[TestMethod] public void M4RootCube() => Test(@"\frac{1}{\sqrt[4]{x}^{3}}", MathS.Pow(x, 3m / -4));
[TestMethod] public void IM1Pow() => Test(@"{\left(-1 + i\right)}^{x}", MathS.Pow(im1, x));
[TestMethod] public void SquareRootAsPow() => Test(@"\sqrt{x}", MathS.Pow(x, 1m / 2));
[TestMethod] public void M2ICubeRoot() => Test(@"\sqrt[3]{-2i}", MathS.Pow(m2i, 1m / 3));
[TestMethod] public void PowM2I() => Test(@"{2}^{-2i}", MathS.Pow(2, m2i));
[TestMethod] public void M2Pow() => Test(@"{-2}^{x}", MathS.Pow(-2, x));
[TestMethod] public void Frac34Pow() => Test(@"{\left(\frac{3}{4}\right)}^{x}", MathS.Pow(frac34, x));
private static Entity Sqrt(Entity a) => MathS.Pow(a, Number.CreateRational(1, 2));
public void TestPatt1()
{
var expr = MathS.Pow(x * 4, 3);
Assert.IsTrue(expr.Simplify() == 64 * MathS.Pow(x, 3));
}
[TestMethod] public void PowPowNumber() => Test(@"{\left({12}^{23}\right)}^{34}", MathS.Pow(MathS.Pow(12, 23), 34));
private static Entity Cubt(Entity a) => MathS.Pow(a, Number.CreateRational(1, 3));
[TestMethod] public void PowPow() => Test("{x}^{{x}^{x}}", MathS.Pow(x, MathS.Pow(x, x)));
[TestMethod] public void PowBase10() => Test("{10}^{x}", MathS.Pow(10, x));
[TestMethod] public void PowBase2() => Test("{2}^{x}", MathS.Pow(2, x));
[TestMethod] public void PowM1() => Test(@"{x}^{-1}", MathS.Pow(x, m1));
/// <summary>
/// Inverts operator and returns a set
/// x^2 = a
/// => x = sqrt(a)
/// x = -sqrt(a)
/// </summary>
/// <param name="func"></param>
/// <param name="value"></param>
/// <param name="x"></param>
/// <returns></returns>
public static Set InvertOperatorEntity(OperatorEntity func, Entity value, Entity x)
{
Entity a, un;
int arg;
if (func.Children[0].FindSubtree(x) != null)
{
a = func.Children[1];
un = func.Children[0];
arg = 0;
}
else
{
a = func.Children[0];
un = func.Children[1];
arg = 1;
}
var n = Utils.FindNextIndex(func + value, "n");
switch (func.Name)
{
case "sumf":
// x + a = value => x = value - a
return(FindInvertExpression(un, value - a, x));
case "minusf":
if (arg == 0)
{
// x - a = value => x = value + a
return(FindInvertExpression(un, value + a, x));
}
else
{
// a - x = value => x = a - value
return(FindInvertExpression(un, a - value, x));
}
case "mulf":
// x * a = value => x = value / a
return(FindInvertExpression(un, value / a, x));
case "divf":
if (arg == 0)
{
// x / a = value => x = a * value
return(FindInvertExpression(un, value * a, x));
}
else
{
// a / x = value => x = a / value
return(FindInvertExpression(un, a / value, x));
}
case "powf":
if (arg == 0)
{
// x ^ a = value => x = value ^ (1/a)
if (a.entType == Entity.EntType.NUMBER && a.GetValue().IsInteger())
{
var res = new Set();
foreach (var root in Number.GetAllRoots(1, a.GetValue().AsIntegerNumber()).FiniteSet())
{
res.AddRange(FindInvertExpression(un, root * MathS.Pow(value, 1 / a), x));
}
return(res);
}
else
{
return(FindInvertExpression(un, MathS.Pow(value, 1 / a), x));
}
}
else
{
// a ^ x = value => x = log(a, value)
return(FindInvertExpression(un, MathS.Log(a, value) + 2 * MathS.i * n * MathS.pi, x));
}
default:
throw new SysException("Unknown operator");
}
}
[TestMethod] public void M1Pow() => Test(@"{-1}^{x}", MathS.Pow(m1, x));
/// <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");
}
}
[TestMethod] public void Frac34CubeRoot() => Test(@"\sqrt[3]{\frac{3}{4}}", MathS.Pow(frac34, 1m / 3));
[TestMethod] public void MPow() => Test(@"{\left(-x\right)}^{x}", MathS.Pow(-x, x));
[TestMethod] public void PowFrac34() => Test(@"\sqrt[4]{2}^{3}", MathS.Pow(2, frac34));
[TestMethod] public void AddPow() => Test(@"{\left(3+4\right)}^{x}", MathS.Pow("3+4", x));
[TestMethod] public void M2IPow() => Test(@"{\left(-2i\right)}^{x}", MathS.Pow(m2i, x));
[TestMethod] public void SubtractPow() => Test(@"{\left(3-4\right)}^{x}", MathS.Pow("3-4", x));
[TestMethod] public void IM1CubeRoot() => Test(@"\sqrt[3]{-1 + i}", MathS.Pow(im1, 1m / 3));
[TestMethod] public void MultiplyPow() => Test(@"{\left(3\times 4\right)}^{x}", MathS.Pow("3*4", x));
[TestMethod] public void PowIM1() => Test(@"{2}^{-1 + i}", MathS.Pow(2, im1));
[TestMethod] public void DividePow() => Test(@"{\left(\frac{3}{4}\right)}^{x}", MathS.Pow("3/4", x));
[TestMethod] public void Cube() => Test(@"{x}^{3}", MathS.Pow(x, 3));
[TestMethod] public void M3Root() => Test(@"\frac{1}{\sqrt[3]{x}}", MathS.Pow(x, 1m / -3));