Use wheel factorization for prime numbers

Certain existing units in the library require very large prime
numbers---so large, in fact, that our naive trial division hits the
_iteration limit_ for `constexpr` loops.  We don't want to force users
to provide a compiler option override, so we'd better find another way.

The solution is to use the "wheel factorization" algorithm:
https://en.wikipedia.org/wiki/Wheel_factorization

This lets us skip most composite numbers in our trial division.  The
implementation presented here is configurable in terms of the size of
the "basis" of primes we use.  Bigger bases let us skip more primes, but
at the cost of storing more numbers.  Fortunately, it turns out that N=3
was good enough for our purposes.
This commit is contained in:
Chip Hogg committed 2022-03-10 23:29:02 +00:00
1 parent 2198c8a403
commit 04b80f0827
5 files changed
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@@ -140,6 +140,15 @@ TEST_CASE("make_ratio performs prime factorization correctly")
// The failure was due to a prime factor which is larger than 2^31.
as_magnitude<ratio(16'605'390'666'050, 10'000'000'000'000)>();
}
SECTION("Can handle prime number which would exceed GCC iteration limit")
{
// GCC 10 has a constexpr loop iteration limit of 262144. A naive algorithm, which performs trial division on 2 and
// all odd numbers up to sqrt(N), will exceed this limit for the following prime. Thus, for this test to pass, we
// need to be using a more efficient algorithm. (We could increase the limit, but we don't want users to have to
// mess with compiler flags just to compile the code.)
as_magnitude<334524384739>();
}
}
TEST_CASE("magnitude converts to numerical value")
@@ -344,15 +353,6 @@ TEST_CASE("int_power computes integer powers") {
TEST_CASE("Prime helper functions")
{
SECTION("find_first_factor()") {
CHECK(find_first_factor(1) == 1);
CHECK(find_first_factor(2) == 2);
CHECK(find_first_factor(4) == 2);
CHECK(find_first_factor(6) == 2);
CHECK(find_first_factor(15) == 3);
CHECK(find_first_factor(17) == 17);
}
SECTION("multiplicity") {
CHECK(multiplicity(2, 8) == 3);
CHECK(multiplicity(2, 1024) == 10);