Mulf(Sinf(var any1a), Cosecantf(var any1)) when any1 == any1a => 1,
Minusf(Powf(Cosf(var any1), Integer(2)), Powf(Sinf(var any1a), Integer(2))) when any1 == any1a =>
Powf(Sinf(var any1a), Integer(2))) when any1 == any1a => 1,
// arcfunc(func(x)) = x Arcsinf(Sinf(var any1)) => any1,
// sin({}) * cos({}) = 1/2 * sin(2{}) Mulf(Sinf(var any1), Cosf(var any1a)) when any1 == any1a => Rational.Create(1, 2) * new Sinf(2 * any1),
// sin(x) -> sin(1 * x) // sin(x) + sin(a * x) // if x * a // x * a => a * x // if ! a * x // x => 1 * x // if ! a * x + b // a * x => a * x + 0 private static Entity ReplaceTrigonometry(Entity expr, VariableEntity variable, VariableEntity replacement) { // SolveLinear should also solve tan and cotan equations, but currently Polynomial solver cannot handle big powers // uncomment lines above when it will be fixed (TODO) var sin = expr.FindSubtree(MathS.Sin(variable)); var cos = expr.FindSubtree(MathS.Cos(variable)); //var tan = expr.FindSubtree(MathS.Tan (variable)); //var cot = expr.FindSubtree(MathS.Cotan(variable)); var sinReplacement = replacement / (2 * MathS.i) - MathS.Pow(replacement, -1) / (2 * MathS.i); var cosReplacement = replacement / 2 + MathS.Pow(replacement, -1) / 2; // var tanReplacement = (1 - MathS.Sqr(replacement)) * MathS.i * MathS.Pow(MathS.Sqr(replacement) + 1, -1); // var cotReplacement = (MathS.Sqr(replacement) + 1) * MathS.i * MathS.Pow(MathS.Sqr(replacement) - 1, -1); TreeAnalyzer.FindAndReplace(ref expr, sin, sinReplacement); TreeAnalyzer.FindAndReplace(ref expr, cos, cosReplacement); // TreeAnalyzer.FindAndReplace(ref expr, tan, tanReplacement); // TreeAnalyzer.FindAndReplace(ref expr, cot, cotReplacement); // workaround for missing Patterns.Variable(x), replace all expressions, false-matching pattern to variables // which will be returned back after all actual replacements var falseReplacements = new List <KeyValuePair <VariableEntity, Entity> >(); string falseReplacementName = "trig"; void ReplaceSinSubExpression(ref Entity expr, Entity toReplace, Entity a, Entity b, VariableEntity replacement) { // sin(ax + b) = (t^a * e^(i*b) - t^(-a) * e^(-i*b)) / (2i) var resultReplacement = (MathS.Pow(replacement, a) * (MathS.Pow(MathS.e, b * MathS.i) / (2 * MathS.i)) - MathS.Pow(replacement, -a) * (MathS.Pow(MathS.e, -b * MathS.i)) / (2 * MathS.i)); TreeAnalyzer.FindAndReplace(ref expr, toReplace, resultReplacement); } void ReplaceCosSubExpression(ref Entity expr, Entity toReplace, Entity a, Entity b, VariableEntity replacement) { // cos(ax + b) = (t^a * e^(i*b) + t^(-a) * e^(-i*b)) / 2 var resultReplacement = (MathS.Pow(replacement, a) * (MathS.Pow(MathS.e, b * MathS.i) / 2) + MathS.Pow(replacement, -a) * (MathS.Pow(MathS.e, -b * MathS.i)) / 2); TreeAnalyzer.FindAndReplace(ref expr, toReplace, resultReplacement); } // checks if pattern-matching succeeded. If not, replace subexpression with some variable bool CheckIfReplacementIsSuitable(ref Entity expr, Entity found, Entity variable, Entity variableToCheckFor) { if (variable != variableToCheckFor) { var repl = Utils.FindNextIndex(expr, falseReplacementName); falseReplacements.Add(new KeyValuePair <VariableEntity, Entity>(repl, found)); TreeAnalyzer.FindAndReplace(ref expr, found, repl); return(false); } return(true); } void MatchSinUntil(Pattern p, Func <Entity, (Entity, Entity, Entity)> variableGetter) { Entity found; while ((found = expr.FindPatternSubtree(p)) != null) { (Entity x, Entity a, Entity b) = variableGetter(found.Children[0]); if (CheckIfReplacementIsSuitable(ref expr, found, x, variable)) { ReplaceSinSubExpression(ref expr, found, a, b, replacement); } } } void MatchCosUntil(Pattern p, Func <Entity, (Entity, Entity, Entity)> variableGetter) { Entity found; while ((found = expr.FindPatternSubtree(p)) != null) { (Entity x, Entity a, Entity b) = variableGetter(found.Children[0]); if (CheckIfReplacementIsSuitable(ref expr, found, x, variable)) { ReplaceCosSubExpression(ref expr, found, a, b, replacement); } } } // arg => (x, a, b) // TODO: refactor this. Move to a list var variablePattern = new Pattern(1000, Entity.PatType.VARIABLE, tree => tree.entType == Entity.EntType.VARIABLE && tree.Name == variable.Name); var pattern1 = Sinf.PHang(Patterns.const1 * variablePattern + Patterns.any1); MatchSinUntil(pattern1, arg => (arg.Children[0].Children[1], arg.Children[0].Children[0], arg.Children[1])); var pattern2 = Sinf.PHang(variablePattern * Patterns.const1 + Patterns.any1); MatchSinUntil(pattern2, arg => (arg.Children[0].Children[0], arg.Children[0].Children[1], arg.Children[1])); var pattern3 = Sinf.PHang(Patterns.any1 + Patterns.const1 * variablePattern); MatchSinUntil(pattern3, arg => (arg.Children[1].Children[1], arg.Children[1].Children[0], arg.Children[0])); var pattern4 = Sinf.PHang(Patterns.any1 + variablePattern * Patterns.const1); MatchSinUntil(pattern4, arg => (arg.Children[1].Children[0], arg.Children[1].Children[1], arg.Children[0])); var pattern5 = Sinf.PHang(Patterns.const1 * variablePattern); MatchSinUntil(pattern5, arg => (arg.Children[1], arg.Children[0], 0)); var pattern6 = Sinf.PHang(variablePattern * Patterns.const1); MatchSinUntil(pattern6, arg => (arg.Children[0], arg.Children[1], 0)); var pattern7 = Sinf.PHang(Patterns.any1 + variablePattern); MatchSinUntil(pattern7, arg => (arg.Children[1], 1, arg.Children[0])); var pattern8 = Sinf.PHang(variablePattern + Patterns.any1); MatchSinUntil(pattern8, arg => (arg.Children[0], 1, arg.Children[1])); var pattern9 = Cosf.PHang(Patterns.const1 * variablePattern + Patterns.any1); MatchCosUntil(pattern9, arg => (arg.Children[0].Children[1], arg.Children[0].Children[0], arg.Children[1])); var pattern10 = Cosf.PHang(variablePattern * Patterns.const1 + Patterns.any1); MatchCosUntil(pattern10, arg => (arg.Children[0].Children[0], arg.Children[0].Children[1], arg.Children[1])); var pattern11 = Cosf.PHang(Patterns.any1 + Patterns.const1 * variablePattern); MatchCosUntil(pattern11, arg => (arg.Children[1].Children[1], arg.Children[1].Children[0], arg.Children[0])); var pattern12 = Cosf.PHang(Patterns.any1 + variablePattern * Patterns.const1); MatchCosUntil(pattern12, arg => (arg.Children[1].Children[0], arg.Children[1].Children[1], arg.Children[0])); var pattern13 = Cosf.PHang(Patterns.const1 * variablePattern); MatchCosUntil(pattern13, arg => (arg.Children[1], arg.Children[0], 0)); var pattern14 = Cosf.PHang(variablePattern * Patterns.const1); MatchCosUntil(pattern14, arg => (arg.Children[0], arg.Children[1], 0)); var pattern15 = Cosf.PHang(Patterns.any1 + variablePattern); MatchCosUntil(pattern15, arg => (arg.Children[1], 1, arg.Children[0])); var pattern16 = Cosf.PHang(variablePattern + Patterns.any1); MatchCosUntil(pattern16, arg => (arg.Children[0], 1, arg.Children[1])); // re-substitute all false replacement to return expression to normal foreach (var repl in falseReplacements) { TreeAnalyzer.FindAndReplace(ref expr, repl.Key, repl.Value); } return(expr); }