Prime Number Functions

.gitignore

@@ -7,3 +7,6 @@ inspect.html
 test/cuda
 *.DS_Store
 /**/*.dSYM/
+build/*
+*.json
+

doc/sf/number_series.qbk

@@ -259,9 +259,63 @@ Passing a value greater than `max_prime` results in a __domain_error being raise
 
 This function is `constexpr` only if the compiler supports C++14 constexpr functions.
 
-[endsect] [/section:primes]
+[endsect] [/section:primes Prime Numbers]
 
-[endsect] [/Number Series]
+[section:prime_sieve Prime Sieve]
+
+[h4 Synopsis]
+
+``
+#include <boost/math/special_functions/prime_sieve.hpp>
+``
+
+namespace boost { namespace math {
+
+    template<class ExecutionPolicy, typename Z, class OutputIterator>
+    auto prime_sieve(ExecutionPolicy&& policy, Z upper_bound, OutputIterator output) -> decltype(output)
+
+    template<class ExecutionPolicy, class Z, class OutputIterator>
+    auto prime_range(ExecutionPolicy&& policy, Z lower_bound, Z upper_bound, OutputIterator output) -> decltype(output)
+
+    template<typename Z, class OutputIterator>
+    auto prime_sieve(Z upper_bound, OutputIterator output) -> decltype(output)
+
+    template<class Z, class OutputIterator>
+    auto prime_range(Z lower_bound, Z upper_bound, OutputIterator output) -> decltype(output)
+
+
+}} // End namespaces
+
+[h4 Description]
+
+There are two sets of functions available `prime_sieve` and `prime_range`. `prime_sieve` will return all primes in the
+range [2, `upper_bound`). `prime_range` will return all the primes in the range [lower_bound, upper_bound).
+
+If you have a C++17 compatible compiler you are able to pass an execution policy to both functions. Any
+policy besides `std::execution::seq` will enable internal multi-threading. If your compiler is not C++17 compatible the
+sequential overloads will be used.
+
+For upper_bound <= 2[super 24] two threads will be used. For any value larger than 2[super 24] `std::thread::hardware_concurrency()` will be called,
+and all available concurrency will be used. Additionally, the memory requirements are `bigo[](sqrt(N))`.
+
+/Nota bene:/ If `std::thread::hardware_concurrency()` returns 0 the max number of threads will be set to 2.
+
+[h4 Examples]
+    // To calculate primes 2 - 1,000,000 in parallel
+    std::vector<int64_t> primes;
+    boost::math::prime_sieve(std::execution::par, 1000000, std::back_inserter(primes));
+
+    // To calculate primes 100 - 1,000 sequentially
+    std::vector<int64_t> primes;
+    boost::math::prime_range(100, 1000, std::back_inserter(primes));
+
+[h4 References]
+* Sorensen, Jonathan [@https://research.cs.wisc.edu/techreports/1990/TR909.pdf An Introduction to Prime Number Sieves]
+* Gries, David and Misra, Jayadev [@https://www.cs.utexas.edu/users/misra/scannedPdf.dir/linearSieve.pdf A Linear Sieve Algorithm for Finding Prime Numbers]
+
+[endsect] [/section:primes Prime Sieve]
+
+[endsect] [/section:number_series Number Series]
 
 [/
   Copyright 2013, 2014 Nikhar Agrawal, Christopher Kormanyos, John Maddock, Paul A. Bristow.

include/boost/math/special_functions/interval_sieve.hpp

@@ -0,0 +1,309 @@
+// Copyright 2020 Matt Borland and Jonathan Sorenson
+//
+// Use, modification and distribution are subject to the
+// Boost Software License, Version 1.0.
+// (See accompanying file LICENSE_1_0.txt
+// or copy at http://www.boost.org/LICENSE_1_0.txt)
+
+#ifndef BOOST_MATH_SPECIAL_FUNCTIONS_INTERVAL_SIEVE_HPP
+#define BOOST_MATH_SPECIAL_FUNCTIONS_INTERVAL_SIEVE_HPP
+
+#include <boost/dynamic_bitset.hpp>
+#include <boost/multiprecision/cpp_int.hpp>
+#include <boost/math/special_functions/prime_wheel.hpp>
+#include <cmath>
+#include <memory>
+#include <array>
+#include <cstdint>
+#include <bitset>
+
+namespace boost::math::detail
+{
+
+#if defined(__MPIR_VERSION) || defined(__GNU_MP_VERSION)
+// GNU GMP C or MPIR
+inline double get_double(const mpz_t &x) noexcept
+{
+    return mpz_get_d(x);
+}
+#endif
+
+#if defined(__GNU_MP_VERSION)
+#if __has_include(<gmpxx.h>)
+// GNU GMP C++ bindings
+inline double get_double(const mpz_class &x) noexcept
+{
+    return x.get_d()
+}
+#endif
+#endif
+
+// boost::multiprecision and POD
+template<class Integer>
+inline double get_double(const Integer &x) noexcept
+{
+    return static_cast<double>(x);
+}
+
+template<class Integer, class PrimeContainer, class Container>
+class IntervalSieve
+{
+#ifdef __SIZEOF_INT128__   // Defined in GCC 4.6+, clang, intel. MSVC does not define. 
+using int_128t = __int128; // One machine word smaller than the boost equivalent
+#else
+using int_128t = boost::multiprecision::int128_t;
+#endif
+
+private:
+    // Table of pseudo-sqares (https://mathworld.wolfram.com/Pseudosquare.html)
+    // This table is from page 421, table 16.3.1, Hugh Williams' book
+    // Last 8 entries added from Wooding's MS thesis, 2003, pp. 92-93
+    struct pssentry
+    {
+        static constexpr std::size_t len {49};
+        static constexpr std::array<std::int_fast16_t, len> prime
+        {
+            3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 67, 71, 79, 83, 101, 103, 107, 113, 131, 149, 157,
+            173, 181, 193, 197, 211, 227, 229, 233, 239, 241, 251, 257, 263, 277, 281, 283, 293, 311, 331, 337, 347, 353
+        };
+
+        static constexpr std::array<int_128t, len> ps
+        {
+            73, 241, 1'009, 2'641, 8'089, 18'001, 53'881, 87'481, 117'049, 515'761, 1'083'289, 3'206'641, 3'818'929,
+            9'257'329, 22'000'801, 48'473'881, 175'244'281, 427'733'329, 898'716'289, 2'805'544'681, 10'310'263'441,
+            23'616'331'489, 85'157'610'409, 196'265'095'009, 2'871'842'842'801, 26'250'887'023'729, 112'434'732'901'969,
+            178'936'222'537'081, 696'161'110'209'049, 2'854'909'648'103'881, 6'450'045'516'630'769, 11'641'399'247'947'921,
+            190'621'428'905'186'449, 196'640'148'121'928'601, 712'624'335'095'093'521, 1'773'855'791'877'850'321,
+            2'327'687'064'124'474'441, 6'384'991'873'059'836'689, 8'019'204'661'305'419'761, 10'198'100'582'046'287'689,
+            69'848'288'320'900'186'969, 208'936'365'799'044'975'961, 533'552'663'339'828'203'681, 936'664'079'266'714'697'089,
+            2'142'202'860'370'269'916'129, 13'649'154'491'558'298'803'281, 34'594'858'801'670'127'778'801, 
+            99'492'945'930'479'213'334'049, 295'363'187'400'900'310'880'401
+        };
+    };
+
+    static constexpr pssentry pss_{};
+    static constexpr boost::math::detail::MOD210Wheel<Integer> w_{};
+    std::size_t tdlimit_;
+
+    Integer delta_;
+    Integer left_;
+    Integer right_;
+
+    // https://www.researchgate.net/publication/220803585_Performance_of_C_bit-vector_implementations
+    boost::dynamic_bitset<> b_;
+
+    const PrimeContainer& primes_;
+    std::int_fast64_t plimit_;
+
+    void Settdlimit() noexcept;
+    void SeiveLength(const Integer d) noexcept;
+    void Sieve() noexcept;
+    bool Psstest(const std::size_t pos) noexcept;
+    void Psstestall() noexcept;
+    void WriteOutput(Container &resultant_primes) noexcept;
+
+public:
+    IntervalSieve(const Integer left, const Integer right, const PrimeContainer &primes, Container &resultant_primes) noexcept;
+    void NewRange(const Integer left, const Integer right, Container &resultant_primes) noexcept;
+};
+
+template<class Integer, class PrimeContainer, class Container>
+void IntervalSieve<Integer, PrimeContainer, Container>::Settdlimit() noexcept
+{
+    const double dr = get_double(right_);
+    const double delta = get_double(delta_);
+    const double tdest = delta * std::log(dr);
+
+    // Small cases
+    if(tdest * tdest >= dr)
+    {
+        tdlimit_ = static_cast<std::size_t>(std::sqrt(dr));
+        plimit_ = 0;
+        return;
+    }
+
+    // First guess
+    if(tdest <= 1ul<<30)
+    {
+        tdlimit_ = static_cast<std::size_t>(tdest);
+    }
+
+    else
+    {
+        tdlimit_ = 1ul<<30;
+    }
+
+    // Find the corresponding prime
+    std::size_t i;
+    for(i = pss_.len - 1; i > 0; --i)
+    {
+        if(static_cast<double>(pss_.ps[i]) * tdlimit_ < dr)
+        {
+            break;
+        }
+    }
+    plimit_ = pss_.prime[i];
+
+    double tdlimit_guess = 1 + std::fmod(dr, pss_.ps[i]);
+    if(tdlimit_guess * tdlimit_guess >= dr)
+    {
+        tdlimit_ = static_cast<std::size_t>(std::sqrt(dr));
+        plimit_ = 0;
+    }
+}
+
+template<class Integer, class PrimeContainer, class Container>
+void IntervalSieve<Integer, PrimeContainer, Container>::SeiveLength(const Integer d) noexcept
+{
+    Integer r {left_ % d};
+    Integer start {0};
+
+    if(r != 0)
+    {
+        start = d - r;
+    }
+
+    for(Integer i {start}; i >= 0 && i < b_.size(); i += d)
+    {
+        b_[static_cast<std::size_t>(i)] = 0;
+    }
+}
+
+template<class Integer, class PrimeContainer, class Container>
+void IntervalSieve<Integer, PrimeContainer, Container>::Sieve() noexcept
+{
+    std::int_fast64_t primes_range {};
+    if(plimit_ <= 10)
+    {
+        primes_range = 10;
+    }
+
+    else
+    {
+        primes_range = plimit_;
+    }
+
+    // Sieve with pre-computed primes and then use the wheel for the remainder    
+    std::size_t i {};
+    for(; primes_[i] < primes_range; ++i)
+    {
+        SeiveLength(primes_[i]);
+    }
+
+    for(Integer j = w_.Next(primes_[--i]); j <= tdlimit_; j = w_.Next(j))
+    {
+        SeiveLength(j);
+    }
+}
+
+template<class Integer, class PrimeContainer, class Container>
+void IntervalSieve<Integer, PrimeContainer, Container>::WriteOutput(Container &resultant_primes) noexcept
+{
+    for(std::size_t i {}; i < b_.size(); ++i)
+    {
+        if(b_[i])
+        {
+            resultant_primes.emplace_back(left_ + i);
+        }
+    }
+}
+
+// Performs the pseduosqaure prime test on n = left + pos
+// return 1 if prime or prime power, 0 otherwise
+// Begins with a base-2 test
+template<class Integer, class PrimeContainer, class Container>
+bool IntervalSieve<Integer, PrimeContainer, Container>::Psstest(const std::size_t pos) noexcept
+{
+    const Integer n {left_ + pos};
+    const Integer exponent {(n - 1) / 2};
+    const std::int_fast64_t nmod8 = n % 8;
+
+    std::int_fast64_t negative_one_count {0};
+
+    for(std::size_t i {}; i < primes_.size(); ++i)
+    {
+        Integer temp = primes_[i];
+        temp = static_cast<Integer>(std::pow(get_double(temp), get_double(n)));
+
+        if(temp == 1)
+        {
+            if(i == 0 && nmod8 == 5)
+            {
+                return false;
+            }
+        }
+
+        else
+        {
+            ++temp;
+            if(temp == n)
+            {
+                if(i > 0)
+                {
+                    ++negative_one_count;
+                }
+            }
+            else
+            {
+                return false;
+            }
+        }
+    }
+
+    return (nmod8 != 1 || negative_one_count > 0);
+}
+
+template<class Integer, class PrimeContainer, class Container>
+void IntervalSieve<Integer, PrimeContainer, Container>::Psstestall() noexcept
+{
+    for(std::size_t i {}; i < b_.size(); ++i)
+    {
+        if(b_[i])
+        {
+            if(!Psstest(i))
+            {
+                b_[i] = 0;
+            }
+        }
+    }
+}
+
+template<class Integer, class PrimeContainer, class Container>
+IntervalSieve<Integer, PrimeContainer, Container>::IntervalSieve(const Integer left, const Integer right, const PrimeContainer &primes, Container &resultant_primes) noexcept : 
+    left_ {left}, right_ {right}, primes_ {primes}
+{
+    delta_ = right_ - left_;
+    b_.resize(static_cast<std::size_t>(delta_), 1);
+    Settdlimit();
+    Sieve();
+
+    if(plimit_ != 0 )
+    {
+        Psstestall();
+    }
+
+    WriteOutput(resultant_primes);
+}
+
+template<class Integer, class PrimeContainer, class Container>
+void IntervalSieve<Integer, PrimeContainer, Container>::NewRange(const Integer left, const Integer right, Container &resultant_primes) noexcept
+{
+    left_ = left;
+    right_ = right;
+    delta_ = right_ - left_;
+
+    b_.resize(static_cast<std::size_t>(delta_));
+    b_.set();
+    Settdlimit();
+    Sieve();
+
+    if(plimit_ != 0)
+    {
+        Psstestall();
+    }
+
+    WriteOutput(resultant_primes);
+}
+}
+
+#endif // BOOST_MATH_SPECIAL_FUNCTIONS_INTERVAL_SIEVE_HPP