Stop presenting a near-rational root as an exact one (#235)

Sources/AngouriMath/Functions/Continuous/Solvers/EquationSolver/AnalyticalEquationSolver.cs

@@ -92,6 +92,17 @@ internal static Set Solve(Entity expr, Variable x, bool compensateSolving = fals
             if (expr == x)
                 return new Entity[] { 0 }.ToSet();
 
+            // Whether a candidate root really is one. The loose tolerance that guesses the
+            // rational must not also be what decides this: at 1e-7, x^41 + 6x + 1 accepted
+            // -1/6, whose residual is -1.25e-32 -- small, but the difference between an
+            // answer and a decoration (https://github.com/asc-community/AngouriMath/issues/235).
+            // Where the residual comes out as an exact ratio, which is the case whenever the
+            // equation and the candidate are both rational, it is required to be exactly zero.
+            // Where it does not -- an equation carrying pi, say -- there is nothing to be exact
+            // about and the ordinary tolerance still decides.
+            static bool IsGenuineRoot(Complex residual)
+                => residual is Rational ratio ? ratio.ERational.IsZero : IsZero(residual);
+
             // Applies an attempt to downcast roots
             static Entity TryDowncast(Entity equation, Variable x, Entity root)
             {
@@ -102,7 +113,7 @@ static Entity TryDowncast(Entity equation, Variable x, Entity root)
                 var downcasted = Complex.Create(preciseValue.RealPart, preciseValue.ImaginaryPart);
                 if (equation.Substitute(x, downcasted).Evaled is not Complex error)
                     return root;
-                return IsZero(error) && downcasted.RealPart is Rational && downcasted.ImaginaryPart is Rational
+                return IsGenuineRoot(error) && downcasted.RealPart is Rational && downcasted.ImaginaryPart is Rational
                        ? downcasted : root.InnerSimplified;
             }
 

Sources/Tests/UnitTests/Algebra/SolveTest/RootDowncastTest.cs

@@ -0,0 +1,64 @@
+//
+// 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 AngouriMath;
+using AngouriMath.Extensions;
+using Xunit;
+
+namespace AngouriMath.Tests.Algebra
+{
+    /// <summary>
+    /// A numeric root that lands near a simple ratio is rewritten as that ratio. These
+    /// pin when that is allowed to happen, since a ratio is read as an exact answer.
+    /// </summary>
+    public sealed class RootDowncastTest
+    {
+        private static Entity SingleRoot(string equation) =>
+            Assert.Single((Entity.Set.FiniteSet)equation.ToEntity().Solve("x"));
+
+        // https://github.com/asc-community/AngouriMath/issues/235
+        // x^41 + 6x + 1 = 0 answered -1/6, which is not a root: it leaves
+        // -1/80204967233062404407033075859456. The tolerance that guesses the ratio was
+        // also deciding whether the guess was right, and at 1e-7 a residual of 1.25e-32
+        // passes for zero.
+        [Fact]
+        public void Issue235_NearRationalRootIsNotPresentedAsExact()
+        {
+            var root = SingleRoot("x + x + x + x + x + x + x41 + 1 = 0");
+            Assert.False(root is Entity.Number.Rational,
+                $"root came back as the exact {root.Stringize()}, which is not a root");
+            // It is still the right root, to the accuracy the numeric solver has.
+            Assert.Equal(-1.0 / 6, root.EvalNumerical().RealPart.EDecimal.ToDouble(), 15);
+        }
+
+        [Fact]
+        public void Issue235_TheRejectedRatioReallyIsNotARoot() =>
+            Assert.NotEqual(Entity.Number.Integer.Create(0),
+                "x + x + x + x + x + x + x41 + 1".ToEntity().Substitute("x", "-1/6".ToEntity()).Evaled);
+
+        // Roots that genuinely are ratios must still come back as ratios, or the fix has
+        // traded one wrong answer for uglier right ones.
+        [Theory]
+        [InlineData("2 * x - 1 = 0", "1/2")]
+        [InlineData("x ^ 2 - 1/4 = 0", "1/2")]
+        [InlineData("3 * x + 2 = 0", "-2/3")]
+        [InlineData("x ^ 2 + 2 * x + 1 = 0", "-1")]
+        public void GenuineRationalRootsStayExact(string equation, string expected)
+        {
+            var roots = (Entity.Set.FiniteSet)equation.ToEntity().Solve("x");
+            Assert.Contains(expected.ToEntity().InnerSimplified, roots);
+        }
+
+        // And irrational ones must not be rounded into a ratio either.
+        [Fact]
+        public void IrrationalRootsStayIrrational()
+        {
+            var roots = (Entity.Set.FiniteSet)"x ^ 2 - 2 = 0".ToEntity().Solve("x");
+            Assert.Contains("sqrt(2)".ToEntity().InnerSimplified, roots);
+        }
+    }
+}