def sfield(exprs, *symbols, **options):
    """Construct a field deriving generators and domain
    from options and input expressions.

    Parameters
    ==========

    exprs : :class:`Expr` or sequence of :class:`Expr` (sympifiable)
    symbols : sequence of :class:`Symbol`/:class:`Expr`
    options : keyword arguments understood by :class:`Options`

    Examples
    ========

    >>> from sympy.core import symbols
    >>> from sympy.functions import exp, log
    >>> from sympy.polys.fields import sfield

    >>> x = symbols("x")
    >>> K, f = sfield((x*log(x) + 4*x**2)*exp(1/x + log(x)/3)/x**2)
    >>> K
    Rational function field in x, exp(1/x), log(x), x**(1/3) over ZZ with lex order
    >>> f
    (4*x**2*(exp(1/x)) + x*(exp(1/x))*(log(x)))/((x**(1/3))**5)
    """
    single = False
    if not is_sequence(exprs):
        exprs, single = [exprs], True

    exprs = list(map(sympify, exprs))
    opt = build_options(symbols, options)
    numdens = []
    for expr in exprs:
        numdens.extend(expr.as_numer_denom())
    reps, opt = _parallel_dict_from_expr(numdens, opt)

    if opt.domain is None:
        # NOTE: this is inefficient because construct_domain() automatically
        # performs conversion to the target domain. It shouldn't do this.
        coeffs = sum([list(rep.values()) for rep in reps], [])
        opt.domain, _ = construct_domain(coeffs, opt=opt)

    _field = FracField(opt.gens, opt.domain, opt.order)
    fracs = []
    for i in range(0, len(reps), 2):
        fracs.append(_field(tuple(reps[i:i+2])))

    if single:
        return (_field, fracs[0])
    else:
        return (_field, fracs)

def _sort_gens(gens, **args):
    """Sort generators in a reasonably intelligent way. """
    opt = build_options(args)

    gens_order, wrt = {}, None

    if opt is not None:
        gens_order, wrt = {}, opt.wrt

        for i, gen in enumerate(opt.sort):
            gens_order[gen] = i + 1

    def order_key(gen):
        gen = str(gen)

        if wrt is not None:
            try:
                return (-len(wrt) + wrt.index(gen), gen, 0)
            except ValueError:
                pass

        name, index = _re_gen.match(gen).groups()

        if index:
            index = int(index)
        else:
            index = 0

        try:
            return ( gens_order[name], name, index)
        except KeyError:
            pass

        try:
            return (_gens_order[name], name, index)
        except KeyError:
            pass

        return (_max_order, name, index)

    try:
        gens = sorted(gens, key=order_key)
    except TypeError:  # pragma: no cover
        pass

    return tuple(gens)

def construct_domain(obj, **args):
    """Construct a minimal domain for the list of coefficients. """
    opt = build_options(args)

    if hasattr(obj, '__iter__'):
        if isinstance(obj, dict):
            if not obj:
                monoms, coeffs = [], []
            else:
                monoms, coeffs = list(zip(*list(obj.items())))
        else:
            coeffs = obj
    else:
        coeffs = [obj]

    coeffs = list(map(sympify, coeffs))
    result = _construct_simple(coeffs, opt)

    if result is not None:
        if result is not False:
            domain, coeffs = result
        else:
            domain, coeffs = _construct_expression(coeffs, opt)
    else:
        if opt.composite is False:
            result = None
        else:
            result = _construct_composite(coeffs, opt)

        if result is not None:
            domain, coeffs = result
        else:
            domain, coeffs = _construct_expression(coeffs, opt)

    if hasattr(obj, '__iter__'):
        if isinstance(obj, dict):
            return domain, dict(list(zip(monoms, coeffs)))
        else:
            return domain, coeffs
    else:
        return domain, coeffs[0]

def dict_from_expr(expr, **args):
    """Transform an expression into a multinomial form. """
    rep, opt = _dict_from_expr(expr, build_options(args))
    return rep, opt.gens

def parallel_dict_from_expr(exprs, **args):
    """Transform expressions into a multinomial form. """
    reps, opt = _parallel_dict_from_expr(exprs, build_options(args))
    return reps, opt.gens