\\[ x^2 – 2ax + a^2 + y^2 \\rightarrow (x-a)^2 + y^2 \\; .\\]<\/p>\n\n\n\n
The use of .has(a,x) (see last post<\/a>) allows the relevant terms to be separated from the polynomial without needing to \u2018see\u2019 them or \u2018spell out\u2019 what they are. The function .as_ordered_terms was used to separate the top level of the Add function. Note that for other types of expressions, other choices, like .as_ordered_factors (for Mul) and args, may be needed, with args being the most generic of the lot. The key point being that the tree was \u2018broken down\u2019 into smaller pieces, some of which were amenable to the kinds of simplification functions discussed in a previous post<\/a>.<\/p>\n\n\n\n There are three often used functions in the second category of built-in methods. On these points, I received an excellent high-level summary from Oscar Benjamin, who is a Sympy developer. I will draw on this summary for what follows with two caveats: first, these methods are nuanced and complex, so I won\u2019t be able to fill in as ably as one of the developers can and, second, any mistakes are solely a consequence of my ignorance.<\/p>\n\n\n\n The three methods are .xreplace, .subs, and .replace, given in order of what seems to be best described as generality and complexity. That is to say that Oscar Benjamin has described them as:<\/p>\n\n\n\n The xreplace function is the fastest and simplest. The replace function is the most flexible. The subs function is the slowest and is the only one that applies any semantic meaning to the substitution.<\/p><\/div>\n\n\n\n The following table summarizes their usage.<\/p>\n\n\n\n Note: {o1:n1, o2:n2, \u2026} is a dictionary specifying how old expressions (o1, o2, etc.) are to be replaced by new terms (n1, n2, etc.).<\/p>\n\n\n\n In order to illustrate the differences between these methods, we\u2019ll start with the master expression<\/p>\n\n\n\n \\[ \u00a0Q = a + x + 2 x^2 + 3 x^3 + 4 x^4 + 5 \\cos(x) + \\\\ 6 \\cos(x^2) + 7 \\cos^2 (x) + 8 \\cos(b) + 9 e^{-x^2} \\; .\\]<\/p>\n\n\n\n We\u2019ll look at all three methods under two term-rewriting rules:<\/p>\n\n\n\n Under the action of .xreplace $Q$ becomes<\/p>\n\n\n\n \\[ Q \\xrightarrow{x^2 \\rightarrow y} a + x + 2 y + 3 x^3 + 4 x^4 + 5 \\cos(x) \\\\ + 6\\cos(y) + 7 \\cos^2(x) + 8 \\cos(b) + 9 e^{-y} \\; ,\\]<\/p>\n\n\n\n showing that .xreplace simply went through $Q$\u2019s tree structure and structural rewrote sub-trees as shown in the following figure.<\/p>\n\n\n Under the action of .subs $Q$ changes more, becoming:<\/p>\n\n\n\n \\[ Q \\xrightarrow{x^2 \\rightarrow y} a + x + 2 y + 3 x^3 + 4 y^2 + 5 \\cos(x) + \\\\ 6\\cos(y) + 7 \\cos^2(x) + 8 \\cos(b) + 9e^{-y} \\; .\\]<\/p>\n\n\n\n Note that .subs \u2018understood\u2019 that $x^2 \\rightarrow y$ properly, and without assumptions, means that $x^4 \\rightarrow y^2$. Odd powers of $x$ remain unchanged since .subs doesn\u2019t know enough to invert the replacement since it would have to choose between $x \\rightarrow +\\sqrt{y}$ and $x \\rightarrow -\\sqrt{y}$.<\/p>\n\n\n\n Finally, under the action of .replace, $Q$ yields<\/p>\n\n\n\n \\[ Q \\xrightarrow{x^2 \\rightarrow y} a + x + 2 y + 3 x^3 + 4 x^4 + 5 \\cos(x) + \\\\ 6\\cos(y) + 7 \\cos^2(x) + 8 \\cos(b) + 9 e^{-y} \\; ,\\]<\/p>\n\n\n\n which is exactly identical to what resulted from the application of .xreplace.<\/p>\n\n\n\n Under the application of .xreplace, $Q$ becomes<\/p>\n\n\n\n \\[ Q \\xrightarrow{\\cos(x) \\rightarrow \\sin(x)} a + x + 2 x^2 + 3 x^3 + 4 x^4 + 5 \\sin(x) + \\\\ 6\\cos(x^2) + 7 \\sin^2(x) + 8 \\cos(b) + 9 e^{-x^2} \\; , \\]<\/p>\n\n\n\n again, as in the example for Rule 1, .xreplace only makes a structural change\/rewrite for only those subexpressions that, in a tree-sense, exactly match the old subexpression.<\/p>\n\n\n\n The application of .subs gives identical expressions for $Q$, as does a na\u00efve use of .replace. However, a much more interesting example using .replace involves using Sympy\u2019s \u2018Wild\u2019 symbol. The following snippet of the Jupyter notebook<\/p>\n\n\nMethod<\/strong><\/strong><\/td> Usage<\/strong><\/strong><\/td> Comment<\/strong><\/strong><\/td><\/tr> .xreplace<\/td> expr.xreplace({o1:n1, o2:n2, \u2026})<\/td> Purely structural replacement which identifies a exact subexpression to be replaced (hence the \u2018x\u2019 in xreplace).<\/td><\/tr> .subs<\/td> expr.sub({o1:n1, o2:n2, \u2026})<\/td> Mostly structural replacement but some mathematical equivalence is used in simplify terms.<\/td><\/tr> .replace<\/td> expr.replace(old_pattern,new_pattern)<\/td> Purely structural replacement which uses a generic pattern to replace all subexpressions that match)<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n \n
Rule 1: $x^2 \\rightarrow y$<\/h2>\n\n\n\n
![\"\"](\"https:\/\/aristotle2digital.blogwyrm.com\/wp-content\/uploads\/2026\/06\/image.png\")
<\/a><\/figure><\/div>\n\n\nRule 2 $\\cos(x) \\rightarrow \\sin(x)$<\/h2>\n\n\n\n
![\"\"](\"https:\/\/aristotle2digital.blogwyrm.com\/wp-content\/uploads\/2026\/06\/image-1.png\")
<\/a><\/figure><\/div>\n\n\n