Consider the following program that computes the area of a disc
whose radius is `10`

:

`3.14159 * 10 * 10`

To make complex expressions more readable we can give meaningful names to intermediate expressions:

```
val radius = 10
val pi = 3.14159
pi * radius * radius
```

Besides making the last expression more readable it also allows us to not repeat the actual value of the radius.

A name is evaluated by replacing it with the right hand side of its definition

Here are the evaluation steps of the above expression:

```
pi * radius * radius
3.14159 * radius * radius
3.14159 * 10 * radius
31.4159 * radius
31.4159 * 10
314.159
```

Definitions can have parameters. For instance:

```
def square(x: Double) = x * x
square(3.0) shouldBe res0
```

Let’s define a method that computes the area of a disc, given its radius:

```
def square(x: Double) = x * x
def area(radius: Double): Double = 3.14159 * square(radius)
area(10) shouldBe res0
```

Separate several parameters with commas:

`def sumOfSquares(x: Double, y: Double) = square(x) + square(y)`

Function parameters come with their type, which is given after a colon

`def power(x: Double, y: Int): Double = ...`

If a return type is given, it follows the parameter list.

The right hand side of a `def`

definition is evaluated on each use.

The right hand side of a `val`

definition is evaluated at the point of the definition
itself. Afterwards, the name refers to the value.

```
val x = 2
val y = square(x)
```

For instance, `y`

above refers to `4`

, not `square(2)`

.

Applications of parametrized functions are evaluated in a similar way as operators:

- Evaluate all function arguments, from left to right
- Replace the function application by the function's right-hand side, and, at the same time
- Replace the formal parameters of the function by the actual arguments.

```
sumOfSquares(3, 2 + 2)
sumOfSquares(3, 4)
square(3) + square(4)
3 * 3 + square(4)
9 + square(4)
9 + 4 * 4
9 + 16
25
```

This scheme of expression evaluation is called the *substitution model*.

The idea underlying this model is that all evaluation does is *reduce
an expression to a value*.

It can be applied to all expressions, as long as they have no side effects.

The substitution model is formalized in the λ-calculus, which gives a foundation for functional programming.

Does every expression reduce to a value (in a finite number of steps)?

No. Here is a counter-example:

```
def loop: Int = loop
loop
```

The difference between `val`

and `def`

becomes apparent when the right
hand side does not terminate. Given

`def loop: Int = loop`

A definition

`def x = loop`

is OK, but a definition

`val x = loop`

will lead to an infinite loop.

The interpreter reduces function arguments to values before rewriting the function application.

One could alternatively apply the function to unreduced arguments.

For instance:

```
sumOfSquares(3, 2 + 2)
square(3) + square(2 + 2)
3 * 3 + square(2 + 2)
9 + square(2 + 2)
9 + (2 + 2) * (2 + 2)
9 + 4 * (2 + 2)
9 + 4 * 4
25
```

The first evaluation strategy is known as *call-by-value*,
the second is is known as *call-by-name*.

Both strategies reduce to the same final values as long as

- the reduced expression consists of pure functions, and
- both evaluations terminate.

Call-by-value has the advantage that it evaluates every function argument only once.

Call-by-name has the advantage that a function argument is not evaluated if the corresponding parameter is unused in the evaluation of the function body.

Scala normally uses call-by-value.

Complete the following definition of the `triangleArea`

function,
which takes a triangle base and height as parameters and returns
its area:

```
def triangleArea(base: Double, height: Double): Double =
base * height / res0
triangleArea(3, 4) shouldBe 6
triangleArea(5, 6) shouldBe res1
```