Apply the definitions in the supplied dictionary to an expression and return the result.

Unlike format this may entail performing calculations and irreversible transformations.

See also [ComputeEngine.evaluate()](#(ComputeEngine%3Aclass).(evaluate%3Ainstance)).

evaluate(expr: Expression, options?: {dictionaries: Readonly<Dictionary>[]}): Promise<Expression | "null">
expr: Expression
dictionaries: Readonly<Dictionary>[];
Promise<Expression | "null">


Transform an expression by applying one or more rewriting rules to it, recursively.

There are many ways to symbolically manipulate an expression, but transformations with form have the following characteristics:

  • they don’t require calculations or assumptions about the domain of free variables or the value of constants
  • the output expression is expressed with more primitive functions, for example subtraction is replaced with addition
format(expr: Expression, forms: Form[], options?: {dictionaries: Readonly<Dictionary>[]}): Expression
expr: Expression
forms: Form[]
dictionaries: Readonly<Dictionary>[];

class ComputeEngine

For best performance when calling repeatedly format() or evaluate(), create an instance of ComputeEngine and call its methods. The constructor of ComputeEngine will compile and optimize the dictionary so that calls of the format() and evaluate() methods will bypass that step. By contrast invoking the format() and evaluate() functions will compile the dictionary each time they are called.

class ComputeEnginenew ComputeEngine

Construct a new ComputeEngine environment.

If no options.dictionaries is provided a default set of dictionaries will be used. The ComputeEngine.getDictionaries() method can be called to access some subset of dictionaries, e.g. for arithmetic, calculus, etc… The order of the dictionaries matter: the definitions from the later ones override the definitions from earlier ones. The first dictionary should be the 'core' dictionary which include some basic definitions such as domains (‘Boolean’, ‘Number’, etc…) that are used by later dictionaries.

new ComputeEngine(options?: {dictionaries: Readonly<Dictionary>[]}): ComputeEngine
dictionaries: Readonly<Dictionary>[];

class ComputeEnginecanonical

Format the expression to the canonical form.

In the canonical form, some operations are simplified (subtractions becomes additions of negative, division become multiplications of inverse, etc…) and terms are ordered using a deglex order. This can make subsequent operations easier.

canonical(expr: Expression): Expression
expr: Expression

class ComputeEnginedomain

Return the domain of the expression

domain(expr: Expression): Expression
expr: Expression

class ComputeEngineevaluate

Evaluate the expression exp asynchronously.

Evaluating some expressions can take a very long time. Some can invole making network queries. Therefore to avoid blocking the main event loop, a promise is returned.

Use result = await engine.evaluate(expr) to get the result without blocking.

class ComputeEngineformat

Format the expression according to the specified forms.

If no form is provided, the expression is formatted with the ‘canonical’ form.

format(expr: Expression, forms?: Form | Form[]): Expression
expr: Expression

class ComputeEnginegetDefinition

getDefinition(name: string): FunctionDefinition
name: string

class ComputeEnginegetFunctionDefinition

getFunctionDefinition(name: string): FunctionDefinition
name: string

class ComputeEnginegetSetDefinition

getSetDefinition(name: string): SetDefinition
name: string

class ComputeEnginegetSymbolDefinition

getSymbolDefinition(name: string): FunctionDefinition
name: string

class ComputeEnginegetVars

Return the variables (free or not) in this expression

getVars(expr: Expression): Set<string>
expr: Expression

class ComputeEngineisSubsetOf

Test if lhs is a subset of rhs.

lhs and rhs can be set expressions, i.e. ["SetMinus", "ComplexNumber", 0]

isSubsetOf(lhs: Expression, rhs: Expression): boolean
lhs: Expression
rhs: Expression

class ComputeEnginepopScope

Remove the topmost scope from the scope stack.

popScope(): void

class ComputeEnginepushScope

Create a new scope and add it to the top of the scope stack

pushScope(dictionary?: Dictionary): void
dictionary: Dictionary

class ComputeEnginesame

Indicate if two expressions are structurally identical, using a literal symbolic identity.

Using a canonical format will result in more positive matches.

Two expressions are the same if:

  • they have the same domain
  • if they are numbers, if their value and domain are identical.
  • if they are symbols, if their names are identical.
  • if they are functions, if the head of the functions are identical, and if all the arguments are identical.
same(["Add", "x", 1], ["Add", 1,  "x"])
// ➔ false

same(canonical(["Add", "x", 1]), canonical(["Add", 1,  "x"]))`
// ➔ true
same(lhs: Expression, rhs: Expression): boolean
lhs: Expression
rhs: Expression

class ComputeEngineshouldContinueExecution

Return false if the execution should stop.

This can occur if:

  • an error has been signaled
  • the time limit or memory limit has been exceeded
shouldContinueExecution(): boolean

class ComputeEnginesignal

Call this function if an unexpected condition occurs during execution a function in the engine.

An ErrorSignal a problem that cannot be recovered from.

A WarningSignal indicate a minor problem that should not prevent the execution to continue.

signal(sig: WarningSignal | ErrorSignal): void

class ComputeEnginegetDictionaries

Return dictionaries suitable for the specified categories, or "all" for all categories ("arithmetic", "algebra", etc…).

A symbol dictionary defines how the symbols and function names in a MathJSON expression should be interpreted, i.e. how to evaluate and manipulate them.

getDictionaries(categories: DictionaryCategory[] | "all"): Readonly<Dictionary>[]

class ComputeEngineComputeEngine.context


The current scope.

A scope is a dictionary that contains the definition of local symbols.

Scopes form a stack, and definitions in more recent scopes can obscure definitions from older scopes.



domain: Domain;
scope: Scope;

The scope this definition belongs to. This field is usually undefined, but its value is set by getDefinition(), getFunctionDefinition() and getSymbolDefinition().

wikidata: string;

A short string indicating an entry in a wikibase. For example "Q167" is the wikidata entry for the Pi constant.


  • BaseDefinition &
  • at: (index: number): Expression;
    countable: boolean;

    If true, the size of the collection is finite.

    indexable: boolean;

    If true, elements of the collection can be accessed with a numerical index with the at() function

    isElementOf: (expr: Expression): boolean;

    A predicate function to determine if an expression is a member of the collection or not (answers “True”, “False” or “Maybe”).

    iterable: boolean;

    If true, the elements of the collection can be iterated over using the `iterator() function

    iterator: {done: (): boolean; next: (): Expression};
    size: (): number;

    Return the number of elements in the collection.


The entries of a CompiledDictionary have been validated and optimized for faster evaluation.

When a new scope is created with pushScope() or when creating a new engine instance, new instances of CompiledDictionary are created as needed.


asyncEvaluate: (scope: {}): Promise<Expression>;
evaluate: (scope: {}): Expression;



A dictionary maps a MathJSON name to a definition.

A named entry in a dictionary can refer to a symbol, as in the expression, "Pi", to a function: “Add” in the expression ["Add", 2, 3], or to a "Set".

The name can be an arbitrary string of Unicode characters, however the following conventions are recommended:

  • Use only letters, digits and -, and the first character should be a letter: /^[a-zA-Z][a-zA-Z0-9-]+/
  • Built-in functions and symbols should start with an uppercase letter

As a shorthand for a numeric symbol definition, a number can be used as well. In that case the domain is determined automatically.

{ "x": 1.0 }
{ "x" : { "domain": "RealNumber", "value": 1.0 } }

[name: string]: number | Definition


A domain such as ‘Number’ or ‘Boolean’ represents a set of values.

Domains can be defined as a union or intersection of domains:

  • ["Union", "Number", "Boolean"] A number or a boolean.
  • ["SetMinus", "Number", 1] Any number except “1”.

Domains are defined in a hierarchy (a lattice).


A given mathematical expression can be represented in multiple equivalent ways as a MathJSON expression.

Form is used to specify a representation:

  • 'full': only transformations applied are those necessary to make it valid JSON (for example making sure that Infinity and NaN are represented as strings)
  • 'flatten': associative functions are combined, e.g. f(f(a, b), c) -> f(a, b, c)
  • 'sorted': the arguments of commutative functions are sorted such that: - numbers are first, sorted numerically - complex numbers are next, sorted numerically by imaginary value - symbols are next, sorted lexicographically - add functions are next - multiply functions are next - power functions are next, sorted by their first argument, then by their second argument - other functions follow, sorted lexicographically
  • 'stripped-metadata': any metadata associated with elements of the expression is removed.
  • 'object-literal': each term of an expression is expressed as an object literal: no shorthand representation is used.
  • 'canonical-add': `addition of 0 is simplified, associativity rules are applied, unnecessary groups are moved, single argument ‘add’ are simplified
  • 'canonical-divide': divide is replaced with multiply and `power’, division by 1 is simplified,
  • 'canonical-exp': exp is replaced with power
  • 'canonical-multiply': multiplication by 1 or -1 is simplified
  • 'canonical-power': power with a first or second argument of 1 is simplified
  • 'canonical-negate': real or complex number is replaced by the negative of that number. Negation of negation is simplified.
  • 'canonical-number': complex numbers with no imaginary compnents are simplified
  • 'canonical-root': root is replaced with power
  • 'canonical-subtract': subtract is replaced with add and negate
  • 'canonical': the following transformations are performed, in this order:
    • ‘canonical-number’, // ➔ simplify number
    • ‘canonical-exp’, // ➔ power
    • ‘canonical-root’, // ➔ power, divide
    • ‘canonical-subtract’, // ➔ add, negate, multiply,
    • ‘canonical-divide’, // ➔ multiply, power
    • ‘canonical-power’, // simplify power
    • ‘canonical-multiply’, // ➔ multiply, power
    • ‘canonical-negate’, // simplify negate
    • ‘canonical-add’, // simplify add
    • ‘flatten’, // simplify associative, idempotent and groups
    • ‘sorted’,
    • ‘full’,

  • | "canonical"
  • | "canonical-add"
  • | "canonical-divide"
  • | "canonical-domain"
  • | "canonical-exp"
  • | "canonical-list"
  • | "canonical-multiply"
  • | "canonical-power"
  • | "canonical-negate"
  • | "canonical-number"
  • | "canonical-root"
  • | "canonical-subtract"
  • | "flatten"
  • | "full"
  • | "object-literal"
  • | "sorted"
  • | "stripped-metadata"
  • | "sum-product"


  • BaseDefinition &
  • Partial<FunctionFeatures> &
  • hold: "none" | "all" | "first" | "rest";
    • ‘none’: Each of the arguments is evaluated.
    • ‘all’: The arguments will not be evaluated and will be passed as is.
    • ‘first’: The first argument is not evaluated, the others are
    • ‘rest’: The first argument is evaluated, the others aren’t
    sequenceHold: boolean;

    If true, Sequence arguments are not automatically spliced in

    signatures: FunctionSignature[];


A function definition can have some flags set indicating specific properties of the function.

additive: boolean;

If true, when the function is univariate, [f, ["Add", x, c]] where c is constant, is simplified to ["Add", [f, x], c].

When the function is multivariate, additivity is considered only on the first argument: [f, ["Add", x, c], y] simplifies to ["Add", [f, x, y], c].

Default: false

associative: boolean;

If true, [f, [f, a], b] simplifies to [f, a, b]

Default: false

commutative: boolean;

If true, [f, a, b] simplifies to [f, b, a]

Default: false

idempotent: boolean;

If true, [f, [f, x]] simplifies to [f, x].

Default: false

involution: boolean;

If true, [f, [f, x]] simplifies to x.

Default: false

multiplicative: boolean;

If true, when the function is univariate, [f, ["Multiply", x, y]] simplifies to ["Multiply", [f, x], [f, y]].

When the function is multivariate, multipicativity is considered only on the first argument: [f, ["Multiply", x, y], z] simplifies to ["Multiply", [f, x, z], [f, y, z]]

Default: false

outtative: boolean;

If true, when the function is univariate, [f, ["Multiply", x, c]] simplifies to ["Multiply", [f, x], c] where c is constant

When the function is multivariate, multiplicativity is considered only on the first argument: [f, ["Multiply", x, y], z] simplifies to ["Multiply", [f, x, z], [f, y, z]]

Default: false

pure: boolean;

If true, invoking the function with a given set of arguments will always return the same value, i.e. ‘Sin’ is pure, ‘Random’ isn’t. This is used to cache the result of the function.

Default: true

threadable: boolean;

If true, the function is applied element by element to lists, matrices and equations.

Default: false


Function signature: definition of the inputs and output of a function.

A function should have at least one signature, but can have several:

  • Add(RealNumber, RealNumber): RealNumber
  • Add(ComplexNumber, ComplexNumber): ComplexNumber
  • etc…

The signature of a function that accepts any input and may output anything is: { rest: "Anything", result: "Anything" }.

args: Domain | [name: string, domain: Domain][];

Input arguments

asyncEvaluate: (engine: ComputeEngine, ...args: Expression[]): Promise<Expression>;

Evaluate the function with the passed in arguments and return a corresponding result.

compile: (engine: ComputeEngine, ...args: CompiledExpression[]): CompiledExpression;

Return a compiled (optimized) function for evaluation

dimension: (engine: ComputeEngine, ...args: Expression[]): Expression;

Dimensional analysis

evaluate: (engine: ComputeEngine, ...args: Expression[]): Expression;

Evaluate the function with the passed in arguments and return a corresponding result.

rest: Domain | [name: string, domain: Domain];

If this signature accepts unlimited additional arguments after the named arguments, they should be of this domain

result: Domain | (engine: ComputeEngine, ...args: Expression[]): Expression;

Domain result computation


  • Scope &
  • deadline: number;

    Absolute time beyond which evaluation should not proceed

    dictionary: CompiledDictionary;
    lowWaterMark: number;

    Free memory should not go below this level for execution to proceed

    origin: {column: number; line: number; name: string};

    The location of the call site that created this scope

    parentScope: RuntimeScope;
    warnings: WarningSignal[];

    Set when one or more warning have been signaled in this scope


A scope is a set of names in a dictionary that are bound (defined) in a MathJSON expression.

Scopes are arranged in a stack structure. When an expression that defined a new scope is evaluated, the new scope is added to the scope stack. Outside of the expression, the scope is removed from the scope stack.

The scope stack is used to resolve symbols, and it is possible for a scope to ‘mask’ definitions from previous scopes.

Scopes are lexical (also called a static scope): they are defined based on where they are in an expression, they are not determined at runtime.

iterationLimit: number;

Signal ‘iteration-limit-exceeded’ when the iteration limit for this scope is exceeded. Default: no limits.

memoryLimit: number;

Signal ‘out-of-memory’ when the memory usage for this scope is exceeded. Memory in Megabytes, default: 1Mb.

recursionLimit: number;

Signal ‘recursion-depth-exceeded’ when the recursion depth for this scope is exceeded.

timeLimit: number;

Signal ‘timeout’ when the execution time for this scope is exceeded. Time in seconds, default 2s.

warn: WarningSignalHandler;

This handler is invoked when exiting this scope if there are any warnings pending.


  • CollectionDefinition &
  • isSubsetOf: (engine: ComputeEngine, lhs: Expression, rhs: Expression): boolean;

    A function that determins if a set is a subset of another. The rhs argument is either the name of the symbol, or a function with the head of the symbol.

    supersets: string[];

    The supersets of this set: they should be symbol with a ‘Set’ domain

    value: Expression;

    If a set can be defined explicitely in relation to other sets, the value represents that relationship. For example “NaturalNumber” = [“Union”, “PrimeNumber”, “CompositeNumber”].



constant: boolean;

If true the value of the symbol is constant.

If false, the symbol is a variable.

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