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///
/// @file PhiTiny.hpp
/// @brief phi_tiny(x, a) counts the numbers <= x that are not
/// divisible by any of the first a primes. phi_tiny(x, a)
/// computes phi(x, a) in constant time for a <= 8 using
/// lookup tables and the formula below.
///
/// phi(x, a) = (x / pp) * φ(pp) + phi(x % pp, a)
/// with pp = 2 * 3 * ... * prime[a]
/// φ(pp) = \prod_{i=1}^{a} (prime[i] - 1)
///
/// Copyright (C) 2021 Kim Walisch, <kim.walisch@gmail.com>
///
/// This file is distributed under the BSD License. See the COPYING
/// file in the top level directory.
///
#ifndef PHITINY_HPP
#define PHITINY_HPP
#include <BitSieve240.hpp>
#include <fast_div.hpp>
#include <imath.hpp>
#include <popcnt.hpp>
#include <stdint.h>
#include <array>
#include <cassert>
#include <limits>
#include <type_traits>
#include <vector>
namespace primecount {
class PhiTiny : public BitSieve240
{
public:
PhiTiny();
/// Uses at most one level of phi(x, a) recursion
/// to ensure that the runtime is O(1).
template <typename T>
T phi_recursive(T x, uint64_t a) const
{
// Unsigned integer division is usually
// faster than signed integer division,
// especially for int128_t.
using UT = typename std::make_unsigned<T>::type;
if (a < max_a())
return phi((UT) x, a);
else
{
assert(a == 8);
// This code path will be executed most of the time.
// In phi7(x) the variable a has been hardcoded to 7
// which makes it run slightly faster than phi(x, a).
// phi(x, 8) = phi(x, 7) - phi(x / prime[8], 7)
return phi7((UT) x) - phi7((UT) x / 19);
}
}
template <typename T>
T phi(T x, uint64_t a) const
{
assert(a < prime_products.size());
auto pp = prime_products[a];
auto remainder = (uint64_t)(x % pp);
T xpp = x / pp;
T sum = xpp * totients[a];
// For prime[a] <= 5 our phi(x % pp, a) lookup table
// is a simple two dimensional array.
if (a < phi_.size())
sum += phi_[a][remainder];
else
{
// For prime[a] > 5 we use a compressed phi(x % pp, a)
// lookup table. Each bit of the sieve array corresponds
// to an integer that is not divisible by 2, 3 and 5.
// Hence the 8 bits of each byte correspond to the offsets
// [ 1, 7, 11, 13, 17, 19, 23, 29 ].
assert(a < sieve_.size());
assert(remainder / 240 < sieve_[a].size());
uint64_t count = sieve_[a][remainder / 240].count;
uint64_t bits = sieve_[a][remainder / 240].bits;
uint64_t bitmask = unset_larger_[remainder % 240];
sum += (T)(count + popcnt64(bits & bitmask));
}
return sum;
}
/// In phi7(x) the variable a has been hardcoded to 7.
/// phi7(x) uses division by a constant instead of regular
/// integer division and hence phi7(x) is expected to run
/// faster than the phi(x, a) implementation above.
///
template <typename T>
T phi7(T x) const
{
constexpr uint32_t a = 7;
constexpr uint32_t pp = 510510;
constexpr uint32_t totient = 92160;
auto remainder = (uint64_t)(x % pp);
T xpp = x / pp;
T sum = xpp * totient;
// For prime[a] > 5 we use a compressed phi(x % pp, a)
// lookup table. Each bit of the sieve array corresponds
// to an integer that is not divisible by 2, 3 and 5.
// Hence the 8 bits of each byte correspond to the offsets
// [ 1, 7, 11, 13, 17, 19, 23, 29 ].
assert(sieve_.size() - 1 == a);
assert(remainder / 240 < sieve_[a].size());
uint64_t count = sieve_[a][remainder / 240].count;
uint64_t bits = sieve_[a][remainder / 240].bits;
uint64_t bitmask = unset_larger_[remainder % 240];
sum += (T)(count + popcnt64(bits & bitmask));
return sum;
}
static uint64_t get_c(uint64_t y)
{
if (y < pi.size())
return pi[y];
else
return max_a();
}
/// In Xavier Gourdon's algorithm the small
/// constant is named k instead of c.
/// k <= PrimePi[min(x_star, sqrt(x / y))]
///
template <typename T>
static uint64_t get_k(T x)
{
return get_c(iroot<4>(x));
}
static constexpr uint64_t max_a()
{
return primes.size();
}
private:
static const std::array<uint32_t, 8> primes;
static const std::array<uint32_t, 8> prime_products;
static const std::array<uint32_t, 8> totients;
static const std::array<uint8_t, 20> pi;
/// Packing sieve_t increases the cache's capacity by 25%
/// which improves performance by up to 10%.
#pragma pack(push, 1)
struct sieve_t
{
uint32_t count = 0;
uint64_t bits = ~0ull;
};
#pragma pack(pop)
/// sieve[a] contains only numbers that are not divisible
/// by any of the the first a primes. sieve[a][i].count
/// contains the count of numbers < i * 240 that are not
/// divisible by any of the first a primes.
std::array<std::vector<sieve_t>, 8> sieve_;
std::array<std::vector<uint8_t>, 4> phi_;
};
extern const PhiTiny phiTiny;
inline bool is_phi_tiny(uint64_t a)
{
return a <= PhiTiny::max_a();
}
template <typename T>
typename std::enable_if<(sizeof(T) == sizeof(typename make_smaller<T>::type)), T>::type
phi_tiny(T x, uint64_t a)
{
return phiTiny.phi_recursive(x, a);
}
template <typename T>
typename std::enable_if<(sizeof(T) > sizeof(typename make_smaller<T>::type)), T>::type
phi_tiny(T x, uint64_t a)
{
using smaller_t = typename make_smaller<T>::type;
// If possible use smaller integer type
// to speed up integer division.
if (x <= std::numeric_limits<smaller_t>::max())
return phiTiny.phi_recursive((smaller_t) x, a);
else
return phiTiny.phi_recursive(x, a);
}
} // namespace
#endif