Simplifying acos(0/1), asin(0,1) and anything + 0

ext/SymbolicsGroebnerExt.jl

@@ -320,13 +320,9 @@ end
 # Helps with precompilation time
 PrecompileTools.@setup_workload begin
     @variables a b c x y z
-    equation1 = a*log(x)^b + c ~ 0
-    equation_actually_polynomial = sin(x^2 +1)^2 + sin(x^2 + 1) + 3
     simple_linear_equations = [x - y, y + 2z]
     equations_intersect_sphere_line = [x^2 + y^2 + z^2 - 9, x - 2y + 3, y - z]
     PrecompileTools.@compile_workload begin
-        symbolic_solve(equation1, x)
-        symbolic_solve(equation_actually_polynomial)
         symbolic_solve(simple_linear_equations, [x, y], warns=false)
         symbolic_solve(equations_intersect_sphere_line, [x, y, z], warns=false)
     end

ext/SymbolicsNemoExt.jl

@@ -61,7 +61,13 @@ end
 PrecompileTools.@setup_workload begin
     @variables a b c x y z
     expr_with_params = expand((x + b)*(x^2 + 2x + 1)*(x^2 - a))
+    equation1 = a*log(x)^b + c ~ 0
+    equation_polynomial = 9^x + 3^x + 2
+    exp_eq = 5*2^(x+1) + 7^(x+3)
     PrecompileTools.@compile_workload begin
+        symbolic_solve(equation1, x)
+        symbolic_solve(equation_polynomial, x)
+        symbolic_solve(exp_eq)
         symbolic_solve(expr_with_params, x, dropmultiplicity=false)
         symbolic_solve(x^10 - a^10, x, dropmultiplicity=false)
     end

src/solver/attract.jl

@@ -197,10 +197,8 @@ function attract_trig(lhs, var)
     r_trig = [@acrule(sin(~x::(contains_var))^2 + cos(~x::(contains_var))^2=>one(~x))
               @acrule(sin(~x::(contains_var))^2 + -1=>-1 * cos(~x)^2)
               @acrule(cos(~x::(contains_var))^2 + -1=>-1 * sin(~x)^2)
-              @acrule(cos(~x::(contains_var))^2 + -1 * sin(~x::(contains_var))^2=>cos(2 *
-                                                                                      ~x))
-              @acrule(sin(~x::(contains_var))^2 + -1 * cos(~x::(contains_var))^2=>-cos(2 *
-                                                                                       ~x))
+              @acrule(cos(~x::(contains_var))^2 + -1 * sin(~x::(contains_var))^2=>cos(2*~x))
+              @acrule(sin(~x::(contains_var))^2 + -1 * cos(~x::(contains_var))^2=>-cos(2*~x))
               @acrule(cos(~x::(contains_var)) * sin(~x::(contains_var))=>sin(2 * ~x) / 2)
               @acrule(tan(~x::(contains_var))^2 + -1 * sec(~x::(contains_var))^2=>one(~x))
               @acrule(-1 * tan(~x::(contains_var))^2 + sec(~x::(contains_var))^2=>one(~x))

src/solver/ia_main.jl

@@ -123,7 +123,7 @@ function isolate(lhs, var; warns=true, conditions=[])
             new_var = (@variables $new_var)[1]
             rhs = map(
                 sol -> term(rev_oper[oper], sol) +
-                       term(*, Base.MathConstants.pi, 2 * new_var),
+                       term(*, Base.MathConstants.pi, new_var),
                 rhs)
             @info string(new_var) * " ϵ" * " Ζ"
 

src/solver/postprocess.jl

@@ -1,4 +1,3 @@
-
 # Alex: make sure `Num`s are not processed here as they'd break it.
 _postprocess_root(x) = x
 
@@ -32,30 +31,30 @@ function _postprocess_root(x::SymbolicUtils.BasicSymbolic)
     !iscall(x) && return x
 
     x = Symbolics.term(operation(x), map(_postprocess_root, arguments(x))...)
+    oper = operation(x)
 
     # sqrt(0), cbrt(0) => 0
     # sqrt(1), cbrt(1) => 1
-    if iscall(x) &&
-       (operation(x) === sqrt || operation(x) === cbrt || operation(x) === ssqrt ||
-        operation(x) === scbrt)
+    if (oper === sqrt || oper === cbrt || oper === ssqrt ||
+        oper === scbrt)
         arg = arguments(x)[1]
         if isequal(arg, 0) || isequal(arg, 1)
             return arg
         end
     end
 
     # (X)^0 => 1
-    if iscall(x) && operation(x) === (^) && isequal(arguments(x)[2], 0)
+    if oper === (^) && isequal(arguments(x)[2], 0)
         return 1
     end
 
     # (X)^1 => X
-    if iscall(x) && operation(x) === (^) && isequal(arguments(x)[2], 1)
+    if oper === (^) && isequal(arguments(x)[2], 1)
         return arguments(x)[1]
     end
 
     # sqrt((N / D)^2 * M) => N / D * sqrt(M)
-    if iscall(x) && (operation(x) === sqrt || operation(x) === ssqrt)
+    if (oper === sqrt || oper === ssqrt)
         function squarefree_decomp(x::Integer)
             square, squarefree = big(1), big(1)
             for (p, d) in collect(Primes.factor(abs(x)))
@@ -90,7 +89,7 @@ function _postprocess_root(x::SymbolicUtils.BasicSymbolic)
     end
 
     # (sqrt(N))^M => N^div(M, 2)*sqrt(N)^(mod(M, 2))
-    if iscall(x) && operation(x) === (^)
+    if oper === (^)
         arg1, arg2 = arguments(x)
         if iscall(arg1) && (operation(arg1) === sqrt || operation(arg1) === ssqrt)
             if arg2 isa Integer
@@ -105,6 +104,19 @@ function _postprocess_root(x::SymbolicUtils.BasicSymbolic)
         end
     end
 
+    x = convert_consts(x)
+
+    if oper === (+)
+        args = arguments(x)
+        for arg in args
+            if isequal(arg, 0)
+                after_removing = setdiff(args, arg)
+                isone(length(after_removing)) && return after_removing[1]
+                return Symbolics.term(+, after_removing)
+            end
+        end
+    end
+
     return x
 end
 
@@ -122,3 +134,54 @@ function postprocess_root(x)
     end
     x # unreachable
 end
+
+
+inv_exacts = [0, Symbolics.term(*, pi),
+        Symbolics.term(/,pi,3),
+        Symbolics.term(/, pi, 2),
+        Symbolics.term(/, Symbolics.term(*, 2, pi), 3),
+        Symbolics.term(/, pi, 6),
+        Symbolics.term(/, Symbolics.term(*, 5, pi), 6),
+        Symbolics.term(/, pi, 4)
+]
+inv_evald = Symbolics.symbolic_to_float.(inv_exacts)
+
+const inv_pairs = collect(zip(inv_exacts, inv_evald))
+"""
+    function convert_consts(x)
+This function takes BasicSymbolic terms as input (x) and attempts
+to simplify these basic symbolic terms using known values.
+Currently, this function only supports inverse trigonometric functions.
+
+## Examples
+```jldoctest
+julia> Symbolics.convert_consts(Symbolics.term(acos, 0))
+π / 2
+
+julia> Symbolics.convert_consts(Symbolics.term(atan, 0))
+0
+
+julia> Symbolics.convert_consts(Symbolics.term(atan, 1))
+π / 4
+```
+"""
+function convert_consts(x)
+    !iscall(x) && return x
+
+    oper = operation(x)
+    inv_opers = [asin, acos, atan]
+
+    if any(isequal(oper, o) for o in inv_opers) && isempty(Symbolics.get_variables(x))
+        val = Symbolics.symbolic_to_float(x)
+        for (exact, evald) in inv_pairs
+            if isapprox(evald, val)
+                return exact
+            elseif isapprox(-evald, val)
+                return -exact
+            end
+        end
+    end
+
+    # add [sin, cos, tan] simplifications in the future?
+    return x
+end

src/solver/solve_helpers.jl

@@ -78,7 +78,7 @@ function check_expr_validity(expr)
     valid_type = false
 
     if type_expr <: Number || type_expr == Num || type_expr == SymbolicUtils.BasicSymbolic{Real} ||
-       type_expr == Complex{Num} || type_expr == ComplexTerm{Real}
+       type_expr == Complex{Num} || type_expr == ComplexTerm{Real} || type_expr == SymbolicUtils.BasicSymbolic{Complex{Real}}
         valid_type = true
     end
     iscall(unwrap(expr)) && @assert !hasderiv(unwrap(expr)) "Differential equations are not currently supported"