Gather a quotient of powers whose exponents a rewrite has already moved (#740)

Sources/AngouriMath/Functions/Simplification/Patterns/Patterns.Power.cs

@@ -48,6 +48,22 @@ expr is Sumf(var any1, Mulf(Real { IsNegative: true } const1, var any2))
             Mulf(Powf(var any1, var any3), Powf(var any2, var any3a)) when any3 == any3a => new Powf(any1 * any2, any3),
             Divf(Powf(var any1, var any3), Powf(var any2, var any3a)) when any3 == any3a => new Powf(any1 / any2, any3),
 
+            // {1} ^ n / {2} ^ (c * n) = ({1} / {2} ^ c) ^ n, and the same the other way up.
+            //
+            // The rule above pairs two powers by their exponents, and the ({}^{})^{} rule
+            // further up rewrites (b^c)^n as b^(c*n) -- on the child, on the way up, so it
+            // has already happened by the time the pair is looked at. Where it applies to
+            // only one of the two, which is whenever only one base is itself a power, the
+            // exponents stop matching and the pair is lost: (a^2)^x / (b^2)^x gathers,
+            // because both sides moved together, while (a^2)^x / b^x does not.
+            // These read that pair back. Restricted to a whole c so that nothing gains a
+            // root it did not have -- b^c goes into the base, rather than the exponent
+            // being divided. https://github.com/asc-community/AngouriMath/issues/740
+            Divf(Powf(var any1, var any3), Powf(var any2, Mulf(Integer { IsPositive: true } const1, var any3a)))
+                when any3 == any3a => new Powf(any1 / new Powf(any2, const1), any3),
+            Divf(Powf(var any1, Mulf(Integer { IsPositive: true } const1, var any3)), Powf(var any2, var any3a))
+                when any3 == any3a => new Powf(new Powf(any1, const1) / any2, any3),
+
             // x / x^n
             Divf(var any1, Powf(var any1a, var any2)) when any1 == any1a => new Powf(any1, 1 - any2),
 

Sources/Tests/UnitTests/Common/PowerQuotientGatheringTest.cs

@@ -0,0 +1,140 @@
+//
+// Copyright (c) 2019-2022 Angouri.
+// AngouriMath is licensed under MIT.
+// Details: https://github.com/asc-community/AngouriMath/blob/master/LICENSE.md.
+// Website: https://am.angouri.org.
+//
+
+using System.Threading.Tasks;
+using AngouriMath;
+using AngouriMath.Extensions;
+using Xunit;
+
+namespace AngouriMath.Tests.Common
+{
+    /// <summary>
+    /// <c>a^p / b^p</c> is gathered into <c>(a/b)^p</c>, but stopped being gathered as soon
+    /// as one of the bases was itself a power: <c>(a^2)^x / (b^2)^x</c> gathers and
+    /// <c>(a^2)^x / b^x</c> did not. The cause is that <c>(b^c)^p</c> is rewritten to
+    /// <c>b^(c*p)</c> on the child, on the way up, so by the time the pair is looked at it
+    /// has already happened -- and where it applies to only one of the two, the exponents
+    /// no longer match. https://github.com/asc-community/AngouriMath/issues/740
+    /// </summary>
+    public sealed class PowerQuotientGatheringTest
+    {
+        private static Entity Simplified(string expr) => expr.ToEntity().Simplify();
+
+        /// <summary>
+        /// The rewrite has to preserve the value, not just the shape. Checked at a point
+        /// where every base is positive, which is where <c>a^p / b^(c*p) = (a/b^c)^p</c>
+        /// holds without a branch argument.
+        /// </summary>
+        private static void AssertSameValueAt(string expr, params (string Variable, string Value)[] at)
+        {
+            var original = expr.ToEntity();
+            var simplified = original.Simplify();
+            Entity Substituted(Entity e)
+            {
+                foreach (var (variable, value) in at)
+                    e = e.Substitute(variable, value.ToEntity());
+                return e;
+            }
+            Assert.Equal(
+                Substituted(original).EvalNumerical().RealPart.EDecimal.ToDouble(),
+                Substituted(simplified).EvalNumerical().RealPart.EDecimal.ToDouble(),
+                9);
+        }
+
+        // A single power is what the gathering is for, and what the limit machinery reads a
+        // 1^oo off. Asserted as "one power" rather than as a string, since which of the
+        // equivalent single powers the complexity metric picks is not the point.
+        private static void AssertGathersIntoOnePower(string expr)
+        {
+            var simplified = Simplified(expr);
+            Assert.True(simplified is Entity.Powf,
+                $"{expr} came back as {simplified.Stringize()}, which is not a single power");
+        }
+
+        [Theory]
+        [InlineData("(a ^ 2 + 1) ^ x / (a ^ 2) ^ x")]
+        [InlineData("(x ^ 2 + 1) ^ x / (x ^ 2) ^ x")]
+        [InlineData("(x ^ 3 + 1) ^ x / (x ^ 3) ^ x")]
+        [InlineData("(a ^ 2) ^ x / b ^ x")]
+        [InlineData("(x ^ 2) ^ x / (x ^ 2 + 1) ^ x")]
+        [InlineData("x ^ (2 * a) / y ^ a")]
+        public void AQuotientOfPowersGathersWhenOneBaseIsItselfAPower(string expr) =>
+            AssertGathersIntoOnePower(expr);
+
+        [Theory]
+        [InlineData("(a ^ 2 + 1) ^ x / (a ^ 2) ^ x")]
+        [InlineData("(a ^ 2) ^ x / b ^ x")]
+        [InlineData("x ^ (2 * a) / y ^ a")]
+        [InlineData("(x ^ 3 + 1) ^ x / (x ^ 3) ^ x")]
+        public void GatheringDoesNotChangeTheValue(string expr) =>
+            AssertSameValueAt(expr, ("a", "17/10"), ("b", "23/10"), ("x", "13/10"), ("y", "31/10"));
+
+        /// <summary>
+        /// Why it is worth gathering. The limit machinery reads a 1^oo off a single power
+        /// and cannot see one in a quotient, so the same function was answered or not
+        /// according only to how it had been written -- and spent five and a half seconds
+        /// not answering.
+        /// </summary>
+        [Theory]
+        [InlineData("(x ^ 2 + 1) ^ x / (x ^ 2) ^ x")]
+        [InlineData("(x ^ 3 + 1) ^ x / (x ^ 3) ^ x")]
+        [InlineData("((x ^ 2 + 1) / x ^ 2) ^ x")]
+        public void TheQuotientFormIsAnsweredAsTheSinglePowerFormIs(string expr)
+        {
+            var limit = Task.Run(() => expr.ToEntity().Limit("x", "+oo").Simplify());
+            Assert.True(limit.Wait(System.TimeSpan.FromSeconds(30)), $"{expr} did not terminate");
+            Assert.Equal(Entity.Number.Integer.Create(1), limit.Result);
+        }
+
+        // What must not change. A numeric exponent is not written as a product, so nothing
+        // here matches it, and x^4 / y^2 keeps the form it had.
+        [Theory]
+        [InlineData("x ^ 4 / y ^ 2")]
+        [InlineData("x ^ 4 / y ^ 3")]
+        [InlineData("a ^ 2 / b")]
+        public void AQuotientWithNumericExponentsIsUnaffected(string expr) =>
+            Assert.False(Simplified(expr) is Entity.Powf,
+                $"{expr} was gathered into {Simplified(expr).Stringize()}");
+
+        // What already worked, and still does.
+        [Theory]
+        [InlineData("(y + 1) ^ x / y ^ x")]
+        [InlineData("a ^ x / b ^ x")]
+        [InlineData("(a ^ 2) ^ x / (b ^ 2) ^ x")]
+        [InlineData("(x + 1) ^ (2 * x) / x ^ (2 * x)")]
+        public void TheQuotientsThatAlreadyGatheredStillDo(string expr) =>
+            AssertGathersIntoOnePower(expr);
+
+        /// <summary>
+        /// The limits #739 fixed go through the same gathering, so they are pinned here as
+        /// well: a change to which quotients gather is a change to which of these are
+        /// answered.
+        /// </summary>
+        [Theory]
+        [InlineData("(x - 5) ^ x / x ^ x", "+oo", "1 / e ^ 5")]
+        [InlineData("(x + 1) ^ x / x ^ x", "+oo", "e")]
+        [InlineData("(x - 5) ^ x / x ^ x", "-oo", "1 / e ^ 5")]
+        public void TheSecondRemarkableLimitStillReadsWhatGatheringGivesIt(
+            string expr, string approach, string expected)
+        {
+            var limit = expr.ToEntity().Limit("x", approach.ToEntity()).Simplify();
+            Assert.Equal(Entity.Number.Integer.Create(0),
+                (limit - expected.ToEntity()).Simplify());
+        }
+
+        /// <summary>
+        /// What is deliberately left. A fractional c would have to divide the exponent
+        /// rather than move into the base, so <c>(sqrt(x) + 1)^x / sqrt(x)^x</c> would
+        /// become <c>((sqrt(x) + 1)^2 / x)^(x/2)</c> -- a squared numerator to buy a
+        /// gathered form. That is a judgement about output rather than the gap this fixes,
+        /// so it is pinned as it stands rather than forced.
+        /// </summary>
+        [Fact]
+        public void AFractionalExponentRatioIsStillNotGathered() =>
+            Assert.False(Simplified("(sqrt(x) + 1) ^ x / sqrt(x) ^ x") is Entity.Powf);
+    }
+}