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223 lines
4.9 KiB
Mathematica
223 lines
4.9 KiB
Mathematica
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%-----------------------------------------------------------------------------%
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% Copyright (C) 1997-1998 The University of Melbourne.
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% This file may only be copied under the terms of the GNU General
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% Public License - see the file COPYING in the Mercury distribution.
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%-----------------------------------------------------------------------------%
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%
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% file: rational.m
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% authors: aet Apr 1998. (with plagiarism from rat.m)
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%
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% Implements a rational number type and a set of basic operations on
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% rational numbers.
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:- module rational.
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:- interface.
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:- import_module integer.
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:- type rational.
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:- pred rational:'<'(rational, rational).
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:- mode rational:'<'(in, in) is semidet.
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:- pred rational:'>'(rational, rational).
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:- mode rational:'>'(in, in) is semidet.
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:- pred rational:'=<'(rational, rational).
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:- mode rational:'=<'(in, in) is semidet.
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:- pred rational:'>='(rational, rational).
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:- mode rational:'>='(in, in) is semidet.
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:- func rational(int, int) = rational.
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:- func rational_from_integers(integer, integer) = rational.
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% :- func float(rational) = float.
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:- func rational:'+'(rational) = rational.
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:- func rational:'-'(rational) = rational.
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:- func rational:'+'(rational, rational) = rational.
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:- func rational:'-'(rational, rational) = rational.
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:- func rational:'*'(rational, rational) = rational.
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:- func rational:'/'(rational, rational) = rational.
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:- func rational__numer(rational) = integer.
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:- func rational__denom(rational) = integer.
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:- func rational__abs(rational) = rational.
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:- func one = rational.
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:- func zero = rational.
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:- implementation.
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:- import_module require.
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% The normal form of a rational number has the following
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% properties:
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% - numerator and denominator have no common factors.
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% - denominator is positive.
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% - denominator is not zero.
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% - if numerator is zero, then denominator is one.
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%
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% These invariants must be preserved by any rational number
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% constructed using this module since the equality predicate
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% on rationals is simply Mercury's default unification
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% predicate =/2. If the invariants were not maintained,
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% we would have pathologies like r(-1,2) \= r(1,-2).
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%
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% The rational_norm/2 function generates rationals in this
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% normal form.
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%
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:- type rational
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---> r(integer, integer).
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rational:'<'(R1, R2) :-
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Cmp = cmp(R1, R2),
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Cmp = lessthan.
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rational:'>'(R1, R2) :-
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Cmp = cmp(R1, R2),
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Cmp = greaterthan.
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rational:'=<'(R1, R2) :-
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Cmp = cmp(R1, R2),
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(Cmp = lessthan ; Cmp = equal).
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rational:'>='(R1, R2) :-
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Cmp = cmp(R1, R2),
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(Cmp = greaterthan ; Cmp = equal).
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rational(Num, Den) = rational_norm(integer(Num), integer(Den)).
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rational_from_integers(Num, Den) = rational_norm(Num, Den).
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%% XXX: There are ways to do this in some cases even if the
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%% float conversions would overflow.
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% rational__float(r(Num, Den)) =
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% float:'/'(integer__float(Num), integer__float(Den)).
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one = r(integer(1), integer(1)).
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zero = r(integer(0), integer(1)).
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rational:'+'(Rat) = Rat.
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rational:'-'(r(Num, Den)) = r(-Num, Den).
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rational:'+'(r(An, Ad), r(Bn, Bd)) = rational_norm(Numer, M) :-
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Numer is An * CA + Bn * CB,
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M = lcm(Ad, Bd),
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CA = M // Ad,
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CB = M // Bd.
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rational:'-'(R1, R2) =
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R1 + (-R2).
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% XXX: need we call rational_norm here?
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rational:'*'(r(An, Ad), r(Bn, Bd)) = rational_norm(Numer, Denom) :-
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Numer is (An//G1) * (Bn//G2),
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Denom is (Ad//G2) * (Bd//G1),
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G1 = gcd(An, Bd),
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G2 = gcd(Ad, Bn).
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rational:'/'(R1, R2) =
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R1 * inverse(R2).
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:- func inverse(rational) = rational.
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inverse(r(Num, Den)) = Rat :-
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( Num = izero ->
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error("rational__inverse: division by zero")
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;
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Rat = r(signum(Num)*Den, abs(Num))
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).
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rational__numer(r(Num, _)) = Num.
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rational__denom(r(_, Den)) = Den.
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rational__abs(r(Num, Den)) = r(abs(Num), Den).
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:- func rational_norm(integer, integer) = rational.
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rational_norm(Num, Den) = Rat :-
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( Den = izero ->
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error("rational__rational_norm: division by zero")
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; Num = izero ->
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Rat = r(izero, ione)
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;
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Rat = r(Num2//G, Den2//G),
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Num2 = Num * signum(Den),
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Den2 = abs(Den),
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G = gcd(Num, Den)
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).
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:- func gcd(integer, integer) = integer.
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gcd(A, B) =
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gcd_2(abs(A), abs(B)).
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:- func gcd_2(integer, integer) = integer.
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gcd_2(A, B) =
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( B = izero -> A
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; gcd_2(B, A rem B)
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).
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:- func lcm(integer, integer) = integer.
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lcm(A, B) =
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( A = izero -> izero
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; B = izero -> izero
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; abs((A // gcd(A, B)) * B)
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).
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:- func izero = integer.
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izero = integer(0).
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:- func ione = integer.
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ione = integer(1).
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:- func signum(integer) = integer.
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signum(N) =
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( N = izero -> izero
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; N < izero -> -ione
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; ione
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).
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:- type comparison
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---> equal
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; lessthan
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; greaterthan.
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:- func cmp(rational, rational) = comparison.
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cmp(R1, R2) = Cmp :-
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Diff = R1 - R2,
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( is_zero(Diff) ->
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Cmp = equal
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; is_negative(Diff) ->
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Cmp = lessthan
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;
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Cmp = greaterthan
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).
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:- pred is_zero(rational).
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:- mode is_zero(in) is semidet.
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is_zero(r(Num, _)) :-
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Num = izero.
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:- pred is_negative(rational).
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:- mode is_negative(in) is semidet.
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is_negative(r(Num, _)) :-
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Zero = izero,
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Num < Zero.
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