Hi all,

I have been playing with my implementation or the Prime class since the my 
previous thread where Antonio Cangiano provided the new and improved Ruby 
1.9 version of mathn.rb.  Once I understood the new version, I found it to 
be very close to the base version of my implentation.  I did find a simple 
way to make it significantly (over 8 times) faster.  The Prime#succ method 
uses the following line to check a number, and add it to the list if it is a 
prime number.

@@primes.push @@next_to_check if @@primes[2..@@ulticheck_index].find 
{|prime| @@next_to_check % prime == 0 }.nil?

I noticed that that creates a new sub array @@primes[2..@@ulticheck_index] 
every time.  Simply extracting that out to another class variable array that 
gets one entry added to it every time ulticheck_index is incremented dropped 
the time to get 100,000 prime numbers from 377 seconds to 46.3 seconds.  I 
have made a few other adjustments to implement a method of skipping more 
non-prime numbers without having to test each case.  The existing Ruby 1.9 
version only checks 2 out of 6 numbers (1 and 5 for N modulo 6, step size of 
4 and 2).  I have it setup to that it skips more numbers using higher 
modulos (6 = 2*3 = first 2 primes; I use more primes to get the product 
value).

As is, the code rebuilds the list of step sizes when the mathn.rb file is 
initially required.  That introduces a startup penalty that becomes 
noticable in the benchmarks when 5 primes are used.  At 7 primes it becomes 
significant (4 seconds).  Even with that overhead included, the 7 primes 
case is faster at getting 100,000 primes than any of the others I have tried 
(I did not try 8 primes).

The step size calculation code could be moved to the initialize method, to 
delay that startup cost until the first prime is calculated.  For lists 
based on products of less than about 6 prime numbers, the calculation can be 
replaced by hard-coded lists.  The product for 5 primes is 2310, and the 
list of step sizes is 480 elements long.  For 6 primes the product is 30030, 
and the list of step sizes jumps to 5760 elements.

Perhaps someone will see a more efficient way of creating that step size 
list.  I did not spend too much time on it, once I found a fairly clean 
'Ruby' way of building it.  It does not seem to be worth it to make the list 
much bigger.  The improvements I get by increasing it are fairly small. 
Going from the initial adjustment that got to 46.3 seconds, to a version 
that uses an array of 2 step values increased the time to 48.4 seconds. 
Increasing the number of primes used in the product to 3,4,5,6,7 changed the 
time to 46.6, 44.8,  44.1, 43.0, 42,4 seconds.  All times are in seconds to 
get 100,008 prime numbers on my 1Ghz Windows XP Pro with 512M.  All times 
include the require line to load the Prime class (not the full mathn).  More 
improvement is possible if the step array can be created more efficiently. 
That final 42.4 second value includes almost 4 seconds of overhead to build 
the array.

Here are the times to get the first 50 prime numbers :: mostly the time for 
the require, which includes the step size array initialization.  version 2b 
uses 2 primes to calculate the product, then calculates the step sizes.  3b, 
4b, 5b, 6b, 7b each used one more prime number in the product.
Starting Prime number search benchmark script for mathn_1p9_2b
                                        user     system      total 
real
mathn_1p9_2b:      1 to     50      0.016000   0.000000   0.016000 ( 
0.016000)
mathn_1p9_3b:      1 to     50      0.000000   0.016000   0.016000 ( 
0.016000)
mathn_1p9_4b:      1 to     50      0.000000   0.016000   0.016000 ( 
0.015000)
mathn_1p9_5b:      1 to     50      0.016000   0.016000   0.032000 ( 
0.032000)
mathn_1p9_6b:      1 to     50      0.219000   0.015000   0.234000 ( 
0.234000)
mathn_1p9_7b:      1 to     50      3.906000   0.032000   3.938000 ( 
3.938000)

Here are the changes to the Prime class.  This shows the version that uses 7 
prime numbers in the product, but all that needs to change for the other 
cases, is to change the initial list of prime numbers.  Everything else is 
calculated from that.  I only made changes to the class variables and the 
succ method.  Unchanged methods not included here.

#
#   Modified Prime class from Ruby 1.9 mathn.rb -
#   by H. Phil Duby (PHrienDly Computer Consulting, Ltd.)
#

class Prime
  include Enumerable
  # These are included as class variables to cache them for later uses.  If 
memory
  #   usage is a problem, they can be put in Prime#initialize as instance 
variables.

  @@primes = [2, 3, 5, 7, 11, 13, 17]
  @@ulticheck_index = @@primes.size - 1
  @@ulticheck_next_squared = @@primes[@@ulticheck_index] ** 2
  @@root_primes = []

  prd = @@primes.inject( 1 ) { |p, n| p * n } # product of first few prime 
numbers
  prime_filter = [ * @@primes.last .. prd + 1 ] # list of numbers modulo prd 
that
        # could be prime numbers
  @@primes.each { |p| prime_filter.delete_if { |n| n % p == 0 }} #remove all 
of
        # numbers in the initial filter that are multiples of the prime 
numbers that
        # are factors of the calculated product.
  accum = 1; @@skip_known = prime_filter.map { |n| ( accum, x = n, n - 
accum ).last }
        # create step sizes to skip values that are known to be multiples of 
the factors
        # of the product of the first few prime numbers.  This starts from 
prd * <n> + 1
        # The sum of the step sizes equals prd, so the steps can be used 
continiously
        # in a circular manner.
  # The above steps to calculate @@skip_known can be replaced by hard-code 
initilization
  # of the step values.  Here are the values to use that match the first few 
products.
=begin
  @@skip_known = [ 2 ]                      # one prime, product = 2
  @@skip_known = [ 4, 2 ]                   # two primes, product = 6
  @@skip_known = [ 6, 4, 2, 4, 2, 4, 6, 2 ] # three primes, product = 30
  @@skip_known = [ 10, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 
6, 8,
    4, 2, 4, 2, 4, 8, 6, 4, 6, 2, 4, 6, 2, 6, 6, 4, 2, 4, 6, 2, 6, 4, 2, 4, 
2, 10, 2 ]
    #                                       # four primes, product = 210
=end
  class << @@skip_known #provide method for getting next step size in 
infinite circle.
    @@skip_i            = @@skip_known.size - 1
    @@skip_n            = [ * 1 .. @@skip_i ] << 0
    def next # sequence through the offsets needed to skip the known prime 
multiples
      return self.at( @@skip_i = @@skip_n.at( @@skip_i )) #circular indexing
    end #def next
  end #class << @@skip_known
  @@next_to_check = 1 #+ @@skip_known.next will get to first possible prime 
after
                      # @@primes[@@ulticheck_index]

  def succ
    @index += 1
    while @index >= @@primes.length
      # Only check for prime factors up to the square root of the potential 
primes,
      #   but without the performance hit of an actual square root 
calculation.
      @@next_to_check  += @@skip_known.next
      if @@next_to_check >= @@ulticheck_next_squared
        @@ulticheck_index += 1
        @@root_primes.push @@primes.at(@@ulticheck_index)
        @@ulticheck_next_squared = @@primes.at(@@ulticheck_index + 1) ** 2
      end
      @@primes.push @@next_to_check if @@root_primes.find {|prime| 
@@next_to_check % prime == 0 }.nil?
    end
    return @@primes[@index]
  end
  alias next succ
end

-- 
Phil
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