Issue with shadowing and binary functions

[rok-cesnovar]

This is a continuation of the discussion with @WardBrian here: #995 (comment)

Long story short, this model's generated code fails to compile:

transformed data {
   real multiply;
   real a = multiply(0, 1);   
}

with

examples/test/test.hpp: In constructor ‘test_model_namespace::test_model::test_model(stan::io::var_context&, unsigned int, std::ostream*)’:
examples/test/test.hpp:66:24: error: expression cannot be used as a function
   66 |       a = multiply(0, 1);
      |                        ^

We allow shadowing and it works with nullary and unary functions. For example this model works fine:

transformed data {
   real log;
   real a = log(10);
   real e;
   real f = e();
   real positive_infinity;
   real g = positive_infinity();  
}

Solutions to this are:
a) add stan::math namespace to a function if there is a variable with the same name used in the model
b) add stan::math namespace to all Stan Math function calls (this probably doesnt work as we may fallback to std:: in some cases)
c) if a function is shadowed it can no longer be used as a function in that model (@WardBrian mentioned this is what Python does)
d) disallow shadowing for binary functions (we have to check functions with 3+ args as well)

This is not a new issue for this release, its been present since we allowed shadowing.

[rok-cesnovar]

For the record, I would go with a) if its easy to implement, if not then c).

[SteveBronder]

I think I would do (b) here, all of the functions available through stan and std are available in stan::math

[rok-cesnovar]

I am not so sure we are supporting all signatures in Math at the moment. At least in the past we had a few things like pow(int, int). But maybe we cleaned that up.

But yeah, there are not that many std:: function and we might as well just check them all.

[WardBrian]

@SteveBronder did you ever get a chance to look into this?

[WardBrian]

We currently have this chunk of code:

stanc3/src/stan_math_backend/Expression_gen.ml

Lines 8 to 18 in 8429968

	let functions_requiring_namespace =
	String.Set.of_list
	[ "e"; "pi"; "log2"; "log10"; "sqrt2"; "not_a_number"; "positive_infinity"
	; "negative_infinity"; "machine_precision"; "abs"; "acos"; "acosh"; "asin"
	; "asinh"; "atan"; "atanh"; "cbrt"; "ceil"; "cos"; "cosh"; "erf"; "erfc"
	; "exp"; "exp2"; "expm1"; "fabs"; "floor"; "lgamma"; "log"; "log1p"; "log2"
	; "log10"; "round"; "sin"; "sinh"; "sqrt"; "tan"; "tanh"; "tgamma"; "trunc"
	; "fdim"; "fmax"; "fmin"; "hypot"; "fma"; "complex" ]
	let stan_namespace_qualify f =
	if Set.mem functions_requiring_namespace f then "stan::math::" ^ f else f

Which I believe is why the unary functions still do compile (though it seems outdated/wrong, complex isn't a function). It seems like instead of a whitelist we could make this just call on all arguments, and if we need some std:: instead we could make a list of those and branch on that?