boost/math/special_functions/detail/bernoulli_details.hpp
///////////////////////////////////////////////////////////////////////////////
// Copyright 2013 John Maddock
// Distributed under 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_BERNOULLI_DETAIL_HPP
#define BOOST_MATH_BERNOULLI_DETAIL_HPP
#include <boost/math/tools/atomic.hpp>
#include <boost/math/tools/toms748_solve.hpp>
#include <boost/math/tools/cxx03_warn.hpp>
#include <boost/math/tools/throw_exception.hpp>
#include <boost/math/tools/config.hpp>
#include <boost/math/special_functions/fpclassify.hpp>
#include <vector>
#include <type_traits>
#if defined(BOOST_MATH_HAS_THREADS) && !defined(BOOST_NO_CXX11_HDR_MUTEX) && !defined(BOOST_MATH_NO_ATOMIC_INT)
#include <mutex>
#else
# define BOOST_MATH_BERNOULLI_NOTHREADS
#endif
namespace boost{ namespace math{ namespace detail{
//
// Asymptotic expansion for B2n due to
// Luschny LogB3 formula (http://www.luschny.de/math/primes/bernincl.html)
//
template <class T, class Policy>
T b2n_asymptotic(int n)
{
BOOST_MATH_STD_USING
const auto nx = static_cast<T>(n);
const T nx2(nx * nx);
const T approximate_log_of_bernoulli_bn =
((boost::math::constants::half<T>() + nx) * log(nx))
+ ((boost::math::constants::half<T>() - nx) * log(boost::math::constants::pi<T>()))
+ (((T(3) / 2) - nx) * boost::math::constants::ln_two<T>())
+ ((nx * (T(2) - (nx2 * 7) * (1 + ((nx2 * 30) * ((nx2 * 12) - 1))))) / (((nx2 * nx2) * nx2) * 2520));
return ((n / 2) & 1 ? 1 : -1) * (approximate_log_of_bernoulli_bn > tools::log_max_value<T>()
? policies::raise_overflow_error<T>("boost::math::bernoulli_b2n<%1%>(std::size_t)", nullptr, nx, Policy())
: static_cast<T>(exp(approximate_log_of_bernoulli_bn)));
}
template <class T, class Policy>
T t2n_asymptotic(int n)
{
BOOST_MATH_STD_USING
// Just get B2n and convert to a Tangent number:
T t2n = fabs(b2n_asymptotic<T, Policy>(2 * n)) / (2 * n);
T p2 = ldexp(T(1), n);
if(tools::max_value<T>() / p2 < t2n)
{
return policies::raise_overflow_error<T>("boost::math::tangent_t2n<%1%>(std::size_t)", nullptr, T(n), Policy());
}
t2n *= p2;
p2 -= 1;
if(tools::max_value<T>() / p2 < t2n)
{
return policies::raise_overflow_error<T>("boost::math::tangent_t2n<%1%>(std::size_t)", nullptr, Policy());
}
t2n *= p2;
return t2n;
}
//
// We need to know the approximate value of /n/ which will
// cause bernoulli_b2n<T>(n) to return infinity - this allows
// us to elude a great deal of runtime checking for values below
// n, and only perform the full overflow checks when we know that we're
// getting close to the point where our calculations will overflow.
// We use Luschny's LogB3 formula (http://www.luschny.de/math/primes/bernincl.html)
// to find the limit, and since we're dealing with the log of the Bernoulli numbers
// we need only perform the calculation at double precision and not with T
// (which may be a multiprecision type). The limit returned is within 1 of the true
// limit for all the types tested. Note that although the code below is basically
// the same as b2n_asymptotic above, it has been recast as a continuous real-valued
// function as this makes the root finding go smoother/faster. It also omits the
// sign of the Bernoulli number.
//
struct max_bernoulli_root_functor
{
explicit max_bernoulli_root_functor(unsigned long long t) : target(static_cast<double>(t)) {}
double operator()(double n) const
{
BOOST_MATH_STD_USING
// Luschny LogB3(n) formula.
const double nx2(n * n);
const double approximate_log_of_bernoulli_bn
= ((boost::math::constants::half<double>() + n) * log(n))
+ ((boost::math::constants::half<double>() - n) * log(boost::math::constants::pi<double>()))
+ (((static_cast<double>(3) / 2) - n) * boost::math::constants::ln_two<double>())
+ ((n * (2 - (nx2 * 7) * (1 + ((nx2 * 30) * ((nx2 * 12) - 1))))) / (((nx2 * nx2) * nx2) * 2520));
return approximate_log_of_bernoulli_bn - target;
}
private:
double target;
};
template <class T, class Policy>
inline std::size_t find_bernoulli_overflow_limit(const std::false_type&)
{
// Set a limit on how large the result can ever be:
static const auto max_result = static_cast<double>((std::numeric_limits<std::size_t>::max)() - 1000u);
unsigned long long t = lltrunc(boost::math::tools::log_max_value<T>());
max_bernoulli_root_functor fun(t);
boost::math::tools::equal_floor tol;
std::uintmax_t max_iter = boost::math::policies::get_max_root_iterations<Policy>();
double result = boost::math::tools::toms748_solve(fun, sqrt(static_cast<double>(t)), static_cast<double>(t), tol, max_iter).first / 2;
if (result > max_result)
{
result = max_result;
}
return static_cast<std::size_t>(result);
}
template <class T, class Policy>
inline std::size_t find_bernoulli_overflow_limit(const std::true_type&)
{
return max_bernoulli_index<bernoulli_imp_variant<T>::value>::value;
}
template <class T, class Policy>
std::size_t b2n_overflow_limit()
{
// This routine is called at program startup if it's called at all:
// that guarantees safe initialization of the static variable.
using tag_type = std::integral_constant<bool, (bernoulli_imp_variant<T>::value >= 1) && (bernoulli_imp_variant<T>::value <= 3)>;
static const std::size_t lim = find_bernoulli_overflow_limit<T, Policy>(tag_type());
return lim;
}
//
// The tangent numbers grow larger much more rapidly than the Bernoulli numbers do....
// so to compute the Bernoulli numbers from the tangent numbers, we need to avoid spurious
// overflow in the calculation, we can do this by scaling all the tangent number by some scale factor:
//
template <class T, typename std::enable_if<std::numeric_limits<T>::is_specialized && (std::numeric_limits<T>::radix == 2), bool>::type = true>
inline T tangent_scale_factor()
{
BOOST_MATH_STD_USING
return ldexp(T(1), std::numeric_limits<T>::min_exponent + 5);
}
template <class T, typename std::enable_if<!std::numeric_limits<T>::is_specialized || !(std::numeric_limits<T>::radix == 2), bool>::type = true>
inline T tangent_scale_factor()
{
return tools::min_value<T>() * 16;
}
//
// We need something to act as a cache for our calculated Bernoulli numbers. In order to
// ensure both fast access and thread safety, we need a stable table which may be extended
// in size, but which never reallocates: that way values already calculated may be accessed
// concurrently with another thread extending the table with new values.
//
// Very very simple vector class that will never allocate more than once, we could use
// boost::container::static_vector here, but that allocates on the stack, which may well
// cause issues for the amount of memory we want in the extreme case...
//
template <class T>
struct fixed_vector : private std::allocator<T>
{
using size_type = unsigned;
using iterator = T*;
using const_iterator = const T*;
fixed_vector() : m_used(0)
{
std::size_t overflow_limit = 5 + b2n_overflow_limit<T, policies::policy<> >();
m_capacity = static_cast<unsigned>((std::min)(overflow_limit, static_cast<std::size_t>(100000u)));
m_data = this->allocate(m_capacity);
}
~fixed_vector()
{
using allocator_type = std::allocator<T>;
using allocator_traits = std::allocator_traits<allocator_type>;
allocator_type& alloc = *this;
for(unsigned i = 0; i < m_used; ++i)
{
allocator_traits::destroy(alloc, &m_data[i]);
}
allocator_traits::deallocate(alloc, m_data, m_capacity);
}
T& operator[](unsigned n) { BOOST_MATH_ASSERT(n < m_used); return m_data[n]; }
const T& operator[](unsigned n)const { BOOST_MATH_ASSERT(n < m_used); return m_data[n]; }
unsigned size()const { return m_used; }
unsigned size() { return m_used; }
bool resize(unsigned n, const T& val)
{
if(n > m_capacity)
{
#ifndef BOOST_MATH_NO_EXCEPTIONS
BOOST_MATH_THROW_EXCEPTION(std::runtime_error("Exhausted storage for Bernoulli numbers."));
#else
return false;
#endif
}
for(unsigned i = m_used; i < n; ++i)
new (m_data + i) T(val);
m_used = n;
return true;
}
bool resize(unsigned n) { return resize(n, T()); }
T* begin() { return m_data; }
T* end() { return m_data + m_used; }
T* begin()const { return m_data; }
T* end()const { return m_data + m_used; }
unsigned capacity()const { return m_capacity; }
void clear() { m_used = 0; }
private:
T* m_data;
unsigned m_used {};
unsigned m_capacity;
};
template <class T, class Policy>
class bernoulli_numbers_cache
{
public:
bernoulli_numbers_cache() : m_overflow_limit((std::numeric_limits<std::size_t>::max)())
, m_counter(0)
, m_current_precision(boost::math::tools::digits<T>())
{}
using container_type = fixed_vector<T>;
bool tangent(std::size_t m)
{
static const std::size_t min_overflow_index = b2n_overflow_limit<T, Policy>() - 1;
if (!tn.resize(static_cast<typename container_type::size_type>(m), T(0U)))
{
return false;
}
BOOST_MATH_INSTRUMENT_VARIABLE(min_overflow_index);
std::size_t prev_size = m_intermediates.size();
m_intermediates.resize(m, T(0U));
if(prev_size == 0)
{
m_intermediates[1] = tangent_scale_factor<T>() /*T(1U)*/;
tn[0U] = T(0U);
tn[1U] = tangent_scale_factor<T>()/* T(1U)*/;
BOOST_MATH_INSTRUMENT_VARIABLE(tn[0]);
BOOST_MATH_INSTRUMENT_VARIABLE(tn[1]);
}
for(std::size_t i = std::max<size_t>(2, prev_size); i < m; i++)
{
bool overflow_check = false;
if(i >= min_overflow_index && (boost::math::tools::max_value<T>() / (i-1) < m_intermediates[1]) )
{
std::fill(tn.begin() + i, tn.end(), boost::math::tools::max_value<T>());
break;
}
m_intermediates[1] = m_intermediates[1] * (i-1);
for(std::size_t j = 2; j <= i; j++)
{
overflow_check =
(i >= min_overflow_index) && (
(boost::math::tools::max_value<T>() / (i - j) < m_intermediates[j])
|| (boost::math::tools::max_value<T>() / (i - j + 2) < m_intermediates[j-1])
|| (boost::math::tools::max_value<T>() - m_intermediates[j] * (i - j) < m_intermediates[j-1] * (i - j + 2))
|| ((boost::math::isinf)(m_intermediates[j]))
);
if(overflow_check)
{
std::fill(tn.begin() + i, tn.end(), boost::math::tools::max_value<T>());
break;
}
m_intermediates[j] = m_intermediates[j] * (i - j) + m_intermediates[j-1] * (i - j + 2);
}
if(overflow_check)
break; // already filled the tn...
tn[static_cast<typename container_type::size_type>(i)] = m_intermediates[i];
BOOST_MATH_INSTRUMENT_VARIABLE(i);
BOOST_MATH_INSTRUMENT_VARIABLE(tn[static_cast<typename container_type::size_type>(i)]);
}
return true;
}
bool tangent_numbers_series(const std::size_t m)
{
BOOST_MATH_STD_USING
static const std::size_t min_overflow_index = b2n_overflow_limit<T, Policy>() - 1;
typename container_type::size_type old_size = bn.size();
if (!tangent(m))
return false;
if (!bn.resize(static_cast<typename container_type::size_type>(m)))
return false;
if(!old_size)
{
bn[0] = 1;
old_size = 1;
}
T power_two(ldexp(T(1), static_cast<int>(2 * old_size)));
for(std::size_t i = old_size; i < m; i++)
{
T b(static_cast<T>(i * 2));
//
// Not only do we need to take care to avoid spurious over/under flow in
// the calculation, but we also need to avoid overflow altogether in case
// we're calculating with a type where "bad things" happen in that case:
//
b = b / (power_two * tangent_scale_factor<T>());
b /= (power_two - 1);
bool overflow_check = (i >= min_overflow_index) && (tools::max_value<T>() / tn[static_cast<typename container_type::size_type>(i)] < b);
if(overflow_check)
{
m_overflow_limit = i;
while(i < m)
{
b = std::numeric_limits<T>::has_infinity ? std::numeric_limits<T>::infinity() : tools::max_value<T>();
bn[static_cast<typename container_type::size_type>(i)] = ((i % 2U) ? b : T(-b));
++i;
}
break;
}
else
{
b *= tn[static_cast<typename container_type::size_type>(i)];
}
power_two = ldexp(power_two, 2);
const bool b_neg = i % 2 == 0;
bn[static_cast<typename container_type::size_type>(i)] = ((!b_neg) ? b : T(-b));
}
return true;
}
template <class OutputIterator>
OutputIterator copy_bernoulli_numbers(OutputIterator out, std::size_t start, std::size_t n, const Policy& pol)
{
//
// There are basically 3 thread safety options:
//
// 1) There are no threads (BOOST_MATH_HAS_THREADS is not defined).
// 2) There are threads, but we do not have a true atomic integer type,
// in this case we just use a mutex to guard against race conditions.
// 3) There are threads, and we have an atomic integer: in this case we can
// use the double-checked locking pattern to avoid thread synchronisation
// when accessing values already in the cache.
//
// First off handle the common case for overflow and/or asymptotic expansion:
//
if(start + n > bn.capacity())
{
if(start < bn.capacity())
{
out = copy_bernoulli_numbers(out, start, bn.capacity() - start, pol);
n -= bn.capacity() - start;
start = static_cast<std::size_t>(bn.capacity());
}
if(start < b2n_overflow_limit<T, Policy>() + 2u)
{
for(; n; ++start, --n)
{
*out = b2n_asymptotic<T, Policy>(static_cast<typename container_type::size_type>(start * 2U));
++out;
}
}
for(; n; ++start, --n)
{
*out = policies::raise_overflow_error<T>("boost::math::bernoulli_b2n<%1%>(std::size_t)", nullptr, T(start), pol);
++out;
}
return out;
}
#if defined(BOOST_MATH_HAS_THREADS) && defined(BOOST_MATH_BERNOULLI_NOTHREADS) && !defined(BOOST_MATH_BERNOULLI_UNTHREADED)
// Add a static_assert on instantiation if we have threads, but no C++11 threading support.
static_assert(sizeof(T) == 1, "Unsupported configuration: your platform appears to have either no atomic integers, or no std::mutex. If you are happy with thread-unsafe code, then you may define BOOST_MATH_BERNOULLI_UNTHREADED to suppress this error.");
#elif defined(BOOST_MATH_BERNOULLI_NOTHREADS)
//
// Single threaded code, very simple:
//
if(m_current_precision < boost::math::tools::digits<T>())
{
bn.clear();
tn.clear();
m_intermediates.clear();
m_current_precision = boost::math::tools::digits<T>();
}
if(start + n >= bn.size())
{
std::size_t new_size = (std::min)((std::max)((std::max)(std::size_t(start + n), std::size_t(bn.size() + 20)), std::size_t(50)), std::size_t(bn.capacity()));
if (!tangent_numbers_series(new_size))
{
return std::fill_n(out, n, policies::raise_evaluation_error<T>("boost::math::bernoulli_b2n<%1%>(std::size_t)", "Unable to allocate Bernoulli numbers cache for %1% values", T(start + n), pol));
}
}
for(std::size_t i = (std::max)(std::size_t(max_bernoulli_b2n<T>::value + 1), start); i < start + n; ++i)
{
*out = (i >= m_overflow_limit) ? policies::raise_overflow_error<T>("boost::math::bernoulli_b2n<%1%>(std::size_t)", nullptr, T(i), pol) : bn[i];
++out;
}
#else
//
// Double-checked locking pattern, lets us access cached already cached values
// without locking:
//
// Get the counter and see if we need to calculate more constants:
//
if((static_cast<std::size_t>(m_counter.load(std::memory_order_consume)) < start + n)
|| (static_cast<int>(m_current_precision.load(std::memory_order_consume)) < boost::math::tools::digits<T>()))
{
std::lock_guard<std::mutex> l(m_mutex);
if((static_cast<std::size_t>(m_counter.load(std::memory_order_consume)) < start + n)
|| (static_cast<int>(m_current_precision.load(std::memory_order_consume)) < boost::math::tools::digits<T>()))
{
if(static_cast<int>(m_current_precision.load(std::memory_order_consume)) < boost::math::tools::digits<T>())
{
bn.clear();
tn.clear();
m_intermediates.clear();
m_counter.store(0, std::memory_order_release);
m_current_precision = boost::math::tools::digits<T>();
}
if(start + n >= bn.size())
{
std::size_t new_size = (std::min)((std::max)((std::max)(std::size_t(start + n), std::size_t(bn.size() + 20)), std::size_t(50)), std::size_t(bn.capacity()));
if (!tangent_numbers_series(new_size))
return std::fill_n(out, n, policies::raise_evaluation_error<T>("boost::math::bernoulli_b2n<%1%>(std::size_t)", "Unable to allocate Bernoulli numbers cache for %1% values", T(new_size), pol));
}
m_counter.store(static_cast<atomic_integer_type>(bn.size()), std::memory_order_release);
}
}
for(std::size_t i = (std::max)(static_cast<std::size_t>(max_bernoulli_b2n<T>::value + 1), start); i < start + n; ++i)
{
*out = (i >= m_overflow_limit) ? policies::raise_overflow_error<T>("boost::math::bernoulli_b2n<%1%>(std::size_t)", nullptr, T(i), pol) : bn[static_cast<typename container_type::size_type>(i)];
++out;
}
#endif // BOOST_MATH_HAS_THREADS
return out;
}
template <class OutputIterator>
OutputIterator copy_tangent_numbers(OutputIterator out, std::size_t start, std::size_t n, const Policy& pol)
{
//
// There are basically 3 thread safety options:
//
// 1) There are no threads (BOOST_MATH_HAS_THREADS is not defined).
// 2) There are threads, but we do not have a true atomic integer type,
// in this case we just use a mutex to guard against race conditions.
// 3) There are threads, and we have an atomic integer: in this case we can
// use the double-checked locking pattern to avoid thread synchronisation
// when accessing values already in the cache.
//
//
// First off handle the common case for overflow and/or asymptotic expansion:
//
if(start + n > bn.capacity())
{
if(start < bn.capacity())
{
out = copy_tangent_numbers(out, start, bn.capacity() - start, pol);
n -= bn.capacity() - start;
start = static_cast<std::size_t>(bn.capacity());
}
if(start < b2n_overflow_limit<T, Policy>() + 2u)
{
for(; n; ++start, --n)
{
*out = t2n_asymptotic<T, Policy>(static_cast<typename container_type::size_type>(start));
++out;
}
}
for(; n; ++start, --n)
{
*out = policies::raise_overflow_error<T>("boost::math::bernoulli_b2n<%1%>(std::size_t)", 0, T(start), pol);
++out;
}
return out;
}
#if defined(BOOST_MATH_BERNOULLI_NOTHREADS)
//
// Single threaded code, very simple:
//
if(m_current_precision < boost::math::tools::digits<T>())
{
bn.clear();
tn.clear();
m_intermediates.clear();
m_current_precision = boost::math::tools::digits<T>();
}
if(start + n >= bn.size())
{
std::size_t new_size = (std::min)((std::max)((std::max)(start + n, std::size_t(bn.size() + 20)), std::size_t(50)), std::size_t(bn.capacity()));
if (!tangent_numbers_series(new_size))
return std::fill_n(out, n, policies::raise_evaluation_error<T>("boost::math::bernoulli_b2n<%1%>(std::size_t)", "Unable to allocate Bernoulli numbers cache for %1% values", T(start + n), pol));
}
for(std::size_t i = start; i < start + n; ++i)
{
if(i >= m_overflow_limit)
*out = policies::raise_overflow_error<T>("boost::math::bernoulli_b2n<%1%>(std::size_t)", nullptr, T(i), pol);
else
{
if(tools::max_value<T>() * tangent_scale_factor<T>() < tn[static_cast<typename container_type::size_type>(i)])
*out = policies::raise_overflow_error<T>("boost::math::bernoulli_b2n<%1%>(std::size_t)", nullptr, T(i), pol);
else
*out = tn[static_cast<typename container_type::size_type>(i)] / tangent_scale_factor<T>();
}
++out;
}
#elif defined(BOOST_MATH_NO_ATOMIC_INT)
static_assert(sizeof(T) == 1, "Unsupported configuration: your platform appears to have no atomic integers. If you are happy with thread-unsafe code, then you may define BOOST_MATH_BERNOULLI_UNTHREADED to suppress this error.");
#else
//
// Double-checked locking pattern, lets us access cached already cached values
// without locking:
//
// Get the counter and see if we need to calculate more constants:
//
if((static_cast<std::size_t>(m_counter.load(std::memory_order_consume)) < start + n)
|| (static_cast<int>(m_current_precision.load(std::memory_order_consume)) < boost::math::tools::digits<T>()))
{
std::lock_guard<std::mutex> l(m_mutex);
if((static_cast<std::size_t>(m_counter.load(std::memory_order_consume)) < start + n)
|| (static_cast<int>(m_current_precision.load(std::memory_order_consume)) < boost::math::tools::digits<T>()))
{
if(static_cast<int>(m_current_precision.load(std::memory_order_consume)) < boost::math::tools::digits<T>())
{
bn.clear();
tn.clear();
m_intermediates.clear();
m_counter.store(0, std::memory_order_release);
m_current_precision = boost::math::tools::digits<T>();
}
if(start + n >= bn.size())
{
std::size_t new_size = (std::min)((std::max)((std::max)(start + n, std::size_t(bn.size() + 20)), std::size_t(50)), std::size_t(bn.capacity()));
if (!tangent_numbers_series(new_size))
return std::fill_n(out, n, policies::raise_evaluation_error<T>("boost::math::bernoulli_b2n<%1%>(std::size_t)", "Unable to allocate Bernoulli numbers cache for %1% values", T(start + n), pol));
}
m_counter.store(static_cast<atomic_integer_type>(bn.size()), std::memory_order_release);
}
}
for(std::size_t i = start; i < start + n; ++i)
{
if(i >= m_overflow_limit)
*out = policies::raise_overflow_error<T>("boost::math::bernoulli_b2n<%1%>(std::size_t)", nullptr, T(i), pol);
else
{
if(tools::max_value<T>() * tangent_scale_factor<T>() < tn[static_cast<typename container_type::size_type>(i)])
*out = policies::raise_overflow_error<T>("boost::math::bernoulli_b2n<%1%>(std::size_t)", nullptr, T(i), pol);
else
*out = tn[static_cast<typename container_type::size_type>(i)] / tangent_scale_factor<T>();
}
++out;
}
#endif // BOOST_MATH_HAS_THREADS
return out;
}
private:
//
// The caches for Bernoulli and tangent numbers, once allocated,
// these must NEVER EVER reallocate as it breaks our thread
// safety guarantees:
//
fixed_vector<T> bn, tn;
std::vector<T> m_intermediates;
// The value at which we know overflow has already occurred for the Bn:
std::size_t m_overflow_limit;
#if !defined(BOOST_MATH_BERNOULLI_NOTHREADS)
std::mutex m_mutex;
atomic_counter_type m_counter, m_current_precision;
#else
int m_counter;
int m_current_precision;
#endif // BOOST_MATH_HAS_THREADS
};
template <class T, class Policy>
inline typename std::enable_if<(std::numeric_limits<T>::digits == 0) || (std::numeric_limits<T>::digits >= INT_MAX), bernoulli_numbers_cache<T, Policy>&>::type get_bernoulli_numbers_cache()
{
//
// When numeric_limits<>::digits is zero, the type has either not specialized numeric_limits at all
// or it's precision can vary at runtime. So make the cache thread_local so that each thread can
// have it's own precision if required:
//
static
#ifndef BOOST_MATH_NO_THREAD_LOCAL_WITH_NON_TRIVIAL_TYPES
BOOST_MATH_THREAD_LOCAL
#endif
bernoulli_numbers_cache<T, Policy> data;
return data;
}
template <class T, class Policy>
inline typename std::enable_if<std::numeric_limits<T>::digits && (std::numeric_limits<T>::digits < INT_MAX), bernoulli_numbers_cache<T, Policy>&>::type get_bernoulli_numbers_cache()
{
//
// Note that we rely on C++11 thread-safe initialization here:
//
static bernoulli_numbers_cache<T, Policy> data;
return data;
}
}}}
#endif // BOOST_MATH_BERNOULLI_DETAIL_HPP