Re: [GP/PARI-2.1.0] arithmetic weirdness

[Gerhard Niklasch <nikl@mathematik.tu-muenchen.de>]

In response to:
> Message-ID: <Pine.LNX.4.33.0104020109490.6239-100000@pev.math.univ-montp2.fr>
> Date: Mon, 2 Apr 2001 01:27:17 +0200 (CEST)
> From: Philippe Elbaz-Vincent <pev@pev.math.univ-montp2.fr>
> To: pari-dev list <pari-dev@list.cr.yp.to>
> 
> Is the following an expected behavior ?

Yes.

> (01:23) gp > 1390.48+171364.53
> time = 0 ms.
> %1 = 172755.0099999999999999999999
> (01:24) gp >

Where's the problem here?

Use \x to check out the actual bit patterns of the summands.
Note that neither .48 nor .53 can be exactly represented as
a finite dyadic fraction (and thus as a PARI t_REAL of finite
precision).  The actual operands are rounded approximations
to the decimal fractions, and it just so happens that their
sum, rounded again  (to get rid of the three extra bits con-
tributed at the least significant end by the smaller summand),
converts back to a decimal fraction which is one unit below
the infinite-precision result in the least significant digit
position.  The result is correct to within the precision used
to represent the operands.

(NB the integer parts don't matter so long as their sizes
conspire to get the right bits of the periodic dyadic ex-
pansions lined up, and get them cut off and rounded at the
same positions within their periods.  1200.48+170000.53 will
show the same behaviour.)

This is not PARI-specific in any way, this is material for
the first week in any numerical analysis course. :)

Cheers, Gerhard