How to update a formula object containing a pipe ("|")

Some packages such as fixest use the pipe to separate explanatory variables from fixed effects in formulas. How can I use update() to modify each of those parts? In the example below, I would like to remove z from the right-hand side but keep the fixed effects fe.

f <- log(y) ~ log(x) + z | fe

# does not work
f |> update(~ . - z)
#> log(y) ~ (log(x) + z | fe)
f |> update(~ . - z | .)
#> log(y) ~ ((log(x) + z | fe) - z | (log(x) + z | fe))

^{Created on 2023-02-26 with reprex v2.0.2}

Apparently, the fixest authors have only created an update() method for model objects that supports this behavior but none for formula objects (see here). Is there any workaround other than fitting a model, updating it and then extracting the formula? In the worst case, I could manipulate the formula as a character string but that feels rather heavy-handed. rlang and modelr don't seem to have any appropriate functions either.

The Formula package provides tools for handling multi-part formulas. @user20650 pointed me to page 9: converting the formula to a Formula before updating:

library(Formula)
update(Formula(f), . ~ . - z | .)

or (f |> Formula |> update(...) if you like pipes).

Previous answer (brute force): ugly but functional.

g <- f[[3]][[2]]  ## f[[3]] is RHS of formula: 
                  ## f[[3]][[2]] is the stuff before the pipe
g_u <- update(as.formula(call("~", g)), ~ . - z)
f[[3]][[2]] <- g_u[[2]]
f
## log(y) ~ log(x) | fe

You could presumably package this into your own my_update() formula that would at least work for a narrow range of use cases:

my_update <- function(f, new, part = c("left", "right")) {
   part <- match.arg(part)
   ind <- if (part == "left") 2 else 3
   g <- f[[3]][[ind]]
   g_u <- update(as.formula(call("~", g)), new)
   f[[3]][[ind]] <- g_u[[2]]
   f
}

Note that the new argument should be a one-sided formula:

my_update(f, ~ . - z)
## log(y) ~ log(x) | fe
my_update(f, ~ . + junk, "right")
## log(y) ~ log(x) + z | fe + junk