primesieve

Fast C/C++ prime number generator

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About

primesieve is a free software program and C/C++ library that generates primes using a highly optimized sieve of Eratosthenes implementation. It counts the primes below 1010 in just 0.45 seconds on an Intel Core i7-6700 CPU (4 x 3.4GHz). primesieve can generate primes and prime k-tuplets up to 264.

primesieve Mac OS X application

Speed comparison

primesieve generates primes about 50 times faster (single-threaded) than an ordinary C/C++ sieve of Eratosthenes implementation and about 10,000 times faster than trial-division. primesieve outperforms the author's older ecprime (fastest open source prime number generator from 2002 to 2010) by about 30 percent and also substantially outperforms primegen the fastest sieve of Atkin implementation on the web. Here is a historical list of fast prime sieve implementations.

Timings

x Prime Count Intel Core i7-6700
(4 x 3.4GHz, 256 KB L2 cache)
AMD Ryzen 7 1700
(8 x 3.0GHz, 512 KB L2 cache)
IBM POWER8
(22 x 3.0GHz, 512 KB L2 cache)
108 5,761,455 0.01s 0.01s 0.01s
109 50,847,534 0.04s 0.03s 0.07s
232 203,280,221 0.18s 0.11s 0.15s
1010 455,052,511 0.44s 0.27s 0.24s
1011 4,118,054,813 5.06s 3.00s 1.72s
1012 37,607,912,018 61.03s 35.20s 19.34s
1013 346,065,536,839 753.21s 413.91s 225.92s


The above benchmarks were run on Linux, for each benchmark primesieve used all available CPU cores and the sieve size was set to the CPU's L2 cache size.

CPU scaling

primesieve CPU scaling

primesieve scales linearly up to a very large number of CPU cores (if the interval is sufficiently large). The above benchmark was run on a system with 8 physical CPU cores, at each start offset the primes inside an interval of size 1011 were counted using different numbers of threads.

Installing primesieve

The primesieve console application can be installed using your operating system's package manager. The primesieve GUI application can be downloaded from http://primesieve.org/downloads.

•••
# Debian/Ubuntu
sudo apt-get install primesieve

# Windows Bash
sudo apt-get install primesieve

# macOS
brew install primesieve

Usage examples

The primesieve console application can generate primes and prime k-tuplets and find the nth prime.

•••
# Count the primes below 1e10 using all CPU cores
primesieve 1e10

# Print the primes below 1000000
primesieve 1000000 --print

# Count the primes within [1e10, 2e10] using 4 threads
primesieve 1e10 2e10 --threads=4

# Print an option summary
primesieve --help

Algorithms

primesieve generates primes using the segmented sieve of Eratosthenes with wheel factorization, this algorithm has a complexity of O(n log log n) operations and uses O(sqrt(n)) space.

Segmentation is currently the best known practical improvement to the sieve of Eratosthenes. Instead of sieving the interval [2, n] at once one subdivides the sieve interval into a number of equal sized segments that are then sieved consecutively. Segmentation drops the memory requirement of the sieve of Eratosthenes from O(n) to O(sqrt(n)). The segment size is usually chosen to fit into the CPU's fast L1 or L2 cache memory which significantly speeds up sieving. A segmented version of the sieve of Eratosthenes was first published by Singleton in 1969 [1]. Here is a simple implementation of the segmented sieve of Eratosthenes.

Wheel factorization is used to skip multiples of small primes. If a kth wheel is added to the sieve of Eratosthenes then only those multiples are crossed off that are coprime to the first k primes, i.e. multiples that are divisible by any of the first k primes are skipped. The 1st wheel considers only odd numbers, the 2nd wheel (modulo 6) skips multiples of 2 and 3, the 3rd wheel (modulo 30) skips multiples of 2, 3, 5 and so on. Pritchard has shown in [2] that the running time of the sieve of Eratosthenes can be reduced by a factor of log log n if the wheel size is sqrt(n) but for cache reasons the sieve of Eratosthenes usually performs best with a modulo 30 or 210 wheel. Sorenson explains wheels in [3].

Additionally primesieve uses Tomás Oliveira e Silva's cache-friendly bucket list algorithm if needed [4]. This algorithm is relatively new, it has been devised by Tomás Oliveira e Silva in 2001 in order to speed up the segmented sieve of Eratosthenes for prime numbers past 32 bits. The idea is to store the sieving primes into lists of buckets with each list being associated with a segment. A list of sieving primes related to a specific segment contains only those primes that have multiple occurrence(s) in that segment. Whilst sieving a segment only the primes of the related list are used for sieving and each prime is reassigned to the list responsible for its next multiple when processed. The benefit of this approach is that it is now possible to use segments (i.e. sieve arrays) smaller than sqrt(n) without deteriorating efficiency, this is important as only small segments that fit into the CPU's L1 or L2 cache provide fast memory access.

Implementation

primesieve is written entirely in C++ and does not depend on external libraries. It's speed is mainly due to the segmentation of the sieve of Eratosthenes which prevents cache misses when crossing off multiples in the sieve array and the use of a bit array instead of a boolean sieve array. primesieve reuses and improves ideas from other great sieve of Eratosthenes implementations, namely Achim Flammenkamp's prime_sieve.c, Tomás Oliveira e Silva's A1 implementation and the author's older ecprime all written in the late '90s and '00s. Furthermore primesieve contains new optimizations to increase instruction-level parallelism and more efficiently uses the larger number of registers in today's CPUs.

Optimizations used in primesieve

  • Uses a bit array with 8 flags each 30 numbers for sieving
  • Pre-sieves multiples of small primes ≤ 19
  • Compresses the sieving primes in order to improve cache efficiency [5]
  • Starts crossing off multiples at the square
  • Uses a modolo 210 wheel that skips multiples of 2, 3, 5 and 7
  • Uses specialized algorithms for small, medium and big sieving primes
  • Uses a custom memory pool (for big sieving primes)
  • Processes multiple sieving primes per loop iteration to increase instruction-level parallelism
  • Multi-threaded using C++11 std::async

Highly optimized inner loop

primesieve's inner sieving loop has been optimized using extreme loop unrolling, on average crossing off a multiple uses just 1.375 instructions on x64 CPUs. Below is the assembly GCC generates for primesieve's inner sieving loop, each andb instruction unsets a bit (crosses off a multiple) in the sieve array.

This README file contains more technical implementation details.

C/C++ library

Below is a C++ example that shows how to iterate over primes using libprimesieve.

libprimesieve natively supports C and C++. For generating an array (or vector) of primes and iterating over primes < 264 libprimesieve is easily the fastest library on the web!

Bindings for other languages

primesieve natively supports C and C++ and has bindings available for:

Python:
Perl:
Ruby:
Rust:
Haskell:

Many thanks to the developers of these bindings!

References and notes

  1. R. C. Singleton, "An efficient prime number generator", Communications of the ACM 12, 563-564, 1969.
  2. Paul Pritchard, "Fast compact prime number sieves (among others)", Journal of Algorithms 4 (1983), 332-344.
  3. Jonathan Sorenson, "An analysis of two prime number sieves", Computer Science Technical Report Vol. 1028, 1991.
  4. Tomás Oliveira e Silva, "Fast implementation of the segmented sieve of Eratosthenes", 2002.
  5. Actually not the sieving primes are compressed but their sieve and wheel indexes.