#!/usr/bin/perl -w
use strict;

#
# Find an approximate value for Pi using the Pythagorean theorem
#

#
# Start with circle inscribed in polygon
#

my @circle_in_gon_pi;

sub b_from_a ($) {
    my ($a) = @_;

    my $b = sqrt((1.0/($a*$a))+1.0) - (1.0/$a);
    return $b;
}

print "**************** Circle inscribed in polygon ****************\n";

my $start_b = 1.0;
my $sides = 4;

my $b = $start_b;
foreach my $i (1..20) {
    my $pi_approx = $b * $sides;
    push @circle_in_gon_pi, $pi_approx;
    print "Sides: $sides; side length/2: $b; pi: $pi_approx\n";
    $b = b_from_a ($b);
    $sides *= 2;
}

print "**************** Polygon inscribed in circle ****************\n";

my @gon_in_circle_pi;

sub b_from_a_inside ($) {
    my ($a) = @_;

    my $b = sqrt((1.0 - sqrt(1.0 - ($a*$a)))/2.0);
    return $b;
}

$start_b = sqrt(2.0)/2.0;
$sides = 4;

$b = $start_b;
foreach my $i (1..20) {
    my $pi_approx = $b * $sides;
    push @gon_in_circle_pi, $pi_approx;
    print "Sides: $sides; side length/2: $b; pi: $pi_approx\n";
    $b = b_from_a_inside ($b);
    $sides *= 2;
}

print "**************** Averages and bounds at each step ****************\n";
print "Format:  sides, interior estimate, exterior estimate, average of them\n";
print "Each entry is formatted as a table line.\n";

my $low = 0.0;
my $high = 5.0;
my $best = 0;

$sides = 4;
foreach my $i (0..20) {
    my $new_high = $circle_in_gon_pi[$i];
    my $new_low = $gon_in_circle_pi[$i];
    my $out_of_range = 0;
    if ($new_low < $low
	|| $new_low > $high) {
	print "Low value for step $i out of range: $new_low\n";
	$out_of_range = 1;
    }

    if ($new_high < $low
	|| $new_high > $high) {
	print "High value for step $i out of range: $new_high\n";
	$out_of_range = 1;
    }

    my $average = ($new_low + $new_high) / 2.0;
    print "<tr>\n";
    print "<td>$sides</td> <td>$new_low</td> <td>$new_high</td> <td>$average</td>\n";
    print "</tr>\n";
    ##print "Step: $i; Sides: $sides; Low: $new_low; High: $new_high; Average: $average\n";
    if ($out_of_range) { last; }
    $best = $average;
    $low = $new_low;
    $high = $new_high;
    $sides *= 2;
}

my $pi = 4.0 * atan2(1,1);
print "pi, by taking 4*arctan(1): $pi\n";
my $error = $best - $pi;
if ($error < 0) { $error *= -1.0; }
print "    Error:  $error\n";