GMP库中是否实现了对数函数?
B^N
远大于 B^{N-1}
) double LogE(mpq_t m_op)
{
// log(a/b) = log(a) - log(b)
// And if a is represented in base B as:
// a = a_N B^N + a_{N-1} B^{N-1} + ... + a_0
// => log(a) \approx log(a_N B^N)
// = log(a_N) + N log(B)
// where B is the base; ie: ULONG_MAX
static double logB = log(ULONG_MAX);
// Undefined logs (should probably return NAN in second case?)
if (mpz_get_ui(mpq_numref(m_op)) == 0 || mpz_sgn(mpq_numref(m_op)) < 0)
return -INFINITY;
// Log of numerator
double lognum = log(mpq_numref(m_op)->_mp_d[abs(mpq_numref(m_op)->_mp_size) - 1]);
lognum += (abs(mpq_numref(m_op)->_mp_size)-1) * logB;
// Subtract log of denominator, if it exists
if (abs(mpq_denref(m_op)->_mp_size) > 0)
{
lognum -= log(mpq_denref(m_op)->_mp_d[abs(mpq_denref(m_op)->_mp_size)-1]);
lognum -= (abs(mpq_denref(m_op)->_mp_size)-1) * logB;
}
return lognum;
}
log(a) = N log(B) + log(a_N)
的核心概念,这里是ia64的glibc实现。在遇到这个问题后,我再次使用了它。从输入参数 x
中提取 2 的指数幂,即使用 mpf_get_d_2exp() 重写 x = d * 2^exp
。
使得从上一步得到的 d
满足不等式 2/3 <= d <= 4/3
,这可以通过将 d
乘以 2 并做 --exp
操作来实现。这确保了 d
和 1 最多相差 1/3
,换句话说,d
向正负两个方向都扩展了相同的距离。
使用 mpf_div_2exp() 和 mpf_mul_2exp() 将 x
除以 2^exp
。
对 x
进行多次(num_sqrt
次)平方根运算,使得 x
接近 1。这将确保 Taylor 级数更快地收敛。因为进行多次平方根运算比在 Taylor 级数的额外迭代中花费更多时间要快。
计算 ln(1 + x) 的 Taylor 级数 直到所需精度(甚至需要数千位的精度)。
因为在步骤 4. 中我们多次进行了平方根运算,现在我们需要将 y
(Taylor 级数的结果)乘以 2^num_sqrt
。
最后,在步骤 1. 中我们提取了 2^exp
,现在我们需要将 ln(2) * exp
加到 y
中。这里的 ln(2)
只需通过同一个实现整个算法的函数进行一次递归调用即可计算出来。
上述步骤源于公式序列:ln(x) = ln(d * 2^exp) = ln(d) + exp * ln(2) = ln(sqrt(...sqrt(d))) * num_sqrt + exp * ln(2)
。
我的实现会自动进行定时(每次程序运行仅一次),以确定需要多少次平方根运算来平衡 Taylor 级数的计算。如果您需要避免定时,则可以将 mpf_ln()
的第三个参数 sqrt_range
传递为 0.001
,而不是零。
main()
函数包含使用示例、正确性测试(通过与低精度 std::log()进行比较)、时间测量和不同详细信息的输出。该函数在π数的前1024位上进行测试。
在调用我的mpf_ln()
函数之前,请不要忘记通过调用mpf_set_default_prec(bits)来设置所需的计算精度,以位为单位。
我的mpf_ln()
的计算时间约为40-90
微秒,适用于1024
位精度。更大的精度将需要更多的时间,这大致成正比于精度位数的数量。
函数的第一次运行需要相对较长的时间,因为它执行时序表的预计算和ln(2)
值的计算。因此建议在程序启动时首先进行单个计算,以避免后面在代码中出现的时间关键区域内的较长计算。
例如在Linux上编译,您必须安装GMP库并发出以下命令:
clang++-14 -std=c++20 -O3 -lgmp -lgmpxx -o main main.cpp && ./main
#include <cstdint>
#include <iomanip>
#include <iostream>
#include <cmath>
#include <chrono>
#include <mutex>
#include <vector>
#include <unordered_map>
#include <gmpxx.h>
double Time() {
static auto const gtb = std::chrono::high_resolution_clock::now();
return std::chrono::duration_cast<std::chrono::duration<double>>(
std::chrono::high_resolution_clock::now() - gtb).count();
}
mpf_class mpf_ln(mpf_class x, bool verbose = false, double sqrt_range = 0) {
auto total_time = verbose ? Time() : 0.0;
int const prec = mpf_get_prec(x.get_mpf_t());
if (sqrt_range == 0) {
static std::mutex mux;
std::lock_guard<std::mutex> lock(mux);
static std::vector<std::pair<size_t, double>> ranges;
if (ranges.empty())
mpf_ln(3.14, false, 0.01);
while (ranges.empty() || ranges.back().first < prec) {
size_t const bits = ranges.empty() ? 64 : ranges.back().first * 3 / 2;
mpf_class x = 3.14;
mpf_set_prec(x.get_mpf_t(), bits);
double sr = 0.35, sr_best = 1, time_best = 1000;
size_t constexpr ntests = 5;
while (true) {
auto tim = Time();
for (size_t i = 0; i < ntests; ++i)
mpf_ln(x, false, sr);
tim = (Time() - tim) / ntests;
bool updated = false;
if (tim < time_best) {
sr_best = sr;
time_best = tim;
updated = true;
}
sr /= 1.5;
if (sr <= 1e-8) {
ranges.push_back(std::make_pair(bits, sr_best));
break;
}
}
}
sqrt_range = std::lower_bound(ranges.begin(), ranges.end(), size_t(prec),
[](auto const & a, auto const & b){
return a.first < b;
})->second;
}
signed long int exp = 0;
// https://gmplib.org/manual/Converting-Floats
double d = mpf_get_d_2exp(&exp, x.get_mpf_t());
if (d < 2.0 / 3) {
d *= 2;
--exp;
}
mpf_class t;
// https://gmplib.org/manual/Float-Arithmetic
if (exp >= 0)
mpf_div_2exp(x.get_mpf_t(), x.get_mpf_t(), exp);
else
mpf_mul_2exp(x.get_mpf_t(), x.get_mpf_t(), -exp);
auto sqrt_time = verbose ? Time() : 0.0;
// Multiple Sqrt of x
int num_sqrt = 0;
if (x >= 1)
while (x >= 1.0 + sqrt_range) {
// https://gmplib.org/manual/Float-Arithmetic
mpf_sqrt(x.get_mpf_t(), x.get_mpf_t());
++num_sqrt;
}
else
while (x <= 1.0 - sqrt_range) {
mpf_sqrt(x.get_mpf_t(), x.get_mpf_t());
++num_sqrt;
}
if (verbose)
sqrt_time = Time() - sqrt_time;
static mpf_class const eps = [&]{
mpf_class eps = 1;
mpf_div_2exp(eps.get_mpf_t(), eps.get_mpf_t(), prec + 8);
return eps;
}(), meps = -eps;
// Taylor Serie for ln(1 + x)
// https://math.stackexchange.com/a/878376/826258
x -= 1;
mpf_class k = x, y = x, mx = -x;
size_t num_iters = 0;
for (int32_t i = 2;; ++i) {
k *= mx;
y += k / i;
// Check if error is small enough
if (meps <= k && k <= eps) {
num_iters = i;
break;
}
}
auto VerboseInfo = [&]{
if (!verbose)
return;
total_time = Time() - total_time;
std::cout << std::fixed << "Sqrt range " << sqrt_range << ", num sqrts "
<< num_sqrt << ", sqrt time " << sqrt_time << " sec" << std::endl;
std::cout << "Ln number of iterations " << num_iters << ", ln time "
<< total_time << " sec" << std::endl;
};
// Correction due to multiple sqrt of x
y *= 1 << num_sqrt;
if (exp == 0) {
VerboseInfo();
return y;
}
mpf_class ln2;
{
static std::mutex mutex;
std::lock_guard<std::mutex> lock(mutex);
static std::unordered_map<size_t, mpf_class> ln2s;
auto it = ln2s.find(size_t(prec));
if (it == ln2s.end()) {
mpf_class sqrt_sqrt_2 = 2;
mpf_sqrt(sqrt_sqrt_2.get_mpf_t(), sqrt_sqrt_2.get_mpf_t());
mpf_sqrt(sqrt_sqrt_2.get_mpf_t(), sqrt_sqrt_2.get_mpf_t());
it = ln2s.insert(std::make_pair(size_t(prec), mpf_class(mpf_ln(sqrt_sqrt_2, false, sqrt_range) * 4))).first;
}
ln2 = it->second;
}
y += ln2 * exp;
VerboseInfo();
return y;
}
std::string mpf_str(mpf_class const & x) {
mp_exp_t exp;
auto s = x.get_str(exp);
return s.substr(0, exp) + "." + s.substr(exp);
}
int main() {
// https://gmplib.org/manual/Initializing-Floats
mpf_set_default_prec(1024); // bit-precision
// http://www.math.com/tables/constants/pi.htm
mpf_class x(
"3."
"1415926535 8979323846 2643383279 5028841971 6939937510 "
"5820974944 5923078164 0628620899 8628034825 3421170679 "
"8214808651 3282306647 0938446095 5058223172 5359408128 "
"4811174502 8410270193 8521105559 6446229489 5493038196 "
"4428810975 6659334461 2847564823 3786783165 2712019091 "
"4564856692 3460348610 4543266482 1339360726 0249141273 "
"7245870066 0631558817 4881520920 9628292540 9171536436 "
);
std::cout << std::boolalpha << std::fixed << std::setprecision(14);
std::cout << "x:" << std::endl << mpf_str(x) << std::endl;
auto cmath_val = std::log(mpf_get_d(x.get_mpf_t()));
std::cout << "cmath ln(x): " << std::endl << cmath_val << std::endl;
auto volatile tmp = mpf_ln(x); // Pre-Compute to heat-up timings table.
auto time_start = Time();
size_t constexpr ntests = 20;
for (size_t i = 0; i < ntests; ++i) {
auto volatile tmp = mpf_ln(x);
}
std::cout << "mpf ln(x) time " << (Time() - time_start) / ntests << " sec" << std::endl;
auto mpf_val = mpf_ln(x, true);
std::cout << "mpf ln(x):" << std::endl << mpf_str(mpf_val) << std::endl;
std::cout << "equal to cmath: " << (std::abs(mpf_get_d(mpf_val.get_mpf_t()) - cmath_val) <= 1e-14) << std::endl;
return 0;
}
输出:
x:
3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067982148086513282306647093844609550582231725359408128481117450284102701938521105559644622948954930381964428810975665933446128475648233786783165271201909145648566923460348610454326648213393607260249141273724587007
cmath ln(x):
1.14472988584940
mpf ln(x) time 0.00004426845000 sec
Sqrt range 0.00000004747981, num sqrts 23, sqrt time 0.00001440000000 sec
Ln number of iterations 42, ln time 0.00003873100000 sec
mpf ln(x):
1.144729885849400174143427351353058711647294812915311571513623071472137769884826079783623270275489707702009812228697989159048205527923456587279081078810286825276393914266345902902484773358869937789203119630824756794011916028217227379888126563178049823697313310695003600064405487263880223270096433504959511813198
equal to cmath: true
bigintegerR.cc
中的函数 biginteger_log
下找到(您首先需要下载源代码(即tar文件))。您还可以在此处查看它:biginteger_log。// Adapted for general use from the original biginteger_log
// xi = di * 2 ^ ex ==> log(xi) = log(di) + ex * log(2)
double biginteger_log_modified(mpz_t x) {
signed long int ex;
const double di = mpz_get_d_2exp(&ex, x);
return log(di) + log(2) * (double) ex;
}
这是它的链接: https://github.com/linas/anant
提供 GNU MP 实数和复数对数、指数、正弦、余弦、伽马函数、反正切函数、平方根、多对数 Riemann 和 Hurwitz zeta 函数、互变超几何函数、拓扑正弦函数等更多功能。
mpz_get_d_2exp
并利用对数的性质,可以获得log值。 - Joseph Wood