Replace a randomly chosen part of an expression

Question

Given an arbitrary Mathematica expression, how can I choose a random part of the expression and replace it with another symbol of my choosing?

For example, given $a x^3 +2y\cos(x)- \tanh(x^{y+3})/(x^4-\sqrt{b})$, I want to select a single randomly chosen part of the expression that involves an $x$ term and replace it with $\mathbf{u}$. This part could be any sub-expression with an $x$ such as $x^3,a x^3, 2y\cos(x),\cos(x),x, x^{y+3},\tanh(x^{y+3}),x^4,(x^4-\sqrt{b})$, or even the whole expression. Examples:

$$ \mathbf{u}\\ \mathbf{u} +2y\cos(x)- \tanh(x^{y+3})/(x^4-\sqrt{b})\\ a x^3 +2y\cos(x)- \tanh(x^{y+3})/\mathbf{u}\\ a x^3 +2y\cos(x)- \tanh(\mathbf{u})/(x^4-\sqrt{b})\\ a x^3 +\mathbf{u}- \tanh(x^{y+3})/(x^4-\sqrt{b})\\ a x^3 +2y\cos(x)- \mathbf{u}/(x^4-\sqrt{b})\\ \mathbf{u} - \tanh(x^{y+3})/(x^4-\sqrt{b})\\ a x^3 + 2y\cos(x) - \mathbf{u}\\ a x^3 + \mathbf{u} $$

Also I'd like to know which subexpression was chosen for replacement with $\mathbf{u}$. So far I've been able to generate random replacements like this:

expr = a x^3 + 2 y Cos[x] - Tanh[x^(y + 3)]/(x^4 - Sqrt[b]);
Table[ReplacePart[expr, 
 RandomChoice[Position[expr, _, Heads -> False]] -> u], 
 1000] // DeleteDuplicates

...but I have not been able to 1) only select expressions involving $x$, and 2) record which subexpression was chosen.

For clarification, the following image shows the expression tree for the example I gave above. Green nodes are eligible for replacement with $\mathbf{u}$ because they contain $x$ or some sub-expression involving $x$. Red nodes do not contain any $x$ and are not eligible for replacement:

Answer 1

Here's one way:

expr = a x^3 + 2 y Cos[x] - Tanh[x^(y + 3)]/(x^4 - Sqrt[b]);
pos = Position[expr, x];
thisPos = RandomChoice[pos]; 
ReplacePart[expr, Drop[thisPos, -RandomInteger[{0, Length[thisPos] - 1}]] -> u]

Each time you execute it, you select a random term containing x from the expression and replace it with u.

Answer 2

Depending on your definition of random the method from bill s might have a problem in that some elements can be selected in more than one way. For example in your expr the element at position {3} can be selected from either {3, 2, 1, 2, 1} or {3, 3, 1, 1}. If we use Position to find all elements directly (rather than Dropping afterward) we avoid this.

expr = a x^3 + 2 y Cos[x] - Tanh[x^(y + 3)]/(x^4 - Sqrt[b]);

pos = Most @ Position[expr, s_ /; ! FreeQ[s, x]]

ReplacePart[expr, RandomChoice[pos] -> u]

---

This never seems to replace the whole expression with $u$, even if I run it 100,000 times. Shouldn't that happen, as the Plus in the top of the TreeForm is eligible for replacement too?

I did not consider the whole expression to be a part. The term {} in the return of Position represents the whole expression, but I dropped it using Most because ReplacePart cannot use this specification.

If you want to include that as a possibility I propose this instead:

pos = Position[expr, s_ /; ! FreeQ[s, x]]

Module[{x = expr}, x[[## & @@ RandomChoice @ pos]] = u; x]