Term representation and simplification

Performance of symbolic simplification depends on the datastructures used to represent terms. Efficient datastructures often have the advantage of automatic simplification, and of efficient storage.

The most basic term representation simply holds a function call and stores the function and the arguments it is called with. This is done by the Term type in SymbolicUtils. Functions that aren't commutative or associative, such as sin or hypot are stored as Terms. Commutatative and associative operations like +, *, and their supporting operations like -, / and ^, when used on terms of type <:Number, stand to gain from the use of more efficient datastrucutres.

All term representations must support operation and arguments functions. And they must define istree to return true when called with an instance of the type. Generic term-manipulation programs such as the rule-based rewriter make use of this interface to inspect expressions. In this way, the interface wins back the generality lost by having a zoo of term representations instead of one. (see interface section for more on this.)

Preliminary representation of arithmetic

Linear combinations such as α1x1+α2x2+...+αnxn\alpha_1 x_1 + \alpha_2 x_2 +...+ \alpha_n x_n are represented by Add(Dict(x₁ => α₁, x₂ => α₂, ..., xₙ => αₙ)). Here, any xix_i may itself be other types mentioned here, except for Add. When an Add is added to an Add, we merge their dictionaries and add up matching coefficients to create a single "flattened" Add.

Similarly, x1m1x2m2...xmnx_1^{m_1}x_2^{m_2}...x_{m_n} is represented by Mul(Dict(x₁ => m₁, x₂ => m₂, ..., xₙ => mₙ)). xix_i may not themselves be Mul, multiplying a Mul with another Mul returns a "flattened" Mul.

Note that Add and Mul types perform a preliminary simplification which suffices to simplify numeric expressions to a large extent during construction.

p/qp / q is represented by Div(p, q). The result of * on Div is maintainted as a Div. For example, Div(p_1, q_1) * Div(p_2, q_2) results in Div(p_1 * p_2, q_1 * q_2) and so on. The effect is, in Div(p, q), p or q or, if they are Mul, any of their multiplicands is not a Div. So Muls must always be nested inside a Div and can never show up immediately wrapping it. This rule sets up an expression so that it helps the simplify_fractions procedure described two sections below.

Polynomial representation

Packages like DynamicPolynomials.jl provide representations that are even more efficient than the Add and Mul types mentioned above. They are designed specifically for multi-variate polynomials. They provide common algorithms such as multi-variate polynomial GCD. The restrictions that make it fast also mean some things are not possible: Firstly, DynamicPolynomials can only represent flat polynomials. For example, (x-3)*(x+5) can only be represented as (x^2) + 15 - 8x. Secondly, DynamicPolynomials does not have ways to represent generic Terms such as sin(x-y) in the tree.

To reconcile these differences while being able to use the efficient algorithms of DynamicPolynomials we have the PolyForm type. This type holds a polynomial and the mappings necessary to present the polynomial as a SymbolicUtils expression (i.e. by defining operation and arguments). The mappings constructed for the conversion are 1) a bijection from DynamicPolynomials PolyVar type to a Symbolics Sym, and 2) a mapping from Syms to non-polynomial terms that the Syms stand-in for. These terms may themselves contain PolyForm if there are polynomials inside them. The mappings are transiently global, that is, when all references to the mappings go out of scope, they are released and re-created.

julia> @syms x y
(x, y)

julia> PolyForm((x-3)*(y-5))
x*y + 15 - 5x - 3y

Terms for which the operation is not +, *, or ^ are replaced with a generated symbol when representing the polynomial, and a mapping from this new symbol to the original expression it stands-in for is maintained as stated above.

julia> p = PolyForm((sin(x) + cos(x))^2)
(cos(x)^2) + 2cos(x)*sin(x) + (sin(x)^2)

julia> p.p # this is the actual DynamicPolynomial stored
cos_3658410937268741549² + 2cos_3658410937268741549sin_10720964503106793468 + sin_10720964503106793468²

By default, polynomials inside non-polynomial terms are not also converted to PolyForm. For example,

julia> PolyForm(sin((x-3)*(y-5)))
sin((x - 3)*(y - 5))

But you can pass in the recurse=true keyword argument to do this.

julia> PolyForm(sin((x-3)*(y-5)), recurse=true)
sin(x*y + 15 - 5x - 3y)

Polynomials are constructed by first turning symbols and non-polynomial terms into DynamicPolynomials-style variables and then applying the +, *, ^ operations on these variables. You can control the list of the polynomial operations with the Fs keyword argument. It is a Union type of the functions to apply. For example, let's say you want to turn terms into polynomials by only applying the + and ^ operations, and want to preserve * operations as-is, you could pass in Fs=Union{typeof(+), typeof(^)}

julia> PolyForm((x+y)^2*(x-y), Fs=Union{typeof(+), typeof(^)}, recurse=true)
((x - (y))*((x^2) + 2x*y + (y^2)))

Note that in this case recurse=true was necessary as otherwise the polynomialization would not descend into the * operation as it is now considered a generic operation.

Simplifying fractions

simplify_fractions(expr) recurses through expr finding Divs and simplifying them using polynomial divison.

First the factors of the numerators and the denominators are converted into PolyForm objects, then numerators and denominators are divided by their respective pairwise GCDs. The conversion of the numerator and denominator into PolyForm is set up so that simplify_fractions does not result in increase in the expression size due to polynomial expansion. Specifically, the factors are individually converted into PolyForm objects, and any powers of polynomial is not expanded, but the divison process repeatedly divides them as many times as the power.

julia> simplify_fractions((x*y+5x+3y+15)/((x+3)*(x-4)))
(5.0 + y) / (x - 4)

julia> simplify_fractions((x*y+5x+3y+15)^2/((x+3)*(x-4)))
((5.0 + y)*(15 + 5x + x*y + 3y)) / (x - 4)

julia> simplify_fractions(3/(x+3) + x/(x+3))

julia> simplify_fractions(sin(x)/cos(x) + cos(x)/sin(x))
(cos(x)^2 + sin(x)^2) / (cos(x)*sin(x))

julia> simplify(ans)
1 / (cos(x)*sin(x))