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Current File : /usr/lib/python3/dist-packages/sympy/simplify/__pycache__/ratsimp.cpython-310.pyc
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�8Va��@s�ddlmZddlmZmZmZddlmZddlm	Z	m
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    Put an expression over a common denominator, cancel and reduce.

    Examples
    ========

    >>> from sympy import ratsimp
    >>> from sympy.abc import x, y
    >>> ratsimp(1/x + 1/y)
    (x + y)/(x*y)
    TF)�field�expand)r�as_numer_denomr
rr)�expr�f�g�Q�r�r�8/usr/lib/python3/dist-packages/sympy/simplify/ratsimp.py�ratsimp	s
�rTF)�quick�
polynomialcs�ddlm�td|�t|���\}}zt||g�g|�Ri|��\}�Wnty1|YSw�j}	|	jr>|	�	��_nt
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    Simplifies a rational expression ``expr`` modulo the prime ideal
    generated by ``G``.  ``G`` should be a Groebner basis of the
    ideal.

    Examples
    ========

    >>> from sympy.simplify.ratsimp import ratsimpmodprime
    >>> from sympy.abc import x, y
    >>> eq = (x + y**5 + y)/(x - y)
    >>> ratsimpmodprime(eq, [x*y**5 - x - y], x, y, order='lex')
    (-x**2 - x*y - x - y)/(-x**2 + x*y)

    If ``polynomial`` is ``False``, the algorithm computes a rational
    simplification which minimizes the sum of the total degrees of
    the numerator and the denominator.

    If ``polynomial`` is ``True``, this function just brings numerator and
    denominator into a canonical form. This is much faster, but has
    potentially worse results.

    References
    ==========

    .. [1] M. Monagan, R. Pearce, Rational Simplification Modulo a Polynomial
        Ideal, http://citeseer.ist.psu.edu/viewdoc/summary?doi=10.1.1.163.6984
        (specifically, the second algorithm)
    r)�solve�ratsimpmodprimez-can't compute rational simplification over %scsg|]}|��j��qSr)�LM�order)�.0r��optrr�
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        Compute all monomials with degree less than ``n`` that are
        not divisible by any element of ``leading_monomials``.
        r�csg|]	}t�|�du�qS)N)r
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        Computes a rational simplification of ``a/b`` which minimizes
        the sum of the total degrees of the numerator and the denominator.

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

        The algorithm proceeds by looking at ``a * d - b * c`` modulo
        the ideal generated by ``G`` for some ``c`` and ``d`` with degree
        less than ``a`` and ``b`` respectively.
        The coefficients of ``c`` and ``d`` are indeterminates and thus
        the coefficients of the normalform of ``a * d - b * c`` are
        linear polynomials in these indeterminates.
        If these linear polynomials, considered as system of
        equations, have a nontrivial solution, then `\frac{a}{b}
        \equiv \frac{c}{d}` modulo the ideal generated by ``G``. So,
        by construction, the degree of ``c`` and ``d`` is less than
        the degree of ``a`` and ``b``, so a simpler representation
        has been found.
        After a simpler representation has been found, the algorithm
        tries to reduce the degree of the numerator and denominator
        and returns the result afterwards.

        As an extension, if quick=False, we look at all possible degrees such
        that the total degree is less than *or equal to* the best current
        solution. We retain a list of all solutions of minimal degree, and try
        to find the best one at the end.
        rr'z%s / %s: %s, %szc:%d)�clszd:%dc�g|]
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&�&���

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