# AngouriMath 1.0.16.1-Beta

Enables to work with formulas built in the code or from a string. Computing, derivating, latex rendering, fast functions, and many more.

This is a prerelease version of AngouriMath.
There is a newer version of this package available.
See the version list below for details.
Install-Package AngouriMath -Version 1.0.16.1-Beta
dotnet add package AngouriMath --version 1.0.16.1-Beta
<PackageReference Include="AngouriMath" Version="1.0.16.1-Beta" />
For projects that support PackageReference, copy this XML node into the project file to reference the package.
paket add AngouriMath --version 1.0.16.1-Beta

## Documentation Nuget: https://www.nuget.org/packages/AngouriMath

### AngouriMath

AngouriMath is an open-source library that enables to work with non-linear multi-variable expressions. Its functionality includes derivation, variable substitution, equation solving, definite integration, formula-to-latex formatting, and some more.

#### Examples

##### Use as a simple calculator
var inp = "1 + 2 * log(9, 3)";
var expr = MathS.FromString(inp);
Console.WriteLine(expr.Eval());
> >> 5

##### Build an expression
var x = MathS.Var("x");
var y = MathS.Var("y");
var c = x * y + x / y;
Console.WriteLine(MathS.Sqr(c));
> >> (x * y + x / y) ^ 2

##### Substitute variables
var x = MathS.Var("x");
var expr = x * 2 + MathS.Sin(x) / MathS.Sin(MathS.Pow(2, x));
var subs = expr.Substitute(x, 0.3);
Console.WriteLine(subs.Simplify());
> >> 0,9134260185941638

##### Find derivatives
var x = MathS.Var("x");
var func = MathS.Sqr(x) + MathS.Ln(MathS.Cos(x) + 3) + 4 * x;
var derivative = func.Derive(x);
Console.WriteLine(derivative.Simplify());
> >> 2 * x + -1 * sin(x) / (cos(x) + 3) + 4

##### Build formulas
var x = MathS.Var("x");
var expr = (x + 3).Pow(x + 4);
Func<NumberEntity, Entity> wow = v => expr.Substitute(x, v).Simplify();
Console.WriteLine(wow(4));
Console.WriteLine(wow(5));
Console.WriteLine(wow(6));
> >> 5764801
> >> 134217728
> >> 3486784401

##### Render latex
var x = MathS.Var("x");
var y = MathS.Var("y");
var expr = x.Pow(y) + MathS.Sqrt(x + y / 4) * (6 / x);
Console.WriteLine(expr.Latexise());
> >> {x}^{y}+\sqrt{x+\frac{y}{4}}*\frac{6}{x}

##### Play with complex numbers
var expr = MathS.Pow(MathS.e, MathS.pi * MathS.i);
Console.WriteLine(expr);
Console.WriteLine(expr.Eval());
> >> 2,718281828459045 ^ 3,141592653589793i
> >> -1

##### Solve eqations

Only numerical solutions with Newton's method is supported yet :(

var x = MathS.Var("x");
var equation = (x - 1) * (x - 2) * (MathS.Sqr(x) + 1);
foreach (var re in equation.SolveNt(x))
Console.Write(re.ToString() + "  ");
> >> 1  2  1i

##### Integrate

Only definite integral over single variable is supported yet :(

var x = MathS.Var("x");
var expr = MathS.Sin(x) + MathS.Sqrt(x) / (MathS.Sqrt(x) + MathS.Cos(x)) + MathS.Pow(x, 3);
Console.WriteLine(expr.DefiniteIntegral(x, -3, 3));
var expr2 = MathS.Sin(x);
Console.WriteLine(expr2.DefiniteIntegral(x, 0, MathS.pi));
> >> 5.56669223384056 + 0.0889406793629381i
> >> 1.98003515236381

##### Compile functions

Compiled functions work 15x faster

var x = MathS.Var("x");
var expr = MathS.Sin(x) + MathS.Sqrt(x) / (MathS.Sqrt(x) + MathS.Cos(x)) + MathS.Pow(x, 3);
var func = expr.Compile(x);
Console.WriteLine(func.Substitute(3));


### Full documentation

#### Entity methods

##### Derivation

expr.Derive(x) - derivation for variable x.

var x = MathS.Var("x");
var expr = MathS.Sqr(x);
Console.WriteLine(expr.Derive(x)); // 2 * x


How it works? We have some rules for derivation which are applied to each node, for example (a + b)' = a' + b'. So, we replace each node according to the appropriate rule.

##### Evaluation & Simplification

expr.Simplify(level) simplifies an expression. Level is number of iterations (relevant for long expressions).

expr.Eval() = expr.Simplify(1) - recommended to use to evaluation substituted expression.

expr.Simplify() = expr.Simplify(2) - use to simplify expressions, a * x + x = (a + 1) * x

var x = MathS.Var("x");
var expr = 3 * x + x;
Console.WriteLine(expr.Simplify()); // 4 * x


How it works? Thanks to the pattern system, now we are able to find subtrees that we know how to simplify. The full list of used patterns presents in file Patterns.cs.

##### Expansion & Collapse

expr.Expand(level=2) - expands the expression trying to remove all the braces (for example, a * (1 + x) = a * x + a * 1). level - number of iterations.

expr.Collapse(level=2) - collapses the expression trying to remove all the powers (for example, x^2 - y^2 = (x - y) * (x + y) ).

##### To string

expr.ToString() - string presentation of an expression.

expr.Latexise() - neat output in LaTeX format.

How it works? For each node we encounter, we use the appropriate latex syntax.

##### Solving equations

By this time, only Newton's method over one variable is available.

expr.SolveNt(from, to, stepCount, precision) - find roots assuming we are solving equation expr=0.

The algorithm iterates on [from.Re; to.Re] for real part and on [from.Im; to.Im] for imaginary part.

The higher stepCount is, the more roots the function can find

Precision - if you get similar roots that you think are equal, you can increase this argument.

You can also decrease MathS.EQUALITY_THRESHOLD which is responsible for comparing Numbers.

##### Integration

By this time, only definite integration over one variable is available.

expr.DefiniteIntegral(x, from, to) - numerically counts integral from from to to. Note that you can specify the two parameters in Complex numbers.

##### Compilation

expr.Compile(a, b, c...) the arguments are arguments of the target function. You should list all the used variables in the order you will then call.

fe.Call(a, b, c...) the arguments are Numbers in the order of variables. Retunrs Number.

var x = MathS.Var("x");
var expr = MathS.Sqr(x) + 2 * x;
var func = expr.Compile(x);
Console.WriteLine(func.Call(3));


Performane improved a lot. Testing on i7-7700HQ and expr=MathS.Sin(x) we get the following report:

| Function | Time per iteration |
| ------------------------------------ | ------------------------------- |
| Substitute(x, 3).Eval() from 1.0.13 | 12000 ns |
| Substitute(x, 3).Eval() from 1.0.15 | 2500 ns |
| Call(3) from 1.0.15 | 54 ns |
| Complex.Sin(3) | 27 ns |

If we take expr=(MathS.Log(x, 3) + MathS.Sqr(x)) * MathS.Sin(x + MathS.Cosec(x)), we get the following performance

| Method | Time per iteration |
|--------------------|-------------------:|
| AM Compiled | 350.767 ns |
| In-code expression | 211.472 ns |

So, for most cases using compilation will save you enough time even though Complex.Sin is still faster.

##### Function list

MathS.
Log(num, base), Pow(base, power), Sqrt(x), Sqr(x), Sin(x), Cos(x), Tan(x), Cotan(x),
Sec(x), Cosec(x), Arcsin(x), Arccos(x), Arctan(x), Arccotan(x), B(x), TB(x)

MathS.FromString(str) - returns Entity Nuget: https://www.nuget.org/packages/AngouriMath

### AngouriMath

AngouriMath is an open-source library that enables to work with non-linear multi-variable expressions. Its functionality includes derivation, variable substitution, equation solving, definite integration, formula-to-latex formatting, and some more.

#### Examples

##### Use as a simple calculator
var inp = "1 + 2 * log(9, 3)";
var expr = MathS.FromString(inp);
Console.WriteLine(expr.Eval());
> >> 5

##### Build an expression
var x = MathS.Var("x");
var y = MathS.Var("y");
var c = x * y + x / y;
Console.WriteLine(MathS.Sqr(c));
> >> (x * y + x / y) ^ 2

##### Substitute variables
var x = MathS.Var("x");
var expr = x * 2 + MathS.Sin(x) / MathS.Sin(MathS.Pow(2, x));
var subs = expr.Substitute(x, 0.3);
Console.WriteLine(subs.Simplify());
> >> 0,9134260185941638

##### Find derivatives
var x = MathS.Var("x");
var func = MathS.Sqr(x) + MathS.Ln(MathS.Cos(x) + 3) + 4 * x;
var derivative = func.Derive(x);
Console.WriteLine(derivative.Simplify());
> >> 2 * x + -1 * sin(x) / (cos(x) + 3) + 4

##### Build formulas
var x = MathS.Var("x");
var expr = (x + 3).Pow(x + 4);
Func<NumberEntity, Entity> wow = v => expr.Substitute(x, v).Simplify();
Console.WriteLine(wow(4));
Console.WriteLine(wow(5));
Console.WriteLine(wow(6));
> >> 5764801
> >> 134217728
> >> 3486784401

##### Render latex
var x = MathS.Var("x");
var y = MathS.Var("y");
var expr = x.Pow(y) + MathS.Sqrt(x + y / 4) * (6 / x);
Console.WriteLine(expr.Latexise());
> >> {x}^{y}+\sqrt{x+\frac{y}{4}}*\frac{6}{x}

##### Play with complex numbers
var expr = MathS.Pow(MathS.e, MathS.pi * MathS.i);
Console.WriteLine(expr);
Console.WriteLine(expr.Eval());
> >> 2,718281828459045 ^ 3,141592653589793i
> >> -1

##### Solve eqations

Only numerical solutions with Newton's method is supported yet :(

var x = MathS.Var("x");
var equation = (x - 1) * (x - 2) * (MathS.Sqr(x) + 1);
foreach (var re in equation.SolveNt(x))
Console.Write(re.ToString() + "  ");
> >> 1  2  1i

##### Integrate

Only definite integral over single variable is supported yet :(

var x = MathS.Var("x");
var expr = MathS.Sin(x) + MathS.Sqrt(x) / (MathS.Sqrt(x) + MathS.Cos(x)) + MathS.Pow(x, 3);
Console.WriteLine(expr.DefiniteIntegral(x, -3, 3));
var expr2 = MathS.Sin(x);
Console.WriteLine(expr2.DefiniteIntegral(x, 0, MathS.pi));
> >> 5.56669223384056 + 0.0889406793629381i
> >> 1.98003515236381

##### Compile functions

Compiled functions work 15x faster

var x = MathS.Var("x");
var expr = MathS.Sin(x) + MathS.Sqrt(x) / (MathS.Sqrt(x) + MathS.Cos(x)) + MathS.Pow(x, 3);
var func = expr.Compile(x);
Console.WriteLine(func.Substitute(3));


### Full documentation

#### Entity methods

##### Derivation

expr.Derive(x) - derivation for variable x.

var x = MathS.Var("x");
var expr = MathS.Sqr(x);
Console.WriteLine(expr.Derive(x)); // 2 * x


How it works? We have some rules for derivation which are applied to each node, for example (a + b)' = a' + b'. So, we replace each node according to the appropriate rule.

##### Evaluation & Simplification

expr.Simplify(level) simplifies an expression. Level is number of iterations (relevant for long expressions).

expr.Eval() = expr.Simplify(1) - recommended to use to evaluation substituted expression.

expr.Simplify() = expr.Simplify(2) - use to simplify expressions, a * x + x = (a + 1) * x

var x = MathS.Var("x");
var expr = 3 * x + x;
Console.WriteLine(expr.Simplify()); // 4 * x


How it works? Thanks to the pattern system, now we are able to find subtrees that we know how to simplify. The full list of used patterns presents in file Patterns.cs.

##### Expansion & Collapse

expr.Expand(level=2) - expands the expression trying to remove all the braces (for example, a * (1 + x) = a * x + a * 1). level - number of iterations.

expr.Collapse(level=2) - collapses the expression trying to remove all the powers (for example, x^2 - y^2 = (x - y) * (x + y) ).

##### To string

expr.ToString() - string presentation of an expression.

expr.Latexise() - neat output in LaTeX format.

How it works? For each node we encounter, we use the appropriate latex syntax.

##### Solving equations

By this time, only Newton's method over one variable is available.

expr.SolveNt(from, to, stepCount, precision) - find roots assuming we are solving equation expr=0.

The algorithm iterates on [from.Re; to.Re] for real part and on [from.Im; to.Im] for imaginary part.

The higher stepCount is, the more roots the function can find

Precision - if you get similar roots that you think are equal, you can increase this argument.

You can also decrease MathS.EQUALITY_THRESHOLD which is responsible for comparing Numbers.

##### Integration

By this time, only definite integration over one variable is available.

expr.DefiniteIntegral(x, from, to) - numerically counts integral from from to to. Note that you can specify the two parameters in Complex numbers.

##### Compilation

expr.Compile(a, b, c...) the arguments are arguments of the target function. You should list all the used variables in the order you will then call.

fe.Call(a, b, c...) the arguments are Numbers in the order of variables. Retunrs Number.

var x = MathS.Var("x");
var expr = MathS.Sqr(x) + 2 * x;
var func = expr.Compile(x);
Console.WriteLine(func.Call(3));


Performane improved a lot. Testing on i7-7700HQ and expr=MathS.Sin(x) we get the following report:

| Function | Time per iteration |
| ------------------------------------ | ------------------------------- |
| Substitute(x, 3).Eval() from 1.0.13 | 12000 ns |
| Substitute(x, 3).Eval() from 1.0.15 | 2500 ns |
| Call(3) from 1.0.15 | 54 ns |
| Complex.Sin(3) | 27 ns |

If we take expr=(MathS.Log(x, 3) + MathS.Sqr(x)) * MathS.Sin(x + MathS.Cosec(x)), we get the following performance

| Method | Time per iteration |
|--------------------|-------------------:|
| AM Compiled | 350.767 ns |
| In-code expression | 211.472 ns |

So, for most cases using compilation will save you enough time even though Complex.Sin is still faster.

##### Function list

MathS.
Log(num, base), Pow(base, power), Sqrt(x), Sqr(x), Sin(x), Cos(x), Tan(x), Cotan(x),
Sec(x), Cosec(x), Arcsin(x), Arccos(x), Arctan(x), Arccotan(x), B(x), TB(x)

MathS.FromString(str) - returns Entity

## Release Notes

New functions added (arc-*), bugs fixed, tests added, compilation speed up, linq support added, and some more

## Dependencies

• #### .NETStandard 2.0

• No dependencies.

## Used By

### GitHub repositories (0)

This package is not used by any popular GitHub repositories.

## Version History

1.1.0.1 42 6/25/2020
1.1.0 46 6/5/2020
1.1.0-Beta 66 6/3/2020
1.0.20 75 4/23/2020
1.0.18-Alpha 120 2/4/2020
1.0.17.2-Beta 101 1/6/2020
1.0.17.1-Beta 92 1/5/2020
1.0.17-Beta 113 1/3/2020
1.0.16.2-Beta 194 1/2/2020
1.0.16.1-Beta 104 12/31/2019
1.0.16-Beta 163 12/31/2019
1.0.15.1 99 12/27/2019
1.0.15 94 12/25/2019
1.0.14 97 12/23/2019
1.0.13.7-Beta 110 12/23/2019
1.0.12 85 12/21/2019
1.0.11 83 12/8/2019
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