public void Setup()
{
exprEasy = x + MathS.Sqr(x) - 3;
exprMedium = MathS.Sin(x + MathS.Cos(x)) + MathS.Sqrt(x + MathS.Sqr(x));
exprHard = MathS.Sin(x + MathS.Arcsin(x)) / (MathS.Sqr(x) + MathS.Cos(x)) * MathS.Arccos(x / 1200 + 0.00032 / MathS.Cotan(x + 43));
exprSolvable = MathS.FromString("3arccos(2x + a)3 + 6arccos(2x + a)2 - a3 + 3");
}
public void Test1()
{
var x = MathS.Var("x");
var expr = x * x + MathS.Sin(x) * 0;
Assert.IsTrue(expr.Substitute(x, 0).Simplify() == 0);
}
public void Test3()
{
var x = MathS.Var("x");
var expr = MathS.Sin(x);
Assert.IsTrue(expr.DefiniteIntegral(x, 0, 3).Real > 1.5);
}
public void TestPatt2()
{
var y = MathS.Var("y");
var expr = (MathS.Sqr(MathS.Sin(x + 2 * y)) + MathS.Sqr(MathS.Cos(x + 2 * y))) / (2 * MathS.Sin(x - y) * MathS.Cos(x - y) + 1);
Assert.IsTrue(expr.Simplify() == 1 / (MathS.Sin(2 * (x - y)) + 1));
}
public void Test2()
{
var x = MathS.Var("x");
var expr = MathS.Sin(x);
Assert.IsTrue(expr.DefiniteIntegral(x, -1, 1) == 0);
}
public void TestCosCustom()
{
var func = MathS.Cos(MathS.Pow(x, 3));
var expected = -3 * MathS.Sin(MathS.Pow(x, 3)) * MathS.Sqr(x);
var actual = func.Derive(x).Simplify();
AssertEqEntity(expected.ToString(), actual.ToString());
}
public CompiledFuncTest()
{
multiFuncNotCompiled = (MathS.Log(3, x) + MathS.Sqr(x)) * MathS.Sin(x + MathS.Cosec(x));
multiFunc = multiFuncNotCompiled.Compile(x);
Expression <Func <Complex, Complex> > expr = x => (Complex.Log(x, 3) + Complex.Pow(x, 2)) * Complex.Sin(x + 1 / Complex.Sin(x));
linqFunc = expr.Compile();
}
/// <summary>
/// 根据旋转轴和旋转角生成单位四元数
/// </summary>
/// <param name="angle"></param>
/// <param name="axisNormal"></param>
/// <returns>单位四元数</returns>
public static Quaternion AngleAxis(float angle, Vector3 axisNormal)
{
float radian = MathS.DegToRad * angle * 0.5f;
float real = MathS.Cos(radian);
Vector3 imag = MathS.Sin(radian) * axisNormal;
return(new Quaternion(imag.x, imag.y, imag.z, real));
}
public void TestCosCustom()
{
var func = MathS.Cos(MathS.Pow(x, 3));
var expected = -3 * MathS.Sin(MathS.Pow(x, 3)) * MathS.Sqr(x);
var actual = func.Differentiate(x).Simplify();
Assert.Equal(expected, actual);
}
public CacheCompiledFunc()
{
notCompiled = MathS.Sin(MathS.Sqr(x)) + MathS.Cos(MathS.Sqr(x)) + MathS.Sqr(x) + MathS.Sin(MathS.Sqr(x));
complexFunc = notCompiled.Compile(x);
Expression <Func <Complex, Complex> > linqExpr = x => Complex.Sin(Complex.Pow(x, 2)) + Complex.Cos(Complex.Pow(x, 2)) + Complex.Pow(x, 2) + Complex.Sin(Complex.Pow(x, 2));
linqComp = linqExpr.Compile();
}
public void Test2()
{
var x = MathS.Var("x");
var y = MathS.Var("y");
var expr = x.Pow(x) - MathS.Sqrt(x - 3) / x + MathS.Sin(x);
var expected = (3 * y).Pow(3 * y) - MathS.Sqrt(3 * y - 3) / (3 * y) + MathS.Sin(3 * y);
var actual = expr.Substitute(x, 3 * y);
Assert.IsTrue(expected == actual);
}
private void EveryFrame(object sender, EventArgs e)
{
var B = MathS.Var("B");
var expr2 = B * MathS.Sin(t + B) * MathS.Pow(MathS.e, MathS.i * B * MathS.Cos(t));
var niceFunc2 = expr2.Compile(B);
plotter.Clear();
plotter.PlotIterativeComplex(niceFunc2, 0, t);
plotter.Render();
t += 0.0005m;
}
public void TestComplicated()
{
Entity subExpr = "(a * x2 + b x) / (c x2 - 3)";
Entity expr = MathS.Sqrt(subExpr * 3 / MathS.Sin(subExpr) + MathS.Sin("d"));
Entity.Variable x = nameof(x);
Entity dest = Real.PositiveInfinity;
var limit = expr.Limit(x, dest, ApproachFrom.Left);
Assert.NotNull(limit);
Assert.Equal("sqrt(a / c * 3 / sin(a / c) + sin(d))", limit?.Stringize());
}
public DerivationTest()
{
IterCount = 10000;
tests = new List <Func <object> > {
() => x.Derive(x),
() => (MathS.Cos(x) * MathS.Sin(x)).Derive(x),
() => (MathS.Sqr(MathS.Sin(x + 2 * y)) + MathS.Sqr(MathS.Cos(x + 2 * y))).Derive(x),
() => (x * MathS.Cos(x) / MathS.Sin(MathS.Sqrt(x / MathS.Ln(x)))
* x * MathS.Cos(x) / MathS.Sin(MathS.Sqrt(x / MathS.Ln(x)))
* x * MathS.Cos(x) / MathS.Sin(MathS.Sqrt(x / MathS.Ln(x)))
* x * MathS.Cos(x) / MathS.Sin(MathS.Sqrt(x / MathS.Ln(x)))).Derive(x)
};
}
public SubsTest()
{
IterCount = 10000;
tests = new List <Func <object> > {
() => (x * MathS.Sin(x)).Substitute(x, 3).Eval(),
() => (MathS.Cos(x) * MathS.Sin(x)).Substitute(x, 3).Eval(),
() => (MathS.Sqr(MathS.Sin(x + 2 * x)) + MathS.Sqr(MathS.Cos(x + 2 * x))).Substitute(x, 3).Eval(),
() => (x * MathS.Cos(x) / MathS.Sin(MathS.Sqrt(x / MathS.Ln(x)))
* x * MathS.Cos(x) / MathS.Sin(MathS.Sqrt(x / MathS.Ln(x)))
* x * MathS.Cos(x) / MathS.Sin(MathS.Sqrt(x / MathS.Ln(x)))
* x * MathS.Cos(x) / MathS.Sin(MathS.Sqrt(x / MathS.Ln(x)))).Substitute(x, 3).Eval()
};
}
public SimplificationTest()
{
IterCount = 1500;
tests = new List <Func <object> > {
() => (x * MathS.Sin(x)).Simplify(),
() => (MathS.Cos(x) * MathS.Sin(x)).Simplify(),
() => (MathS.Sqr(MathS.Sin(x + 2 * y)) + MathS.Sqr(MathS.Cos(x + 2 * y))).Simplify(),
() => (x * MathS.Cos(x) / MathS.Sin(MathS.Sqrt(x / MathS.Ln(x)))
* x * MathS.Cos(x) / MathS.Sin(MathS.Sqrt(x / MathS.Ln(x)))
* x * MathS.Cos(x) / MathS.Sin(MathS.Sqrt(x / MathS.Ln(x)))
* x * MathS.Cos(x) / MathS.Sin(MathS.Sqrt(x / MathS.Ln(x)))).Simplify()
};
}
/// <summary>
/// 球面插值
/// </summary>
/// <param name="q1">单位四元数</param>
/// <param name="q2">单位四元数</param>
/// <param name="t">0--1</param>
/// <returns>单位四元数</returns>
public static Quaternion Slerp(Quaternion q1, Quaternion q2, float t)
{
Quaternion uq = q1;
t = MathS.Clamp(t, 0, 1);
float dot = Quaternion.Dot(q1, q2);
if ((1 - dot) > Threshold)
{
float radian = MathS.Acos(dot);
if (radian < 0)
{
radian = MathS.PI - radian;
}
float sin = MathS.Sin(radian);
float t1 = MathS.Sin((1 - t) * radian) / sin;
float t2 = MathS.Sin(t * radian) / sin;
uq = q1 * t1 + q2 * t2;
}
return(uq);
}
public void TestCoTan()
{
var func = MathS.Cotan(2 * x);
AssertEqEntity(func.Derive(x).Simplify(), -2 / MathS.Pow(MathS.Sin(2 * x), 2));
}
public void TestCusfunc()
{
var func = MathS.Sin(x).Pow(2);
AssertEqEntity(func.Derive(x).Simplify(3), MathS.Sin(2 * x));
}
public void Test3()
{
var expr = MathS.Sin(x);
Assert.True(MathS.Compute.DefiniteIntegral(expr, x, 0, 3).RealPart > 1.5);
}
public void TestSin()
{
var func = MathS.Sin(x);
AssertEqEntity(func.Derive(x).Simplify(), MathS.Cos(x));
}
public void Test2()
{
var func = (MathS.Sin(x) + MathS.Cos(x)).Compile(x);
Assert.IsTrue(func.Substitute(0) == 1);
}
[TestMethod] public void Set3() => Test(@"\left\{\sqrt{x},{x}^{2},\sin\left(x\right)\right\}", MathS.Sets.Finite(MathS.Sqrt(x), MathS.Sqr(x), MathS.Sin(x)));
public void TestCoTan()
{
var func = MathS.Cotan(2 * x);
Assert.Equal(-2 / MathS.Pow(MathS.Sin(2 * x), 2), func.Differentiate(x).Simplify());
}
public void TestSin()
{
var func = MathS.Sin(x);
Assert.Equal(MathS.Cos(x), func.Differentiate(x).Simplify());
}
/// <summary>
/// Returns a set of possible roots of a function, e. g.
/// sin(x) = a =>
/// x = arcsin(a) + 2 pi n
/// x = pi - arcsin(a) + 2 pi n
/// </summary>
/// <param name="func"></param>
/// <param name="value"></param>
/// <param name="x"></param>
/// <returns></returns>
public static Set InvertFunctionEntity(FunctionEntity func, Entity value, Entity x)
{
Entity a = func.Children[0];
Entity b = func.Children.Count == 2 ? func.Children[1] : null;
int arg = func.Children.Count == 2 && func.Children[1].FindSubtree(x) != null ? 1 : 0;
var n = Utils.FindNextIndex(func + value, "n");
var res = new Set();
var pi = MathS.pi;
Set GetNotNullEntites(Set set)
{
return(set.FiniteWhere(el => el.entType != Entity.EntType.NUMBER || el.GetValue().IsDefinite()));
}
switch (func.Name)
{
// Consider case when sin(sin(x)) where double-mention of n occures
case "sinf":
{
// sin(x) = value => x = arcsin(value) + 2pi * n
res.AddRange(GetNotNullEntites(FindInvertExpression(a, MathS.Arcsin(value) + 2 * pi * n, x)));
// sin(x) = value => x = pi - arcsin(value) + 2pi * n
res.AddRange(GetNotNullEntites(FindInvertExpression(a, pi - MathS.Arcsin(value) + 2 * pi * n, x)));
return(res);
}
case "cosf":
{
// cos(x) = value => x = arccos(value) + 2pi * n
res.AddRange(GetNotNullEntites(FindInvertExpression(a, MathS.Arccos(value) + 2 * pi * n, x)));
// cos(x) = value => x = -arccos(value) + 2pi * n
res.AddRange(GetNotNullEntites(FindInvertExpression(a, -MathS.Arccos(value) - 2 * pi * n, x)));
return(res);
}
case "tanf":
{
var inverted = FindInvertExpression(a, MathS.Arctan(value) + pi * n, x);
// tan(x) = value => x = arctan(value) + pi * n
res.AddRange(GetNotNullEntites(inverted));
return(res);
}
case "cotanf":
{
var inverted = FindInvertExpression(a, MathS.Arccotan(value) + pi * n, x);
// cotan(x) = value => x = arccotan(value)
res.AddRange(GetNotNullEntites(inverted));
return(res);
}
case "arcsinf":
// arcsin(x) = value => x = sin(value)
if (EntityInBounds(value, ArcsinFrom, ArcsinTo))
{
return(GetNotNullEntites(FindInvertExpression(a, MathS.Sin(value), x)));
}
else
{
return(Empty);
}
case "arccosf":
// arccos(x) = value => x = cos(value)
if (EntityInBounds(value, ArccosFrom, ArccosTo))
{
return(GetNotNullEntites(FindInvertExpression(a, MathS.Cos(value), x)));
}
else
{
return(Empty);
}
case "arctanf":
// arctan(x) = value => x = tan(value)
return(GetNotNullEntites(FindInvertExpression(a, MathS.Tan(value), x)));
case "arccotanf":
// arccotan(x) = value => x = cotan(value)
return(GetNotNullEntites(FindInvertExpression(a, MathS.Cotan(value), x)));
case "logf":
if (arg != 0)
{
// log(x, a) = value => x = a ^ value
return(GetNotNullEntites(FindInvertExpression(b, MathS.Pow(a, value), x)));
}
else
{
// log(a, x) = value => a = x ^ value => x = a ^ (1 / value)
return(GetNotNullEntites(FindInvertExpression(a, MathS.Pow(b, 1 / value), x)));
}
default:
throw new SysException("Unknown function");
}
}
public void Test7() => Assert.Equal(3 * x, MathS.Sin(MathS.Arcsin(x * 3)).Simplify());
public void Test4() => MathS.FromString((MathS.Sin(x) / MathS.Cos(x)).Differentiate(x).Stringize() ?? "");
[TestMethod] public void Trig() => TestSimplify(@"\sin\left(\cos\left(\tan\left(\cot\left(x\right)\right)\right)\right)", MathS.Sin(MathS.Cos(MathS.Tan(MathS.Cotan(x)))));
public void TestCusfunc()
{
var func = MathS.Sin(x).Pow(2);
Assert.Equal(MathS.Sin(2 * x), func.Differentiate(x).Simplify(3));
}