I am wondering if there is a name for the operation $f$ that separates a multivariate rational function into fractions, wherein neither numerator nor denominator share common symbols: given $y=\frac{x^2+2x+y}{(y-1)(x-1)z}$ the result would be $f(y) = \frac{x + 3}{z(y - 1)} + \frac{1}{z(x - 1)} + \frac{4}{z(x - 1)(y - 1)}$
For $y = \frac{x^{2} + 2 x y - y}{x - 2}$, $f(y) = x + 2 y + 2 + \frac{3 y + 4}{x - 2}$.
But this is not always possible. The algorithm that I am experimenting with repeatedly divides the numerator by a factor in the denominator $d_i$ to produce $n/d_1 = w + r/d_1$ and then dividing each by the next $d_2$. In the following case, the numerator and denominator of the terms share a common symbol: $y = \frac{x^{3} + x y - 3}{y \left(x^{2} + 1\right)}$ gives $f(y) = \frac{x}{x^{2} + 1} + \frac{x}{y} + \frac{- x - 3}{y \left(x^{2} + 1\right)}$. Perhaps this will happen when the degree of a symbol in the denominator is greater than that in the numerator. In any case, that is the procedure I am working with and inquiring about. It's not really partial fraction decomposition, that's why I am wondering if this is a procedure that anyone recognizes.