Let a numeric factor reach the coefficient when collecting like terms (#855) (#862)

CollectLikeTerms failed on the expression its own documentation cites:
(x+1)^2 * (x+1)^2 came back with two x^2 terms side by side.

(2 * x)^2 reduces to 4 * x^2, a product and not a power, so it fell to the
exponent-of-one branch and the 4 was baked into the monomial key instead of
the coefficient. A factor that reduces to a product is now split again, and an
integer power of a product is distributed over its factors.

Only for polynomial-shaped factors. Opening a product is what lets like terms
meet and also what stops one cancelling as a whole: lifting the 1/2 out of
(1/2 * sin(2t))^2 * csc(t)^2 costs the cancellation to cos(t)^2, which
DoubleAngleOverSineTest is there to catch.

Splitting only products that carry a number does not help, since 1/2 is one;
and choosing between the collected and expanded forms by complexity cannot
work, because the payoff of leaving a product intact appears only after passes
that run later.

Author: Rafael-SOWNet

Sources/AngouriMath/Functions/Evaluation/Evaluation.Definition.cs

@@ -239,6 +239,22 @@ static Entity Expand_(Entity e, int level) =>
         /// product of powers, and terms whose products agree are added together --
         /// enough to collect like terms and nothing more.
         /// </remarks>
+        /// <summary>
+        /// Whether every node of the expression is one that monomial collection describes:
+        /// numbers, variables, and the operations that build a polynomial out of them.
+        /// </summary>
+        /// <remarks>
+        /// Opening a product to reach its numeric factor is what lets like terms meet, and
+        /// it is also what stops a product cancelling as a whole -- lifting the 1/2 out of
+        /// <c>(1/2 * sin(2t))^2 * csc(t)^2</c> costs the cancellation to <c>cos(t)^2</c>.
+        /// Collection is a statement about polynomials, so it opens polynomials and leaves
+        /// anything carrying a function intact for the rules that do know it.
+        /// https://github.com/asc-community/AngouriMath/issues/855
+        /// </remarks>
+        private static bool IsPolynomialShaped(Entity expression)
+            => expression.Nodes.All(node =>
+                node is Number or Variable or Mulf or Powf or Sumf or Minusf or Divf);
+
         private static Entity CollectLikeTerms(Entity expanded)
         {
             if (expanded is not Sumf and not Minusf)
@@ -261,16 +277,39 @@ private static Entity CollectLikeTerms(Entity expanded)
                 var exponents = new Dictionary<string, Entity>();
                 var bases = new Dictionary<string, Entity>();
 
-                foreach (var factor in Mulf.LinearChildren(term))
+                var pending = new Stack<Entity>(Mulf.LinearChildren(term));
+                while (pending.Count > 0)
                 {
                     // PowerRules folds (x^2)^2 into x^4, without which the two would be
                     // counted as different monomials.
-                    var reduced = factor.Rewrite(RewriteRules.Power).InnerSimplified;
+                    var reduced = pending.Pop().Rewrite(RewriteRules.Power).InnerSimplified;
                     if (reduced is Number)
                     {
                         coefficient = (coefficient * reduced).InnerSimplified;
                         continue;
                     }
+                    // Reducing a factor can turn it into a product -- (2 * x)^2 becomes
+                    // 4 * x^2 -- and taken whole that is a monomial of its own, keyed on
+                    // "4 * x ^ 2" and unable to meet the plain x^2 terms it belongs with.
+                    // Split it again so the 4 reaches the coefficient. Each child is
+                    // smaller than the product it came from, so this terminates.
+                    // https://github.com/asc-community/AngouriMath/issues/855
+                    if (reduced is Mulf && IsPolynomialShaped(reduced))
+                    {
+                        foreach (var inner in Mulf.LinearChildren(reduced))
+                            pending.Push(inner);
+                        continue;
+                    }
+                    // The same one shape out: (2 * x * y)^2 is a power of a product that
+                    // the rules above do not distribute. Only for an integer exponent,
+                    // where (a * b)^n = a^n * b^n holds for every a and b; for any other
+                    // exponent it is a statement about branches.
+                    if (reduced is Powf(Mulf product, Integer wholePower) && IsPolynomialShaped(product))
+                    {
+                        foreach (var inner in Mulf.LinearChildren(product))
+                            pending.Push(inner.Pow(wholePower));
+                        continue;
+                    }
                     var (@base, exponent) = reduced is Powf(var b, var e) ? (b, e) : (reduced, (Entity)1);
                     var key = @base.Stringize();
                     bases[key] = @base;
@@ -327,6 +366,7 @@ private static Entity CollectLikeTerms(Entity expanded)
             if (collected.Nodes.Any(node => node == MathS.NaN)
                 && !expanded.Nodes.Any(node => node == MathS.NaN))
                 return expanded;
+
             return collected;
         }
 

Sources/Tests/UnitTests/Common/ExpandCollectsTermsTest.cs

@@ -91,6 +91,40 @@ public void ATermThatTakesNoPowerIsNotLost(string input)
             Assert.NotEqual(MathS.NaN, input.ToEntity().Simplify());
         }
 
+        // Collecting keys a term on its monomial, so a numeric factor has to reach the
+        // coefficient rather than stay inside the key. A factor that reduces to a product
+        // -- (2 * x)^2 becomes 4 * x^2 -- or that is a power of one -- (2 * x * y)^2 --
+        // was taken whole, keyed on "4 * x ^ 2", and could not meet the plain x^2 terms it
+        // belonged with. https://github.com/asc-community/AngouriMath/issues/855
+        [Theory]
+        [InlineData("(x+1)^2 * (x+1)^2", "1 + 4 * x + 6 * x ^ 2 + 4 * x ^ 3 + x ^ 4")]
+        [InlineData("(x+1)^4", "1 + 4 * x + 6 * x ^ 2 + 4 * x ^ 3 + x ^ 4")]
+        [InlineData("(2x+1)^2 * (2x+1)^2", "1 + 8 * x + 24 * x ^ 2 + 32 * x ^ 3 + 16 * x ^ 4")]
+        [InlineData("(3x)^2 + x^2", "10 * x ^ 2")]
+        public void ANumericFactorReachesTheCoefficient(string input, string expected) =>
+            Assert.Equal(expected.ToEntity(), input.ToEntity().Expand());
+
+        // Compared as printed rather than as trees: the two agree in value and in every
+        // coefficient and differ only in how the products associate.
+        [Fact]
+        public void APowerOfAProductIsSplitOverItsFactors() =>
+            Assert.Equal("y ^ 4 + 4 * x * y ^ 3 + 6 * x ^ 2 * y ^ 2 + 4 * x ^ 3 * y + x ^ 4",
+                "(x+y)^2 * (x+y)^2".ToEntity().Expand().Stringize());
+
+        // A term carrying a function is left whole, because opening it costs a cancellation
+        // the trigonometric rules would otherwise make: (1/2 * sin(2t))^2 * csc(t)^2
+        // collapses to cos(t)^2 while it is intact and does not once the 1/2 is lifted out.
+        [Fact]
+        public void ATermCarryingAFunctionIsLeftWhole()
+        {
+            // The domain condition is stripped the way DoubleAngleOverSineTest does: the
+            // cosecant makes this undefined where sin(x) is 0, which is a true statement
+            // about the expression and not what this test is about.
+            var simplified = "(sin(2 * x) * cosec(x)) ^ 2 / 4 - cos(2 * x) - sin(x) ^ 2".ToEntity().Simplify();
+            while (simplified is Entity.Providedf(var inner, _)) simplified = inner;
+            Assert.Equal(Entity.Number.Integer.Create(0), simplified);
+        }
+
         // The sets that can be shifted elementwise still are -- the guard above must not
         // turn collecting off for them.
         [Theory]