# for review: big rats

Bert Thompson aet at hydra.cs.mu.oz.au
Wed Apr 8 17:35:18 AEST 1998

```Peoples,

Thanks,
Bert
----------------------------------------
Estimated hours taken: 2

Implementation of arbitrary precision rational numbers.

library/rational.m

%-----------------------------------------------------------------------------%
% Copyright (C) 1997-1998 The University of Melbourne.
% This file may only be copied under the terms of the GNU General
% Public License - see the file COPYING in the Mercury distribution.
%-----------------------------------------------------------------------------%
%
% file: rational.m
% authors: aet Apr 1998. (with plagiarism from rat.m)
%
% Implements a rational number type and a set of basic operations on
% rational numbers.

:- module rational.

:- interface.

:- import_module integer.

:- type rational.

:- pred rational:'<'(rational, rational).
:- mode rational:'<'(in, in) is semidet.

:- pred rational:'>'(rational, rational).
:- mode rational:'>'(in, in) is semidet.

:- pred rational:'=<'(rational, rational).
:- mode rational:'=<'(in, in) is semidet.

:- pred rational:'>='(rational, rational).
:- mode rational:'>='(in, in) is semidet.

:- pred rational:'='(rational, rational).
:- mode rational:'='(in, in) is semidet.

:- func rational(int, int) = rational.

:- func rational_from_integers(integer, integer) = rational.

% :- func float(rational) = float.

:- func rational:'+'(rational) = rational.

:- func rational:'-'(rational) = rational.

:- func rational:'+'(rational, rational) = rational.

:- func rational:'-'(rational, rational) = rational.

:- func rational:'*'(rational, rational) = rational.

:- func rational:'/'(rational, rational) = rational.

:- func rational:numer(rational) = integer.

:- func rational:denom(rational) = integer.

:- func rational:abs(rational) = rational.

:- func one = rational.

:- func zero = rational.

:- implementation.

:- import_module require.

:- type rational
--->	r(integer, integer).

rational:'<'(R1, R2) :-
cmp(R1, R2) = lessthan.

rational:'>'(R1, R2) :-
cmp(R1, R2) = greaterthan.

rational:'=<'(R1, R2) :-
Cmp = cmp(R1, R2),
(Cmp = lessthan; Cmp = equal).

rational:'>='(R1, R2) :-
Cmp = cmp(R1, R2),
(Cmp = greaterthan; Cmp = equal).

rational(Num, Den) = rational_norm(r(integer(Num), integer(Den))).

rational_from_integers(Num, Den) = rational_norm(r(Num, Den)).

% We needn't call rational_norm since rationals can be
% created only by the rational constructors or by rational
% operations, the results of which must be normalised.
An = Bn, Ad = Bd.

%% XXX: There are ways to do this in some cases even if the
%% float conversions would overflow.
% rational:float(r(Num, Den)) =
%	float:'/'(integer:float(Num), integer:float(Den)).

one = r(integer(1), integer(1)).

zero = r(integer(0), integer(1)).

rational:'+'(Rat) = Rat.

rational:'-'(r(Num, Den)) = r(-Num, Den).

rational_norm(r(An*CA + Bn*CB, M)) :-
CB = M / Bd.

rational:'-'(R1, R2) =
R1 + (-R2).

% XXX: need we call rational_norm here?
G1 = gcd(An,Bd),

rational:'/'(R1, R2) =
R1 * inverse(R2).

:- func inverse(rational) = rational.
inverse(r(Num, Den)) = Rat :-
(Num = izero ->
error("rational:inverse: division by zero")
;
Rat = r(signum(Num)*Den,abs(Num))
).

rational:numer(r(Num, _)) = Num.

rational:denom(r(_, Den)) = Den.

rational:abs(r(Num,Den)) = r(abs(Num),Den).

% The normal form of a rational number has the following
% properties:
%	- numerator and denominator have no common factors.
%	- denominator is positive.
%	- denominator is not zero.
%	- if numerator is zero, then denominator is one.
:- func rational_norm(rational) = rational.
rational_norm(r(Num, Den)) = Rat :-
( Den = izero ->
error("rational_norm: division by zero")
; Num = izero ->
Rat = r(izero,ione)
;
Rat = r(Num2//G, Den2//G),
Num2 = Num * signum(Den),
Den2 = abs(Den),
G = gcd(Num,Den)
).

:- func gcd(integer, integer) = integer.
gcd(A, B) =
gcd_2(abs(A), abs(B)).

:- func gcd_2(integer, integer) = integer.
gcd_2(A, B) =
( B = izero -> A
; gcd_2(B, A rem B)
).

:- func lcm(integer, integer) = integer.
lcm(A, B) =
( A = izero -> izero
; B = izero -> izero
; abs((A // gcd(A, B)) * B)
).

:- func izero = integer.
izero = integer(0).

:- func ione = integer.
ione = integer(1).

:- func signum(integer) = integer.
signum(N) =
( N = izero -> izero
; N < izero -> -ione
; ione
).

:- type comparison
--->	equal
;	lessthan
;	greaterthan.

:- func cmp(rational, rational) = comparison.
cmp(R1, R2) = Cmp :-
Diff = R1 - R2,
( is_zero(Diff) ->
Cmp = equal
; is_negative(Diff) ->
Cmp = lessthan
;
Cmp = greaterthan
).

:- pred is_zero(rational).
:- mode is_zero(in) is semidet.
is_zero(r(Num,_)) :-
Num = izero.

:- pred is_negative(rational).
:- mode is_negative(in) is semidet.
is_negative(r(Num,_)) :-
Num < izero.

```