To express choosing between two alternatives, Scala
has a conditional expression `if-else`

.

It looks like a `if-else`

in Java, but is used for expressions, not statements.

Example:

`def abs(x: Double) = if (x >= 0) x else -x`

`x >= 0`

is a *predicate*, of type `Boolean`

.

Boolean expressions `b`

can be composed of

```
true false // Constants
!b // Negation
b && b // Conjunction
b || b // Disjunction
```

and of the usual comparison operations:

`e <= e, e >= e, e < e, e > e, e == e, e != e`

Here are reduction rules for Boolean expressions (`e`

is an arbitrary expression):

```
!true --> false
!false --> true
true && e --> e
false && e --> false
true || e --> true
false || e --> e
```

Note that `&&`

and `||`

do not always need their right operand to be evaluated.

We say these expressions use “short-circuit evaluation”.

We will define in this section a method

```
/** Calculates the square root of parameter x */
def sqrt(x: Double): Double = ...
```

The classical way to achieve this is by successive approximations using Newton's method.

To compute `sqrt(x)`

:

- Start with an initial
*estimate*`y`

(let's pick`y = 1`

). - Repeatedly improve the estimate by taking the mean of
`y`

and`x/y`

.

Example: Evaluation of the square root of 2 (x = 2):

```
Estimation Quotient Mean
1 2 / 1 = 2 1.5
1.5 2 / 1.5 = 1.333 1.4167
1.4167 2 / 1.4167 = 1.4118 1.4142
1.4142 ... ...
```

First, we define a method which computes one iteration step:

```
def sqrtIter(guess: Double, x: Double): Double =
if (isGoodEnough(guess, x)) guess
else sqrtIter(improve(guess, x), x)
```

Note that `sqrtIter`

is *recursive*, its right-hand side calls itself.

Recursive methods need an explicit return type in Scala.

For non-recursive methods, the return type is optional.

Second, we define a method `improve`

to improve an estimate and a test to check for termination:

```
def improve(guess: Double, x: Double) =
(guess + x / guess) / 2
def isGoodEnough(guess: Double, x: Double) =
abs(guess * guess - x) < 0.001
```

Third, we define the `sqrt`

function:

`def sqrt(x: Double) = sqrtIter(1.0, x)`

You have seen simple elements of functional programing in Scala.

- arithmetic and boolean expressions
- conditional expressions if-else
- functions with recursion

You have learned the difference between the call-by-name and call-by-value evaluation strategies.

You have learned a way to reason about program execution: reduce expressions using the substitution model.

Complete the following method definition that computes the factorial of a number:

```
def factorial(n: Int): Int =
if (n == res0) res1
else factorial(n - 1) * n
factorial(3) shouldBe 6
factorial(4) shouldBe 24
```