ChangeLog

Changes in primecount-6.5, 2021-04-02

In primecout-6.4 (and earlier) the AC formula scales very badly on
servers with a large number of CPU cores. The root cause of this
scaling issue is the SegmentedPiTable lookup table which is accessed
randomly and which is much too large to fit into the CPU's cache. In
primecount-6.5 the AC algorithm as been slightly modified to allow a
much smaller SegmentedPiTable without deteriorating the runtime
complexity of the algorithm.

* AC.cpp: Decrease SegmentedPiTable by a factor > log(x).
* AC_libdivide.cpp: Decrease SegmentedPiTable by a factor > log(x).
* SegmentedPiTable.cpp: Increase minimum segment size to 2 MiB.

Changes in primecount-6.4, 2021-03-20

This version fixes a critical integer overflow in the B formula
which is part of Xavier Gourdon's prime counting algorithm. The
integer overflow occurred when computing B(x) with x > 1e25, this bug
has been present in primecount-6.1, 6.2 and 6.3. This version also
adds a workaround for a severe scaling issue in Clang's OpenMP 
library (Clang 11, 2021) that occurs when using dynamic thread
scheduling combined with long running computations. Because of this
issue primecount-6.3 compiled with Clang takes 2.5x longer to
compute AC(1e24) than primecount-6.3 compiled with GCC on my
dual-socket AMD EPYC Rome server. I have submitted a bug report
to LLVM which contains more information:
https://bugs.llvm.org/show_bug.cgi?id=49588

* B.cpp: Fix integer overflow (thanks to David Baugh for reporting it)
* B_mpi.cpp: Fix integer overflow (same as in B.cpp).
* P2.cpp: Fix integer overflow (same as in B.cpp).
* for_atomic.hpp: Lock-free for loop built with std::atomic.
* AC.cpp: Fix Clang/OpenMP scaling issue.
* AC.cpp: Decrease size of SegmentedPiTable by 1.5x.
* AC_libdivide.cpp: Fix Clang/OpenMP scaling issue.
* AC_libdivide.cpp: Decrease size of SegmentedPiTable by 1.5x.
* S2_easy.cpp: Fix Clang/OpenMP scaling issue.
* S2_easy_libdivide.cpp: Fix Clang/OpenMP scaling issue.
* SegmentedPiTable.cpp: Increase minimum segment size.
* SegmentedPiTable.cpp: Annotate pi(x) with ALWAYS_INLINE.
* Sieve.cpp: Simplify counters_dist initialization.
* PiTable.cpp: Annotate pi(x) with ALWAYS_INLINE.
* LoadBalancerP2.cpp: Fix last iteration detection.

Changes in primecount-6.3, 2021-03-05

  The memory usage of the PiTable and SegmentedPiTable has been
  reduced by about 2x, this speeds up the computation of the easy
  special leaves (S2_easy & AC formulas) by up to 30% for large pi(x)
  computations with x >= 1e23. Furthermore the partial sieve function
  a.k.a. phi(x, a) has been improved significantly, it now runs 3 to
  5x faster for most input. phi(x, a) is used extensively by many
  other functions such as pi_legendre(x) and pi_meissel(x) which now
  also run up to 5x faster.

  * PiTable.cpp: Reduce memory usage by 2x.
  * SegmentedPiTable.cpp: Reduce memory usage by 2x.
  * Sieve.cpp: Reduce memory usage of counters array by 2x.
  * phi.cpp: Fixed integer overflow #39.
  * phi.cpp: Cache 40x more phi(x, a) results.
  * phi.cpp: Use pi(x) if a > pi(sqrt(x)).
  * phi.cpp: Use larger c constant if phi(x, larger_c) is cached.
  * generate_phi.hpp: Same optimizations as phi.cpp.
  * LoadBalancer.cpp: Tune for new phi(x, a) implementation.
  * FactorTable.hpp: Reduce number of branches.
  * DFactorTable.hpp: Reduce number of branches.

Changes in primecount-6.2, 2021-01-07

  The sieving algorithm has been improved by annotating branches
  with likely/unlikely and __builtin_unreachable(). This improves
  the performance of the S2_hard and D formulas by up to 5%. GCC
  benefits most from these changes (Clang's performance was
  already very good before). With this release there is also a new
  primecount-backup version (last primecount-backup release was 6.0).

  In order to reduce the amount of work for myself there will be no
  precompiled binaries anymore going forward. You can now install
  primecount using your operating system's package manager (if it
  is available there) or by compiling it from source yourself.

  * Use Appveyor-CI instead of Travis-CI for testing.
  * README.md: primecount can now be installed using package managers.
  * Sieve.cpp: Annotate branches with unlikely, up to 5% speedup.
  * Sieve.cpp: Annotate switch cases with fallthrough.
  * AC.cpp: Reduce number of function paramaters.
  * AC_libdivide.cpp: Reduce number of function paramaters.
  * B.cpp: Improve GCC performance.
  * P2.cpp: Improve GCC performance.
  * popcnt.hpp: Improve GCC performance.

Changes in primecount-6.1, 2020-09-12

  The main focus of this release has been on polishing the code
  and improving the documentation. I also tried many things to
  improve the scaling on servers with a large number of CPU cores,
  however I only achieved minor speed ups. The only meaningful
  improvement is that the same threads are now reused throughout
  the entire AC computation. This improves the scaling for small
  to medium sized computations up to 10^20. GCC benefits most
  from this change whereas Clang performance is mostly unchanged.

  * Xavier Gourdon's algorithm has been distributed using MPI
    so that computations can now run on HPC clusters.
  * CMakeLists.txt: New WITH_JEMALLOC option (default OFF).
  * AC.cpp: Reuse the same threads throughout the computation.
  * AC.cpp: Improve upper bound of C2 formula.
  * AC.cpp: Avoid branch inside hot loop of A formula.
  * SegmentedPiTable.cpp: Reuse threads from AC.cpp.
  * LoadBalancerP2.cpp: New load balancer for P2 & B.
  * phi.cpp: Reduce caching for tiny numbers.
  * generate_phi.hpp: Reduce caching for tiny numbers.
  * pod_vector.hpp: Like std::vector, but without default
    initialization (useful when allocating 100s of GiB).
  * PiTable.cpp: Multi-threaded initialization.
  * Status.cpp: Avoid thread synchronization when printing
    in order to improve scaling of AC and S2_easy.
  * Status.cpp: Improve S2_hard & D status accuracy.
  * StatusAC.cpp: More accurate status for AC formula.
  * cmdoptions.cpp: Add -B & -D options.
  * Fixed #32: primecount-backup prints incorrect time elapsed.

Changes in primecount-6.0, 2020-03-16

  This version fixes a scaling issue that has been present in
  primecount since the first version. The computation of the
  hard special leaves (S2_hard & D formulas) used a flawed
  optimization that deteriorated the runtime complexity of the
  algorithm. The doc/Special-Leaves.md document contains more
  information. The new version should run much faster for large
  computations >= 10^23.

  * Sieve.cpp: Implemented O(sqrt(sieve_size)) counting,
    previously counting was O(sieve_size).
  * AC_libdivide.cpp: Up to 20% faster using GCC.
  * S2_easy_libdivide.cpp: Up to 20% faster using GCC.
  * primecount.h: New C API. primecount now has both a C API
    (primecount.h) and a C++ API (primecount.hpp).
  * LoadBalancer.cpp: Refactor using new ThreadSettings struct.
  * find_optimal_alpha_*.sh: New scripts that find optimal
    alpha tuning factors using the Linux perf tool.

Changes in primecount-5.3, 2020-01-18

  libprimesieve has been updated to the latest version which
  provides a minor speedup for all formulas that use
  primesieve::iterator e.g. P2, B, pi_lehmer, ...

  * Sieve.hpp: Use NOINLINE.
  * S2Status.hpp: Use NOINLINE.
  * D4.cpp: Simplify bounds.
  * S2_hard.cpp: Simplify bounds.
  * LoadBalancer.cpp: Simplify bounds.
  * RiemannR.cpp: Add support for __float128 type.
  * popcnt.hpp: Faster ARM64 popcount implementation.
  * fast_div.hpp: Add ENABLE_DIV32 macro.
  * S2Status.cpp: Fix non-critical data race.
  * LoadBalancer.cpp: Improve thread load balancing.

Changes in primecount-5.2, 2019-11-17

  I have now implemented Xavier Gourdon's algorithm in the
  primecount-backup version (branch backup3). Other than that
  there have been documentation improvements and build system
  improvements which should make it easier to package
  primecount for e.g. Linux distros (see BUILD.md).

  * Sieve.cpp: Fix vector out of bounds.
  * cmdoptions.cpp: Support options of type: --option VALUE.
  * doc/BUILD.md: Detailed build instructions. Contains new
    section: "Packaging primecount" for Linux distros.
  * doc/primecount.1: Add primecount man page.
  * doc/primecount.txt: AsciiDoc, used to generate primecount.1.
  * CMakeLists.txt: Generate man page using a2x program.
  * CMakeLists.txt: New option: -DBUILD_LIBPRIMESIEVE=OFF.
  * Get rid of underscores in command-line options:

      --deleglise_rivat -> --deleglise-rivat
      --S2_trivial      -> --S2-trivial
      --S2_easy         -> --S2-easy
      --S2_hard         -> --S2-hard

Changes in primecount-5.1, 2019-09-02

  The memory usage of Xavier Gourdon's algorithm has been
  reduced from O(x^(1/2)) to O(x^(1/3) * log^3 x). Xavier
  Gourdon's algorithm is now fully implemented and luckily it
  scales well up to a very large number of CPU cores. Xavier
  Gourdon's algorithm is now enabled by default for
  numbers > 10^7.

  * main.cpp: New options: --AC, --B, --D, --Phi0, --Sigma.
  * SegmentedPiTable.cpp: New PiTable implementation.
  * AC.cpp: Combine the A & C formulas to improve scaling.
  * D4.cpp: Tighter bounds, minor speedup.
  * Sigma.cpp: Reduce initialization overhead.
  * Sieve.cpp: Improved pre-sieving.
  * S2_hard.cpp: Improved pre-sieving.
  * print.cpp: Updated for Gourdon's algorithm.

  === C++ API Changes ===

  In order to simplify the API the following functions have
  been removed from the primecount.hpp header:

  int64_t pi_deleglise_rivat(int64_t x);
  int64_t pi_legendre(int64_t x);
  int64_t pi_lehmer(int64_t x);
  int64_t pi_lmo(int64_t x);
  int64_t pi_meissel(int64_t x);
  int64_t pi_primesieve(int64_t x);

  You should use pi(x) instead which is an alias for the
  fastest prime counting function implementation in
  primecount.

Changes in primecount-5.0, 2019-08-13

  I have finally implemented Xavier Gourdon's prime counting
  algorithm! Xavier Gourdon's algorithm is an improved
  version of the Deleglise-Rivat algorithm. According to my
  benchmarks Gourdon's algorithm runs up to 2x faster than
  the Deleglise-Rivat algorithm.

  * src/gourdon: Implementation of Xavier Gourdon's algorithm.
  * LoadBalancer.cpp: Updated for Gourdon's algorithm.
  * primecount.cpp: Updated for Gourdon's algorithm.
  * print.cpp: Updated for Gourdon's algorithm.
  * generate.cpp: Generate array with largest prime factors.
  * test.cpp: Test Gourdon's algorithm.
  * README.md: Update benchmarks.

Changes in primecount-4.8, 2019-07-14

  This version reduces the memory usage of S2_easy(n) by
  15% and the memory usage of S2_hard(n) by 10%.
  I have also improved the documentation of the code so
  that others and myself can easier understand it.

  * PiTable.cpp: Reduce memory usage by 2x.
  * FactorTable.hpp: Reduce memory usage by 10%.
  * LoadBalancer.cpp: Improve scaling.
  * MpiLoadBalancer.cpp: Improve scaling.
  * fast_div.hpp: x64 assembly for (128-bit / 64-bit) = 64-bit.
  * phi_tiny.hpp: Speedup for 128-bit integers.
  * S2_trivial.cpp: Implement O(1) math formula.
  * libdivide.h: Update to libdivide-2.0.
  * appveyor.yml: Treat compiler warnings as errors.

Changes in primecount-4.7, 2019-04-16

  This version fixes an issue with the new MD5 checksum
  feature introduced in primecount-backup-4.6 that I
  found unfortunately 1 day after releasing primecount-4.6.

  * json.hpp: Fix incorrect MD5 formatting, bytes with the
    first 4 bits masked off must be prefixed with '0'.

Changes in primecount-4.6, 2019-04-13

  This version fixes 2 bugs in the CMakeLists.txt build script
  and improves the primecount backup version that stores
  intermediate results to hard disk. Note that
  primecount.backup files produced by previous primecount
  releases are incompatible with primecount-4.6.

  * CMakeLists.txt: Fix libatomic issue.
  * CMakeLists.txt: Require CMake 3.4 instead of 3.9.
  * LoadBalancer.cpp: Increase number of backups.
  * S2_easy.cpp: Allow resuming using fewer/more threads.
  * S2_hard.cpp: Allow resuming using fewer/more threads.
  * primecount.backup: Add MD5 checksum.

Changes in primecount-4.5, 2019-02-20

  This is a maintenance release that contains minor
  improvements e.g. tests should run 10% faster.

  * lib/primesieve: upgrade to version 7.4.
  * travis.yml: Test using many different compiler versions.
  * calculator.hpp: Silence clang-cl -Wdeprecated warning.

Changes in primecount-4.4, 2018-05-09

  The computation of the second partial sieve function
  P2(x, a) has been sped up by 30% due to an update to the
  latest primesieve-7.0 library.

  * CMakeLists.txt: Add OpenMP support for LLVM/Clang.
  * CMakeLists.txt: Add Intel C++ compiler support.
  * nth_prime.cpp: Fix non-critical data race.
  * pi_legendre.cpp: Fix non-critical data race.
  * pi_primesieve.cpp: Fix non-critical data race.
  * make test: Runs twice as fast.

Changes in primecount-4.3, 2018-03-18

  * Support big-endian CPUs.
  * Speed up x86 without POPCNT.
  * libdivide.h: update to libdivide 1.0.
  * calculator.hpp: Fix integer overflow in pow().
  * Required CMake version is now 3.4 (previously 3.1).
  * CMakeLists.txt: Fix make install issue.
  * CMakeLists.txt: Get rid of int128_t, __int128_t checks.
  * isqrt_constexpr.cpp: Add test for compile time square root.

Changes in primecount-4.2, 2017-12-02

  The computation of the second partial sieve function
  P2(x, a) has been sped up by 10% due to an upgrade to the
  latest primesieve-6.3 library.

  * lib/primesieve: upgrade to version 6.3.
  * src/backup.cpp: Speed up resume from backup.
  * test/RiemannR.cpp: Fix bash/ubuntu on Windows issue.
  * CMakeLists.txt: Silence GCC 7 warnings.
  * .travis.yml: Update to Ubuntu 14.

Changes in primecount-4.1, 2017-07-19

  This version improves the backup & resume functionality
  and uses up to 8% less memory due to a reduced alpha
  tuning factor.

  * S2_easy.cpp: Fix severe backup scaling issues.
  * S2_easy_libdivide.cpp: Fix severe backup scaling issues.
  * S2_hard.cpp: Faster resume.
  * LoadBalancer.cpp: Simplify backup.
  * results.txt: Fix incorrect time elapsed.
  * primecount.cpp: smaller alpha tuning factor.

Changes in primecount-4.0, 2017-07-06

  This version features a new highly optimized sieve of
  Eratosthenes implementation for the computation of the
  hard special leaves that speeds up the S2_hard algorithm
  by up to 40%, giving an overall speed up of up to 20%.

  * Sieve.cpp: New sieve of Eratosthenes.
  * S2_hard.cpp: Use new sieve.
  * S2_hard_mpi.cpp: Use new sieve.
  * lmo5.cpp: Use new sieve.
  * pi_lmo_parallel.cpp: Use new sieve.
  * LoadBalancer.cpp: L1 cache size optimization.
  * MpiLoadBalancer.cpp: L1 cache size optimization.

Changes in primecount-3.8, 2017-06-27

  This version reduces the memory usage by a factor of 2
  above 10^20! Furthermore the S2_hard algorithm for
  computing the hard special leaves has been improved: The
  binary indexed tree data structure (random memory access
  pattern) has been removed. This fixes the scaling issues
  above 10^24 primecount had previously.

  * libdivide.h: Reduce memory usage, pack structs.
  * BitSieve.hpp: Reduce memory usage, store odd integers.
  * S2_hard.cpp: Remove binary indexed tree.
  * S2_hard_mpi.cpp: Remove binary indexed tree.
  * primecount.cpp: Update alpha tuning factor formula.

Changes in primecount-3.7, 2017-06-07

  This version features a new multi-threading load balancer
  for the computation of the hard special leaves. The new
  load balancer requires no synchronization between threads
  and achieves 100% CPU cores usage. The new load balancer
  is based on ideas from Xavier Gourdon and Douglas Staple.

  * LoadBalancer.cpp: New multi-threading load balancer.
  * generate_phi.hpp: Part of new load balancer.
  * S2_hard.cpp: Use new load balancer.
  * BitSieve.cpp: Get rid of AVX2 (use POPCNT).
  * Li.cpp: Riemann R and inverse Riemann R implementations.
  * fast_div.hpp: Refactor using template metaprogramming.
  * src/test: Added 27 unit tests.
  * phi(x, a) now scales > 8 threads.
  * BinaryIndexedTree.hpp: Do not store multiples of 2.
  * CMakeLists.txt: Faster C++ compilation.
  * S2Status.cpp: Reduce --status overhead.

Changes in primecount-3.6, 2017-03-04

  This version features a new AVX2 popcount algorithm which
  computes the hard special leaves up to 15% faster on x86 CPUs
  with AVX2 support (2013 or later).

  * BitSieve-popcnt.cpp: New AVX2 popcount algorithm.
  * popcnt.hpp: Fix clang performance bug.
  * primecount.cpp: Fix clang time measuring.
  * CMakeLists.txt: Add AVX2 check.

Changes in primecount-3.5, 2016-12-16

  * CMake: Use CMake build system instead of Autotools.
  * README.md: Update build instructions.

Changes in primecount-3.4, 2016-08-06

  * Makfile.msvc: Use libdivide also with MSVC compiler.
  * S2LoadBalancer.cpp: Improved load balancing.
  * P2.cpp: Improved load balancing.
  * P2_mpi.cpp: Improved load balancing.
  * .travis.yml: Add static C++ code analysis using cppcheck.

Changes in primecount-3.3, 2016-07-16

  * README.md: Fix error in "C++ API" section.
  * S2_hard.cpp: Refactoring.
  * S2_hard_mpi.cpp: Refactoring.
  * P2.cpp: Refactoring.
  * P2_mpi.cpp: Refactoring.

Changes in primecount-3.2, 2016-05-09

  This version runs up to 5% faster due to an improved P2(x, a)
  implementation.

  * P2.cpp: 30% faster.
  * P2_mpi.cpp: 30% faster.
  * libdivide.h: Update to latest version.
  * autogen.sh: Print error msg if Autotools is not installed.

Changes in primecount-3.1, 2016-04-02

  This version runs up to 20% faster! The speed up is due to
  the usage of libdivide in S2_easy, libdivide allows to replace
  expensive integer divides with comparatively cheap
  multiplication and bitshifts.

  * S2_easy_libdivide.cpp: 40% speed up due to libdivide.
  * S2_easy_mpi_libdivide.cpp: 40% speed up due to libdivide.
  * BitSieve.cpp: Fix broken Big-Endian CPU support.

Changes in primecount-3.0, 2016-03-12

  In this release I have merged the MPI (Message Passing Interface)
  branch into the master branch. primecount can now distribute
  computations onto cluster nodes if MPI is enabled in the build
  process (--enable-mpi).

  * doc/primecount-MPI.md: primecount MPI documentation.
  * src/mpi: primecount MPI source code.
  * include/FactorTable.hpp: Multi-threaded initialization.
  * build.sh: Improved build script with support for primecount MPI.
  * README.md: Updated build instructions.

Changes in primecount-2.6, 2016-02-14

  primecount has been distributed using MPI (Message Passing Interface)
  and can now run computations on large clusters!
  The distributed version of primecount is named primecount-mpi and
  the code can be found at:

  https://github.com/kimwalisch/primecount/tree/mpi

  * src/phi.cpp: 2x speed up due to pi(x) lookup table optimization.
  * scripts/build.sh: Easily build primecount.

Changes in primecount-2.5, 2016-01-24

  This version adds logging to primecount-backup and fixes 2
  integer overflow bugs in primecount-backup.

  * --log: Logs results into results.txt and primecount.log.
  * scripts/worktodo.sh: Script for batch processing via worktodo.txt.
  * src/S1.cpp: Fix integer overflow bug (in backup branch).
  * src/deleglise-rivat/S2_trivial.cpp: Fix integer overflow bug (in backup branch).

Changes in primecount-2.4, 2015-12-27

  * README.md: Simplified build instructions.
  * appveyor.yml: Automated Windows (MSVC++) testing.
  * configure.ac: Removed usage of buggy ax_gcc_builtin.m4 macro.
  * src/P2.cpp: Port to primesieve-5.5.0 library.
  * src/test.cpp: Faster nth prime testing.

Changes in primecount-2.3, 2015-10-07

  This version runs about 10% faster for x <= 10^21. Because of the
  sieving optimizations implemented in primecount-2.2 the sieving
  limit has now been increased which reduces the time to compute the
  easy special leaves.

  I have now also officially published primecount-backup binaries
  which save intermediate results to a file once per hour and which
  can resume from these files after e.g. a crash:
  https://github.com/kimwalisch/primecount/tree/backup

  * Renamed to --P2, --S1, --S2_trivial, --S2_easy, --S2_hard.
  * find_fastest_alpha.sh: Benchmark which finds fastest alpha factors.
  * src/primecount.cpp: Alpha factor tuning.
  * src/P2.cpp: Use multi-threading for initialization.
  * src/S2Status.cpp: --status[=N], N digits after decimal point.
  * src/pi_lehmer.cpp: Remove pi_lehmer2(x).

Changes in primecount-2.2, 2015-09-19

  This version runs about 10% faster due to newly added pre-sieving
  of small primes and wheel factorization.

  * src/primecount.cpp: Increase pi(x) limit to 10^31 (previously 10^27).
  * src/BitSieve.cpp: Add pre-sieving of small primes.
  * src/P2.cpp: Use pre-sieving and wheel factorization.
  * src/deleglise-rivat/*: Use pre-sieving and wheel factorization.
  * src/lmo/*: Use pre-sieving and wheel factorization.
  * include/popcount.hpp: Faster popcount algorithm for CPUs without POPCNT.
  * include/Wheel.hpp: New wheel factorization data structures.

Changes in primecount-2.1, 2015-04-12

  * src/S1.cpp: Fix Windows performance regression.
  * src/S2LoadBalancer.cpp: Refactoring.
  * Makefile.am: Add autogen.sh to EXTRA_DIST.

Changes in primecount-2.0, 2015-03-26

  This version improves the POPCNT algorithm in the computation of the
  hard special leaves and contains a new algorithm for the computation
  of the ordinary leaves which uses half as much memory.

  * src/S1.cpp: New implementation based on Douglas Staple's algorithm.
  * src/S2LoadBalancer.cpp: Improved load balancing.
  * src/deleglise-rivat/S2_hard.cpp: Also use POPCNT if high < y.
  * src/lmo/pi_lmo5.cpp: Optimized sieving, up to 10% faster.
  * src/lmo/pi_lmo_parallel3.cpp: Optimized sieving, up to 10% faster.
  * include/BitSieve.hpp: Add count backwards optimization.

Changes in primecount-1.9, 2015-03-08

  This version introduces new command-line options for calculating
  intermediate formulas of the Deleglise-Rivat algorithm. This allows
  to distribute the computation of large pi(x) values on multiple
  systems. This version also fixes two non-critical bugs.

  * src/app: New options: --p2, --s1, --s2_trivial, --s2_easy, --s2_hard.
  * src/deleglise-rivat/S2_trivial.cpp: Fix off by 1 bug.
  * src/deleglise-rivat/*: Bugfix, set 1 and unset 2 in sieve.
  * src/deleglise-rivat/S2_hard.cpp: Improved scaling for large x.
  * src/deleglise-rivat/*: Alpha factor tuning.
  * src/lmo/*: Bugfix, set 1 and unset 2 in sieve.
  * src/lmo/*: Alpha factor tuning.
  * src/primecount.cpp: Improved alpha formula.
  * src/print.cpp: New print functions for terminal output.

Changes in primecount-1.8, 2015-02-28

  This version reduces the memory usage of the Deleglise-Rivat
  implementation by up to 30 percent. Instead of using the same set of
  memory intense data structures for all formulas (P2, S1, S2_trvial,
  S2_easy, S2_hard), each individual formula now generates its own set
  of data structures and the size of each data structure is the
  smallest possible for the given formula.

  * -a<N>, --alpha=<N>: Manually set the tuning factor.
  * src/primecount.cpp: Improved alpha factor formula.
  * src/deleglise-rivat/*: Reduce memory usage.
  * src/lmo/*: Update S1(x) function calls.
  * src/nth_prime.cpp: Add optimization for small primes.
  * src/app/*: Add --alpha option.
  * avoid_128bit_div.patch: merged into master branch.

Changes in primecount-1.7.1, 2015-01-26

  * Makefile.am: Add avoid_128bit_div.patch to EXTRA_DIST
  * avoid_128bit_div.patch: Renamed

Changes in primecount-1.7, 2015-01-24

  This version runs to 20% faster on Windows for numbers >= 2^63 by
  using 64-bit divisions instead of 128-bit divisions whenever
  possible. While primecount-1.6 has already been very fast on Linux
  it ran slower on Windows, primecount-1.7 now achieves the same
  speed on both Windows and Linux.

  * fast_div.patch: Avoid 128-bit divisions.

Changes in primecount-1.6, 2015-01-05

  This version calculates hard special leaves much faster, above a
  certain threshold the algorithm switches to POPCNT for counting the
  number of unsieved elements (instead of Tomás Oliveira's special
  tree data structure) which dramatically improves memory efficiency.
  This version also fixes a serious bug in P2(x, a) for x > 10^21,
  thanks to Dana Jacobsen for reporting it.

  * src/deleglise-rivat/S2_hard.cpp: Add POPCNT optimization.
  * src/lmo/pi_lmo_parallel3: Add POPCNT optimization.
  * src/lmo/pi_lmo5: Add POPCNT optimization.
  * src/P2.cpp: Bug fix for numbers > 10^21.
  * src/BitSieve.hpp: Improved memory efficiency.
  * include/isqrt.hpp: Fixed C++11 bug in isqrt(x).
  * include/min.hpp: Refactoring.
  * include/int128.hpp: Add int128_t trait functions.

Changes in primecount-1.5, 2014-12-27

  This version runs up to 30 percent faster than primecount-1.4 for
  numbers > 2^64. The speed up is achieved using a clever trick:
  Instead of calculating xn = x / (primes[b] * primes[l]) which uses
  a 128-bit division, x2 = x / primes[b] is calculated upfront and
  then xn = x2 / primes[l]. If x2 is < 2^64 then xn can be calculated
  using a 64-bit division which is much faster.

  * src/deleglise-rivat/S2_*.cpp: Avoid 128-bit divisions.
  * src/S2Status.cpp: Improved status precision.
  * src/app/cmdoptions.cpp: Set -l = --lmo instead of --lehmer.
  * README.md: Add command-line options summary.

Changes in primecount-1.4, 2014-12-15

  * src/deleglise-rivat/*.cpp: Improved computation of special leaves.
  * src/P2.cpp: Use unsigned division.
  * src/S1.cpp: Use unsigned division.
  * src/S2LoadBalancer.cpp: Update documentation.
  * src/PhiCache.cpp: Update documentation.
  * include/PiTable.hpp: Update documentation.

Changes in primecount-1.3, 2014-11-04

  * Fixed bug in m4-ax_popcnt.m4 for QEMU virtual machines.
  * Limit number of threads in phi.cpp to 8.
  * Improve scaling of P2_lehmer(x, a).
  * Improve scaling of P3(x, a).

Changes in primecount-1.2, 2014-08-24

  * Add --status (-s) command-line option.
  * src/S2LoadBalancer.cpp: New faster load balancer.
  * src/S2Status.cpp: Print S2 progress.
  * Fixed integer overflow bugs in: Li_inverse(x), isqrt(x), iroot(x)

Changes in primecount-1.1, 2014-08-06

  * pi_deleglise_rivat4.cpp: 128-bit implementation.
  * pi_deleglise_rivat_parallel4.cpp: 128-bit implementation.
  * Add --time option to print elapsed seconds.
  * include/S1.hpp: Added multi-threading and templates.
  * include/FactorTable.hpp: template implementation.
  * src/PiTable.cpp: Faster initialization.
  * src/balance_S2_load.cpp: Improved load balancer.
  * src/generate.cpp: Contains all generate_*() functions.

Changes in primecount-1.0, 2014-07-19

  * src/BitSieve.cpp: Fix big-endian CPU bug.
  * src/lmo/*.cpp: Improved alpha factor.
  * src/deleglise-rivat/*.cpp: Improved alpha factor.
  * include/pmath.hpp: Add max3(a, b, c).
  * m4/m4-ax_popcnt.m4: Support non x86 CPU architectures.
  * Readme.md: Update documentation.