Better simplification.

expreduce/resources/power.m

@@ -59,6 +59,7 @@
 Power[-1, 1/2] := I;
 4^(-1/2) := 1/2;
 16^(-1/2) := 1/4;
+16^(1/2) := 4;
 Power[Rational[a_?Positive,b_?Positive], 1/2] := Power[a, 1/2] * Power[b, -1/2];
 Power[Power[x_, y_Rational], -1] := Power[x, -y];
 (*We may want to deprecate this in favor of the general definition.*)
@@ -201,6 +202,7 @@
 Attributes[Sqrt] = {Listable, NumericFunction, Protected};
 Sqrt[a_Integer?Negative] := I*Sqrt[-a];
 Sqrt[-a_?NumberQ] := I*Sqrt[a];
+Sqrt[a_Integer*b_Plus] := Sqrt[Abs[a]]*Sqrt[(a/Abs[a])*b] /; a != 0;
 Sqrt[a_Real?Positive] := a^.5;
 Sqrt[x_] := Which[
     (*Normally we would define these directly, but right now "x_" is
@@ -216,6 +218,8 @@
         ESameTest[I * Sqrt[3], Sqrt[-3]],
         ESameTest[1, Sqrt[1]],
         ESameTest[0, Sqrt[0]]
+    ], ETests[
+        ESameTest[Sqrt[2] Sqrt[-2-x^y], (-2)(x^y+2)//Sqrt],
     ]
 };
 
@@ -259,8 +263,8 @@
     ]
 };
 
-ExpandAll::usage = "`Expand[expr]` attempts to expand `expr` at all levels.";
-ExpandAll[a]:= Map[Expand, a, {0, Infinity}];
+ExpandAll::usage = "`ExpandAll[expr]` attempts to expand `expr` at all levels.";
+ExpandAll[a_] := Map[Expand, a, {0, Infinity}];
 Attributes[ExpandAll] = {Protected};
 Tests`ExpandAll = {
     ESimpleExamples[

expreduce/resources/simplify.m

@@ -35,12 +35,14 @@
     And[x1___, a_, x2___, Or[x3___, !a_, x4___], x5___] :> And[a, x1, x2, Or[x3, x4], x5],
     And[x1___, Or[x2___, !a_, x3___], x4___, a_, x5___] :> And[a, x1, Or[x2, x3], x4, x5]
 };
-Simplify[exp_] := Module[{e = exp, expanded},
+simplifyInner[exp_] := Module[{e = exp, tryVal},
     e = booleanSimplify[e];
-    expanded = e // Expand;
-    If[LeafCount[expanded] < LeafCount[e], e = expanded];
+    tryVal = e // Expand;
+    If[LeafCount[tryVal] < LeafCount[e], e = tryVal];
+    e = Replace[e, Sqrt[inner_] :> Sqrt[FactorTerms[inner]]];
     e
 ];
+Simplify[exp_] := Map[simplifyInner, exp, {0, Infinity}];
 Attributes[Simplify] = {Protected};
 Tests`Simplify = {
     ESimpleExamples[
@@ -79,7 +81,8 @@
         ESameTest[a || c || Not[b], c || a || Not[b] // Simplify // Sort],
         ESameTest[False, And[x1, a, x2, Not[Or[x3, a, x4]], x5] // Simplify],
         ESameTest[a && x1 && x2 && x5, And[x1, a, x2, Or[x3, a, x4], x5] // Simplify],
-        ESameTest[a && b, a&&b&&a//Simplify]
+        ESameTest[a && b, a&&b&&a//Simplify],
+        ESameTest[(-32+32 x-16 x^2)^9, (16 + 8 (-6 + 4 x - 2 x^2))^9 // Simplify],
     ]
 };
 

expreduce/resources/solve.m

@@ -122,7 +122,7 @@
    ];
 isolateInEqn[eqn_Equal, var_Symbol] := Module[{isolated},
   isolated = {#}& /@ isolate[Rule @@ eqn, var];
-  If[AllTrue[isolated, (Head[#[[1]]] == Rule)&], Return[isolated//Sort]];
+  If[AllTrue[isolated, (Head[#[[1]]] == Rule)&], Return[isolated//Simplify//Sort]];
   Print["isolation procedure failed"];
   isolated
 ];
@@ -149,7 +149,7 @@
   collected
 ];
 
-solveQuadratic[a_.*x_^2 + b_.*x_ + c_., x_] := {{x->(-b-Sqrt[b^2-4 a c])/(2 a)},{x->(-b+Sqrt[b^2-4 a c])/(2 a)}};
+solveQuadratic[a_.*x_^2 + b_.*x_ + c_., x_] := {{x->(-b-Sqrt[b^2-4 a c])/(2 a)},{x->(-b+Sqrt[b^2-4 a c])/(2 a)}}/;FreeQ[{a,b,c},x];
 
 (* Following method described in: *)
 (*Sterling, L, Bundy, A, Byrd, L, O'Keefe, R & Silver, B 1982, Solving Symbolic Equations with PRESS. in*)
@@ -163,7 +163,7 @@
    poly = eqn[[1]]-eqn[[2]];
    If[PolynomialQ[poly, var],
     degree = Exponent[poly, var];
-    If[degree === 2, Return[solveQuadratic[poly, var]]];
+    If[degree === 2, Return[solveQuadratic[poly//Expand, var]//Simplify//Sort]];
    ];
 
    collected = collect[eqn, var];
@@ -189,6 +189,7 @@
     ESimpleExamples[
         ESameTest[{{x->Log[y]/Log[a+b]}}, Solve[(a+b)^x==y,x]],
         ESameTest[{{x -> -Sqrt[-3 + y]}, {x -> Sqrt[-3 + y]}}, Solve[y == x^2 + 3, x]],
+        ESameTest[{{y->1-Sqrt[-2+2 x-x^2]},{y->1+Sqrt[-2+2 x-x^2]}}, Solve[2==x^2+y^2+(x-2)^2+(y-2)^2,y]],
         (*ESameTest[{{x -> (-b - Sqrt[b^2 - 4 a c])/(2 a)}, {x -> (-b + Sqrt[b^2 - 4 a c])/(2 a)}}, Solve[a*x^2 + b*x + c == 0, x]],*)
         (*ESameTest[{{x -> (-b - Sqrt[b^2 - 4 a c + 4 a d])/(2 a)}, {x -> (-b + Sqrt[b^2 - 4 a c + 4 a d])/(2 a)}}, Print[a,b,c,d,x];Solve[a*x^2 + b*x + c == d, x]]*)
     ], ETests[