[English Translation of H. Barendregt's press release]
World Championship of Exact Arithmetic
Nijmegen, 4 October 2005
In the Department of computer Science of the Radboud University on 3
and 4 October an international competition was held for calculating
mathematical quantities with an arbitrarily given degree of
accuracy. There were 9 participating teams and two classes of problems
were posed. The winners were in the basic problems the team of Wolfram
Research (USA), known for their computer program Mathematica, and in
the intermediate problems a team working at the academic institute
LORIA (France).
Numbers such as Pi=3.141... have infinitely many digits after the
decimal point without a periodic pattern, unlike the case with
22/7=3,142857142857142857... and all other fractions of
integers. Throughout the ages it was considered as a sport to find
more and more digits after the decimal point of such numbers. Around
the year 1600 Ludolph van Ceulen, after many years of work, calculated
Pi=3,1415926535897932384626433832795..., correct up to 31
digits. Around 1670 Newton could calculate that
Pi=3,1415926535897928... where he could use the help of his just
developed new mathematics (differential and integral calculus). For
that, Newton needed only one afternoon. However, he did make a mistake
in the last two decimals `28' (it should be:`32'), while van Ceulen
had verified all his obtained decimals. Later, in the 20th century
computers have accurately calculated Pi up to many billions of digits.
Also in the 21st century this sport is still being practised, not with
a chosen number such as Pi but with arbitrary mathematical
expressions. The international competition between the computer
programs that was held in Nijmegen in 3 and 4 October, was organised
by Milad Niqui and Freek Wiedijk and its aim was to correctly
calculate a number of mathematical expressions in 1,000,000 and more
digits as fast as possible. In the basic problems there were
expressions such as sin(tan(cos 1)) and in the intermediate problems
x_1000000, where x_0=0,5 and x_{n+1}= 3,999 x_n(1-x_n).
There is the question why this sport is of any
use. There is a good mathematical tradition that prevents the rounding
errors in the accumulated calculations from getting out of hand. In
order to calculate a number in 14 digits after the decimal point,
sometimes one needs more and more digits for the intermediate
results. This is related to the well-known butterfly in Brazil from
Chaos Theory, which by moving his wings can cause a storm in
Europe. It is actually possible to work with an arbitrary precision
where the number of decimal digits can be freely chosen. In principal,
this can cause a compromise in the speed. The competition has shown
that for calculating a typical set of mathematical expressions one
doesn't have to sacrifice precision for speed. For the problems and
results see .