cp-library
競技プログラミング用 C++ ライブラリ
test/kth_root.test.cpp
test/kth_root.test.cpp · GitHub で見る
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コード
#define PROBLEM "https://judge.yosupo.jp/problem/kth_root_integer"
#include "math/kth_root.hpp"
#include "components/fast_io.hpp"
int main() {
FastInput input;
FastOutput output;
int t = 0;
input.read(t);
while (t--) {
std::uint64_t n = 0, k = 0;
input.read(n), input.read(k);
output.println(kth_root(n, k));
}
}
#line 1 "test/kth_root.test.cpp"
#define PROBLEM "https://judge.yosupo.jp/problem/kth_root_integer"
#line 1 "math/kth_root.hpp"
#include <array>
#include <algorithm>
#include <limits>
#include <bit>
#include <cassert>
#include <cmath>
#include <cstdint>
namespace kth_root_detail {
inline constexpr auto thresholds = [] {
std::array<std::array<std::uint64_t, 13>, 25> table{};
for (unsigned k = 16; k <= 40; ++k) for (unsigned a = 3; a <= 15; ++a) {
std::uint64_t p = 1;
for (unsigned i = 0; i < k; ++i) {
if (p > UINT64_MAX / a) { p = 0; break; }
p *= a;
}
table[k - 16][a - 3] = p - 1;
}
return table;
}();
inline std::uint64_t cube(std::uint64_t n) {
if constexpr (std::numeric_limits<double>::is_iec559 && std::numeric_limits<double>::digits == 53 && sizeof(double) == sizeof(std::uint64_t)) {
static constexpr auto inverse_roots = [] {
std::array<double, 64> result{};
for (int i = 0; i < 64; ++i) {
double a = 1 + (i + 0.5) / 64, x = 1;
for (int j = 0; j < 8; ++j) x += x * (1 - a * x * x * x) / 3;
result[i] = x;
}
return result;
}();
double a = double(n);
std::uint64_t bits = std::bit_cast<std::uint64_t>(a);
unsigned exponent = unsigned(bits >> 52) - 1023;
constexpr double factor[] = {1, 0.79370052598409973738, 0.62996052494743658238};
double x = inverse_roots[(bits >> 46) & 63] * factor[exponent % 3]
* std::bit_cast<double>(std::uint64_t(1023 - exponent / 3) << 52);
{
double h = 1 - a * x * x * x;
x += x * h * (1.0 / 3 + h * (2.0 / 9 + h * (14.0 / 81)));
}
std::uint64_t r = std::min(std::uint64_t(1 / x), std::uint64_t(2642245));
while (r * r * r > n) --r;
while (r < 2642245 && (r + 1) * (r + 1) * (r + 1) <= n) ++r;
return r;
} else {
std::uint64_t r = std::cbrt(double(n));
while (__uint128_t(r) * r * r > n) --r;
while (__uint128_t(r + 1) * (r + 1) * (r + 1) <= n) ++r;
return r;
}
}
template<unsigned K>
constexpr double power(double x) {
double value = 1;
for (unsigned k = K;; x *= x) {
if (k & 1) value *= x;
if (!(k >>= 1)) return value;
}
}
template<unsigned K>
inline double estimate(std::uint64_t n) {
if constexpr (std::numeric_limits<double>::is_iec559 && std::numeric_limits<double>::digits == 53 && sizeof(double) == sizeof(std::uint64_t)) {
static constexpr auto roots = [] {
std::array<double, 64> values{};
for (int i = 0; i < 64; ++i) {
double a = 1 + (i + 0.5) / 64, x = 1;
for (int j = 0; j < 10; ++j) x += x * (1 - a * power<K>(x)) / K;
values[i] = x;
}
return values;
}();
static constexpr auto factors = [] {
std::array<double, K> values{};
double x = 1;
for (int j = 0; j < 10; ++j) x += x * (1 - 2 * power<K>(x)) / K;
values[0] = 1;
for (unsigned j = 1; j < K; ++j) values[j] = values[j - 1] * x;
return values;
}();
double a = double(n);
auto bits = std::bit_cast<std::uint64_t>(a);
unsigned exponent = unsigned(bits >> 52) - 1023;
double x = roots[(bits >> 46) & 63] * factors[exponent % K]
* std::bit_cast<double>(std::uint64_t(1023 - exponent / K) << 52);
double h = 1 - a * power<K>(x);
constexpr double c = 1.0 / K;
x += x * h * (c + h * (c * (c + 1) / 2 + h * (c * (c + 1) * (c + 2) / 6)));
return 1 / x;
} else return std::pow(double(n), 1.0 / K);
}
template <unsigned K>
inline std::uint64_t small(std::uint64_t n) {
auto leq = [n](std::uint64_t x) {
std::uint64_t a = x, r = 1;
for (unsigned k = K;;) {
if ((k & 1) && __builtin_mul_overflow(r, a, &r)) return false;
if (!(k >>= 1)) return r <= n;
if (__builtin_mul_overflow(a, a, &a)) return false;
}
};
std::uint64_t lo = 1, hi = std::uint64_t(1) << ((std::bit_width(n) - 1) / K + 1);
double estimate;
if constexpr (K == 4) estimate = std::sqrt(std::sqrt(double(n)));
else if constexpr (K == 8) estimate = std::sqrt(std::sqrt(std::sqrt(double(n))));
else estimate = kth_root_detail::estimate<K>(n);
std::uint64_t r = std::uint64_t(estimate);
if (r >= hi) r = hi - 1;
if (r < lo) r = lo;
if (leq(r)) {
if (!leq(r + 1)) return r;
lo = r + 1;
} else {
if (leq(r - 1)) return r - 1;
hi = r - 1;
}
while (lo + 1 < hi) {
std::uint64_t mid = (lo + hi) / 2;
if (leq(mid)) lo = mid;
else hi = mid;
}
return lo;
}
}
inline std::uint64_t kth_root(std::uint64_t n, std::uint64_t k) {
assert(k > 0);
if (k == 1 || n <= 1) return n;
if (k >= unsigned(std::bit_width(n))) return 1;
if (k == 2) {
std::uint64_t r = std::sqrt(double(n));
if (__uint128_t(r) * r > n) --r;
if (__uint128_t(r + 1) * (r + 1) <= n) ++r;
return r;
}
if (k >= 16) {
if (k > 40) return 2;
unsigned lo = 3, hi = 1U << (63 / k + 1);
while (lo < hi) {
unsigned mid = (lo + hi) / 2;
if (n > kth_root_detail::thresholds[k - 16][mid - 3]) lo = mid + 1;
else hi = mid;
}
return lo - 1;
}
switch (k) {
case 3: return kth_root_detail::cube(n);
case 4: return kth_root_detail::small<4>(n);
case 5: return kth_root_detail::small<5>(n);
case 6: return kth_root_detail::small<6>(n);
case 7: return kth_root_detail::small<7>(n);
case 8: return kth_root_detail::small<8>(n);
case 9: return kth_root_detail::small<9>(n);
case 10: return kth_root_detail::small<10>(n);
case 11: return kth_root_detail::small<11>(n);
case 12: return kth_root_detail::small<12>(n);
case 13: return kth_root_detail::small<13>(n);
case 14: return kth_root_detail::small<14>(n);
default: return kth_root_detail::small<15>(n);
}
}
#line 1 "components/fast_io.hpp"
#line 8 "components/fast_io.hpp"
#include <charconv>
#include <cstddef>
#line 11 "components/fast_io.hpp"
#include <cstdio>
#include <cstring>
#include <memory>
#include <stdexcept>
#include <string>
#include <string_view>
#include <type_traits>
#if defined(__unix__) || defined(__APPLE__)
#include <sys/mman.h>
#include <sys/stat.h>
#include <unistd.h>
#endif
namespace fast_io_detail { struct BulkInput; struct BulkOutput; }
class FastInput {
friend struct fast_io_detail::BulkInput;
static constexpr std::size_t capacity = 1 << 16;
FILE *file = nullptr;
std::unique_ptr<char[]> buffer;
const char *current = "", *end = current;
std::string scratch;
void *mapping = nullptr;
std::size_t mapped_size = 0;
bool refill() {
if (!file) return false;
std::size_t n = std::fread(buffer.get(), 1, capacity, file);
current = buffer.get();
end = current + n;
return n;
}
static const char *scan(const char *p, const char *end) {
while (end - p >= 8) {
std::uint64_t x;
std::memcpy(&x, p, 8);
if ((x - 0x2121212121212121ULL) & ~x & 0x8080808080808080ULL) break;
p += 8;
}
while (p != end && static_cast<unsigned char>(*p) > ' ') ++p;
return p;
}
public:
explicit FastInput(FILE *input = stdin) {
#if defined(__unix__) || defined(__APPLE__)
struct stat status;
int fd = fileno(input);
auto offset = ftello(input);
if (fd >= 0 && offset >= 0 && !fstat(fd, &status) && S_ISREG(status.st_mode) && status.st_size > offset) {
mapped_size = status.st_size;
mapping = mmap(nullptr, mapped_size, PROT_READ, MAP_PRIVATE, fd, 0);
if (mapping != MAP_FAILED) {
current = static_cast<const char *>(mapping) + offset;
end = static_cast<const char *>(mapping) + mapped_size;
return;
}
mapping = nullptr;
}
#endif
file = input;
buffer = std::make_unique_for_overwrite<char[]>(capacity);
}
explicit FastInput(std::string_view input) : current(input.empty() ? "" : input.data()), end(current + input.size()) {}
FastInput(const FastInput &) = delete;
FastInput &operator=(const FastInput &) = delete;
~FastInput() {
#if defined(__unix__) || defined(__APPLE__)
if (mapping) munmap(mapping, mapped_size);
#endif
}
std::string_view token() {
while (true) {
while (current != end && static_cast<unsigned char>(*current) <= ' ') ++current;
if (current != end) break;
if (!refill()) return {};
}
const char *begin = current;
current = scan(current, end);
if (current != end || !file) return {begin, std::size_t(current - begin)};
scratch.assign(begin, end);
while (refill()) {
current = scan(current, end);
scratch.append(buffer.get(), std::size_t(current - buffer.get()));
if (current != end) break;
}
return scratch;
}
template <class T>
bool read(T &value) {
if constexpr (std::is_integral_v<T> && !std::is_same_v<T, bool>) {
while (true) {
while (current != end && static_cast<unsigned char>(*current) <= ' ') ++current;
if (current != end) break;
if (!refill()) return false;
}
if constexpr (std::is_unsigned_v<T> && sizeof(T) == 4 && std::endian::native == std::endian::little) {
if (end - current >= 11) {
std::uint64_t x;
std::memcpy(&x, current, 8);
if (!(((x + 0x4646464646464646ULL) | (x - 0x3030303030303030ULL)) & 0x8080808080808080ULL)) {
x &= 0x0f0f0f0f0f0f0f0fULL;
x = (x * 10 + (x >> 8)) & 0x00ff00ff00ff00ffULL;
x = (x * 100 + (x >> 16)) & 0x0000ffff0000ffffULL;
std::uint64_t v = std::uint32_t(x * 10000 + (x >> 32));
int n = 8;
for (; n < 10 && static_cast<unsigned char>(current[n]) - unsigned('0') < 10; ++n)
v = v * 10 + (current[n] - '0');
if (static_cast<unsigned char>(current[n]) <= ' ' && v <= 0xffffffffULL) {
current += n;
value = T(v);
return true;
}
}
}
}
T parsed{};
auto result = std::from_chars(current, end, parsed);
if (result.ec == std::errc{} && (result.ptr != end ? static_cast<unsigned char>(*result.ptr) <= ' ' : !file)) {
current = result.ptr;
value = parsed;
return true;
}
}
auto s = token();
if (s.empty()) return false;
if constexpr (std::is_same_v<T, bool>) {
assert(s == "0" || s == "1");
value = s[0] == '1';
} else if constexpr (std::is_integral_v<T>) {
auto result = std::from_chars(s.data(), s.data() + s.size(), value);
assert(result.ec == std::errc{} && result.ptr == s.data() + s.size());
(void)result;
} else value = T(s);
return true;
}
};
class FastOutput {
friend struct fast_io_detail::BulkOutput;
inline static constexpr auto quads = [] {
std::array<std::array<char, 4>, 10000> table{};
for (int i = 0; i < 10000; ++i)
table[i] = {char('0' + i / 1000), char('0' + i / 100 % 10), char('0' + i / 10 % 10), char('0' + i % 10)};
return table;
}();
FILE *file;
std::unique_ptr<char[]> buffer = std::make_unique_for_overwrite<char[]>(1 << 16);
std::size_t capacity = 1 << 16, length = 0;
char *reserve(std::size_t n) {
if (n > capacity - length) {
if (!flush()) throw std::runtime_error("output failed");
if (n > capacity) {
std::size_t next = std::max(n, 2 * capacity);
buffer = std::make_unique_for_overwrite<char[]>(next);
capacity = next;
}
}
return buffer.get() + length;
}
public:
explicit FastOutput(FILE *output = stdout) : file(output) {}
FastOutput(const FastOutput &) = delete;
FastOutput &operator=(const FastOutput &) = delete;
~FastOutput() { flush(); }
bool flush() {
std::size_t offset = 0;
while (offset < length) {
auto n = std::fwrite(buffer.get() + offset, 1, length - offset, file);
if (!n) {
std::memmove(buffer.get(), buffer.get() + offset, length - offset);
length -= offset;
return false;
}
offset += n;
}
length = 0;
return std::fflush(file) == 0;
}
void write(char c) { *reserve(1) = c; ++length; }
void write(__uint128_t value) {
char digits[40];
char *end = digits + sizeof(digits), *begin = end;
do { *--begin = char('0' + value % 10); value /= 10; } while (value);
write(std::string_view(begin, end - begin));
}
void write(__int128_t value) {
__uint128_t magnitude = __uint128_t(value);
if (value < 0) write('-'), magnitude = 0 - magnitude;
write(magnitude);
}
void write(std::string_view s) {
char *out = reserve(s.size());
if (!s.empty()) std::memcpy(out, s.data(), s.size());
length += s.size();
}
template <class T> requires std::is_integral_v<T>
void write(T value) {
char *out = reserve(32);
if constexpr (std::is_same_v<T, bool>) *out = '0' + value, ++length;
else if constexpr (std::is_unsigned_v<T> && sizeof(T) == 4) {
char *p = out;
if (value >= 100000000) {
auto high = value / 100000000, low = value % 100000000;
p = std::to_chars(p, p + 2, high).ptr;
std::memcpy(p, quads[low / 10000].data(), 4);
std::memcpy(p + 4, quads[low % 10000].data(), 4);
p += 8;
} else if (value >= 10000) {
p = std::to_chars(p, p + 4, value / 10000).ptr;
std::memcpy(p, quads[value % 10000].data(), 4);
p += 4;
} else p = std::to_chars(p, p + 4, value).ptr;
length += p - out;
} else length += std::to_chars(out, out + 32, value).ptr - out;
}
template <class T> requires requires(const T &value, char *out) { value.decimal_capacity(); value.to_chars(out); }
void write(const T &value) {
char *out = reserve(value.decimal_capacity());
length += value.to_chars(out) - out;
}
template <class T>
void println(const T &value) { write(value); write('\n'); }
};
#line 4 "test/kth_root.test.cpp"
int main() {
FastInput input;
FastOutput output;
int t = 0;
input.read(t);
while (t--) {
std::uint64_t n = 0, k = 0;
input.read(n), input.read(k);
output.println(kth_root(n, k));
}
}