(*
** Course: Concepts of Programming Languages (BU CAS CS 320)
** Semester: Summer I, 2010
** Instructor: Hongwei Xi (hwxi AT cs DOT bu DOT edu)
*)

//
// Assignment Three (Due: Wednesday, June 9, 2010)
//
// Total points: 120 + 20 (extra)
//

(* ****** ****** *)

#define ATS_STALOADFLAG 0 // no staloading at run-time

(* ****** ****** *)
//
// assgn3ex1: 10 points
// It is assumed that [f] is continous on [x0, x1] and f(x0) * f(x1) <= 0
//
fun solve_bisection (
    f: double -<cloref1> double
  , x0: double, x1: double
  , relprec: double // relative precision
  ) : double

(* ****** ****** *)
//
// assgn3ex2: 10 points
// It is assumed that [f1] is the derivative of [f]
//
fun solve_NewtonRaphson (
    f: double -<cloref1> double
  , f1: double -<cloref1> double
  , x0: double
  , relprec: double // relative precision
  ) : double

(* ****** ****** *)

exception Empty of ()
exception Subscript of ()

(* ****** ****** *)
//
// assgn3ex3: 60 points + 20 points (extra)
//
abstype bignum_t

(* implementation is given *)
fun bignum_make_int (i: int): bignum_t

(* implementation is given *)
fun neg_bignum (x: bignum_t): bignum_t

// 5 points
fun succ_bignum (x: bignum_t): bignum_t

// 5 points
fun pred_bignum (x: bignum_t): bignum_t

(* implementation is given *)
fun add_bignum_bignum (x1: bignum_t, x2: bignum_t): bignum_t

// 10 points
fun sub_bignum_bignum (x1: bignum_t, x2: bignum_t): bignum_t

// 10 points
fun mul_bignum_bignum (x1: bignum_t, x2: bignum_t): bignum_t

// 15 points
fun div_bignum_bignum (x1: bignum_t, x2: bignum_t): bignum_t
fun divrem_bignum_bignum (x1: bignum_t, x2: bignum_t): @(bignum_t, bignum_t)

// 15 points
fun sqrt_bignum (x: bignum_t): bignum_t

(* implementation is given *)
fun eq_bignum_bignum (x1: bignum_t, x2: bignum_t): bool

(*

// 20 points (extra)

Assume X, Y and Z are rational numbers defined as follows:

X = 411340519227716149383203 / 21666555693714761309610
Y = 6803298487826435051217540 / 411340519227716149383203
Z = 224403517704336969924557513090674863160948472041 / 8912332268928859588025535178967163570016480830

Please use the above [bignum] implementation to verify that X^2 + Y^2 = Z^2 holds.

*)

(* ****** ****** *)

(*
//
// assgn3ex4: 40 points
//
The following is a description of {\em Game-of-24}.

Given four natural numbers $n_1,n_2,n_3$ and $n_4$, one chooses two of them
and generates a {\em rational} number $r_1$ using either addition,
subtraction, multiplication or division; one mixes $r_1$ with the remaining
two numbers and chooses two of them to generate a rational number $r_2$
using either addition, subtraction, multiplication or division; one then
takes $r_2$ and the last remaining number to get a rational number $r_3$
using addition, subtraction, multiplication, or division; if there is a way
to make $r_3$ equal to $24$, then we say that $(n_1,n_2,n_3,n_4)$ is a good
quad.  For instance, $(10,10,4,4)$ is a good quad since we have

\[(10 * 10 - 4) / 4 = 24\]

Similarly, $(5,7,7,11)$ is a good quad since we have

\[(5 - 11 / 7) * 7 = 24\]

Game-of-24 is a game that determines whether four given natural numbers are
a good quad.

Please implement a program in ATS that takes four given natural numbers and
returns a value indicating whether the four given natural numbers are a
good quad. If they are, the returned value should represent an arithmetic
expression that attests to their being a good quad.

*)

datatype exp =
  | EXPnum of double
  | EXPadd of (exp, exp)
  | EXPsub of (exp, exp)
  | EXPmul of (exp, exp)
  | EXPdiv of (exp, exp)
// end of [exp]

datatype expopt = EXPOPTsome of exp | EXPOPTnone of ()

fun play24 (n1: int, n2: int, n3: int, n4: int): expopt

(* ****** ****** *)

(* end of [assignment3.sats] *)