Rules behave differently for some trigonometric functions

[pmc4]

Hello. As described in this Julia Discourse post, when defining custom rules for some trigonometric identities, the behaviour is not consistent. It looks like some rules are missing or that they are not being applied. Let me show two MWE with two different trigonometric identities:

Problem 1

Here we define a custom rule for the sine of the double angle:

using SymbolicUtils

@syms a
r = @acrule(sin(2*~x) => 2*sin(~x)*cos(~x))

# This works and returns 2cos(a)*sin(a)
simplify(sin(2a), r)

# This doesn't work and returns (1//4)*sin(2a)
simplify(sin(2a) / 4, r)

The last call should return cos(a)*sin(a)//2, but instead does nothing. We can add an extra rule to solve this issue:

using SymbolicUtils

@syms a

rules = [
    @acrule(sin(2*~x) => 2 * sin(~x) * cos(~x))
    @acrule(~a * sin(2*~x) => ~a * 2 * sin(~x) * cos(~x))
]

r = RestartedChain(rules)

# Now it works and returns (1//2)*cos(a)*sin(a)
simplify(sin(2a) / 4, r)

However this is not efficient, I would like to simplify sin(2a) independently of whatever is being multiplied or whichever place the sine appears. For example, sin(2a) / sin(2b) does not get simplified. I've tried adding something like

@acrule(sin(~~x) / sin(~~y) => sin((~~x)...) / sin((~~y)...))

but it doesn't work either.

Problem 2

Now we define a custom rule for the product of two sine functions and perform the same naïve computation as we did with sin(2a) / 4 in Problem 1.

using SymbolicUtils

@syms λ a₁ a₂ a₃ a₄

r = @acrule cos(~x) * cos(~y) => (cos(~x - ~y) + cos(~x + ~y)) / 2

# Returns the correct result
# (1//32)*(cos(a₁*λ + a₂*λ) + cos(a₁*λ - a₂*λ))*(cos(a₃*λ + a₄*λ) + cos(-a₃*λ + a₄*λ))
simplify(cos(a₁ * λ) * cos(a₂ * λ) * cos(a₃ * λ) * cos(a₄ * λ) / 8, r)

In this case it is giving the correct result when we divide the expression by a number. It even detects that there are two pairs of cosines being multiplied. Nonetheless, if we now divide by λ instead of by 8, it no longer works:

# Returns again (cos(a₂*λ)*cos(a₃*λ)*cos(a₄*λ)*cos(a₁*λ)) / (λ^2)
simplify(cos(a₁ * λ) * cos(a₂ * λ) * cos(a₃ * λ) * cos(a₄ * λ) / λ^2, r)

---

I don't exactly know which kind of rules are missing or if this is a bug. I'd expect bot examples to fail the same way.

[pmc4]

I've read a bit SymbolicUtils.jl/src/simplify_rules.jl at v3.29.0 · JuliaSymbolics/SymbolicUtils.jl and the first two rules that appear in TRIG_EXP_RULES are these

@acrule(~r*~x::has_trig_exp + ~r*~y => ~r*(~x + ~y))
@acrule(~r*~x::has_trig_exp + -1*~r*~y => ~r*(~x - ~y))

Maybe we need something like this?:

@acrule(~r*~x::has_trig_exp => ~r*~x)