# Numerical Evaluation

To obtain a numeric approximation of the value of an expression, call the expr.N() function.

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

console.log(ce.parse('3 + 5 + x').N().latex);
// ➔ "8 + x"


If the expression is pure, the value of the expression can be obtained with expr.numericValue.

console.log(ce.parse('\\sqrt{5} + 7^3').N().latex);
// ➔ "345.2360679774998"

console.log(ce.parse('\\sqrt{5} + 7^3').numericValue?.latex);
// ➔ "345.2360679774998"

console.log(ce.parse('\\sqrt{x} + 7^3').N().latex);
// ➔ "\sqrt{x} + 343"

console.log(ce.parse('\\sqrt{x} + 7^3').numericValue?.latex);
// ➔ undefined


## Numeric Modes

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

### 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.

### Decimal Numeric Mode

In the decimal numeric mode, numbers are represented in Decimal format, using base-10 and a variable amount of memory depending on the number of significant digits (precision) desired.

Numbers in the Decimal format have a minimum value of $$\pm 10^{-9000000000000000}$$ and a maximum value of $$\pm9.99999\ldot\times 10^{+9000000000000000}$$.

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

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


When using the decimal 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 decimal mode, set the engine.precision property.

The precision property affects how the computations are performed. To change how numbers are displayed when 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 decimal 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 machine numbers if the precision is 15 or less. If the precision is more than 15, Decimal numbers are used.

Computations with a result in the Complex domain will return a Complex number.

## 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:

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 dictionaries below can provide numeric evaluations for their numeric functions:

Arithmetic Add Multiply Sqrt Log Abs Round
Trigonometry Sin Cos Tan Sinh Arcsin
Special Functions Erf Gamma Factorial