a symbolic math expression evaluator for javascript
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README.md

Nerdamer

As of version 0.5.0, the library is split into the core and optional add-ons which can be loaded after the core has been loaded.

Getting started with Nerdamer

Load the library in your html page

<!-- assuming you've saved the file in the root of course -->
<script src="nerdamer.core.js"></script> 
<!-- LOAD ADD-ONS -->
<!-- again assuming you've saved the files in root -->
<script src="Algebra.js"></script>
<script src="Calculus.js"></script>

Some functions have dependencies from other add-ons.

You can see nerdamer in action at http://www.nerdamer.com/demo

All operations are done using the 'nerdamer' object.

To add an expression just add it to the nerdamer object which will return a expression object.

var e = nerdamer('x^2+2*(cos(x)+x*x)');
console.log(e.text());

//result: 
//2*cos(x)+3*x^2

You can also pass in an object with known values as the second parameter.

var e = nerdamer('x^2+2*(cos(x)+x*x)',{x:6});
console.log(e.text());

//result:
//108+2*cos(6)

As you can see only the substitution is performed. To evaluate the result just call evaluate. Note that evaluate returns a text string or a number not an object.

var e = nerdamer('x^2+2*(cos(x)+x*x)',{x:6}).evaluate();
console.log(e.text());

//result:
//109.9203405733006

To get back the text as a fraction, call the text method and pass in the string 'fractions'.

var e = nerdamer('x^2+2*(cos(x)+x*x)',{x:6}).evaluate();
console.log(e.text('fractions'));

//result:
//429607273/3908351

You can get your expression back as LaTeX by calling the toTeX method

var LaTeX = nerdamer('x^2+2*(cos(x)+x*x)',{x:0.25}).toTeX();
console.log(LaTeX);

//result:
//2 \cdot \mathrm{cos}\left(\frac{1}{4}\right)+\frac{3}{16}

To have numbers returned as decimals pass in the string 'decimals' to the toTeX method

var LaTeX = nerdamer('x^2+2*(cos(x)+x*x)',{x:0.25}).toTeX('decimal');
console.log(LaTeX);

//result:
//2 \cdot \mathrm{cos}\left(0.25\right)+0.1875

Alternatively you can pass an object containing known values into evaluate method instead. The values passed in don't have to be number they can be another expression if needed.

var e = nerdamer('x^2+2*(cos(x)+x*x)',{x:'x^2+1'});
console.log(e.text());

//result:
//2*cos(1+x^2)+3*(1+x^2)^2

Every time you parse an expression it's stored in nerdamer. To get a list of all the expressions you just call nerdamer.expressions().

var knownValues = {x:'x^2+1'};
nerdamer('x^2+2*(cos(x)+x*x)').evaluate(knownValues);
nerdamer('sin(x)^2+cos(x)^2').evaluate(knownValues);

console.log(nerdamer.expressions());

//result:
//[ 46.692712758272776, 1 ]

You can request it as an object as well by passing in true. This can be convenient in some situations as the numbering starts at 1;

var knownValues = {x:'x^2+1'};
nerdamer('x^2+2*(cos(x)+x*x)', knownValues );
nerdamer('sin(x)^2+cos(x)^2', knownValues );

console.log(nerdamer.expressions(true));

//{ '1': '2*cos(1+x^(2))+3*(1+x^(2))^(2)',
//'2': 'cos(1+x^(2))^(2)+sin(1+x^(2))^(2)' }

Functions aren't always immediately parsed to numbers. For example

var result = nerdamer('cos(x)',{x:6});
console.log(result.text());
//cos(6)

will only subsitute out the variable name. To change this behaviour numer should be passed in as the 3rd argument.

var result = nerdamer('cos(x)',{x:6}, 'numer');
console.log(result.text());
//0.960170286650366

or alternatively

var result = nerdamer('cos(x)').evaluate({x:6});
console.log(result.text());
//0.960170286650366

The difference however is that the first option directly substitutes the variables while the second first evaluates the expression and then makes the substitutions. This library utilizes native javascript functions as much as possible. As a result it inherits whatever rounding errors they possess. One major change with version 0.6.0 however, is dealing with floating point issues.

var result = nerdamer('sqrt(x)*sqrt(x)-2', {x: 2});
console.log(result.text());
//0

The above expample now returns zero whereas in previous version the result would be 4.440892098500626e-16. Same goes for 0.1+0.2.

An expression can be replaced directly by passing in the index of which expression to override. For example

nerdamer('cos(x)',{x:6}, 'numer');
nerdamer('sin(x)+y',{x:6}, null, 1);
console.log(nerdamer.expressions());
//[ 'sin(6)+y' ]

If multiple modifier options need to be passed into nerdamer you can do so using an array. For example ...

var e = nerdamer('cos(x)+(y-x)^2', {x:7}, ['expand', 'numer']);
console.log(e.text());
//-14*y+y^2+49.7539022543433

If you need the code as LaTeX you can pass in true as the second parameter when requesting the expressions.

nerdamer('x^2+2*(cos(x)+x*x)');
nerdamer('sin(x)^0.25+cos(x)^0.5' );
var asObject = true;
var asLaTeX = true;
console.log(nerdamer.expressions(asObject, asLaTeX));

/*{ '1': '2 \\cdot \\mathrm{cos}\\left(x\\right)+3 \\cdot x^{2}',
  '2': '\\sqrt{\\mathrm{cos}\\left(x\\right)}+\\mathrm{sin}\\left(x\\right)^{\\frac{1}{4}}' }*/

You can specify a particular location when adding an expression, which is specified with the third parameter.

nerdamer('x^2+2*(cos(x)+x*x)');
nerdamer('sin(x)^0.25+cos(x)^0.5' );
nerdamer('expr-override', undefined, 2 );
var asObject = false;
var asLaTeX = true;
console.log(nerdamer.expressions(asObject, asLaTeX));

/* [ '2 \\cdot \\mathrm{cos}\\left(x\\right)+3 \\cdot x^{2}',
  '\\sqrt{\\mathrm{cos}\\left(x\\right)}+\\mathrm{sin}\\left(x\\right)^{\\frac{1}{4}}',
  'expr-override' ]
 */

Here's an example of reserved variable and function names.

var reserved = nerdamer.reserved();
console.log(reserved);
//result:
/* csc, sec, cot, erf, fact, mod, GCD, QGCD, LCM, pow, PI, E, cos, sin, tan, acos, asin, atan, sinh, cosh, tanh, asinh, acosh, atanh, exp, min, max, floor, ceil, round, vector, matrix, parens, sqrt, log, expand, abs, invert, transpose, dot */

//or as an array

var reserved = nerdamer.reserved(true);
console.log(reserved);
//result:
/* [ 'csc', 'sec', 'cot', 'erf', 'fact', 'mod', 'GCD', 'QGCD', 'LCM', 'pow', 'PI', 'E', 'cos', 'sin', 'tan', 'acos', 'asin', 'atan', 'sinh', 'cosh', 'tanh', 'asinh', 'acosh', 'atanh', 'exp', 'min', 'max', 'floor', 'ceil', 'round', 'vector', 'matrix',
  'parens', 'sqrt', 'log', 'expand', 'abs', 'invert', 'transpose', 'dot' ]  */

Most math functions are passed in as part of the expression. If you want to differentiate for instance you just use the function diff which is located in the Calculus add-on as of version 0.5.0

var e = nerdamer('diff(x^2+2*(cos(x)+x*x),x)');

console.log(e.text());

//result: 
//-2*sin(x)+6*x

Nerdamer can also handle runtime functions. To do this use the method setFunction. The runtime functions do have symbolic capabilities and support for imaginary numbers. The setfunction method is used as follows:

nerdamer.setFunction( function_name, parameter_array, function_body )

For Example:

//generate some points
var f = function(x) { return 5*x-1; }
console.log(f(1)); //4
console.log(f(2)); //9 - value to be found
console.log(f(7)); //34

nerdamer.setFunction('interpolate',['y0','x0','y1','x1','x'],'y0+(y1-y0)*((x-x0)/(x1-x0))')
var answer = nerdamer('interpolate(4,1,34,7,2)').evaluate();

console.log(answer);

//result: 9

Custom functions alternatively be set in following manner.

nerdamer('hyp(a, b) := sqrt(a^2 + b^2) ');
var result = nerdamer('hyp(3, 4)').evaluate().text();
console.log(result);
//result: 5

If you need to add a constant use the setConstant method

nerdamer.setConstant( 'g', 9.81);
var weight = nerdamer('100*g').text();
console.log(weight);
//result:
//981

To delete just set it to delete

nerdamer.setConstant( 'g', 9.81);
var weight = nerdamer('100*g').text();
console.log(weight);
//981
nerdamer.setConstant( 'g', 'delete');
var weight = nerdamer('100*g').text();
console.log(weight);
//100*g

You also have the option of exporting your function to a javascript function which can be useful if you need some filtering from user input. Do keep in mind that the parameters are sorted alphabetically for more than one parameter. To use it add the expression to nerdamer and use the buildFunction method.

var f = nerdamer('x^2+5').buildFunction();
console.log(f(9));

//result:
//86

If you have a particular order in which you need the parameters to be set, then you pass in an array with the variables in the order in which you want them for instance:

var f = nerdamer('z+x^2+y').buildFunction(['y', 'x', 'z']);
console.log(f(9,2,1));
//result
//14

Every time you add an expression to nerdamer it's stored. To list the expressions currently in nerdamer call the 'expressions' method. To delete an expression use the 'clear' method and pass in the expression you want to delete. To clear everything pass in the string 'all'.

nerdamer('n*R*T/v');
nerdamer('mc^2');
nerdamer('G*m1*m2/d^2');

nerdamer.clear(2);

console.log(nerdamer.expressions(true));

//result:
//{ '1': 'R*T*n*v^(-1)', '2': 'G*d^(-2)*m1*m2' }

nerdamer.clear('all');
console.log(nerdamer.expressions(true));
//result:
//{}

If you need go get the variables of an expression use the variables method. This method can be called after nerdamer was provided an expression. For example

var variables = nerdamer('csc(x*cos(y))-no_boring_x').variables();
console.log(variables);
//result:
//[ 'no_boring_x', 'x', 'y' ]

The order in which the variables appear require a little bit of knowledge of how nerdamer organizes symbols. For the sake of simplicity we'll just assume that there is no particular order


Functions, Operations, and Chaining

You can access the built-in functions directly given its name. For example

var ans = nerdamer.tan('pi/4');
console.log(ans.toString()); // 1

Or this if the Calculus add-on is loaded

var ans = nerdamer.diff('log(x)/x', 'x');
console.log(ans.toString()); // -log(x)*x^(-2)+x^(-2)

Or expand your expression

var ans = nerdamer.expand('(x+1)^3');
console.log(ans.toString()); // 1+3*x+3*x^2+x^3

You can see the list of built-in functions here

Additionally the expressions can be chained

var ans = nerdamer('a+b').multiply(2).pow(3);
console.log(ans.toString()); // 8*(a+b)^3

The supported operations are add, subtract, multiply, divide, pow


Using the solver

To solve equations first load Solve.js. Just remember that Solve also required Algebra.js and Calculus.js to be loaded. You can then solve equations using nerdamer. Important: State the variable for which you are trying to solve.

var sol = nerdamer.solveEquations('x^3+8=x^2+6','x');
console.log(sol.toString());
//1+i,-i+1,-1

Notice that we use toString rather than text as this returns a javascript array.

You can also solve an expression

var e = nerdamer.solveEquations('x^2+4-y', 'y');
console.log(e[0].text());
//4+x^2

You can also solve multivariate equations

var sol = nerdamer.solveEquations('x^2+8+y=x+6','x');
console.log(sol.toString());
//0.5*((-4*y-7)^0.5+1),0.5*(-(-4*y-7)^0.5+1)

You can do up to 3rd order polynomials for multivariate polynomials

Additionally you can try for equations containing functions. This is more of a hit or miss approach unlike single variable polynomials (which uses Mr. David Binner's Jenkins-Traub port - http://www.akiti.ca/PolyRootRe.html) but it's there if you want to give it a try.

var sol = nerdamer.solveEquations('cos(x)+cos(3*x)=1','x');
console.log(sol.toString());
//5.7981235959208695,0.4850617112587174

To solve a system of linear equations pass them in as an array. For example

var sol = nerdamer.solveEquations(['x+y=1', '2*x=6', '4*z+y=6']);
console.log(sol);
//[ [ 'x', 3 ], [ 'y', -2 ], [ 'z', 2 ] ]

In version 0.7.2 and up the solver can additionally be used in the following way

//first parse the equation
var x = nerdamer('x^2+2=y-7*a');
//You can make substitutions to the equation
x = x.evaluate({a: 'x^2-3'});
console.log(x.toString()); //2+x^2=-7*x^2+21+y
var solutions = x.solveFor('x');
console.log(solutions.toString()); //(1/16)*sqrt(32*y+608),(-1/16)*sqrt(32*y+608)

If no solutions could be found then the array will be empty

The Core

To use the core simply request it using the getCore method.

var core = nerdamer.getCore();

The core contains:

  • The various classes -> Symbol, Fraction, Parser, Expression, Latex, Vector
  • The groups into which the Symbol classes is the divided -> groups
  • The utility functions -> Utils
  • The extended math functions -> Math2
  • And an instance of the current parser being used -> PARSER

Additionally the core enables you to use nerdamer's classes directly. You could for instance do this after getting the core

var core = nerdamer.getCore();
var x1 = new core.Symbol('x');
var x2 = new core.Symbol('x');
var product = core.PARSER.multiply(x1,x2);

console.log(product.valueOf());
//x^2

This may or may not be useful but do keep in mind that when doing this one or both symbols may be modified. To prevent this call a clone if you intent to reuse the symbol. So the above example would be

var core = nerdamer.getCore();
var x = new core.Symbol('x');

var product = core.PARSER.multiply(x.clone(),x.clone());

console.log(product.valueOf());
//x^2

This leaves x unchanged.

EXTENDING THE CORE

To add a function to nerdamer use the register method is used

var some_function = {
  //Algebra for example
  parent: 'Algebra',
  name: 'some_function',
  //this determines if the functions can be used by the user when passing a string. 
  visibility: true, 
  //the minimum number of arguments your function requires. 
  //defaults to 1
  numargs: some_integer, 
  //the constructor. This is the environment under which your function is built. 
  build: function(){
    //this refers to the Parser
    //return the function which need to be set
    return function(/*args*/) {
      //body
      
    }
  }
};

//and then register it
nerdamer.register(some_function);

parent: The parent is optional. If this object does not exist it will be created. If it does then it is added. Omitting the parent does not result in it being added to the core directly.

numargs: tells nerdamer how many arguments are allowed and can be either an integer, a range in the form of an array [min arguments, max arguments], or -1 which enables any amount of arguments.

visibility: enables nerdamer's parse function to see the function and allows the user to use it.

build: is the constructor and is within the core's scope. It returns a function which is within the scope of the PARSER (the parser instance which nerdamer uses). The arguments for the return function are Symbols. This means that any algorithm can be directly applied using nerdamer's abstraction layer using the functions, add, subtract, divide, multiply, and pow.

Example 1

nerdamer.register({
    parent: 'Algebra',
    name: 'quad',
    visible: true,
    numargs: 3,
    build: function(){
        var core = this; //get the core
        //I use underscore because it makes the algorithm easier to read but choose what you prefer
        var _ = core.PARSER; 
        var Symbol = core.Symbol;//grab the symbol class or use it directly
        return function(a, b, c) {
            //apply algorithm (-b+sqrt(b^2-4ac))/2a
            var det = _.subtract(_.pow(b.clone(), new Symbol(2)),
                _.multiply(_.multiply(a.clone(), c), new Symbol(4)));
            return _.divide(_.add(b.negate(), _.pow(det, new Symbol(0.5))),
                        _.multiply(new Symbol(2), a));
        };
    }
});

After this the function can be called through nerdamer.

//get one of the roots for x^2+2x+1
var e = nerdamer('quad(1,11,24)');
console.log(e.text());
//-3

Notice the use of the clone method when the symbol is used more than once. When parsing one or more symbols drop per operation. On the next cycle a fresh new Symbol is created. To minimize the creation of new Symbols nerdamer reuses one of the symbols supplied so the return symbol is usually a modified version of one of the parameters. This usually does not cause a problem when the parsing is in a linear fashion but it creates a problem when applying algorithms in which symbols get called again. This is one of the issues I'll be tackling in the future but for now either use a safe block or call the clone method on symbols which get reused.

In this example the layer is used directly. When an algorithm is applied in this fashion the additional cost is negligible. You could alternatively use the parse method as in example 2.

Example 2

nerdamer.register({
    parent: 'Algebra',
    name: 'quad',
    visible: true,
    numargs: 3,
    build: function(){
        var core = this; //get the core
        return function(a, b, c) {
            //apply algorithm (-b+sqrt(b^2-4ac))/2a
            var exp = core.Utils.format('(-{1}+sqrt({1}^2-4*{0}*{2}))/2*{0}', a, b, c);
            return core.PARSER.parse(exp);
        };
    }
});

Although shorter and easier to read it generally comes at a higher cost so choose wisely.

THE PARSER

The core comes with a parser instance fired up and ready to go. It is wise to keep using that parser instance as this is the instance that get extended and modified. To modify the parser instance call the extend method of the parser and supply the name of the function you're extending and the function you are extending it with. The extended function gets called with 3 parameters, the first symbol, the second symbol, and the original function. Here's an example of how it's done in the LinAlg add-on.

var core = nerdamer.getCore();
core.PARSER.extend('add', function(symbol1, symbol2, add){
    //do stuff with your new types
});