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#################################################################################
# This script requires Math::GMPq, Math::GMPz, and Math::MPFR.                  #
# It calculates the euler number e (2.7182818...), correct to $ARGV[0] bits.    #
# The calculated value is displayed unless $ARGV[1] is both provided and false. #
# With each iteration of the for{} loop (below) we get closer and closer to     #
# the actual value of e. Furthermore, with successive iterations of the for{}   #
# loop, the values alternate between "less than e" and "greater than e".        #
# Hence the actual (irrational) value of e is always between the values         #
# calculated by successive iterations of the for{} loop.                        #
#                                                                               #
# Of course, the simplest and most efficient way to get the value of e, to      #
# $ARGV[0] bits is simply to do:                                                #
#   Rmpfr_exp($rop, Math::MPFR->new(1), MPFR_RNDN)                              #
# where $rop is a $ARGV[0]-bit precision Math::MPFR object.                     #
# But doing it that way is a bit less interesting.                              #
#                                                                               #
# The same for{} loop can be also used to calculate the exact probabilities of  #
# "winning" at a simplistic solitaire-type card game. See demos/solitaire.p     #
# in the Math::GMPz source distro.                                              #
#################################################################################

use strict;
use warnings;
use Math::GMPz qw(:mpz);
use Math::GMPq qw(:mpq);
use Math::MPFR qw(:mpfr);

die "Usage: perl euler.pl bits [True|False]" unless @ARGV;
my $bits = shift;
Rmpfr_set_default_prec($bits);

my $display_value;
$display_value = defined($ARGV[0]) ? shift : 1;

#################################################################################
# For the sanity checks (below), set $e_big_p to e, correct to $bits+100 bits.  #
# Then convert $e_big_p exactly to a rational, $e_q (a Math::GMPq object).      #
#################################################################################
my $e_q = Math::GMPq->new();                                                    #
my $e_big_p = Rmpfr_init2($bits + 100);                                         #
Rmpfr_exp($e_big_p, Math::MPFR->new(1), MPFR_RNDN);                             #
Rmpfr_get_q($e_q, $e_big_p);                                                    #
#################################################################################

#################################################################
# Create some variables, and assign some initial values         #
#################################################################
my $first = Math::GMPz->new(1);                                 #
my $second = Math::GMPz->new(0);                                #
my $current_items = 2;                                          #
my $factorial = Math::GMPz->new(1);                             #
my $e_check = Math::GMPq->new();                                #
my ($e, $e_first_fr, $e_second_fr) = (Math::MPFR->new(),        #
                                      Math::MPFR->new(),        #
                                      Math::MPFR->new(),        #
                                     );                         #
my $chance; # becomes a MATH::GMPz object on assignment         #
my $e_first = Math::GMPq->new(3);                               #
my $e_second = Math::GMPq->new();                               #
my $count = 0;                                                  #
my $t = Math::GMPq->new();                                      #
my $save = Math::GMPq->new(4);                                  #
#################################################################

#################################################################
# Set $e to a $bits-bit approximation of the euler number,      #
# rounded to nearest.                                           #
# This should exactly equal the number that we calculate.       #
#################################################################