Take the real root in the sampled checks where the library's power takes it

Sources/AngouriMath/Numerics/IntervalEvaluation.cs

@@ -375,6 +375,16 @@ private static ComplexInterval Evaluate(Entity expr)
             }
         }
 
+        /// <summary>
+        /// The numerator of <paramref name="exponent"/> where it is an exact fraction of odd denominator below 10, to which
+        /// <c>Number.Pow</c> raises a negative real as its real root; <see langword="null"/> otherwise.
+        /// </summary>
+        internal static EInteger? TheRealRootIsTaken(Entity exponent)
+            => exponent is Number.Rational { ERational: var exact } and not Number.Integer
+                && exact.Denominator.Abs() is var q && !q.IsEven && q.CompareTo(EInteger.FromInt32(10)) < 0
+                ? (exact.Denominator.Sign < 0 ? exact.Numerator.Negate() : exact.Numerator)
+                : null;
+
         [System.Runtime.CompilerServices.MethodImpl(System.Runtime.CompilerServices.MethodImplOptions.NoInlining)]
         private static ComplexInterval EvaluateOtherThanArithmetic(Entity expr)
         {
@@ -387,9 +397,21 @@ private static ComplexInterval EvaluateOtherThanArithmetic(Entity expr)
                 case Variable variable when variable == MathS.e:
                     return ComplexInterval.Real(Interval.Around(Math.E));
                 case Powf(var @base, var exponent):
+                {
                     if (@base == MathS.e)
                         return Evaluate(exponent).Exp();
-                    return Evaluate(@base).Pow(Evaluate(exponent));
+                    var evaluatedBase = Evaluate(@base);
+                    // The real root where the library's power takes it: a negative real base and an exact exponent of
+                    // odd denominator below 10, as (-8)^(1/3) is -2. Principal there, the derivative of (-x^3)^(1/3)
+                    // was -e^(-2 i pi/3) at x > 0 where it is -1, and the check turned the right answer away.
+                    // https://github.com/asc-community/AngouriMath/issues/2089
+                    if (TheRealRootIsTaken(exponent) is { } numerator && evaluatedBase.IsReal && evaluatedBase.Re.High < 0)
+                    {
+                        var modulus = ComplexInterval.Real(-evaluatedBase.Re).Pow(Evaluate(exponent));
+                        return numerator.IsEven ? modulus : -modulus;
+                    }
+                    return evaluatedBase.Pow(Evaluate(exponent));
+                }
                 case Sinf(var argument):
                     return Evaluate(argument).Sin();
                 case Cosf(var argument):

Sources/AngouriMath/Numerics/PreciseEvaluation.cs

@@ -456,7 +456,19 @@ private PreciseComplexInterval EvaluateOtherThanArithmetic(Entity expr)
                 case Variable variable when variable == MathS.e:
                     return Real(E());
                 case Powf(var @base, var exponent):
-                    return @base == MathS.e ? Exp(Evaluate(exponent)) : Pow(Evaluate(@base), Evaluate(exponent));
+                {
+                    if (@base == MathS.e)
+                        return Exp(Evaluate(exponent));
+                    var evaluatedBase = Evaluate(@base);
+                    // The real root where the library's power takes it, as in the doubles.
+                    // https://github.com/asc-community/AngouriMath/issues/2089
+                    if (IntervalEvaluation.TheRealRootIsTaken(exponent) is { } numerator && evaluatedBase.IsReal && evaluatedBase.Re.High.Sign < 0)
+                    {
+                        var modulus = Pow(Real(evaluatedBase.Re.Negate()), Evaluate(exponent));
+                        return numerator.IsEven ? modulus : Negate(modulus);
+                    }
+                    return Pow(evaluatedBase, Evaluate(exponent));
+                }
                 case Sinf(var argument):
                     return Sin(Evaluate(argument));
                 case Cosf(var argument):

Sources/Tests/UnitTests/Core/IntervalEvaluationTest.cs

@@ -40,6 +40,8 @@ public sealed class IntervalEvaluationTest
         [InlineData("arcsec(3)")]
         [InlineData("arccsc(-2)")]
         [InlineData("(2/3)^(1/3)")]
+        [InlineData("(-8)^(1/3) + (-2)^(2/5)")]
+        [InlineData("(-3/2)^(-4/7) + (-5)^(1/2)")]
         [InlineData("(1 + 2i)^3")]
         [InlineData("1/(3 + 4i)")]
         [InlineData("e^(i * pi/4)")]
@@ -182,6 +184,7 @@ public void TheSizeOfALargeComplexValueIsFinite()
         [InlineData("sin(1 + i) * cos(2 - 3i)")]
         [InlineData("ln(-2 + i) + arctan(2) - arcsin(1/3)")]
         [InlineData("(2/3)^(1/3) + sec(1) + csc(2)")]
+        [InlineData("(-8)^(1/3) + (-2)^(2/5) + (-3/2)^(-4/7) + (-5)^(1/2)")]
         [InlineData("arccot(-2) + arcsec(3) + arccsc(-2) + arccot(0)")]
         [InlineData("sin(10^6) + cos(10^6)")]
         [InlineData("sqrt(2) provided 1 < 2")]
@@ -265,6 +268,21 @@ public void ADerivativeKeepingASpecialFunctionIsChecked(string antiderivative, s
             Assert.False(Functions.PartialFractions.HoldsAtSampledPoints(derivative, (integrand + " + 1/1000").ToEntity(), x));
         }
 
+        /// <summary>
+        /// The real root where the library's power takes it, as <c>(-8)^(1/3)</c> is <c>-2</c>: the derivative of
+        /// <c>(-x^3)^(1/3)</c>, <c>-x^2 (-x^3)^(-2/3)</c>, is <c>-1</c>, and at the points right of 0 the check read it on
+        /// the principal branch, <c>-e^(-2 i pi/3)</c>, and said it was not.
+        /// <a href="https://github.com/asc-community/AngouriMath/issues/2089">#2089</a>
+        /// </summary>
+        [Fact]
+        public void TheRealRootIsTheOneThePowerTakes()
+        {
+            var x = MathS.Var("x");
+            var derivative = "(-x^3)^(1/3)".ToEntity().Differentiate(x);
+            Assert.True(Functions.PartialFractions.HoldsAtSampledPoints(derivative, "-1".ToEntity(), x));
+            Assert.False(Functions.PartialFractions.HoldsAtSampledPoints(derivative, "-1 + 1/1000".ToEntity(), x));
+        }
+
         [Fact]
         public void AgreementHasThreeOutcomes()
         {