#!/usr/bin/env perl

# Preexperiments to rational geometry

use strict;
use warnings;

use Math::Trig;

# generate rational approximations to nth parts of an arc represented by sin and cos
# approach leads to values in principal usable as first iteration, but has vastly to big numbers
sub nthArc {
  my $n = shift;
  # basis triangle with half the angle 
  my $prex = $n;
  my $prey = 1;
  dumpangles("pre", $prex, $prey, 45/$n);
  # complex multiplication, double the angle
  my $x = $prex*$prex - $prey*$prey;
  my $y = 2*$prex*$prey;
  dumpangles("square", $x, $y, 90/$n);
  my $npowx = 1;
  my $npowy = 0;
  for(1..$n) {
    my $nnpowx = $x*$npowx - $y*$npowy;
    my $nnpowy = $x*$npowy + $y*$npowx;
    $npowx = $nnpowx;
    $npowy = $nnpowy;
  }
  dumpangles("n-powered", $npowx, $npowy, 90);
}

sub dumpangles {
  my ($name, $x, $y, $compare) = @_;
  my $goal = "";
  $goal = " (goal: $compare)" if $compare;;
  my $phi = atan2($y,$x)/pi*180;
  print "[$name] $x $y -- $phi$goal\n";
}

# exp is
# sum_n=0..inf  x^n/n!

# 1/0! - x^2/2! + x^4/4! - x^6/6!
sub preCos {
  my ($steps, $x) = @_;
  my $mxsq = -$x*$x;
  my $sum = 0;
  my $fa = 1;
  for(1..$steps) {
    my $step2 = $_ * 2;
    $sum += $fa;
    $fa *= $mxsq/($step2*($step2-1))
  }
  return $sum
}

# x/1! - x^3/3! + x^5/5! - x^7/7!
sub preSin {
  my ($steps, $x) = @_;
  my $mxsq = -$x*$x;
  my $sum = 0;
  my $fa = $x;
  for(1..$steps) {
    my $step2 = $_ * 2;
    $sum += $fa;
    $fa *= $mxsq/($step2*($step2+1))
  }
  return $sum
}


while(<>) {
#  if(m#^([1-9][0-9]*)$#) {
#    nthArc($1)
#  } else {
#    print "Bad parameter\n";
#  }
  if(m#^([1-9][0-9]*)\s+([0-9\.]+)$#) {
    my ($st, $x) = ($1, $2);
    my $c = cos($x);
    my $cappr = preCos($st, $x);
    print "$x -- c:$c  cappr:$cappr\n";
    my $s = sin($x);
    my $sappr = preSin($st, $x);
    print "$x -- s:$s  sappr:$sappr\n";
  } elsif(m#pi\s+([0-9]+)\s+([0-9]+)#) {
    my ($bigit, $smallit) = ($1, $2);
    my $pi = 22/7;
    for(1..$bigit) {
      $pi = $pi + preSin($smallit, $pi);
    }
    print "piappr: $pi  -- truepu: ", pi, "\n"
  } else {
    print "Bad parameters\n";
  }
}