Canonical Form
Many mathematical objects can be represented by several equivalent expressions.
For example, the expressions in each row below represent the same mathematical object:
$$215.3465$$  $$2.15346\mathrm{e}2$$  $$2.15346 \times 10^2$$ 
$$1  x$$  $$x + 1$$  $$1 + (x)$$ 
$$2x^{1}$$  $$\frac{2}{x}$$  $$\frac{2}{x}$$ 
The Compute Engine stores expressions internally in a canonical form to simplify the implementation of some algorithms.
The value of expr.simplify()
, expr.evaluate()
and expr.N()
are canonical
expressions.
The ce.box()
and ce.parse()
functions return a canonical expression by
default, which is the desirable behavior in most cases.
To get a noncanonical version of an experssion set the canonical
option
of ce.parse()
or ce.box()
to false
.
The noncanonical version will be closer to the literal LaTeX input, which may be desirable to compare a “raw” user input with an expected answer.
ce.parse('\\frac{3}{5}');
// ➔ ["Rational", 3, 5]
// The canonical version moves the sign to the numerator
ce.parse('\\frac{3}{5}', { canonical: false });
// ➔ ["Divide", 3, 5]
// The noncanonical version does not change the arguments, so this is
// interpreted as a regular fraction ("Divide"), not a rational.
The value of expr.json
may not be strictly in canonical form: some “sugaring”
is applied to the internal representation before being returned, for example
["Add", 1, "x"]
may be returned as ["Subtract", "x ", 1]
.
You can further customize how an expression is interpreted by using
ce.jsonSerializationOptions
.
ce.parse('\\frac{3}{5}', { canonical: false });
// ➔ ["Rational", 3, 5]
// This is a rational without modifying the arguments, so a `["Rational"]`
// expression is returned
ce.jsonSerializationOptions = { exclude: ['Rational'] };
ce.parse('\\frac{3}{5}', { canonical: false });
// ➔ ["Divide", 3, 5]
// We've excluded `["Rational"]` expressions, so it is interepreted as a
// division instead.
The canonical form of an expression is always the same when used with a given Compute Engine instance. However, do not rely on the canonical form as future versions of the Compute Engine could provide a different result.
To obtain the canonical representation of an noncanonical expression, use
the expr.canonical
property.
console.log(ce.box(['Add', 2, 'x', 3]).canonical);
// ➔ ["Add", 5, "x"]
To check if an expression is canonical use expr.isCanonical
.
If the expression is already canonical, expr.canonical
immediately returns
expr
.
Canonical Form Transformations
The canonical form used by the Compute Engine follows common conventions. It is not always “the simplest” way to represent an expression.
Calculating the canonical form of an expression is applying some rewriting rules
to an expression to put sums, products, numbers, roots, etc… in canonical
form. In that sense, it is similar to simplifying an expression with
expr.simplify()
, but it is more conservative in the transformations it
applies, and it will not take into account any assumptions about symbols or
their value.
Below is a list of some of the transformations applied to obtain the canonical form:

Idempotency: \( f(f(x)) \to f(x) \)

Involution: \( f(f(x)) \to x \)

Associativity: \( f(a, f(b, c)) \to f(a, b, c) \)

Literals
 Rationals are reduced, e.g. \[(\frac{6}{4} \longrightarrow \frac{3}{2}\]
 The denominator of rationals is made positive, e.g. \[(\frac{5}{11} \longrightarrow \frac{5}{11}\]
 A rational with a denominator of 1 is replaced with a number, e.g. \[(\frac{19}{1} \longrightarrow 19\]
 Square roots of rationals have their perfect squared factored out, e.g. \[(\sqrt{63} \longrightarrow 3\sqrt{7}\]
 Complex numbers with no imaginary component are replaced with a real number

Abs
 The absolute value of literals is evaluated

Add
 Arguments are sorted
 Literal
0
is removed  Sum of a literal and the product of a literal with the imaginary unit are replaced with a complex number.

Multiply
 Arguments are sorted
 $x \times x$ is replaced with
["Square", x]
 The product of two integers literals is evaluated But not exact literals?
I.e. rationals or Square Root of fractional?
 If any argument is
NaN
orUndefined
evaluates toNaN
Might be too aggressive
 If any argument is

Divide

Power
 $x^{\tilde\infty} \longrightarrow \operatorname{NaN}$
 $x^0 \longrightarrow 1$
 $x^1 \longrightarrow x$
 $(\pm 1)^{1} \longrightarrow 1$
 $(\pm\infty)^{1} \longrightarrow 0$
 $0^{\infty} \longrightarrow \tilde\infty$
 $(\pm 1)^{\pm \infty} \longrightarrow \operatorname{NaN}$
 $\infty^{\infty} \longrightarrow \infty$
 $\infty^{\infty} \longrightarrow 0$
 $(\infty)^{\pm \infty} \longrightarrow \operatorname{NaN}$

Square

Sqrt

Root

Subtract

Negate

For
Multiply
, literal 1 is removed, small integers and small rations are multiplied together. 
For
Divide
, replaced byMultiply
/Power

For
Subtract
, replaced byAdd

For
Sqrt
andRoot
, replaced byPower