Function Registration and Tracing

Direct Tracing

Because Symbolics expressions respect Julia semantics, one way to generate symbolic expressions is to simply place Symbolics variables as inputs into existing Julia code. For example, the following uses the standard Julia function for the Lorenz equations to generate the symbolic expression for the Lorenz equations:

using Symbolics
function lorenz(du,u,p,t)
 du[1] = 10.0(u[2]-u[1])
 du[2] = u[1]*(28.0-u[3]) - u[2]
 du[3] = u[1]*u[2] - (8/3)*u[3]
@variables t p[1:3] u(t)[1:3]
du = Array{Any}(undef, 3)
3-element Vector{Any}:
                         10.0(-(u(t))[1] + (u(t))[2])
 -(u(t))[2] + (u(t))[1]*(28.0 - (u(t))[3])
                         -2.6666666666666665(u(t))[3] + (u(t))[1]*(u(t))[2]

Or similarly:

@variables t x(t) y(t) z(t) dx(t) dy(t) dz(t) σ ρ β
du = [dx,dy,dz]
u = [x,y,z]
p = [σ,ρ,β]

\[ \begin{equation} \left[ \begin{array}{c} 10 \left( - x\left( t \right) + y\left( t \right) \right) \\ - y\left( t \right) + x\left( t \right) \left( 28 - z\left( t \right) \right) \\ - 2.6667 z\left( t \right) + x\left( t \right) y\left( t \right) \\ \end{array} \right] \end{equation} \]

Registering Functions

The Symbolics graph only allows registered Julia functions within its type. All other functions are automatically traced down to registered functions. By default, Symbolics.jl pre-registers the common functions utilized in SymbolicUtils.jl and pre-defines their derivatives. However, the user can utilize the @register_symbolic macro to add their function to allowed functions of the computation graph.

@register_symbolic(expr, define_promotion = true, Ts = [Real])

Overload appropriate methods so that Symbolics can stop tracing into the registered function. If define_promotion is true, then a promotion method in the form of

SymbolicUtils.promote_symtype(::typeof(f_registered), args...) = Real # or the annotated return type

is defined for the register function. Note that when defining multiple register overloads for one function, all the rest of the registers must set define_promotion to false except for the first one, to avoid method overwriting.


@register_symbolic foo(x, y)
@register_symbolic foo(x, y::Bool) false # do not overload a duplicate promotion rule
@register_symbolic goo(x, y::Int) # `y` is not overloaded to take symbolic objects
@register_symbolic hoo(x, y)::Int # `hoo` returns `Int`