algebraically exact values of sine function about mathnet-symbolics HOT 5 OPEN

There are many rules we can apply to elementary functions such as exp(x), sin(x) and sinh(x). Here, I would like to ask which of the following is appropriate for the purpose of this library. I'd like to send a PR for selected rules.

(1) auto-simplification of argument: sin(x + y - (x + y)) = sin(0) = 0
(2) undefined and infinities : sin(infinity) = undefined
(3) zero, one, pi, and j: sin(pi) = 0, sin(j) = j*sinh(1)
(4) known rational multiples of pi: sin(1/4*pi) = 1/sqrt(2), ...
(5) multiples of j: sin(j*x) = j*sinh(x)
(6) remove pi term: sin(2*pi + x) = sin(x), sin(pi + x) = -sin(x), sin(1/2*pi + x) = cos(x), sin(3/2*pi + x) = -cos(x)
(7) negative argument: sin(-x) = -sin(x)
(8) negative first term: sin(-x + y) = -sin(x - y)
(9) forward-inverse identities: sin(asin(x)) = x
and more....

(2), (3) and (7) are already partially implemented.

from mathnet-symbolics.

Regarding (1): in general, auto-simplification should not expand expressions. However, in this case it might be a bug that the inner expression is not auto-simplified to zero already (need to rethink this). So, once fixed, this should work out of the box.

from mathnet-symbolics.

@cdrnet, I think so. Now it is clear -(a + b) is not auto-simplified to -a - b. In that respect, returns in the below table are very acceptable, except (d). For consistency, how about to allow only minus or -1 to be distributed inside the parenthesis. i.e. -(a + b) = -a - b. I think the last column may be the correct behavior.

from mathnet-symbolics.

Yes, we'd only distribute -1, otherwise the product is not just a technical artifact of our internal representation (-x is (-1)*x internally). d) would no longer auto-simplify to a+b, but that may be ok.

I agree, I'd also expect the auto-simplified form of the last column.

from mathnet-symbolics.

I can get the last column by simply adding a rule(| Sum ax as x' ...) to the valueMul function as shown below.

and multiply x y =
      ...
      let rec valueMul (v:Value) x =
            if Value.isZero v then zero else
            match x with
            ...
            | Product ax -> if Value.isOne v then x else Product (Values.unpack v::ax)
            | Sum ax as x'-> if Value.isOne v then x' // intentionally distribute -1, e.g. -(a + b) = -a - b
                             else if Value.isMinusOne v then ax |> List.map (function xi -> valueMul v xi) |> Sum
                             else Product [Values.unpack v; x']
            | x -> if Value.isOne v then x else Product [Values.unpack v; x]

• Just now, I sent new PR. Please let me know if there is a better way...

from mathnet-symbolics.