library
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Big Integer(Float)
(Math/bigint.hpp)
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- Last update: 2024-04-26 03:18:17+09:00
- Include:
#include "Math/bigint.hpp"
Depends on
Arbitrary Mod Convolution
(Convolution/arbitrary.hpp)
- Number Theoretic Transform
(Convolution/ntt.hpp)
- Modint
(Math/modint.hpp)
Code
#pragma once
#include "Convolution/arbitrary.hpp"
template <int D> struct bigint {
using u128 = __uint128_t;
static const int B = pow(10, D);
int sign = 0;
vector<int> v;
static int get_D() { return D; }
static int get_B() { return B; }
bigint() {}
bigint(const vector<int> &_v, bool _s = false) : sign(_s), v(_v) {}
bigint(ll x) {
if (x < 0)
x *= -1, sign = 1;
while (x) {
v.push_back(x % B);
x /= B;
}
}
bigint(string s) {
if (s[0] == '-')
s.erase(s.begin()), sign = 1;
int add = 0, cnt = 0, base = 1;
while (s.size()) {
if (cnt == D) {
v.push_back(add);
cnt = 0;
add = 0;
base = 1;
}
add = (s.back() - '0') * base + add;
cnt++;
base *= 10;
s.pop_back();
}
if (add)
v.push_back(add);
}
bigint operator-() const {
bigint res = *this;
res.sign ^= 1;
return res;
}
bigint abs() const {
bigint res = *this;
res.sign = 0;
return res;
}
int &operator[](const int i) { return v[i]; }
int size() const { return v.size(); }
void norm() {
rep(i, 0, v.size() - 1) {
if (v[i] >= 0) {
v[i + 1] += v[i] / B;
v[i] %= B;
} else {
int c = (-v[i] + B - 1) / B;
v[i] += c * B;
v[i + 1] -= c;
}
}
while (!v.empty() and v.back() >= B) {
int c = v.back() / B;
v.back() %= B;
v.push_back(c);
}
while (!v.empty() and v.back() == 0)
v.pop_back();
}
string to_str() const {
string res;
if (v.empty())
return "0";
if (sign)
res += '-';
res += to_string(v.back());
for (int i = v.size() - 2; i >= 0; i--) {
string add;
int w = v[i];
rep(_, 0, D) {
add += ('0' + (w % 10));
w /= 10;
}
reverse(ALL(add));
res += add;
}
return res;
}
friend istream &operator>>(istream &is, bigint<D> &x) {
string tmp;
is >> tmp;
x = bigint(tmp);
return is;
}
friend ostream &operator<<(ostream &os, bigint<D> x) {
os << x.to_str();
return os;
}
bigint &operator<<=(const int &x) {
if (!v.empty()) {
vector<int> add(x, 0);
v.insert(v.begin(), ALL(add));
}
return *this;
}
bigint &operator>>=(const int &x) {
v = vector<int>(v.begin() + min(x, (int)v.size()), v.end());
return *this;
}
bigint &operator+=(const bigint &x) {
if (sign != x.sign) {
*this -= (-x);
return *this;
}
if ((int)v.size() < (int)x.size())
v.resize(x.size(), 0);
rep(i, 0, x.size()) { v[i] += x.v[i]; }
norm();
return *this;
}
bigint &operator-=(const bigint &x) {
if (sign != x.sign) {
*this += (-x);
return *this;
}
if (abs() < x.abs()) {
*this = x - (*this);
sign ^= 1;
return *this;
}
rep(i, 0, x.size()) { v[i] -= x.v[i]; }
norm();
return *this;
}
bigint &operator*=(const bigint &x) {
sign ^= x.sign;
auto v1 = ArbitraryMult<u128>(v, x.v);
u128 add = 0;
v.clear();
v.reserve(v1.size() + 3);
for (int i = 0;; i++) {
if (i >= int(v1.size()) and add == 0)
break;
if (i < int(v1.size()))
add += v1[i];
v.push_back(add % B);
add /= B;
}
norm();
return *this;
}
bigint div_naive(const bigint &a, const bigint &b) {
if (SZ(b) == 1)
return a.div(b.v.back());
if (a < b)
return bigint();
int norm = B / (b.v.back() + 1);
bigint x = a.mul(norm), y = b.mul(norm);
int yb = y.v.back();
bigint quo, rem;
quo.v.resize(x.size() - y.size() + 1);
rem.v = {x.v.end() - y.size(), x.v.end()};
for (int i = x.size() - y.size(); i >= 0; i--) {
if (rem.size() == y.size()) {
if (rem >= y) {
quo[i] = 1;
rem -= y;
}
} else if (rem.size() > y.size()) {
ll rb = ll(rem.v.back()) * B + rem[rem.size() - 2];
int q = rb / yb;
bigint yq = y.mul(q);
while (rem < yq) {
q--;
yq -= y;
}
rem -= yq;
while (rem >= y) {
q++;
rem -= y;
}
quo[i] = q;
}
if (i)
rem.v.insert(rem.v.begin(), x[i - 1]);
}
return quo;
}
bigint &operator/=(const bigint &x) {
bigint a = abs(), b = x.abs();
sign ^= x.sign;
if (a < b)
return *this = bigint();
if (b.size() == 1)
return *this = a.div(b.v.back());
int deg = a.size() - b.size() + 2, k = deg;
while (k > 64)
k = (k + 1) >> 1;
bigint base(1);
base <<= (b.size() + k);
bigint inv(div_naive(base, b));
while (k < deg) {
bigint y = inv.square();
y.v.insert(y.v.begin(), 0);
int l = min(SZ(b), k * 2 + 1);
bigint pref;
pref.v = {b.v.end() - l, b.v.end()};
y *= pref;
y >>= l;
inv = inv + inv;
inv <<= k + 1;
inv -= y;
inv.v.erase(inv.v.begin());
k <<= 1;
}
inv >>= (k - deg);
(*this) = a * inv;
(*this) >>= int(a.size() + 2);
bigint mul = (*this) * b;
while (mul + b <= a) {
(*this) += bigint(1);
mul += b;
}
while (mul > a) {
(*this) -= bigint(1);
mul -= b;
}
return *this;
}
bigint &operator%=(const bigint &x) {
bigint div = (*this) / x;
(*this) -= div * x;
return *this;
}
bigint square() const {
bigint ret;
auto v1 = ArbitraryMult<u128>(v, v);
u128 add = 0;
ret.v.reserve(v1.size() + 3);
for (int i = 0;; i++) {
if (i >= int(v1.size()) and add == 0)
break;
if (i < int(v1.size()))
add += v1[i];
ret.v.push_back(add % B);
add /= B;
}
return ret;
}
bigint mul(ll x) const {
bigint res;
if (x < 0)
res.sign ^= 1, x *= -1;
u128 add = 0;
res.v.reserve(v.size() + 3);
for (int i = 0;; i++) {
if (i >= int(v.size()) and add == 0)
break;
if (i < int(v.size()))
add += ll(v[i]) * x;
res.v.push_back(add % B);
add /= B;
}
return res;
}
bigint div(ll x) const {
bigint res = *this;
if (x < 0)
res.sign ^= 1, x *= -1;
ll add = 0;
for (int i = res.v.size() - 1; i >= 0; i--) {
add = add * B + res.v[i];
int q = add / x, r = add % x;
res.v[i] = q, add = r;
}
res.norm();
return res;
}
bigint operator<<(const int &x) const { return bigint(*this) <<= x; }
bigint operator>>(const int &x) const { return bigint(*this) >>= x; }
bigint operator+(const bigint &x) const { return bigint(*this) += x; }
bigint operator-(const bigint &x) const { return bigint(*this) -= x; }
bigint operator*(const bigint &x) const { return bigint(*this) *= x; }
bigint operator/(const bigint &x) const { return bigint(*this) /= x; }
bigint operator%(const bigint &x) const { return bigint(*this) %= x; }
bool operator<(const bigint &x) const {
if (sign != x.sign)
return sign > x.sign;
if ((int)v.size() != (int)x.size()) {
if (sign)
return (int)x.size() < (int)v.size();
else
return (int)v.size() < (int)x.size();
}
for (int i = v.size() - 1; i >= 0; i--)
if (v[i] != x.v[i]) {
if (sign)
return x.v[i] < v[i];
else
return v[i] < x.v[i];
}
return false;
}
bool operator>(const bigint &x) const { return x < *this; }
bool operator<=(const bigint &x) const { return !(*this > x); }
bool operator>=(const bigint &x) const { return !(*this < x); }
bool operator==(const bigint &x) const {
return !(*this < x) and !(*this > x);
}
bool operator!=(const bigint &x) const { return !(*this == x); }
};
typedef bigint<9> Bigint;
struct Bigfloat {
Bigint v;
int p = 0;
Bigfloat() {}
Bigfloat(const ll &_v) { v = Bigint(_v); }
Bigfloat(const Bigint &_v, int _p = 0) : v(_v), p(_p) {}
void set(int _p) {
if (p < _p) {
v >>= (_p - p);
} else {
v <<= (p - _p);
}
p = _p;
}
Bigint to_int() const {
if (p < 0)
return v >> (-p);
else
return v << p;
}
Bigfloat &operator+=(const Bigfloat &x) {
if (p > x.p)
set(x.p), v += x.v;
else
v += x.v << (x.p - p);
return *this;
}
Bigfloat &operator-=(const Bigfloat &x) {
if (p > x.p)
set(x.p), v -= x.v;
else
v -= x.v << (x.p - p);
return *this;
}
Bigfloat square() {
Bigfloat res = *this;
res.p *= 2;
res.v = res.v.square();
return res;
}
Bigfloat mul(ll x) {
Bigfloat res = *this;
res.v = v.mul(x);
return res;
}
Bigfloat div(ll x) {
Bigfloat res = *this;
res.v = v.div(x);
return res;
}
Bigfloat &operator*=(const Bigfloat &x) {
p += x.p;
v *= x.v;
return *this;
}
Bigfloat &operator/=(const Bigfloat &x) {
p -= x.p;
v /= x.v;
return *this;
}
Bigfloat operator+(const Bigfloat &x) const { return Bigfloat(*this) += x; }
Bigfloat operator-(const Bigfloat &x) const { return Bigfloat(*this) -= x; }
Bigfloat operator*(const Bigfloat &x) const { return Bigfloat(*this) *= x; }
Bigfloat operator/(const Bigfloat &x) const { return Bigfloat(*this) /= x; }
string to_str() {
string res = v.abs().to_str();
int d = Bigint::get_D();
if (p * d > 0)
res += string(p * d, '0');
else if (-p * d >= 1 and -p * d < (int)res.size()) {
res = res.substr(0, (int)res.size() + p * d) + '.' +
res.substr((int)res.size() + p * d);
} else if (-p * d >= (int)res.size())
res = "0." + string(-p * d - (int)res.size(), '0') + res;
if (v.sign) {
res.insert(res.begin(), '-');
}
return res;
}
friend ostream &operator<<(ostream &os, Bigfloat x) {
os << x.to_str();
return os;
}
};
Bigfloat sqrt(ll n, int d) {
Bigfloat res(Bigint((ll)sqrt(1LL * Bigint::get_B() * Bigint::get_B() / n)),
-1),
pre;
int cur = 1;
while (pre.v != res.v) {
cur = min(cur << 1, d);
pre = res;
Bigfloat add = Bigfloat(1) - res.square().mul(n);
add.set(-cur);
res += (res * add).div(2);
res.set(-cur);
}
return res.mul(n);
}
/**
* @brief Big Integer(Float)
*/#line 2 "Convolution/ntt.hpp"
template <typename T> struct NTT {
static constexpr int rank2 = __builtin_ctzll(T::get_mod() - 1);
std::array<T, rank2 + 1> root; // root[i]^(2^i) == 1
std::array<T, rank2 + 1> iroot; // root[i] * iroot[i] == 1
std::array<T, std::max(0, rank2 - 2 + 1)> rate2;
std::array<T, std::max(0, rank2 - 2 + 1)> irate2;
std::array<T, std::max(0, rank2 - 3 + 1)> rate3;
std::array<T, std::max(0, rank2 - 3 + 1)> irate3;
NTT() {
T g = 2;
while (g.pow((T::get_mod() - 1) >> 1) == 1) {
g += 1;
}
root[rank2] = g.pow((T::get_mod() - 1) >> rank2);
iroot[rank2] = root[rank2].inv();
for (int i = rank2 - 1; i >= 0; i--) {
root[i] = root[i + 1] * root[i + 1];
iroot[i] = iroot[i + 1] * iroot[i + 1];
}
{
T prod = 1, iprod = 1;
for (int i = 0; i <= rank2 - 2; i++) {
rate2[i] = root[i + 2] * prod;
irate2[i] = iroot[i + 2] * iprod;
prod *= iroot[i + 2];
iprod *= root[i + 2];
}
}
{
T prod = 1, iprod = 1;
for (int i = 0; i <= rank2 - 3; i++) {
rate3[i] = root[i + 3] * prod;
irate3[i] = iroot[i + 3] * iprod;
prod *= iroot[i + 3];
iprod *= root[i + 3];
}
}
}
void ntt(std::vector<T> &a, bool type = 0) {
int n = int(a.size());
int h = __builtin_ctzll((unsigned int)n);
a.resize(1 << h);
if (type) {
int len = h; // a[i, i+(n>>len), i+2*(n>>len), ..] is transformed
while (len) {
if (len == 1) {
int p = 1 << (h - len);
T irot = 1;
for (int s = 0; s < (1 << (len - 1)); s++) {
int offset = s << (h - len + 1);
for (int i = 0; i < p; i++) {
auto l = a[i + offset];
auto r = a[i + offset + p];
a[i + offset] = l + r;
a[i + offset + p] =
(unsigned long long)(T::get_mod() + l.v - r.v) *
irot.v;
;
}
if (s + 1 != (1 << (len - 1)))
irot *= irate2[__builtin_ctzll(~(unsigned int)(s))];
}
len--;
} else {
// 4-base
int p = 1 << (h - len);
T irot = 1, iimag = iroot[2];
for (int s = 0; s < (1 << (len - 2)); s++) {
T irot2 = irot * irot;
T irot3 = irot2 * irot;
int offset = s << (h - len + 2);
for (int i = 0; i < p; i++) {
auto a0 = 1ULL * a[i + offset + 0 * p].v;
auto a1 = 1ULL * a[i + offset + 1 * p].v;
auto a2 = 1ULL * a[i + offset + 2 * p].v;
auto a3 = 1ULL * a[i + offset + 3 * p].v;
auto a2na3iimag =
1ULL * T((T::get_mod() + a2 - a3) * iimag.v).v;
a[i + offset] = a0 + a1 + a2 + a3;
a[i + offset + 1 * p] =
(a0 + (T::get_mod() - a1) + a2na3iimag) *
irot.v;
a[i + offset + 2 * p] =
(a0 + a1 + (T::get_mod() - a2) +
(T::get_mod() - a3)) *
irot2.v;
a[i + offset + 3 * p] =
(a0 + (T::get_mod() - a1) +
(T::get_mod() - a2na3iimag)) *
irot3.v;
}
if (s + 1 != (1 << (len - 2)))
irot *= irate3[__builtin_ctzll(~(unsigned int)(s))];
}
len -= 2;
}
}
T e = T(n).inv();
for (auto &x : a)
x *= e;
} else {
int len = 0; // a[i, i+(n>>len), i+2*(n>>len), ..] is transformed
while (len < h) {
if (h - len == 1) {
int p = 1 << (h - len - 1);
T rot = 1;
for (int s = 0; s < (1 << len); s++) {
int offset = s << (h - len);
for (int i = 0; i < p; i++) {
auto l = a[i + offset];
auto r = a[i + offset + p] * rot;
a[i + offset] = l + r;
a[i + offset + p] = l - r;
}
if (s + 1 != (1 << len))
rot *= rate2[__builtin_ctzll(~(unsigned int)(s))];
}
len++;
} else {
// 4-base
int p = 1 << (h - len - 2);
T rot = 1, imag = root[2];
for (int s = 0; s < (1 << len); s++) {
T rot2 = rot * rot;
T rot3 = rot2 * rot;
int offset = s << (h - len);
for (int i = 0; i < p; i++) {
auto mod2 = 1ULL * T::get_mod() * T::get_mod();
auto a0 = 1ULL * a[i + offset].v;
auto a1 = 1ULL * a[i + offset + p].v * rot.v;
auto a2 = 1ULL * a[i + offset + 2 * p].v * rot2.v;
auto a3 = 1ULL * a[i + offset + 3 * p].v * rot3.v;
auto a1na3imag =
1ULL * T(a1 + mod2 - a3).v * imag.v;
auto na2 = mod2 - a2;
a[i + offset] = a0 + a2 + a1 + a3;
a[i + offset + 1 * p] =
a0 + a2 + (2 * mod2 - (a1 + a3));
a[i + offset + 2 * p] = a0 + na2 + a1na3imag;
a[i + offset + 3 * p] =
a0 + na2 + (mod2 - a1na3imag);
}
if (s + 1 != (1 << len))
rot *= rate3[__builtin_ctzll(~(unsigned int)(s))];
}
len += 2;
}
}
}
}
vector<T> mult(const vector<T> &a, const vector<T> &b) {
if (a.empty() or b.empty())
return vector<T>();
int as = a.size(), bs = b.size();
int n = as + bs - 1;
if (as <= 30 or bs <= 30) {
if (as > 30)
return mult(b, a);
vector<T> res(n);
rep(i, 0, as) rep(j, 0, bs) res[i + j] += a[i] * b[j];
return res;
}
int m = 1;
while (m < n)
m <<= 1;
vector<T> res(m);
rep(i, 0, as) res[i] = a[i];
ntt(res);
if (a == b)
rep(i, 0, m) res[i] *= res[i];
else {
vector<T> c(m);
rep(i, 0, bs) c[i] = b[i];
ntt(c);
rep(i, 0, m) res[i] *= c[i];
}
ntt(res, 1);
res.resize(n);
return res;
}
};
/**
* @brief Number Theoretic Transform
*/
#line 2 "Math/modint.hpp"
template <unsigned mod = 1000000007> struct fp {
unsigned v;
static constexpr int get_mod() {
return mod;
}
constexpr unsigned inv() const {
assert(v != 0);
int x = v, y = mod, p = 1, q = 0, t = 0, tmp = 0;
while (y > 0) {
t = x / y;
x -= t * y, p -= t * q;
tmp = x, x = y, y = tmp;
tmp = p, p = q, q = tmp;
}
if (p < 0)
p += mod;
return p;
}
constexpr fp(ll x = 0) : v(x >= 0 ? x % mod : (mod - (-x) % mod) % mod) {}
fp operator-() const {
return fp() - *this;
}
fp pow(ull t) {
fp res = 1, b = *this;
while (t) {
if (t & 1)
res *= b;
b *= b;
t >>= 1;
}
return res;
}
fp &operator+=(const fp &x) {
if ((v += x.v) >= mod)
v -= mod;
return *this;
}
fp &operator-=(const fp &x) {
if ((v += mod - x.v) >= mod)
v -= mod;
return *this;
}
fp &operator*=(const fp &x) {
v = ull(v) * x.v % mod;
return *this;
}
fp &operator/=(const fp &x) {
v = ull(v) * x.inv() % mod;
return *this;
}
fp operator+(const fp &x) const {
return fp(*this) += x;
}
fp operator-(const fp &x) const {
return fp(*this) -= x;
}
fp operator*(const fp &x) const {
return fp(*this) *= x;
}
fp operator/(const fp &x) const {
return fp(*this) /= x;
}
bool operator==(const fp &x) const {
return v == x.v;
}
bool operator!=(const fp &x) const {
return v != x.v;
}
friend istream &operator>>(istream &is, fp &x) {
return is >> x.v;
}
friend ostream &operator<<(ostream &os, const fp &x) {
return os << x.v;
}
};
template <unsigned mod> void rd(fp<mod> &x) {
fastio::rd(x.v);
}
template <unsigned mod> void wt(fp<mod> x) {
fastio::wt(x.v);
}
template <typename T> T Inv(ll n) {
static const int md = T::get_mod();
static vector<T> buf({0, 1});
assert(n > 0);
n %= md;
while (SZ(buf) <= n) {
int k = SZ(buf), q = (md + k - 1) / k;
buf.push_back(buf[k * q - md] * q);
}
return buf[n];
}
template <typename T> T Fact(ll n, bool inv = 0) {
static const int md = T::get_mod();
static vector<T> buf({1, 1}), ibuf({1, 1});
assert(n >= 0 and n < md);
while (SZ(buf) <= n) {
buf.push_back(buf.back() * SZ(buf));
ibuf.push_back(ibuf.back() * Inv<T>(SZ(ibuf)));
}
return inv ? ibuf[n] : buf[n];
}
template <typename T> T nPr(int n, int r, bool inv = 0) {
if (n < 0 || n < r || r < 0)
return 0;
return Fact<T>(n, inv) * Fact<T>(n - r, inv ^ 1);
}
template <typename T> T nCr(int n, int r, bool inv = 0) {
if (n < 0 || n < r || r < 0)
return 0;
return Fact<T>(n, inv) * Fact<T>(r, inv ^ 1) * Fact<T>(n - r, inv ^ 1);
}
template <typename T> T nHr(int n, int r, bool inv = 0) {
return nCr<T>(n + r - 1, r, inv);
}
/**
* @brief Modint
*/
#line 4 "Convolution/arbitrary.hpp"
using M1 = fp<1045430273>;
using M2 = fp<1051721729>;
using M3 = fp<1053818881>;
NTT<M1> N1;
NTT<M2> N2;
NTT<M3> N3;
constexpr int r_12 = M2(M1::get_mod()).inv();
constexpr int r_13 = M3(M1::get_mod()).inv();
constexpr int r_23 = M3(M2::get_mod()).inv();
constexpr int r_1323 = M3(ll(r_13) * r_23).v;
constexpr ll w1 = M1::get_mod();
constexpr ll w2 = ll(w1) * M2::get_mod();
template <typename T>
vector<T> ArbitraryMult(const vector<int> &a, const vector<int> &b) {
if (a.empty() or b.empty())
return vector<T>();
int n = a.size() + b.size() - 1;
vector<T> res(n);
if (min(a.size(), b.size()) <= 60) {
rep(i, 0, a.size()) rep(j, 0, b.size()) res[i + j] += T(a[i]) * b[j];
return res;
}
vector<int> vals[3];
vector<M1> a1(ALL(a)), b1(ALL(b)), c1 = N1.mult(a1, b1);
vector<M2> a2(ALL(a)), b2(ALL(b)), c2 = N2.mult(a2, b2);
vector<M3> a3(ALL(a)), b3(ALL(b)), c3 = N3.mult(a3, b3);
for (M1 x : c1)
vals[0].push_back(x.v);
for (M2 x : c2)
vals[1].push_back(x.v);
for (M3 x : c3)
vals[2].push_back(x.v);
rep(i, 0, n) {
ll p = vals[0][i];
ll q = (vals[1][i] + M2::get_mod() - p) * r_12 % M2::get_mod();
ll r = ((vals[2][i] + M3::get_mod() - p) * r_1323 +
(M3::get_mod() - q) * r_23) %
M3::get_mod();
res[i] = (T(r) * w2 + q * w1 + p);
}
return res;
}
template <typename T>
vector<T> ArbitraryMult(const vector<T> &a, const vector<T> &b) {
vector<int> A, B;
for (auto &x : a)
A.push_back(x.v);
for (auto &x : b)
B.push_back(x.v);
return ArbitraryMult<T>(A, B);
}
/**
* @brief Arbitrary Mod Convolution
*/
#line 3 "Math/bigint.hpp"
template <int D> struct bigint {
using u128 = __uint128_t;
static const int B = pow(10, D);
int sign = 0;
vector<int> v;
static int get_D() { return D; }
static int get_B() { return B; }
bigint() {}
bigint(const vector<int> &_v, bool _s = false) : sign(_s), v(_v) {}
bigint(ll x) {
if (x < 0)
x *= -1, sign = 1;
while (x) {
v.push_back(x % B);
x /= B;
}
}
bigint(string s) {
if (s[0] == '-')
s.erase(s.begin()), sign = 1;
int add = 0, cnt = 0, base = 1;
while (s.size()) {
if (cnt == D) {
v.push_back(add);
cnt = 0;
add = 0;
base = 1;
}
add = (s.back() - '0') * base + add;
cnt++;
base *= 10;
s.pop_back();
}
if (add)
v.push_back(add);
}
bigint operator-() const {
bigint res = *this;
res.sign ^= 1;
return res;
}
bigint abs() const {
bigint res = *this;
res.sign = 0;
return res;
}
int &operator[](const int i) { return v[i]; }
int size() const { return v.size(); }
void norm() {
rep(i, 0, v.size() - 1) {
if (v[i] >= 0) {
v[i + 1] += v[i] / B;
v[i] %= B;
} else {
int c = (-v[i] + B - 1) / B;
v[i] += c * B;
v[i + 1] -= c;
}
}
while (!v.empty() and v.back() >= B) {
int c = v.back() / B;
v.back() %= B;
v.push_back(c);
}
while (!v.empty() and v.back() == 0)
v.pop_back();
}
string to_str() const {
string res;
if (v.empty())
return "0";
if (sign)
res += '-';
res += to_string(v.back());
for (int i = v.size() - 2; i >= 0; i--) {
string add;
int w = v[i];
rep(_, 0, D) {
add += ('0' + (w % 10));
w /= 10;
}
reverse(ALL(add));
res += add;
}
return res;
}
friend istream &operator>>(istream &is, bigint<D> &x) {
string tmp;
is >> tmp;
x = bigint(tmp);
return is;
}
friend ostream &operator<<(ostream &os, bigint<D> x) {
os << x.to_str();
return os;
}
bigint &operator<<=(const int &x) {
if (!v.empty()) {
vector<int> add(x, 0);
v.insert(v.begin(), ALL(add));
}
return *this;
}
bigint &operator>>=(const int &x) {
v = vector<int>(v.begin() + min(x, (int)v.size()), v.end());
return *this;
}
bigint &operator+=(const bigint &x) {
if (sign != x.sign) {
*this -= (-x);
return *this;
}
if ((int)v.size() < (int)x.size())
v.resize(x.size(), 0);
rep(i, 0, x.size()) { v[i] += x.v[i]; }
norm();
return *this;
}
bigint &operator-=(const bigint &x) {
if (sign != x.sign) {
*this += (-x);
return *this;
}
if (abs() < x.abs()) {
*this = x - (*this);
sign ^= 1;
return *this;
}
rep(i, 0, x.size()) { v[i] -= x.v[i]; }
norm();
return *this;
}
bigint &operator*=(const bigint &x) {
sign ^= x.sign;
auto v1 = ArbitraryMult<u128>(v, x.v);
u128 add = 0;
v.clear();
v.reserve(v1.size() + 3);
for (int i = 0;; i++) {
if (i >= int(v1.size()) and add == 0)
break;
if (i < int(v1.size()))
add += v1[i];
v.push_back(add % B);
add /= B;
}
norm();
return *this;
}
bigint div_naive(const bigint &a, const bigint &b) {
if (SZ(b) == 1)
return a.div(b.v.back());
if (a < b)
return bigint();
int norm = B / (b.v.back() + 1);
bigint x = a.mul(norm), y = b.mul(norm);
int yb = y.v.back();
bigint quo, rem;
quo.v.resize(x.size() - y.size() + 1);
rem.v = {x.v.end() - y.size(), x.v.end()};
for (int i = x.size() - y.size(); i >= 0; i--) {
if (rem.size() == y.size()) {
if (rem >= y) {
quo[i] = 1;
rem -= y;
}
} else if (rem.size() > y.size()) {
ll rb = ll(rem.v.back()) * B + rem[rem.size() - 2];
int q = rb / yb;
bigint yq = y.mul(q);
while (rem < yq) {
q--;
yq -= y;
}
rem -= yq;
while (rem >= y) {
q++;
rem -= y;
}
quo[i] = q;
}
if (i)
rem.v.insert(rem.v.begin(), x[i - 1]);
}
return quo;
}
bigint &operator/=(const bigint &x) {
bigint a = abs(), b = x.abs();
sign ^= x.sign;
if (a < b)
return *this = bigint();
if (b.size() == 1)
return *this = a.div(b.v.back());
int deg = a.size() - b.size() + 2, k = deg;
while (k > 64)
k = (k + 1) >> 1;
bigint base(1);
base <<= (b.size() + k);
bigint inv(div_naive(base, b));
while (k < deg) {
bigint y = inv.square();
y.v.insert(y.v.begin(), 0);
int l = min(SZ(b), k * 2 + 1);
bigint pref;
pref.v = {b.v.end() - l, b.v.end()};
y *= pref;
y >>= l;
inv = inv + inv;
inv <<= k + 1;
inv -= y;
inv.v.erase(inv.v.begin());
k <<= 1;
}
inv >>= (k - deg);
(*this) = a * inv;
(*this) >>= int(a.size() + 2);
bigint mul = (*this) * b;
while (mul + b <= a) {
(*this) += bigint(1);
mul += b;
}
while (mul > a) {
(*this) -= bigint(1);
mul -= b;
}
return *this;
}
bigint &operator%=(const bigint &x) {
bigint div = (*this) / x;
(*this) -= div * x;
return *this;
}
bigint square() const {
bigint ret;
auto v1 = ArbitraryMult<u128>(v, v);
u128 add = 0;
ret.v.reserve(v1.size() + 3);
for (int i = 0;; i++) {
if (i >= int(v1.size()) and add == 0)
break;
if (i < int(v1.size()))
add += v1[i];
ret.v.push_back(add % B);
add /= B;
}
return ret;
}
bigint mul(ll x) const {
bigint res;
if (x < 0)
res.sign ^= 1, x *= -1;
u128 add = 0;
res.v.reserve(v.size() + 3);
for (int i = 0;; i++) {
if (i >= int(v.size()) and add == 0)
break;
if (i < int(v.size()))
add += ll(v[i]) * x;
res.v.push_back(add % B);
add /= B;
}
return res;
}
bigint div(ll x) const {
bigint res = *this;
if (x < 0)
res.sign ^= 1, x *= -1;
ll add = 0;
for (int i = res.v.size() - 1; i >= 0; i--) {
add = add * B + res.v[i];
int q = add / x, r = add % x;
res.v[i] = q, add = r;
}
res.norm();
return res;
}
bigint operator<<(const int &x) const { return bigint(*this) <<= x; }
bigint operator>>(const int &x) const { return bigint(*this) >>= x; }
bigint operator+(const bigint &x) const { return bigint(*this) += x; }
bigint operator-(const bigint &x) const { return bigint(*this) -= x; }
bigint operator*(const bigint &x) const { return bigint(*this) *= x; }
bigint operator/(const bigint &x) const { return bigint(*this) /= x; }
bigint operator%(const bigint &x) const { return bigint(*this) %= x; }
bool operator<(const bigint &x) const {
if (sign != x.sign)
return sign > x.sign;
if ((int)v.size() != (int)x.size()) {
if (sign)
return (int)x.size() < (int)v.size();
else
return (int)v.size() < (int)x.size();
}
for (int i = v.size() - 1; i >= 0; i--)
if (v[i] != x.v[i]) {
if (sign)
return x.v[i] < v[i];
else
return v[i] < x.v[i];
}
return false;
}
bool operator>(const bigint &x) const { return x < *this; }
bool operator<=(const bigint &x) const { return !(*this > x); }
bool operator>=(const bigint &x) const { return !(*this < x); }
bool operator==(const bigint &x) const {
return !(*this < x) and !(*this > x);
}
bool operator!=(const bigint &x) const { return !(*this == x); }
};
typedef bigint<9> Bigint;
struct Bigfloat {
Bigint v;
int p = 0;
Bigfloat() {}
Bigfloat(const ll &_v) { v = Bigint(_v); }
Bigfloat(const Bigint &_v, int _p = 0) : v(_v), p(_p) {}
void set(int _p) {
if (p < _p) {
v >>= (_p - p);
} else {
v <<= (p - _p);
}
p = _p;
}
Bigint to_int() const {
if (p < 0)
return v >> (-p);
else
return v << p;
}
Bigfloat &operator+=(const Bigfloat &x) {
if (p > x.p)
set(x.p), v += x.v;
else
v += x.v << (x.p - p);
return *this;
}
Bigfloat &operator-=(const Bigfloat &x) {
if (p > x.p)
set(x.p), v -= x.v;
else
v -= x.v << (x.p - p);
return *this;
}
Bigfloat square() {
Bigfloat res = *this;
res.p *= 2;
res.v = res.v.square();
return res;
}
Bigfloat mul(ll x) {
Bigfloat res = *this;
res.v = v.mul(x);
return res;
}
Bigfloat div(ll x) {
Bigfloat res = *this;
res.v = v.div(x);
return res;
}
Bigfloat &operator*=(const Bigfloat &x) {
p += x.p;
v *= x.v;
return *this;
}
Bigfloat &operator/=(const Bigfloat &x) {
p -= x.p;
v /= x.v;
return *this;
}
Bigfloat operator+(const Bigfloat &x) const { return Bigfloat(*this) += x; }
Bigfloat operator-(const Bigfloat &x) const { return Bigfloat(*this) -= x; }
Bigfloat operator*(const Bigfloat &x) const { return Bigfloat(*this) *= x; }
Bigfloat operator/(const Bigfloat &x) const { return Bigfloat(*this) /= x; }
string to_str() {
string res = v.abs().to_str();
int d = Bigint::get_D();
if (p * d > 0)
res += string(p * d, '0');
else if (-p * d >= 1 and -p * d < (int)res.size()) {
res = res.substr(0, (int)res.size() + p * d) + '.' +
res.substr((int)res.size() + p * d);
} else if (-p * d >= (int)res.size())
res = "0." + string(-p * d - (int)res.size(), '0') + res;
if (v.sign) {
res.insert(res.begin(), '-');
}
return res;
}
friend ostream &operator<<(ostream &os, Bigfloat x) {
os << x.to_str();
return os;
}
};
Bigfloat sqrt(ll n, int d) {
Bigfloat res(Bigint((ll)sqrt(1LL * Bigint::get_B() * Bigint::get_B() / n)),
-1),
pre;
int cur = 1;
while (pre.v != res.v) {
cur = min(cur << 1, d);
pre = res;
Bigfloat add = Bigfloat(1) - res.square().mul(n);
add.set(-cur);
res += (res * add).div(2);
res.set(-cur);
}
return res.mul(n);
}
/**
* @brief Big Integer(Float)
*/