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Simplify.jl
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Algebraic simplification in Julia
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Simplify.jl
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# Simplify.jl [](https://travis-ci.org/HarrisonGrodin/Simplify.jl) [](https://ci.appveyor.com/project/HarrisonGrodin/simplify-jl/branch/master) [](https://coveralls.io/github/HarrisonGrodin/Simplify.jl?branch=master) **Simplify.jl** implements methods for symbolic algebraic simplification in the Julia language. ## Examples Normalization involves determining the unique normal form of an expression ("simplest" equivalent expression) through repeated application of rules. *Simplify.jl* will use its [internal set of algebraic rules](./src/rules.jl) by default, which includes trigonometry, logarithms, differentiation (based on [DiffRules.jl](https://github.com/JuliaDiff/DiffRules.jl)), and more. ```julia julia> @syms x y b θ; julia> normalize(@term(1 / (sin(-θ) / cos(-θ)))) @term(1 / (-(sin(θ)) / cos(θ))) julia> normalize(@term(log(b, 1 / (b^abs(x^2))))) @term(log(b, 1 / b ^ abs(x ^ 2))) julia> normalize(@term(diff(sin(2x) - log(x+y), x))) @term(1 * -(inv(x + y) * (1 * diff(y, x) + 1 * one(x))) + 1 * cos(2x) * (2 * one(x) + x * 0)) julia> normalize(@term(!x & x | (y & (y | true)))) @term(!x & x | (y | true) & y) julia> normalize(@term(y^(6 - 3log(x, x^2)))) @term(y ^ (-(6 * log(x, x)) + 6)) ``` In many cases, it is useful to specify entirely custom rules by passing a Term Rewriting System as the second argument to `normalize`. This may be done either by manually constructing a `Rules` object or by using the `RULES` strategy for `@term`. ```julia julia> @syms f g h; @vars x y; julia> normalize(@term(f(x, f(y, y))), @term RULES [ f(x, x) => 1 f(x, 1) => x ]) @term(x) julia> normalize(@term(f(g(f(1), h()))), Rules( @term(f(x)) => @term(x), @term(h()) => @term(3), )) @term(g(1, 3)) julia> using Simplify: EvalRule julia> normalize(@term(f(g(f(1), h()))), Rules( @term(f(x)) => @term(x), @term(h()) => @term(3), EvalRule(g, (a, b) -> 2a + b) )) @term(5) ``` Variables may contain information about their domain, which may result in more specific normalizations. ```julia julia> using SpecialSets julia> @syms x y z; julia> ctx = [get_context(); Image(y, GreaterThan(3)); Image(z, Even ∩ LessThan(0))]; julia> with_context(ctx) do normalize(@term(abs(x))) end @term(abs(x)) julia> with_context(ctx) do normalize(@term(abs(y))) end @term(y) julia> with_context(ctx) do normalize(@term(abs(z))) end @term(-z) ``` ```julia julia> ctx = [get_context(); Image(x, TypeSet(Int)); Image(y, TypeSet(Int))]; julia> with_context(ctx) do normalize(@term(diff(sin(2x) - log(x + y), x))) end @term(cos(2x) * 2 + -(inv(x + y) * (diff(y, x) + 1))) ``` ## Acknowledgements - [*Term Rewriting and All That*](https://www21.in.tum.de/~nipkow/TRaAT/) - [@MasonProtter](https://github.com/MasonProtter) - [@Yijia-Chen](https://github.com/Yijia-Chen)
