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Last active May 12, 2022
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# This Crystal source file is a multiple threaded implementation to perform an
# extremely fast Segmented Sieve of Zakiya (SSoZ) to find Twin Primes <= N.
# Inputs are single values N, or ranges N1 and N2, of 64-bits, 0 -- 2^64 - 1.
# Output is the number of twin primes <= N, or in range N1 to N2; the last
# twin prime value for the range; and the total time of execution.
# This code was developed on a System76 laptop with an Intel I7 6700HQ cpu,
# 2.6-3.5 GHz clock, with 8 threads, and 16GB of memory. Parameter tuning
# probably needed to optimize for other hardware systems (ARM, PowerPC, etc).
# Compile as: $ crystal build twinprimes_ssozgist.cr -Dpreview_mt --release
# To reduce binary size do: $ strip twinprimes_ssoz
# Thread workers default to 4, set to system max for optimum performance.
# Single val: $ CRYSTAL_WORKERS=8 ./twinprimes_ssoz val1
# Range vals: $ CRYSTAL_WORKERS=8 ./twinprimes_ssoz val1 val2
# Mathematical and technical basis for implementation are explained here:
# https://www.academia.edu/37952623/The_Use_of_Prime_Generators_to_Implement_Fast_
# Twin_Primes_Sieve_of_Zakiya_SoZ_Applications_to_Number_Theory_and_Implications_
# for_the_Riemann_Hypotheses
# https://www.academia.edu/7583194/The_Segmented_Sieve_of_Zakiya_SSoZ_
# https://www.academia.edu/19786419/PRIMES-UTILS_HANDBOOK
# This source code, and its updates, can be found here:
# https://gist.github.com/jzakiya/2b65b609f091dcbb6f792f16c63a8ac4
# This code is provided free and subject to copyright and terms of the
# GNU General Public License Version 3, GPLv3, or greater.
# License copy/terms are here: http://www.gnu.org/licenses/
# Copyright (c) 2017-2022; Jabari Zakiya -- jzakiya at gmail dot com
# Last update: 2022/04/16
# Modified on 2022/05/12 by Benjamin Wade
# Customized gcd for prime generators; n > m; m odd
def gcd(m, n)
while m | 1 != 1
t = m; m = n % m; n = t
end
m
end
# Compute modular inverse a^-1 to base m, e.g. a*(a^-1) mod m = 1
def modinv(a0, m0)
return 1 if m0 == 1
a, m = a0, m0
x0, inv = 0, 1
while a > 1
inv -= (a // m) * x0
a, m = m, a % m
x0, inv = inv, x0
end
inv += m0 if inv < 0
inv
end
def gen_pg_parameters(prime)
# Create prime generator parameters for given Pn
puts "using Prime Generator parameters for P#{prime}"
primes = [2, 3, 5, 7, 11, 13, 17, 19, 23]
modpg, res_0 = 1, 0 # compute Pn's modulus and res_0 value
primes.each { |prm| res_0 = prm; break if prm > prime; modpg *= prm }
restwins = [] of Int32 # save upper twinpair residues here
inverses = Array.new(modpg + 2, 0) # save Pn's residues inverses here
pc, inc, res = 5, 2, 0 # use P3's PGS to generate pcs
while pc < (modpg >> 1) # find PG's 1st half residues
if gcd(pc, modpg) == 1 # if pc a residue
mc = modpg - pc # create its modular complement
inverses[pc] = modinv(pc, modpg) # save pc and mc inverses
inverses[mc] = modinv(mc, modpg) # if in twinpair save both hi residues
restwins << pc << mc + 2 if res + 2 == pc
res = pc # save current found residue
end
pc += inc; inc ^= 0b110 # create next P3 sequence pc: 5 7 11 13 17 19 ...
end
restwins.sort!; restwins << (modpg + 1) # last residue is last hi_tp
inverses[modpg + 1] = 1; inverses[modpg - 1] = modpg - 1 # last 2 residues are self inverses
{modpg, res_0, restwins.size, restwins, inverses}
end
def set_sieve_parameters(start_num, end_num)
# Select at runtime best PG and segment size parameters for input values.
# These are good estimates derived from PG data profiling. Can be improved.
nrange = end_num - start_num
bn, pg = 0, 3
if end_num < 49
bn = 1; pg = 3
elsif nrange < 77_000_000
bn = 16; pg = 5
elsif nrange < 1_100_000_000
bn = 32; pg = 7
elsif nrange < 35_500_000_000
bn = 64; pg = 11
elsif nrange < 14_000_000_000_000
pg = 13
if nrange > 7_000_000_000_000
bn = 384
elsif nrange > 2_500_000_000_000
bn = 320
elsif nrange > 250_000_000_000
bn = 196
else
bn = 128
end
else
bn = 384; pg = 17
end
modpg, res_0, pairscnt, restwins, resinvrs = gen_pg_parameters(pg)
kmin = (start_num - 2) // modpg + 1 # number of resgroups to start_num
kmax = (end_num - 2) // modpg + 1 # number of resgroups to end_num
krange = kmax - kmin + 1 # number of resgroups in range, at least 1
n = krange < 37_500_000_000_000 ? 4 : (krange < 975_000_000_000_000 ? 6 : 8)
b = bn * 1024 * n # set seg size to optimize for selected PG
ks = krange < b ? krange : b # segments resgroups size
puts "segment size = #{ks} resgroups for seg bitarray"
maxpairs = krange * pairscnt # maximum number of twinprime pcs
puts "twinprime candidates = #{maxpairs}; resgroups = #{krange}"
{modpg, res_0, ks, kmin, kmax, krange, pairscnt, restwins, resinvrs}
end
def sozpg(val, res_0, start_num, end_num)
# Compute the primes r0..sqrt(input_num) and store in 'primes' array.
# Any algorithm (fast|small) is usable. Here the SoZ for P5 is used.
md, rscnt = 30u64, 8 # P5's modulus and residues count
res = [7, 11, 13, 17, 19, 23, 29, 31] # P5's residues
bitn = [0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 4, 0, 0, 0, 8, 0, 16, 0, 0, 0, 32, 0, 0, 0, 0, 0, 64, 0, 128]
kmax = (val - 2) // md + 1 # number of resgroups upto input value
prms = Array(UInt8).new(kmax, 0) # byte array of prime candidates, init '0'
modk, r, k = 0, -1, 0 # initialize residue parameters
loop do # for r0..sqrtN primes mark their multiples
if (r += 1) == rscnt
r = 0; modk += md; k += 1
end # resgroup parameters
next if prms[k] & (1 << r) != 0 # skip pc if not prime
prm_r = res[r] # if prime save its residue value
prime = modk + prm_r # numerate the prime value
break if prime > Math.isqrt(val) # exit loop when it's > sqrtN
res.each do |ri| # mark prime's multiples in prms
kn, rn = (prm_r * ri - 2).divmod md # cross-product resgroup|residue
bit_r = bitn[rn] # bit mask for prod's residue
kpm = k * (prime + ri) + kn # resgroup for 1st prime mult
while kpm < kmax
prms[kpm] |= bit_r; kpm += prime
end
end
end
# prms now contains the nonprime positions for the prime candidates r0..N
# extract only primes that are in inputs range into array 'primes'
primes = [] of UInt64 # create empty dynamic array for primes
prms.each_with_index do |resgroup, k| # for each kth residue group
res.each_with_index do |r_i, i| # check for each ith residue in resgroup
if resgroup & (1 << i) == 0 # if bit location a prime
prime = md * k + r_i # numerate its value, store if in range
# check if prime has multiple in range, if so keep it, if not don't
n, rem = start_num.divmod prime # if rem 0 then start_num is multiple of prime
primes << prime if (res_0 <= prime <= val) && (prime * (n + 1) <= end_num || rem == 0)
end
end
end
primes
end
def nextp_init(rhi, kmin, modpg, primes, resinvrs)
# Initialize 'nextp' array for twinpair upper residue rhi in 'restwins'.
# Compute 1st prime multiple resgroups for each prime r0..sqrt(N) and
# store consecutively as lo_tp|hi_tp pairs for their restracks.
nextp = Slice(UInt64).new(primes.size*2) # 1st mults array for twinpair
r_hi, r_lo = rhi, rhi - 2 # upper|lower twinpair residue values
primes.each_with_index do |prime, j| # for each prime r0..sqrt(N)
k = (prime - 2) // modpg # find the resgroup it's in
r = (prime - 2) % modpg + 2 # and its residue value
r_inv = resinvrs[r].to_u64 # and residue inverse
rl = (r_inv * r_lo - 2) % modpg + 2 # compute r's ri for r_lo
rh = (r_inv * r_hi - 2) % modpg + 2 # compute r's ri for r_hi
kl = k * (prime + rl) + (r * rl - 2) // modpg # kl 1st mult resgroup
kh = k * (prime + rh) + (r * rh - 2) // modpg # kh 1st mult resgroup
kl < kmin ? (kl = (kmin - kl) % prime; kl = prime - kl if kl > 0) : (kl -= kmin)
kh < kmin ? (kh = (kmin - kh) % prime; kh = prime - kh if kh > 0) : (kh -= kmin)
nextp[j * 2] = kl.to_u64 # prime's 1st mult lo_tp resgroup val in range
nextp[j * 2 | 1] = kh.to_u64 # prime's 1st mult hi_tp resgroup val in range
end
nextp
end
def twins_sieve(r_hi, kmin, kmax, ks, start_num, end_num, modpg, primes, resinvrs)
# Perform in thread the ssoz for given twinpair residues for kmax resgroups.
# First create|init 'nextp' array of 1st prime mults for given twinpair,
# stored consequtively in 'nextp', and init seg array for ks resgroups.
# For sieve, mark resgroup bits to '1' if either twinpair restrack is nonprime
# for primes mults resgroups, and update 'nextp' restrack slices acccordingly.
# Return the last twinprime|sum for the range for this twinpair residues.
s = 6 # shift value for 64 bits
bmask = (1 << s) - 1 # bitmask val for 64 bits
sum, ki, kn = 0_u64, kmin - 1, ks # init these parameters
hi_tp, k_max = 0_u64, kmax # max twinprime|resgroup
seg = Slice(UInt64).new(((ks - 1) >> s) + 1) # seg array of ks resgroups
ki += 1 if ((ki * modpg) + r_hi - 2) < start_num # ensure lo tp in range
k_max -= 1 if ((k_max - 1) * modpg + r_hi) > end_num # ensure hi tp in range
nextp = nextp_init(r_hi, ki, modpg, primes, resinvrs) # init nextp array
while ki < k_max # for ks size slices upto kmax
kn = k_max - ki if ks > (k_max - ki) # adjust kn size for last seg
primes.each_with_index do |prime, j| # for each prime r0..sqrt(N)
# for lower twinpair residue track
k = nextp.to_unsafe[j * 2] # starting from this resgroup in seg
while k < kn # mark primenth resgroup bits prime mults
seg.to_unsafe[k >> s] |= 1_u64 << (k & bmask)
k += prime
end # set resgroup for prime's next multiple
nextp.to_unsafe[j * 2] = k - kn # save 1st resgroup in next eligible seg
# for upper twinpair residue track
k = nextp.to_unsafe[j * 2 | 1] # starting from this resgroup in seg
while k < kn # mark primenth resgroup bits prime mults
seg.to_unsafe[k >> s] |= 1_u64 << (k & bmask)
k += prime
end # set resgroup for prime's next multiple
nextp.to_unsafe[j * 2 | 1] = k - kn # save 1st resgroup in next eligible seg
end # set as nonprime unused bits in last seg[n]
# so fast, do for every seg[i]
seg.to_unsafe[(kn - 1) >> s] |= ~1u64 << ((kn - 1) & bmask)
cnt = 0 # count the twinprimes in the segment
seg[0..(kn - 1) >> s].each { |m| cnt += (~m).popcount }
if cnt > 0 # if segment has twinprimes
sum += cnt # add segment count to total range count
upk = kn - 1 # from end of seg, count back to largest tp
while seg.to_unsafe[upk >> s] & (1_u64 << (upk & bmask)) != 0
upk -= 1
end
hi_tp = ki + upk # set its full range resgroup value
end
ki += ks # set 1st resgroup val of next seg slice
seg.fill(0) if ki < k_max # set next seg to all primes if in range
end # when sieve done, numerate largest twin
# for ranges w/o twins set largest to 1
hi_tp = (r_hi > end_num || sum == 0) ? 1 : hi_tp * modpg + r_hi
{hi_tp.to_u64, sum.to_u64} # return largest twinprime|twins count
end
def twinprimes_ssoz
end_num = {UInt64.new(ARGV[0], underscore: true), 3u64}.max
start_num = ARGV.size > 1 ? {UInt64.new(ARGV[1], underscore: true), 3u64}.max : 3u64
start_num, end_num = end_num, start_num if start_num > end_num
start_num = end_num = 7 if end_num - start_num < 2
puts "threads = #{System.cpu_count}"
ts = Time.monotonic # start timing sieve setup execution
start_num |= 1 # if start_num even increase by 1
end_num = (end_num - 1) | 1 # if end_num even decrease by 1
# select Pn, set sieving params for inputs
modpg, res_0, ks, kmin, kmax, krange, pairscnt, restwins, resinvrs = set_sieve_parameters(start_num, end_num)
# create sieve primes <= sqrt(end_num), only use those whose multiples within inputs range
primes = end_num < 49 ? [5] : sozpg(Math.isqrt(end_num), res_0, start_num, end_num)
puts "each of #{pairscnt} threads has nextp[2 x #{primes.size}] array"
lo_range = restwins[0] - 3 # lo_range = lo_tp - 1
twinscnt = 0_u64 # determine count of 1st 4 twins if in range for used Pn
twinscnt += [3, 5, 11, 17].select { |tp| start_num <= tp <= lo_range }.size unless end_num == 3
te = (Time.monotonic - ts).total_seconds.round(6)
puts "setup time = #{te} secs" # display sieve setup time
puts "perform twinprimes ssoz sieve"
t1 = Time.monotonic # start timing ssoz sieve execution
case ENV["PARALLEL_IMPL"].downcase
when "original"
puts "Using the original implementation of parallelization"
cnts, lastwins = parallel_sieves_original(pairscnt, restwins, kmin, kmax, ks, start_num, end_num, modpg, primes, resinvrs)
when "yield"
puts "Using the 'yield' implementation of parallelization"
cnts, lastwins = parallel_sieves_yield(pairscnt, restwins, kmin, kmax, ks, start_num, end_num, modpg, primes, resinvrs)
when "workers"
puts "Using the 'workers' implementation of parallelization"
cnts, lastwins = parallel_sieves_workers(pairscnt, restwins, kmin, kmax, ks, start_num, end_num, modpg, primes, resinvrs)
else
puts "Using the default (original) implementation of parallelization"
cnts, lastwins = parallel_sieves_original(pairscnt, restwins, kmin, kmax, ks, start_num, end_num, modpg, primes, resinvrs)
end
last_twin = lastwins.max # find largest hi_tp twinprime in range
twinscnt += cnts.sum # compute number of twinprimes in range
last_twin = 5 if end_num == 5 && twinscnt == 1
kn = krange % ks # set number of resgroups in last slice
kn = ks if kn == 0 # if multiple of seg size set to seg size
t2 = (Time.monotonic - t1).total_seconds # sieve execution time
puts "\nsieve time = #{t2.round(6)} secs" # ssoz sieve time
puts "total time = #{(t2 + te).round(6)} secs" # setup + sieve time
puts "last segment = #{kn} resgroups; segment slices = #{(krange - 1)//ks + 1}"
puts "total twins = #{twinscnt}; last twin = #{last_twin - 1}+/-1"
end
def parallel_sieves_original(pairscnt, restwins, kmin, kmax, ks, start_num, end_num, modpg, primes, resinvrs)
cnts = Array(UInt64).new(pairscnt, 0) # number of twinprimes found per thread
lastwins = Array(UInt64).new(pairscnt, 0) # largest twinprime val for each thread
done = Channel(Nil).new(pairscnt)
threadscnt = Atomic.new(0) # count of finished threads
restwins.each_with_index do |r_hi, i| # sieve twinpair restracks
spawn do
lastwins[i], cnts[i] = twins_sieve(r_hi, kmin, kmax, ks, start_num, end_num, modpg, primes, resinvrs)
print "\r#{threadscnt.add(1)} of #{pairscnt} twinpairs done"
done.send(nil)
end
end
pairscnt.times { done.receive } # wait for all threads to finish
print "\r#{pairscnt} of #{pairscnt} twinpairs done"
{cnts, lastwins}
end
def parallel_sieves_yield(pairscnt, restwins, kmin, kmax, ks, start_num, end_num, modpg, primes, resinvrs)
cnts = Array(UInt64).new(pairscnt, 0) # number of twinprimes found per thread
lastwins = Array(UInt64).new(pairscnt, 0) # largest twinprime val for each thread
sieve_outputs = Channel(Tuple(Int32, UInt64, UInt64)).new(pairscnt)
done = Channel(Nil).new
spawn do
finished_sieves = 0
until finished_sieves == pairscnt
sieve_index, largest_twinprime, twinprime_count = sieve_outputs.receive
cnts[sieve_index] = twinprime_count
lastwins[sieve_index] = largest_twinprime
finished_sieves += 1
print "\r#{finished_sieves} of #{pairscnt} twinpairs done"
end
done.send(nil)
end
restwins.each_with_index do |r_hi, i| # sieve twinpair restracks
spawn do
sieve_index = i
largest_twinprime, twinprime_count = twins_sieve(r_hi, kmin, kmax, ks, start_num, end_num, modpg, primes, resinvrs)
sieve_outputs.send({sieve_index, largest_twinprime, twinprime_count})
end
Fiber.yield # this is necessary, but it's not even a guarantee that our status fiber will be executed
end
done.receive
{cnts, lastwins}
end
def parallel_sieves_workers(pairscnt, restwins, kmin, kmax, ks, start_num, end_num, modpg, primes, resinvrs)
cnts = Array(UInt64).new(pairscnt, 0) # number of twinprimes found per thread
lastwins = Array(UInt64).new(pairscnt, 0) # largest twinprime val for each thread
sieve_outputs = Channel(Tuple(Int32, UInt64, UInt64)).new(pairscnt)
# start monitor fiber
done = Channel(Nil).new
spawn(same_thread: true) do
finished_sieves = 0
until finished_sieves == pairscnt
sieve_index, largest_twinprime, twinprime_count = sieve_outputs.receive
cnts[sieve_index] = twinprime_count
lastwins[sieve_index] = largest_twinprime
finished_sieves += 1
print "\r#{finished_sieves} of #{pairscnt} twinpairs done"
end
done.send(nil)
end
sieve_inputs = Channel(Tuple(Int32, Int32)).new(pairscnt)
# start worker fibers
worker_count = (ENV["CRYSTAL_WORKERS"].to_i? || 4) - 1 # we want to avoid putting any workers on this thread
worker_count.times do
spawn do
while (inputs = sieve_inputs.receive?)
r_hi, sieve_index = inputs
largest_twinprime, twinprime_count = twins_sieve(r_hi, kmin, kmax, ks, start_num, end_num, modpg, primes, resinvrs)
sieve_outputs.send({sieve_index, largest_twinprime, twinprime_count})
end
end
end
# send inputs
restwins.each_with_index do |r_hi, i| # sieve twinpair restracks
sieve_inputs.send({r_hi, i})
end
done.receive
sieve_inputs.close
{cnts, lastwins}
end
twinprimes_ssoz
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