## Exercise 2.2 (Power)

```Ocaml
(*First version ; 6 multiplications*)
# let power28(x) =
	let x2 = x + x in
	let x4 = x2*x2 in
	let x8 = x4*x4 in
	let x16 = x8*x8 in
	x16*x8*x4 ;;

(*Second version ; 27 multiplications*)
# let power28(x) = x*x*x*x*x*x...*x ;;

(*Third version ; 11 multiplications*)
# let power28(x) = 
	let sq(x) = x*x in
	let pow4(x) = sq(sq(x)) in
	pow4(pow4(x))*sq(pow4(x))*pow4(x);;

(*Fourth version ; 7 multiplications*)
# let power28(x)=
	let sq(x) = x+x in
	let p4=sq(sq(x)) in
	sq(sq(p4))*sq(p4)*p4;;
```

## Exercise 2.3

```Ocaml
# (*my verison*)
# let mirror(n) = let diz = n/10 and uni = n mod 10 in uni*10 + diz;;
val mirror : int -> int = <fun>

# (*teatcher version*)
# let mirror n = 10 *(n mod 10)+n/10;;
val mirror : int -> int = <fun>
```

```Ocaml
# let abba(n) = n*100 + mirror(n) ;;
val abba: int -> int = <fun>
```

```Ocaml
# let stammer(n) = abba(mirror(n)) * 10 000 + abba(n) ;;
val stammer: int -> int = <fun>
```



## Exercice 2.6
```Ocaml
let sec_of_time h m s = 
	h*3600 + m*60 + s ;;

let time_of_sec s = 
	let hours = s/3600 in let minutes = (s - hours*3600)/60 in let seconds = s - hours *3600 - minutes * 60 in (hours, minutes, seconds);;

let add_times h1 m1 s1 h2 m2 s2 = 
	let sec1 = sec_of_time h1 m1 s1 and sec2 = sec_of_time h2 m2 s2 in let resultsec = sec1 + sec2 in time_of_sec resultsec ;;

```

## Exercise 3.1
```Ocaml
let add a b = 
	if a > b then 
		a-b + a/b
	else
		b-a + b/a

let test x y = (if x> y then x else y ) *(x+y);;


let f a b c =
	let g x y = (x+y)*(x-y) in 
	if a > b then 
		if b > c then (a+b)*(a-b) else (c+a) *(a-c)
	else
		if a > c then (a+b)*(a-b) else (b+c) * (b-c);;

let f a b c =
	(if a > b && if b > c then
		a + b else c + a
	 else if a > c then a + b else b + c)*
	 (if a > b && b > c then
		 a - b else a - c
	 else if a > c then a-b else b-c);;
```

### Exercise 3.2
```OCaml
(*logical and*)
# let and_if a b = 
	if a then
		b
	else false;;
val and_if: bool -> bool -> bool = <fun>

(*logical or*)
# let or_if a b =
	if a then true
	else b;;
	
val or_if: bool -> bool -> bool = <fun>


(*logical implication*)
# let imply_if a b =
	if a then b
	else true;;
val imply_if: bool -> bool -> bool = <fun>

(*logical exclusive or*)
let xor a b = 
	if a then
		if b then false else true
	else
		if b then true else false

(*logical equivalence = a b*)
let equiv a b =
	if a then b
	else
		if b then false else true (*not b*)

```

### Exercise 3.3
```Ocaml
# let max2 number1 number2 = if number1 > number2 then number1 else number2
val max2 : 'a -> 'a -> 'a = <fun>

# let min2 number1 number2 = if number1 > number2 then number2 else number1
val min2 : 'a -> 'a -> 'a = <fun>

# let max3 x y z =
	let high = if x > y then x else y in
		if high > z then
			high
		else z;;
val max3 : 'a -> 'a -> 'a -> 'a = <fun>

# let min3 x y z =
	let low = if x < y then x else y in
		if low < z then
			low
		else z;;
val min3 : 'a -> 'a -> 'a -> 'a = <fun>

# let middle3 x y z = 
	x + y + z - min3 x y z - max x y z ;;
val middle3 = int -> int -> int = <fun>

#let max4 number1 number2 number3 number4 = 
	let nb1 = max2(number1 number2) and nb2 = max2(number3 number4) in max2(nb1 nb2)

# let max4 x y z t= max2(max2 x y)(max2 z t)
val max4 : 'a -> 'a -> 'a -> 'a -> 'a = <fun>
# let min4 x y z t= min2(min2 x y)(min2 z t)

```
### Exercise 3.4
```Ocaml
let highest_square_sum x y z = 
	let sq x = x*x in 
	sq(max3 x y z) + sq(middle3 x y z)
val highest_square_sum : int -> int -> int -> int = <fun>

```

## Exercise 3.7
```Ocaml
# let rate_eco kg = match kg with
	x when x <= 500 -> 3.40
	| x when x <=1000 -> 4.60
	| x when x <= 2000 -> 5.10
	| x when x <= 3000 -> 6.90
	| _ -> invalid_arg "Cannot be more than 3000g";;
val rate_eco : int -> float = <fun>

# let rate_standard kg = match kg with
	x when x <= 500 -> 4.60
	| x when x <=1000 -> 5.90
	| x when x <= 2000 -> 6.50
	| x when x <= 3000 -> 7.20
	| _ -> invalid_arg "Cannot be more than 3000g";;
val rate_standart : int -> float = <fun>

# let rate_express kg = match kg with
	x when x <= 500 -> 9.10
	| x when x <=1000 -> 11.
	| x when x <= 2000 -> 13.5
	| x when x <= 3000 -> 14.2;
	| _ -> invalid_arg "Cannot be more than 3000g";;
val rate_express : int -> float = <fun>

# let rate rt kg = match rt with
	| ^x when x = "economic" -> rate_eco(kg)
	| x when x = "standard" -> rate_standard(kg)
	| x when x = "express" -> rate_express(kg)
	| _ -> invalid_arg "Bad type of shipping class";;
val rate : string -> int -> float = <fun>

(*4th question*)
let price w (p1, p2, p3, p4) = 
	if w <=500 then p1
	else if w <=1000 then p2
	else if w <=2000 then p3
	else if w <=3000 then p4
	else failwith "Too heavy";;
val price = int -> 'a * 'a * 'a * 'a -> 'a = <fun>
```

## Exercise 3.9
```Ocaml
let apm = function 
	| (0, 0) -> 2
	| (0, y) -> 1
	| (x, 0) -> 1
	| (x, y) -> 0
	| _ -> invalid_arg "Error";;
val amp: int*int -> int = <fun>

let strange = function
	| (0, n) -> 0
	| (m, 0) -> 2*m
	| (m, n) -> m*n ;;
val strange: int*int -> int = <fun>

let or3 = function
	| (true, _, _) -> true
	| (_, true, _) -> true
	| (_, _, true) -> true
	| _ -> false
val or3: bool*bool*bool -> bool = <fun>

let or3-simple = function
	| (false, false, false) -> false
	| _ -> true
val or3-simple: bool*bool*bool -> bool = <fun>
```

### Exercise 3.10 
```Ocaml
let time_difference (d1, md1, sd1, pos1) (d2, md2, sd2, pos2)
```

## Exercise 4.2
```Ocaml
let rec sequence = function 
	| 0 -> 1
	| n -> 4* sequence(n-1) - 1;;
val sequence : int -> int = <fun>
```

```mermaid
flowchart LR

A[seq 3] --> B[4*seq2-1] --> C[4*seq1-1] --> D[4*seq0-1] --> E[1] --> F[4*1-1] --> G[4*3-1] --> H[4*11-1]
```

## Exercise 4.3
```Ocaml
# let geometric n u0 q =
	let rec geo = function
	| 0 -> u0
	|n -> q*geo(n-1) in geo n;;
```
## Exercise 4.4
```Ocaml
# let rec gcd a b = 
	if a mod b = 0 then
		b
	else
		gcd b (a mod b);;
```
## Exercise 4.5
```Ocaml
let rec add a = function
	| 0 -> a
	| b -> 1 + add a (b-1);;
```

## Exercise 4.6
```Ocaml
let rec mult a b = 
	if a = 0 || b = 0 then 
		0 
	else if b > 0 then 
		a + mult a (b - 1) 
	else 
		-mult a (-b) ;;
```

## Exercise 4.7
```Ocaml
let rec quo a b = 
	if a < b || b = 0 then
		0
	else if b = 1 then
		a
	else 
		1 + quo (a-b) b ;;
```

## Exercise 4.9
```Ocaml
let rec reverse n = 
	let str_n = string_of_int n in 
	let len = String.length str_n in 
	let rec reverse_helper index = 
		if index < 0 then 
			"" 
		else 
			String.make 1 str_n.[index] ^ reverse_helper (index - 1) 
	in 
	reverse_helper (len - 1) ;;

let rec reverse_int n = 
	let rec reverse_helper acc remaining = 
	if remaining = 0 then 
		acc 
	else 
		let last_digit = remaining mod 10 in 
		let new_acc = acc * 10 + last_digit in 
		let new_remaining = remaining / 10 
	in reverse_helper new_acc new_remaining in reverse_helper 0 n ;;

```

## Exercise 4.9 - Correction
```Ocaml
# let reverse = function
		| 0 -> " "
		| n -> string_of_int(n mod 10) ^ reverse(n/10);;
val reverse : int -> string = <fun>

# let reverse_int =
	let rec rev inv = function
		| 0 -> inv
		| n -> rev (inv*10 + n mod 10)(n/10)
	in rev 0 n;;
```

## Exercise 4.10

```Ocaml
let rec multiply x y =
  if x = 0 || y = 0 then
    0
  else if x mod 2 = 0 then
    multiply (x / 2) (y * 2)
  else
    y + multiply (x / 2) (y * 2)

(*correction*)
# let egypt a b =
	let (a, b) = if a > b then (a,b) else (b,a) in 
	let rec eg = function
		| 0 -> 0
		| b -> 2*eg(b/2) + (if b mod 2 = 0 then 0 else a)
	in
	eg b;;
	
```

## Exercise 4.11
```Ocaml
let rec puissance x n =
	if n = 0 then
		1
	else
		x*puissance x (n-1);;

let rec puissance_better x n =
	if n = 0 then
		1
	else if n mod 2 = 0 then
		let pb = puissance_better x (n/2) in pb*pb
	else
		let pb_odd = puissance_better x (n/2) * n in pb_odd*pb_odd;;

(*Correction v1*)
let power x n = match n with
	| 0 -> (match x with
		| 0. -> failwith "power 0^0 impossible"
		| _ -> 1.)
	| _ -> (match x with 
		| 1. -> 1.
		| 0. -> 0.
		| -1. -> if n mod 2 = 0 then 1. else -1.)
		| _ -> (let rec p = function
			| 0 -> 1.
			| n -> x*.p(n-1) in p n)
;;

(*Correction v2*)
let power x n = match n with
	| 0 -> (match x with
		| 0. -> failwith "power 0^0 impossible"
		| _ -> 1.)
	| _ -> (match x with 
		| 1. -> 1.
		| 0. -> 0.
		| -1. -> if n mod 2 = 0 then 1. else -1.)
		| _ -> (let rec p = function
			| 0 -> 1.
			| n -> (let res. = p x (n/2) in 
					res *. res *. (if n mod 2 = 1 then x else 1.)
			in p x n)
;;

(*Correction v3*)
let power x n = match n with
	| 0 -> (match x with
		| 0. -> failwith "power 0^0 impossible"
		| _ -> 1.)
	| _ -> (match x with 
		| 1. -> 1.
		| 0. -> 0.
		| -1. -> if n mod 2 = 0 then 1. else -1.)
		| _ -> (let rec p x = function
			| 0 -> 1.
			| n -> p (x*x) (n/2) *. (if n mod 2 = 1 then x else 1.)
			in p x n)
;;
```

Complexity :
$$
x^0 -> 1 \ | \ 
x^1 -> (x^0)^2 -> 2 \ | \ 
x^2 -> x^1 -> 3 \ | \ 
x^4 -> x^2 -> 4 \ | \ 
x^8 -> x^4 -> 5 \ | \ 
x^16 -> x^8 -> 6 \ | \ 
x^(2^k) -> x^(2^k-1) -> k+2 \approx{k} \ | \ 
n = 2^k -> K*log(n) \ | \ 
log(n) = log(2^k) = k 
$$
$$
= O(log(n))
$$
##  Exercise 4.12 - Prime number
```OCaml
(* V1 *)
let prime n =
	if n < 1 then
		invalid_args "n should not be inferior to zero"
	else if x = 2 then true
	else 
		let rec pr n k =
			if n = k then
				true
			else if n mod k = 0 then
				false
			else
				check n (k + 1)
		in pr n 2;;
(* V2 *)
let prime n =
	if n < 1 then
		invalid_args "n should not be inferior to zero"
	else if x = 2 then true
	else 
		let rec pr n k =
			if n = k then
				true
			else if n mod k = 0 then
				false
			else
				check n (k + 1)
		in (n = 2) || (n mod 2 = 1 and check n 3)
```