Numeric Evaluation

To obtain an exact numeric evaluation of an expression use expr.evaluate(). To obtain a numeric approximation use expr.N().

An evaluation with expr.evaluate() preserves exact values.

Exact values are:

• integers and rationals
• square roots of integers and rationals
• constants such as ExponentialE and Pi

If one of the arguments is not an exact value the expression is evaluated as a numeric approximation.

To obtain a numeric approximation, use expr.N(). If expr.N() cannot provide a numeric evaluation, a symbolic representation of the partially evaluated expression is returned.

The value of N() is a boxed expression. The numericValue property is either a machine number, a Decimal object or a Complex object, depending on the numericMode of the compute engine, or null if the result is not a number.

To check if the numericValue is a Decimal use ce.isBignum(expr.N().numericValue).

To check if the numericValue is a Complex use ce.isComplex(expr.N().numericValue).

To access a JavaScript machine number approximation of the result use expr.value. If numericValue is a machine number, a Decimal object, or a rational, expr.value will return a machine number approximation.

console.log(ce.parse('11 + \\sqrt{x}').value);
// ➔ "["Add",11,["Sqrt","x"]]"
// Note: if the result is not a number, value returns a string
// representation of the expression

const expr = ce.parse('\\sqrt{5} + 7^3').N();

console.log(expr.value);
// ➔ 345.2360679774998
// If the result is a number, value returns a machine number approximation

console.log(expr.latex);
// ➔ "345.236\,067\,977\,499\,8,"
// Note: the LaTeX representation of the numeric value is rounded to the
// display precision

console.log(expr.numericValue);
// ➔ [Decimal]
// Note: depending on the numeric mode, this may be a machine number,
// a Decimal object or a Complex object

if (ce.isBignum(expr.numericValue)) {
  console.log(
    'The numeric value is a Decimal object',
    expr.numericValue.toNumber()
  );
} else if (ce.isComplex(expr.numericValue)) {
  console.log(
    'The numeric value is a Complex object',
    expr.numericValue.re,
    expr.numericValue.im
  );
} else if (Array.isArray(expr.numericValue)) {
  console.log(
    'The numeric value is a rational',
    expr.numericValue[0],
    expr.numericValue[1]
  );
} else {
  console.log('The numeric value is a machine number', expr.numericValue);
}

Repeated Evaluation

To repeatedly evaluate an expression use ce.assign() to change the value of variables. ce.assign() changes the value associated with one or more variables in the current scope.

const expr = ce.parse('3x^2+4x+2');

for (const x = 0; x < 1; x += 0.01) {
  ce.assign('x', x);
  console.log(`f(${x}) = ${expr.value}`);
}

You can also use expr.subs(), but this will create a brand new expression on each iteration, and will be much slower.

const expr = ce.parse('3x^2+4x+2');

for (const x = 0; x < 1; x += 0.01) {
  console.log(`f(${x}) = ${expr.subs({ x: x }).value}`);
}

To reset a variable to be unbound to a value use ce.assign()

ce.assign('x', null);

console.log(expr.N().latex);
// ➔ "3x^2+4x+c"

You can change the value of a variable by setting its value property:

ce.symbol('x').value = 5;

ce.symbol('x').value = undefined;

If performance is important, you can compile the expression to a JavaScript function.

Compiling

To get a compiled version of an expression use the expr.compile() method:

const expr = ce.parse('3x^2+4x+2');
const fn = expr.compile();
for (const x = 0; x < 1; x += 0.01) console.log(fn({ x }));

The syntax {x} is a shortcut for {"x": x}, in other words it defines an argument named "x" (which is used the expression expr) as having the value of the JavaScript variable x (which is used in the for loop).

This will usually result in a much faster evaluation than using expr.N() but this approach has some limitations.

Numeric Modes

Four numeric modes may be used to perform numeric evaluations with the Compute Engine: "machine" "bignum" "complex" and "auto". The default mode is "auto".

Numbers are represented internally in one of the following format:

• number: a 64-bit float
• complex: a pair of 64-bit float for the real and imaginary part
• bignum: an arbitrary precision floating point number
• rational: a pair of 64-bit float for the numerator and denominator
• big rational: a pair of arbitrary precision floating point numbers for the numerator and denominator

Depending on the current numeric mode, this is what happens to calculations involving the specified number types:

• indicate that no transformation is done
• upgraded indicate that a transformation is done without loss of precision
• downgraded indicate that a transformation is done with may result in a loss of precision, a rounding towards 0 if underflow occurs, or a rounding towards \( \pm\infty \) if overflow occurs.

Machine Numeric Mode

Calculations in the machine numeric mode use a 64-bit binary floating point format.

This format is implemented in hardware and well suited to do fast computations. It uses a fixed amount of memory and represent significant digits in base-2 with about 15 digits of precision and with a minimum value of \( \pm5\times 10^{-324} \) and a maximum value of \( \pm1.7976931348623157\times 10^{+308} \)

To change the numeric mode to the machine mode, use engine.numericMode = "machine".

Changing the numeric mode to machine automatically sets the precision to 15.

Calculations that have a complex value, for example \( \sqrt{-1} \) will return NaN. Some calculations that have a value very close to 0 may return 0. Some calculations that have a value greater than the maximum value representable by a machine number may return \( \pm\infty \).

Warning Some numeric evaluations using machine numbers cannot produce exact results…

ce.numericMode = 'machine';
console.log(ce.parse('0.1 + 0.2').N().latex);
// ➔ "0.30000000000000004"

While \(0.1\) and \(0.2\) look like “round numbers” in base-10, they can only be represented by an approximation in base-2, which introduces cascading errors when manipulating them.

Bignum Numeric Mode

In the bignum numeric mode, numbers are represented as a string of base-10 digits and an exponent.

Bignum numbers have a minimum value of \( \pm 10^{-9\,000\,000\,000\,000\,000} \) and a maximum value of \( \pm9.99999\ldots \times 10^{+9\,000\,000\,000\,000\,000} \).

To change the numeric mode to the bignum mode, use engine.numericMode = "bignum".

ce.numericMode = 'bignum';
console.log(ce.parse('0.1 + 0.2').N().latex);
// ➔ "0.3"

When using the bignum mode, the precision of computation (number of significant digits used) can be changed. By default, the precision is 100.

Trigonometric operations are accurate for precision up to 1,000.

To change the precision of calculations in bignum mode, set the engine.precision property.

The precision property affects how the computations are performed, but not how they are serialized. To change how numbers are serialized to LaTeX, use engine.latexOptions = { precision: 6 } to set it to 6 significant digits, for example.

The LaTeX precision is adjusted automatically when the precision is changed so that the display precision is never greater than the computation precision.

When using the bignum mode, the return value of expr.N().json may be a MathJSON number that looks like this:

{
  "num": "3.141592653589793238462643383279502884197169399375105820974944592307
  8164062862089986280348253421170679821480865132823066470938446095505822317253
  5940812848111745028410270193852110555964462294895493038196442881097566593344
  6128475648233786783165271201909145648566923460348610454326648213393607260249
  1412737245870066063155881748815209209628292540917153643678925903600113305305
  4882046652138414695194151160943305727036575959195309218611738193261179310511
  8548074462379962749567351885752724891227938183011949129833673362440656643086
  0213949463952247371907021798609437027705392171762931767523846748184676694051
  3200056812714526356082778577134275778960917363717872146844090122495343014654
  9585371050792279689258923542019956112129021960864034418159813629774771309960
  5187072113499999983729780499510597317328160963185950244594553469083026425223
  0825334468503526193118817101000313783875288658753320838142061717766914730359
  8253490428755468731159562863882353787593751957781857780532171226806613001927
  876611195909216420199"
}

Complex Numeric Mode

The complex numeric mode can represent complex numbers as a pair of real and imaginary components. The real and imaginary components are stored as 64-bit floating point numbers and have thus the same limitations as the machine format.

The complex number \(1 + 2\imaginaryI\) is represented as ["Complex", 1, 2]. This is a convenient shorthand for ["Add", 1, ["Multiply", 2, "ImaginaryUnit"]].

To change the numeric mode to the complex mode, use engine.numericMode = "complex".

Changing the numeric mode to complex automatically sets the precision to 15.

Auto Numeric Mode

When using the auto numeric mode, calculations are performed using bignum numbers.

Computations which result in a complex number will return a complex number as a Complex object.

To check the type of the result, use ce.isComplex(expr.N().numericValue) and ce.isBignum(expr.N().numericValue).

Simplifying Before Evaluating

When using expr.N(), no rewriting of the expression is done before it is evaluated.

Because of the limitations of machine numbers, this may produce surprising results.

For example, when numericMode = "machine":

const x = ce.parse('0.1 + 0.2').N();
console.log(ce.box(['Subtract', x, x]).N());
// ➔ 2.7755575615628914e-17

However, the result of \( x - x \) from ce.simplify() is \( 0 \) since the simplification is done symbolically, before any floating point calculations are made.

const x = ce.parse('0.1 + 0.2').N();
console.log(ce.parse('x - x').simplify());
// ➔ 0

In some cases, it may be advantageous to invoke expr.simplify() before using expr.N().

Tolerance

Two numbers that are sufficiently close to each other are considered equal.

To control how close two numbers have to be before they are considered equal, set the tolerance property of a ComputeEngine instance.

By default, the tolerance is \( 10^{-10} \).

The tolerance is accounted for by the Chop function to determine when to replace a number of a small magnitude with the exact integer 0.

It is also used when doing some comparison to zero: a number whose absolute value is smaller than the tolerance will be considered equal to 0.

Numeric Functions

The topics below from the MathJSON Standard Library can provide numeric evaluations for their numeric functions: