/// <summary>Solves ax^3 + bx^2 + cx + d</summary>
/// <param name="a">Coefficient of x^3</param>
/// <param name="b">Coefficient of x^2</param>
/// <param name="c">Coefficient of x</param>
/// <param name="d">Free coefficient</param>
/// <returns>Set of roots</returns>
internal static IEnumerable <Entity> SolveCubic(Entity a, Entity b, Entity c, Entity d)
{
// en: https://en.wikipedia.org/wiki/Cubic_equation
// ru: https://ru.wikipedia.org/wiki/%D0%A4%D0%BE%D1%80%D0%BC%D1%83%D0%BB%D0%B0_%D0%9A%D0%B0%D1%80%D0%B4%D0%B0%D0%BD%D0%BE
if (TreeAnalyzer.IsZero(d))
{
return(SolveQuadratic(a, b, c).Append(0));
}
if (TreeAnalyzer.IsZero(a))
{
return(SolveQuadratic(b, c, d));
}
var coeff = MathS.i * MathS.Sqrt(3) / 2;
var u1 = Integer.Create(1);
var u2 = Rational.Create(-1, 2) + coeff;
var u3 = Rational.Create(-1, 2) - coeff;
var D0 = MathS.Sqr(b) - 3 * a * c;
var D1 = (2 * MathS.Pow(b, 3) - 9 * a * b * c + 27 * MathS.Sqr(a) * d).InnerSimplified;
var C = MathS.Pow((D1 + MathS.Sqrt(MathS.Sqr(D1) - 4 * MathS.Pow(D0, 3))) / 2, Rational.Create(1, 3));
return(new[] { u1, u2, u3 }.Select(uk =>
C.Evaled == 0 && D0.Evaled == 0 ? -(b + uk * C) / 3 / a : -(b + uk * C + D0 / C / uk) / 3 / a));
}
public void Test6()
{
var expr = (MathS.Sqr(x) + MathS.Sqr(x)) / MathS.Sqr(x) + MathS.Sqrt(x);
var func = expr.Compile(x);
Assert.IsTrue(func.Call(4) == 4);
}
public void TestInner()
{
var expr = MathS.FromString("sqrt(x + sqrt(x))");
var expected = MathS.Sqrt(x + MathS.Sqrt(x));
Assert.IsTrue(expr == expected);
}
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 Test6()
{
// Caching with one value
var expr = (MathS.Sqr(x) + MathS.Sqr(x)) / MathS.Sqr(x) + MathS.Sqrt(x);
var func = expr.Compile(x);
Assert.Equal(4, func.Call(4));
}
public void TestLong()
{
// Caching with multiple values
var expr = (MathS.Sqr(x) + MathS.Sqr(x)) / MathS.Sqr(x)
+ MathS.Sqrt(x) + MathS.Cbrt(x) * MathS.Cbrt(x) + MathS.Sqrt(x);
var func = expr.Compile(x);
Assert.Equal(34, func.Call(64));
}
public void TestComplex()
{
var expr = MathS.FromString("ln(x) + sqrt(x) + tan(x) + sec(x) + cosec(x) + cotan(x)");
var expected = MathS.Ln(x) + MathS.Sqrt(x) +
MathS.Tan(x) + MathS.Sec(x) + MathS.Cosec(x) +
MathS.Cotan(x);
Assert.IsTrue(expr == expected);
}
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);
}
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 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 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 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>
/// solves ax3 + bx2 + cx + d
/// </summary>
/// <param name="a">
/// Coefficient of x^3
/// </param>
/// <param name="b">
/// Coefficient of x^2
/// </param>
/// <param name="c">
/// Coefficient of x
/// </param>
/// <param name="d">
/// Free coefficient
/// </param>
/// <returns>
/// Set of roots
/// </returns>
internal static Set SolveCubic(Entity a, Entity b, Entity c, Entity d)
{
// en: https://en.wikipedia.org/wiki/Cubic_equation
// ru: https://ru.wikipedia.org/wiki/%D0%A4%D0%BE%D1%80%D0%BC%D1%83%D0%BB%D0%B0_%D0%9A%D0%B0%D1%80%D0%B4%D0%B0%D0%BD%D0%BE
// TODO (to remove sympy code!)
Set res;
if (TreeAnalyzer.IsZero(d))
{
res = SolveQuadratic(a, b, c);
res.Add(0);
return(res);
}
if (TreeAnalyzer.IsZero(a))
{
return(SolveQuadratic(b, c, d));
}
res = new Set();
var coeff = MathS.i * MathS.Sqrt(3) / 2;
var u1 = new NumberEntity(1);
var u2 = SySyn.Rational(-1, 2) + coeff;
var u3 = SySyn.Rational(-1, 2) - coeff;
var D0 = MathS.Sqr(b) - 3 * a * c;
var D1 = (2 * MathS.Pow(b, 3) - 9 * a * b * c + 27 * MathS.Sqr(a) * d).InnerSimplify();
var C = MathS.Pow((D1 + MathS.Sqrt(MathS.Sqr(D1) - 4 * MathS.Pow(D0, 3))) / 2, Number.CreateRational(1, 3));
foreach (var uk in new List <Entity> {
u1, u2, u3
})
{
Entity r;
if (Const.EvalIfCan(C) == 0 && Const.EvalIfCan(D0) == 0)
{
r = -(b + uk * C) / 3 / a;
}
else
{
r = -(b + uk * C + D0 / C / uk) / 3 / a;
}
res.Add(r);
}
return(res);
}
/// <summary>Solves ax^2 + bx + c</summary>
/// <param name="a">Coefficient of x^2</param>
/// <param name="b">Coefficient of x</param>
/// <param name="c">Free coefficient</param>
/// <returns>Set of roots</returns>
internal static IEnumerable <Entity> SolveQuadratic(Entity a, Entity b, Entity c)
{
if (TreeAnalyzer.IsZero(c))
{
return(SolveLinear(a, b).Append(0));
}
if (TreeAnalyzer.IsZero(a))
{
return(SolveLinear(b, c));
}
var D = MathS.Sqr(b) - 4 * a * c;
return(new[] { ((-b - MathS.Sqrt(D)) / (2 * a)).InnerSimplified,
((-b + MathS.Sqrt(D)) / (2 * a)).InnerSimplified });
}
internal static bool PullSin(ComplexNumber arg, out Entity res)
{
if (TryPulling(TableSin, arg, out res))
{
return(true);
}
if (TryPulling(TableSin, MathS.DecimalConst.pi - arg, out res))
{
return(true);
}
if (TryPulling(TableCos, arg * 2, out res))
{
res = MathS.Sqrt((1 - res) / 2);
return(true);
}
return(false);
}
internal static bool PullCos(ComplexNumber arg, out Entity res)
{
if (TryPulling(TableCos, arg, out res))
{
return(true);
}
if (TryPulling(TableCos, -1 * arg, out res))
{
return(true);
}
if (TryPulling(TableCos, arg * 2, out res))
{
res = MathS.Sqrt((1 + res) / 2);
return(true);
}
return(false);
}
internal static bool PullTan(ComplexNumber arg, out Entity res)
{
if (TryPulling(TableTan, arg, out res))
{
return(true);
}
if (TryPulling(TableTan, (RealNumber)MathS.DecimalConst.pi - arg, out res))
{
res *= -1;
return(true);
}
if (TryPulling(TableCos, arg * 2, out res))
{
res = MathS.Sqrt((1 - res) / (1 + res));
return(true);
}
return(false);
}
/// <summary>Solves ax^4 + bx^3 + cx^2 + dx + e</summary>
/// <param name="a">Coefficient of x^4</param>
/// <param name="b">Coefficient of x^3</param>
/// <param name="c">Coefficient of x^2</param>
/// <param name="d">Coefficient of x</param>
/// <param name="e">Free coefficient</param>
/// <returns>Set of roots</returns>
internal static IEnumerable <Entity> SolveQuartic(Entity a, Entity b, Entity c, Entity d, Entity e)
{
// en: https://en.wikipedia.org/wiki/Quartic_function
// ru: https://ru.wikipedia.org/wiki/%D0%9C%D0%B5%D1%82%D0%BE%D0%B4_%D0%A4%D0%B5%D1%80%D1%80%D0%B0%D1%80%D0%B8
if (TreeAnalyzer.IsZero(e))
{
return(SolveCubic(a, b, c, d).Append(0));
}
if (TreeAnalyzer.IsZero(a))
{
return(SolveCubic(b, c, d, e));
}
var alpha = (-3 * MathS.Sqr(b) / (8 * MathS.Sqr(a)) + c / a)
.InnerSimplified;
var beta = (MathS.Pow(b, 3) / (8 * MathS.Pow(a, 3)) - (b * c) / (2 * MathS.Sqr(a)) + d / a)
.InnerSimplified;
var gamma = (-3 * MathS.Pow(b, 4) / (256 * MathS.Pow(a, 4)) + MathS.Sqr(b) * c / (16 * MathS.Pow(a, 3)) - (b * d) / (4 * MathS.Sqr(a)) + e / a)
.InnerSimplified;
if (beta.Evaled == 0)
{
return(sqrtsOf1.SelectMany(_ => sqrtsOf1,
(s, t) => - b / 4 * a + s * MathS.Sqrt((-alpha + t * MathS.Sqrt(MathS.Sqr(alpha) - 4 * gamma)) / 2)));
}
var oneThird = Rational.Create(1, 3);
var P = (-MathS.Sqr(alpha) / 12 - gamma)
.InnerSimplified;
var Q = (-MathS.Pow(alpha, 3) / 108 + alpha * gamma / 3 - MathS.Sqr(beta) / 8)
.InnerSimplified;
var R = -Q / 2 + MathS.Sqrt(MathS.Sqr(Q) / 4 + MathS.Pow(P, 3) / 27);
var U = MathS.Pow(R, oneThird)
.InnerSimplified;
var y = (Rational.Create(-5, 6) * alpha + U + (U.Evaled == 0 ? -MathS.Pow(Q, oneThird) : -P / (3 * U)))
.InnerSimplified;
var W = MathS.Sqrt(alpha + 2 * y)
.InnerSimplified;
// Now we need to permutate all four combinations
return(sqrtsOf1.SelectMany(_ => sqrtsOf1,
(s, t) => - b / (4 * a) + (s * W + t * MathS.Sqrt(-(3 * alpha + 2 * y + s * 2 * beta / W))) / 2));
}
/// <summary>
/// 旋转单位向量From到另外一个单位向量To
/// </summary>
/// <param name="from">单位向量</param>
/// <param name="to">单位向量</param>
/// <returns>单位四元数</returns>
public static Quaternion FromTo(Vector3 from, Vector3 to)
{
Quaternion uq = Quaternion.identity;
float squar = 2 * (1 + Vector3.Dot(from, to));
if (MathS.Abs(squar) <= Threshold)
{
Vector3 perp = from.Perpendicular().normalized;
uq = new Quaternion(perp.x, perp.y, perp.z, 0);
}
else
{
float root = MathS.Sqrt(squar);
Vector3 imag = from.Cross(to) / root;
float real = root * 0.5f;
uq = new Quaternion(imag.x, imag.y, imag.z, real);
}
return(uq);
}
internal static bool PullSin(ComplexNumber arg, out Entity res)
{
if (TryPulling(TableSin, arg, out res))
{
return(true);
}
if (TryPulling(TableSin, (RealNumber)MathS.DecimalConst.pi - arg, out res))
{
return(true);
}
if (TryPulling(TableCos, arg * 2, out res))
{
res = MathS.Sqrt((1 - res) / 2);
if (EDecimalWrapper.IsLess((Number.Sin(arg) as RealNumber).Value, 0))
{
res *= -1;
}
return(true);
}
return(false);
}
internal static bool PullCos(ComplexNumber arg, out Entity res)
{
if (TryPulling(TableCos, arg, out res))
{
return(true);
}
if (TryPulling(TableCos, -1 * arg, out res))
{
return(true);
}
if (TryPulling(TableCos, arg * 2, out res))
{
res = MathS.Sqrt((1 + res) / 2);
if ((Number.Cos(arg) as RealNumber).Value < 0)
{
res *= -1;
}
return(true);
}
return(false);
}
/// <summary>
/// solves ax2 + bx + c
/// </summary>
/// <param name="a">
/// Coefficient of x^2
/// </param>
/// <param name="b">
/// Coefficient of x
/// </param>
/// <param name="c">
/// Free coefficient
/// </param>
/// <returns>
/// Set of roots
/// </returns>
internal static Set SolveQuadratic(Entity a, Entity b, Entity c)
{
Set res;
if (TreeAnalyzer.IsZero(c))
{
res = SolveLinear(a, b);
res.Add(0);
return(res);
}
if (TreeAnalyzer.IsZero(a))
{
return(SolveLinear(b, c));
}
res = new Set();
var D = MathS.Sqr(b) - 4 * a * c;
res.Add(((-b - MathS.Sqrt(D)) / (2 * a)).InnerSimplify());
res.Add(((-b + MathS.Sqrt(D)) / (2 * a)).InnerSimplify());
return(res);
}
/// <summary>
/// solves ax4 + bx3 + cx2 + dx + e
/// </summary>
/// <param name="a">
/// Coefficient of x^4
/// </param>
/// <param name="b">
/// Coefficient of x^3
/// </param>
/// <param name="c">
/// Coefficient of x^2
/// </param>
/// <param name="d">
/// Coefficient of x
/// </param>
/// <param name="e">
/// Free coefficient
/// </param>
/// <returns>
/// Set of roots
/// </returns>
internal static Set SolveQuartic(Entity a, Entity b, Entity c, Entity d, Entity e)
{
// en: https://en.wikipedia.org/wiki/Quartic_function
// ru: https://ru.wikipedia.org/wiki/%D0%9C%D0%B5%D1%82%D0%BE%D0%B4_%D0%A4%D0%B5%D1%80%D1%80%D0%B0%D1%80%D0%B8
Set res;
if (TreeAnalyzer.IsZero(e))
{
res = SolveCubic(a, b, c, d);
res.Add(0);
return(res);
}
if (TreeAnalyzer.IsZero(a))
{
return(SolveCubic(b, c, d, e));
}
res = new Set();
var alpha = (-3 * MathS.Sqr(b) / (8 * MathS.Sqr(a)) + c / a)
.InnerSimplify();
var beta = (MathS.Pow(b, 3) / (8 * MathS.Pow(a, 3)) - (b * c) / (2 * MathS.Sqr(a)) + d / a)
.InnerSimplify();
var gamma = (-3 * MathS.Pow(b, 4) / (256 * MathS.Pow(a, 4)) + MathS.Sqr(b) * c / (16 * MathS.Pow(a, 3)) - (b * d) / (4 * MathS.Sqr(a)) + e / a)
.InnerSimplify();
if (Const.EvalIfCan(beta) == 0)
{
res.FastAddingMode = true;
for (int s = -1; s <= 1; s += 2)
{
for (int t = -1; t <= 1; t += 2)
{
var x = -b / 4 * a + s * MathS.Sqrt((-alpha + t * MathS.Sqrt(MathS.Sqr(alpha) - 4 * gamma)) / 2);
res.Add(x);
}
}
res.FastAddingMode = false;
return(res);
}
var oneThird = Number.CreateRational(1, 3);
var P = (-MathS.Sqr(alpha) / 12 - gamma)
.InnerSimplify();
var Q = (-MathS.Pow(alpha, 3) / 108 + alpha * gamma / 3 - MathS.Sqr(beta) / 8)
.InnerSimplify();
var R = -Q / 2 + MathS.Sqrt(MathS.Sqr(Q) / 4 + MathS.Pow(P, 3) / 27);
var U = MathS.Pow(R, oneThird)
.InnerSimplify();
var y = (Number.CreateRational(-5, 6) * alpha + U + (Const.EvalIfCan(U) == 0 ? -MathS.Pow(Q, oneThird) : -P / (3 * U)))
.InnerSimplify();
var W = MathS.Sqrt(alpha + 2 * y)
.InnerSimplify();
// Now we need to permutate all four combinations
res.FastAddingMode = true; /* we are sure that there's no such root yet */
for (int s = -1; s <= 1; s += 2)
{
for (int t = -1; t <= 1; t += 2)
{
var x = -b / (4 * a) + (s * W + t * MathS.Sqrt(-(3 * alpha + 2 * y + s * 2 * beta / W))) / 2;
res.Add(x);
}
}
res.FastAddingMode = false;
return(res);
}
public void TestArc3()
{
var func = MathS.Arccos(2 * x);
AssertEqEntity(func.Derive(x).Simplify(), (-2) / MathS.Sqrt(1 + (-4) * MathS.Sqr(x)));
}
public void TestArc2()
{
var func = MathS.Arcsin(2 * x);
AssertEqEntity(func.Derive(x).Simplify(), 2 / MathS.Sqrt(1 + (-4) * MathS.Sqr(x)));
}
public void TestArc1()
{
var func = MathS.Arcsin(x);
AssertEqEntity(func.Derive(x).Simplify(), 1 / MathS.Sqrt(1 - MathS.Sqr(x)));
}
[Fact] public void TestFormula19() => Assert.Equal(x * MathS.Sqrt(3), FromString("x sqrt(3)"));
public void Test1()
{
var func = (x + MathS.Sqrt(x)).Compile(x);
Assert.IsTrue(func.Substitute(4) == 6);
}
[TestMethod] public void SquareRoot() => Test(@"\sqrt{x}", MathS.Sqrt(x));
[TestMethod] public void SetDuplicates() =>
Test(@"\left\{\left[x-i,2-x\times i\right],2,\begin{pmatrix}a & i\\\pi & e\end{pmatrix},4,\left[x-i,3-x\times i\right],i\right\}",
new Set(MathS.Sets.Interval(x - MathS.i, 2 - x * MathS.i), 2, MathS.Matrices.Matrix(2, 2, "a", "i", "pi", "e"), 2 + 2, MathS.Sets.Interval(x - MathS.i, 2 - x * MathS.i), MathS.Sqrt(16), MathS.Sets.Interval(x - MathS.i, 3 - x * MathS.i), MathS.Matrices.Matrix(2, 2, "a", "i", "pi", "e"), MathS.i));
