# Re: Is this OK?

From: Xavier Leroy (Xavier.Leroy@inria.fr)
Date: Mon May 18 1998 - 14:51:54 MET DST

Date: Mon, 18 May 1998 14:51:54 +0200
From: Xavier Leroy <Xavier.Leroy@inria.fr>
To: N Hur <mapnh@maths.bath.ac.uk>, caml-list@inria.fr
Subject: Re: Is this OK?
In-Reply-To: <Pine.SUN.3.91.980515135653.27393A-100000@thor.maths.bath.ac.uk>; from N Hur on Fri, May 15, 1998 at 02:02:24PM +0100

> Why are the two answers below different?
> (* a camllight session *)
> #(-234)/4
> ;;
> - : int = -58
>
> #div_big_int (big_int_of_string "-234") (big_int_of_string "4");;
> - : big_int = -59
> Thanks for advance for any explanation.

In computer arithmetic, integer division has never been well specified
for non-positive arguments. There are at least two sensible behaviors:

(a) round towards zero e.g. (-1) / 2 = 0
(b) round towards -infinity e.g. (-1) / 2 = -1

Each of these two behaviors satisfies two of the following three
desirable properties of integer division and modulus, but not all
three:

(1) (-a) / b = a / (-b) = - (a/b)
(2) a = (a/b) * b + (a mod b)
(3) 0 <= a mod b < abs(b)

More precisely, (a) satisfies (1) and (2) but not (3) (the modulus can
be negative), while (b) satisfies (2) and (3) but fails (1).

It is easy to see that (1), (2) and (3) cannot be satisfied
simultaneously anyway, so behaviors (a) and (b) are as good as you'll
ever get.

Most processors implement behavior (a), and therefore most C programs
also follow (a) (though the C reference manual does not fully specify
integer division and modulus for negative arguments, and therefore (a)
and (b) are both legal). Since the Caml bytecode interpreter is a
C program, that's also the behavior you get.

The bignum library uses its own signed arithmetic over big integers,
and elected to implement behavior (b), i.e. round towards -infinity,
thus ensuring that the modulus is always positive. The reason for
this choice, I guess, is that (b) is much better than (a) if you're
doing modulo arithmetic.

So, there is basically no solution to this problem, except providing
two division operations and two modulus operations, one set following
(a) and the other following (b).

- Xavier Leroy

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