An interval under a function monotone on it is the interval between the images of its ends, and a product or quotient of two intervals is an interval

BREAKING-CHANGES.md

@@ -632,6 +632,9 @@ they had.
 | | `"[2; 3] / 2".ToEntity().Simplify()` | `[2; 3] / 2` | `[1; 3/2]` |
 | **Silent** | `"[2; 3) * (-1)".ToEntity().Simplify()` | `[2; 3) * (-1)` | `(-3; -2]` — reflected, ends and openness both |
 | | `"0 - [0; 1)".ToEntity().Simplify()` | `-[0; 1)` | `(-1; 0]` |
+| | `"ln((0; 1))".ToEntity().Simplify()`, and every interval under a function monotone on it — `ln`, `log`, `e^`, `b^`, whole and unit-fraction powers, `abs`, `arctan`, `arcsin`, `arccos` | `ln((0; 1))` — left alone | `(-oo; 0)`; `e^[0; 1]` is `[1; e]`, `(-2; 1]^2` is `[0; 4)`, `abs((-1; 2])` is `[0; 2]`, `arctan([0; 1])` is `[0; pi / 4]`, `sqrt([0; 4])` is `[0; 2]` |
+| | `"(1; 2) * (3; 4)".ToEntity().Simplify()`, and every product or quotient of two intervals with finite numeric ends | `(1; 2) * (3; 4)` — left alone | `(3; 8)`; `(1; 2) / (3; 4)` is `(1/4; 2/3)`; a divisor containing zero stays as written |
+| | `"2 / (0; 1)".ToEntity().Simplify()`, and every constant over an interval that does not contain zero | `2 / (0; 1)` — left alone | `(2; +oo)` |
 | | `"2 * x + 4 * a".ToEntity().Factorize()`, and every sum whose whole coefficients share a divisor | `2 * x + 4 * a` — left alone | `2 * (x + 2 * a)` |
 | | `Transformation.Factorization.Name` | `… then polynomial-factorization` | `… then polynomial-factorization then numeric-content` |
 | | `"sin(x)^2 + 2 sin(x) + 1".ToEntity().Factorize()`, and every polynomial in one subexpression of its variable | `sin(x) ^ 2 + 2 * sin(x) + 1` — left alone | `(sin(x) + 1) ^ 2`; `sin(x) + sin(x)^3` is `sin(x) * (sin(x) ^ 2 + 1)` |

Sources/.editorconfig

@@ -267,6 +267,12 @@ file_header_template=\nCopyright (c) 2019-2026 Angouri.\nAngouriMath is licensed
 [AngouriMath/Functions/Continuous/ExtremumOverSet.cs]
 file_header_template=\nCopyright (c) 2019-2026 Angouri.\nAngouriMath is licensed under MIT.\nDetails: https://github.com/asc-community/AngouriMath/blob/master/LICENSE.md.\nWebsite: https://am.angouri.org.\n
 
+[AngouriMath/Functions/IntervalArithmetic.cs]
+file_header_template=\nCopyright (c) 2019-2026 Angouri.\nAngouriMath is licensed under MIT.\nDetails: https://github.com/asc-community/AngouriMath/blob/master/LICENSE.md.\nWebsite: https://am.angouri.org.\n
+
+[Tests/UnitTests/Core/Sets/IntervalImageTest.cs]
+file_header_template=\nCopyright (c) 2019-2026 Angouri.\nAngouriMath is licensed under MIT.\nDetails: https://github.com/asc-community/AngouriMath/blob/master/LICENSE.md.\nWebsite: https://am.angouri.org.\n
+
 [AngouriMath/Functions/Boolean/Quantifiers.cs]
 file_header_template=\nCopyright (c) 2019-2026 Angouri.\nAngouriMath is licensed under MIT.\nDetails: https://github.com/asc-community/AngouriMath/blob/master/LICENSE.md.\nWebsite: https://am.angouri.org.\n
 

Sources/AngouriMath/Functions/Evaluation/Evaluation.Continuous/Evaluation.Continuous.Arithmetics.Classes.cs

@@ -143,6 +143,7 @@ protected override Entity InnerSimplify(bool isExact) =>
                         (var n1, var n2) when n1 == n2 => new Powf(n1, 2).InnerSimplified(isExact),
                         // After the zero arms above, which answer a multiplication by zero for
                         // every Entity and not only for an interval.
+                        (Interval one, Interval another) => IntervalArithmetic.Product(one, another, isExact),
                         (Interval inter, var n2) when n2 is not Set => ScaledInterval(inter, n2, isExact),
                         (var n2, Interval inter) when n2 is not Set => ScaledInterval(inter, n2, isExact),
                         _ => null
@@ -183,6 +184,11 @@ n0.Evaled is Complex c
                     // and is left alone rather than answered wrongly.
                     (Interval inter, var n2) when n2 is not Set
                         => ScaledInterval(inter, (1 / n2).InnerSimplified(isExact), isExact),
+                    // And the other direction where the divisor does not contain zero, which is
+                    // when its reciprocal is an interval. https://github.com/asc-community/AngouriMath/issues/322
+                    (Interval dividend, Interval divisor) => IntervalArithmetic.Quotient(dividend, divisor, isExact),
+                    (var n1, Interval divisor) when n1 is not Set && IntervalArithmetic.Reciprocal(divisor, isExact) is Interval reciprocal
+                        => ScaledInterval(reciprocal, n1, isExact),
                     _ => null
                 },
                 (@this, a, b) => ((Divf)@this).New(a, b), isExact);
@@ -271,6 +277,11 @@ protected override Entity InnerSimplify(bool isExact) =>
                 (a, b) => (a, b) switch
                 {
                     (Matrix m, Integer(var exp)) when exp is { } expNotNull && TryPower(m, expNotNull, out var res) => res.InnerSimplified(isExact),
+                    // An interval raised to a number, and a number raised to an interval, are
+                    // the images of the interval under a monotone function where the function
+                    // is one. https://github.com/asc-community/AngouriMath/issues/322
+                    (Interval interval, var exponent) when exponent is not Set && IntervalArithmetic.Power(interval, exponent, isExact) is { } image => image,
+                    (var @base, Interval interval) when @base is not Set && IntervalArithmetic.Exponential(@base, interval, isExact) is { } image => image,
                     (Integer(0), var x) =>
                         (isExact ? x.Evaled : x) is Complex c
                         ? c.RealPart.IsPositive
@@ -327,6 +338,9 @@ protected override Entity InnerSimplify(bool isExact) =>
                     (a, b) => (a, b) switch
                     {
                         (Complex n1, Complex n2) when !isExact => Number.Log(n1, n2),
+                        // The image of an interval of positive numbers, monotone for any base.
+                        // https://github.com/asc-community/AngouriMath/issues/322
+                        (var @base, Interval interval) when @base is not Set && IntervalArithmetic.Logarithm(@base, interval, isExact) is { } image => image,
 
                         // log_b(0) is ln(0)/ln(b), that is -oo/ln(b), so the sign of the answer
                         // is the sign of ln(b): -oo for a base above 1 and +oo for one between
@@ -611,6 +625,9 @@ protected override Entity InnerSimplify(bool isExact)
                         Matrix m when m.IsVector => VectorNorm(m, isExact),
                         Complex n when !isExact => Number.Abs(n),
                         Absf abs => abs,
+                        // The interval itself from zero on, its reflection up to zero, and
+                        // folded across zero otherwise. https://github.com/asc-community/AngouriMath/issues/322
+                        Interval interval when IntervalArithmetic.Absolute(interval, isExact) is { } image => image,
                         // |sgn(z)| has unit modulus wherever z is nonzero and is 0 at zero, the
                         // same question as sgn(|z|) above and with the same answer. See there for
                         // why this decides rather than attaching a condition.

Sources/AngouriMath/Functions/Evaluation/Evaluation.Continuous/Evaluation.Continuous.Trigonometry.Classes.cs

@@ -193,6 +193,8 @@ protected override Entity InnerSimplify(bool isExact) =>
                 {
                     Complex n when !isExact => Number.Arcsin(n),
                     { Evaled: Complex n } when isExact && InverseTrigonometricTableValues.PullArcsin(n, out var res) => res,
+                    // Increasing on [-1; 1]. https://github.com/asc-community/AngouriMath/issues/322
+                    Set.Interval interval when IntervalArithmetic.MonotoneOn(interval, -1, 1, increasing: true, MathS.Arcsin, isExact) is { } image => image,
                     _ => null
                 },
                 (@this, a) => ((Arcsinf)@this).New(a), isExact);
@@ -211,6 +213,8 @@ protected override Entity InnerSimplify(bool isExact) =>
                 {
                     Complex n when !isExact => Number.Arccos(n),
                     { Evaled: Complex n } when isExact && InverseTrigonometricTableValues.PullArccos(n, out var res) => res,
+                    // Decreasing on [-1; 1]. https://github.com/asc-community/AngouriMath/issues/322
+                    Set.Interval interval when IntervalArithmetic.MonotoneOn(interval, -1, 1, increasing: false, MathS.Arccos, isExact) is { } image => image,
                     _ => null
                 },
                 (@this, a) => ((Arccosf)@this).New(a), isExact);
@@ -235,6 +239,8 @@ Real r when r.EDecimal.IsNegativeInfinity() => -MathS.pi / 2,
                     Integer(1) => MathS.pi / 4,
                     Integer(-1) => -MathS.pi / 4,
                     { Evaled: Complex n } when isExact && InverseTrigonometricTableValues.PullArctan(n, out var res) => res,
+                    // Increasing everywhere. https://github.com/asc-community/AngouriMath/issues/322
+                    Set.Interval interval when IntervalArithmetic.Monotone(interval, increasing: true, MathS.Arctan, isExact) is { } image => image,
                     _ => null
                 },
                 (@this, a) => ((Arctanf)@this).New(a), isExact);