Editing Standard ML (section)

===Partial application===
Curried functions have many applications, such as eliminating redundant code. For example, a module may require functions of type {{code|lang=sml|a -> b}}, but it is more convenient to write functions of type {{code|lang=sml|a * c -> b}} where there is a fixed relationship between the objects of type {{code|a}} and {{code|c}}. A function of type {{code|lang=sml|c -> (a * c -> b) -> a -> b}} can factor out this commonality. This is an example of the [[adapter pattern]].{{Citation needed|date=May 2015}}

In this example, {{code|lang=sml|fun d}} computes the numerical derivative of a given function {{code|f}} at point {{code|x}}:

<syntaxhighlight lang="sml" highlight="1">
- fun d delta f x = (f (x + delta) - f (x - delta)) / (2.0 * delta)
val d = fn : real -> (real -> real) -> real -> real
</syntaxhighlight>

The type of {{code|lang=sml|fun d}} indicates that it maps a "float" onto a function with the type {{code|lang=sml|(real -> real) -> real -> real}}. This allows us to partially apply arguments, known as [[currying]]. In this case, function {{code|d}} can be specialised by partially applying it with the argument {{code|delta}}. A good choice for {{code|delta}} when using this algorithm is the cube root of the [[machine epsilon]].{{Citation needed|date=August 2008}}

<syntaxhighlight lang="sml" highlight="1">
- val d' = d 1E~8;
val d' = fn : (real -> real) -> real -> real
</syntaxhighlight>

The inferred type indicates that {{code|d'}} expects a function with the type {{code|lang=sml|real -> real}} as its first argument. We can compute an approximation to the derivative of <math>f(x) = x^3-x-1</math> at <math>x=3</math>. The correct answer is <math>f'(3) = 27-1 = 26</math>.

<syntaxhighlight lang="sml" highlight="1">
- d' (fn x => x * x * x - x - 1.0) 3.0;
val it = 25.9999996644 : real
</syntaxhighlight>