== Auxiliary functions ========================
    def _overlaps(args):
        # Calculate a list of lists m such that m[i][j] contains the lengths
        # of all possible overlaps between args[:i+1] and args[i+1+j:].
        # An overlap is a suffix of the prefix that matches a prefix
        # of the suffix.
        # For example, let expr=c*a*b*a*b*a*b*a*b. Then m[3][0] contains
        # the lengths of overlaps of c*a*b*a*b with a*b*a*b. The overlaps
        # are a*b*a*b, a*b and the empty word so that m[3][0]=[4,2,0].
        # All overlaps rather than only the longest one are recorded
        # because this information helps calculate other overlap lengths.
        m = [[([1, 0] if a == args[0] else [0]) for a in args[1:]]]
        for i in range(1, len(args)):
            overlaps = []
            j = 0
            for j in range(len(args) - i - 1):
                overlap = []
                for v in m[i-1][j+1]:
                    if j + i + 1 + v < len(args) and args[i] == args[j+i+1+v]:
                        overlap.append(v + 1)
                overlap += [0]
                overlaps.append(overlap)
            m.append(overlaps)
        return m

    def _reduce_inverses(_args):
        # replace consecutive negative powers by an inverse
        # of a product of positive powers, e.g. a**-1*b**-1*c
        # will simplify to (a*b)**-1*c;
        # return that new args list and the number of negative
        # powers in it (inv_tot)
        inv_tot = 0 # total number of inverses
        inverses = []
        args = []
        for arg in _args:
            if isinstance(arg, _Pow) and arg.args[1].is_extended_negative:
                inverses = [arg**-1] + inverses
                inv_tot += 1
            else:
                if len(inverses) == 1:
                    args.append(inverses[0]**-1)
                elif len(inverses) > 1:
                    args.append(_Pow(_Mul(*inverses), -1))
                    inv_tot -= len(inverses) - 1
                inverses = []
                args.append(arg)
        if inverses:
            args.append(_Pow(_Mul(*inverses), -1))
            inv_tot -= len(inverses) - 1
        return inv_tot, tuple(args)

    def get_score(s):
        # compute the number of arguments of s
        # (including in nested expressions) overall
        # but ignore exponents
        if isinstance(s, _Pow):
            return get_score(s.args[0])
        elif isinstance(s, (_Add, _Mul)):
            return sum(get_score(a) for a in s.args)
        return 1

    def compare(s, alt_s):
        # compare two possible simplifications and return a
        # "better" one
        if s != alt_s and get_score(alt_s) < get_score(s):
            return alt_s
        return s
    # ========================================================

    if not isinstance(expr, (_Add, _Mul, _Pow)) or expr.is_commutative:
        return expr
    args = expr.args[:]
    if isinstance(expr, _Pow):
        if deep:
            return _Pow(nc_simplify(args[0]), args[1]).doit()
        else:
            return expr
    elif isinstance(expr, _Add):
        return _Add(*[nc_simplify(a, deep=deep) for a in args]).doit()
    else:
        # get the non-commutative part
        c_args, args = expr.args_cnc()
        com_coeff = Mul(*c_args)
        if not equal_valued(com_coeff, 1):
            return com_coeff*nc_simplify(expr/com_coeff, deep=deep)

    inv_tot, args = _reduce_inverses(args)
    # if most arguments are negative, work with the inverse
    # of the expression, e.g. a**-1*b*a**-1*c**-1 will become
    # (c*a*b**-1*a)**-1 at the end so can work with c*a*b**-1*a
    invert = False
    if inv_tot > len(args)/2:
        invert = True
        args = [a**-1 for a in args[::-1]]

    if deep:
        args = tuple(nc_simplify(a) for a in args)

    m = _overlaps(args)

    # simps will be {subterm: end} where `end` is the ending
    # index of a sequence of repetitions of subterm;
    # this is for not wasting time with subterms that are part
    # of longer, already considered sequences
    simps = {}

    post = 1
    pre = 1

    # the simplification coefficient is the number of
    # arguments by which contracting a given sequence
    # would reduce the word; e.g. in a*b*a*b*c*a*b*c,
    # contracting a*b*a*b to (a*b)**2 removes 3 arguments
    # while a*b*c*a*b*c to (a*b*c)**2 removes 6. It's
    # better to contract the latter so simplification
    # with a maximum simplification coefficient will be chosen
    max_simp_coeff = 0
    simp = None # information about future simplification

    for i in range(1, len(args)):
        simp_coeff = 0
        l = 0 # length of a subterm
        p = 0 # the power of a subterm
        if i < len(args) - 1:
            rep = m[i][0]
        start = i # starting index of the repeated sequence
        end = i+1 # ending index of the repeated sequence
        if i == len(args)-1 or rep == [0]:
            # no subterm is repeated at this stage, at least as
            # far as the arguments are concerned - there may be
            # a repetition if powers are taken into account
            if (isinstance(args[i], _Pow) and
                            not isinstance(args[i].args[0], _Symbol)):
                subterm = args[i].args[0].args
                l = len(subterm)
                if args[i-l:i] == subterm:
                    # e.g. a*b in a*b*(a*b)**2 is not repeated
                    # in args (= [a, b, (a*b)**2]) but it
                    # can be matched here
                    p += 1
                    start -= l
                if args[i+1:i+1+l] == subterm:
                    # e.g. a*b in (a*b)**2*a*b
                    p += 1
                    end += l
            if p:
                p += args[i].args[1]
            else:
                continue
        else:
            l = rep[0] # length of the longest repeated subterm at this point
            start -= l - 1
            subterm = args[start:end]
            p = 2
            end += l

        if subterm in simps and simps[subterm] >= start:
            # the subterm is part of a sequence that
            # has already been considered
            continue

        # count how many times it's repeated
        while end < len(args):
            if l in m[end-1][0]:
                p += 1
                end += l
            elif isinstance(args[end], _Pow) and args[end].args[0].args == subterm:
                # for cases like a*b*a*b*(a*b)**2*a*b
                p += args[end].args[1]
                end += 1
            else:
                break

        # see if another match can be made, e.g.
        # for b*a**2 in b*a**2*b*a**3 or a*b in
        # a**2*b*a*b

        pre_exp = 0
        pre_arg = 1
        if start - l >= 0 and args[start-l+1:start] == subterm[1:]:
            if isinstance(subterm[0], _Pow):
                pre_arg = subterm[0].args[0]
                exp = subterm[0].args[1]
            else:
                pre_arg = subterm[0]
                exp = 1
            if isinstance(args[start-l], _Pow) and args[start-l].args[0] == pre_arg:
                pre_exp = args[start-l].args[1] - exp
                start -= l
                p += 1
            elif args[start-l] == pre_arg:
                pre_exp = 1 - exp
                start -= l
                p += 1

        post_exp = 0
        post_arg = 1
        if end + l - 1 < len(args) and args[end:end+l-1] == subterm[:-1]:
            if isinstance(subterm[-1], _Pow):
                post_arg = subterm[-1].args[0]
                exp = subterm[-1].args[1]
            else:
                post_arg = subterm[-1]
                exp = 1
            if isinstance(args[end+l-1], _Pow) and args[end+l-1].args[0] == post_arg:
                post_exp = args[end+l-1].args[1] - exp
                end += l
                p += 1
            elif args[end+l-1] == post_arg:
                post_exp = 1 - exp
                end += l
                p += 1

        # Consider a*b*a**2*b*a**2*b*a:
        # b*a**2 is explicitly repeated, but note
        # that in this case a*b*a is also repeated
        # so there are two possible simplifications:
        # a*(b*a**2)**3*a**-1 or (a*b*a)**3
        # The latter is obviously simpler.
        # But in a*b*a**2*b**2*a**2 the simplifications are
        # a*(b*a**2)**2 and (a*b*a)**3*a in which case
        # it's better to stick with the shorter subterm
        if post_exp and exp % 2 == 0 and start > 0:
            exp = exp/2
            _pre_exp = 1
            _post_exp = 1
            if isinstance(args[start-1], _Pow) and args[start-1].args[0] == post_arg:
                _post_exp = post_exp + exp
                _pre_exp = args[start-1].args[1] - exp
            elif args[start-1] == post_arg:
                _post_exp = post_exp + exp
                _pre_exp = 1 - exp
            if _pre_exp == 0 or _post_exp == 0:
                if not pre_exp:
                    start -= 1
                post_exp = _post_exp
                pre_exp = _pre_exp
                pre_arg = post_arg
                subterm = (post_arg**exp,) + subterm[:-1] + (post_arg**exp,)

        simp_coeff += end-start

        if post_exp:
            simp_coeff -= 1
        if pre_exp:
            simp_coeff -= 1

        simps[subterm] = end

        if simp_coeff > max_simp_coeff:
            max_simp_coeff = simp_coeff
            simp = (start, _Mul(*subterm), p, end, l)
            pre = pre_arg**pre_exp
            post = post_arg**post_exp

    if simp:
        subterm = _Pow(nc_simplify(simp[1], deep=deep), simp[2])
        pre = nc_simplify(_Mul(*args[:simp[0]])*pre, deep=deep)
        post = post*nc_simplify(_Mul(*args[simp[3]:]), deep=deep)
        simp = pre*subterm*post
        if pre != 1 or post != 1:
            # new simplifications may be possible but no need
            # to recurse over arguments
            simp = nc_simplify(simp, deep=False)
    else:
        simp = _Mul(*args)

    if invert:
        simp = _Pow(simp, -1)

    # see if factor_nc(expr) is simplified better
    if not isinstance(expr, MatrixExpr):
        f_expr = factor_nc(expr)
        if f_expr != expr:
            alt_simp = nc_simplify(f_expr, deep=deep)
            simp = compare(simp, alt_simp)
    else:
        simp = simp.doit(inv_expand=False)
    return simp


def dotprodsimp(expr, withsimp=False):
    """Simplification for a sum of products targeted at the kind of blowup that
    occurs during summation of products. Intended to reduce expression blowup
    during matrix multiplication or other similar operations. Only works with
    algebraic expressions and does not recurse into non.

    Parameters
    ==========

    withsimp : bool, optional
        Specifies whether a flag should be returned along with the expression
        to indicate roughly whether simplification was successful. It is used
        in ``MatrixArithmetic._eval_pow_by_recursion`` to avoid attempting to
        simplify an expression repetitively which does not simplify.
    """

    def count_ops_alg(expr):
        """Optimized count algebraic operations with no recursion into
        non-algebraic args that ``core.function.count_ops`` does. Also returns
        whether rational functions may be present according to negative
        exponents of powers or non-number fractions.

        Returns
        =======

        ops, ratfunc : int, bool
            ``ops`` is the number of algebraic operations starting at the top
            level expression (not recursing into non-alg children). ``ratfunc``
            specifies whether the expression MAY contain rational functions
            which ``cancel`` MIGHT optimize.
        """

        ops     = 0
        args    = [expr]
        ratfunc = False

        while args:
            a = args.pop()

            if not isinstance(a, Basic):
                continue

            if a.is_Rational:
                if a is not S.One: # -1/3 = NEG + DIV
                    ops += bool (a.p < 0) + bool (a.q != 1)

            elif a.is_Mul:
                if a.could_extract_minus_sign():
                    ops += 1
                    if a.args[0] is S.NegativeOne:
                        a = a.as_two_terms()[1]
                    else:
                        a = -a

                n, d = fraction(a)

                if n.is_Integer:
                    ops += 1 + bool (n < 0)
                    args.append(d) # won't be -Mul but could be Add

                elif d is not S.One:
                    if not d.is_Integer:
                        args.append(d)
                        ratfunc=True

                    ops += 1
                    args.append(n) # could be -Mul

                else:
                    ops += len(a.args) - 1
                    args.extend(a.args)

            elif a.is_Add:
                laargs = len(a.args)
                negs   = 0

                for ai in a.args:
                    if ai.could_extract_minus_sign():
                        negs += 1
                        ai    = -ai
                    args.append(ai)

                ops += laargs - (negs != laargs) # -x - y = NEG + SUB

            elif a.is_Pow:
                ops += 1
                args.append(a.base)

                if not ratfunc:
                    ratfunc = a.exp.is_negative is not False

        return ops, ratfunc

    def nonalg_subs_dummies(expr, dummies):
        """Substitute dummy variables for non-algebraic expressions to avoid
        evaluation of non-algebraic terms that ``polys.polytools.cancel`` does.
        """

        if not expr.args:
            return expr

        if expr.is_Add or expr.is_Mul or expr.is_Pow:
            args = None

            for i, a in enumerate(expr.args):
                c = nonalg_subs_dummies(a, dummies)

                if c is a:
                    continue

                if args is None:
                    args = list(expr.args)

                args[i] = c

            if args is None:
                return expr

            return expr.func(*args)

        return dummies.setdefault(expr, Dummy())

    simplified = False # doesn't really mean simplified, rather "can simplify again"

    if isinstance(expr, Basic) and (expr.is_Add or expr.is_Mul or expr.is_Pow):
        expr2 = expr.expand(deep=True, modulus=None, power_base=False,
            power_exp=False, mul=True, log=False, multinomial=True, basic=False)

        if expr2 != expr:
            expr       = expr2
            simplified = True

        exprops, ratfunc = count_ops_alg(expr)

        if exprops >= 6: # empirically tested cutoff for expensive simplification
            if ratfunc:
                dummies = {}
                expr2   = nonalg_subs_dummies(expr, dummies)

                if expr2 is expr or count_ops_alg(expr2)[0] >= 6: # check again after substitution
                    expr3 = cancel(expr2)

                    if expr3 != expr2:
                        expr       = expr3.subs([(d, e) for e, d in dummies.items()])
                        simplified = True

        # very special case: x/(x-1) - 1/(x-1) -> 1
        elif (exprops == 5 and expr.is_Add and expr.args [0].is_Mul and
                expr.args [1].is_Mul and expr.args [0].args [-1].is_Pow and
                expr.args [1].args [-1].is_Pow and
                expr.args [0].args [-1].exp is S.NegativeOne and
                expr.args [1].args [-1].exp is S.NegativeOne):

            expr2    = together (expr)
            expr2ops = count_ops_alg(expr2)[0]

            if expr2ops < exprops:
                expr       = expr2
                simplified = True

        else:
            simplified = True

    return (expr, simplified) if withsimp else expr


bottom_up = deprecated(
    """
    Using bottom_up from the sympy.simplify.simplify submodule is
    deprecated.

    Instead, use bottom_up from the top-level sympy namespace, like

        sympy.bottom_up
    """,
    deprecated_since_version="1.10",
    active_deprecations_target="deprecated-traversal-functions-moved",
)(_bottom_up)


# XXX: This function really should either be private API or exported in the
# top-level sympy/__init__.py
walk = deprecated(
    """
    Using walk from the sympy.simplify.simplify submodule is
    deprecated.

    Instead, use walk from sympy.core.traversal.walk
    """,
    deprecated_since_version="1.10",
    active_deprecations_target="deprecated-traversal-functions-moved",
)(_walk)