# 高精度

## 无符号整数

vector 自带大小比较为字典序比较， `!=``==` 运算，其余需要时一定记得重载！

``````struct Wint : vector<int> //继承vector
{
static const int width = 9, base = 1e9;
Wint(unsigned long long n = 0) //普通初始化，当整型数和Wint同时运算时会提升至Wint
{
for (; n; n /= base)
push_back(n % base);
}
explicit Wint(const string &s) //字符串初始化函数，未判断字符串合法情况
{
for (int len = int(s.size() - 1) / width + 1, b, e, i = 0; i < len; ++i)
for (e = s.size() - i * width, b = max(0, e - width), push_back(0); b != e; ++b)
back() = back() * 10 + s[b] - '0';
trim(0);
}
Wint &trim(bool up = 1) //去前导0，是否需要进位，很常用的小函数，为方便返回自身
{
for (int i = 1; up && i < size(); ++i)
{
if ((*this)[i - 1] < 0)
--(*this)[i], (*this)[i - 1] += base;
if ((*this)[i - 1] >= base)
(*this)[i] += (*this)[i - 1] / base, (*this)[i - 1] %= base;
}
while (!empty() && back() <= 0)
pop_back();
for (; up && !empty() && back() >= base; (*this)[size() - 2] %= base)
push_back(back() / base);
return *this;
}
friend istream &operator>>(istream &is, Wint &n)
{
string s; //懒
return is >> s, n = Wint(s), is;
}
friend ostream &operator<<(ostream &os, const Wint &n)
{
if (n.empty())
return os.put('0');
os << n.back();
char ch = os.fill('0');
for (int i = n.size() - 2; ~i; --i)
os.width(n.width), os << n[i];
return os.fill(ch), os;
}
friend bool operator<(const Wint &a, const Wint &b)
{
if (a.size() != b.size())
return a.size() < b.size();
for (int i = a.size() - 1; ~i; --i)
if (a[i] != b[i])
return a[i] < b[i];
return 0;
}
friend bool operator>(const Wint &a, const Wint &b) { return b < a; }
friend bool operator<=(const Wint &a, const Wint &b) { return !(a > b); }
friend bool operator>=(const Wint &a, const Wint &b) { return !(a < b); }
Wint &operator+=(const Wint &b)
{
if (size() < b.size())
resize(b.size()); //保证有足够的位数
for (int i = 0; i < b.size(); ++i)
(*this)[i] += b[i];
return trim(); //单独进位防自运算
}
friend Wint operator+(Wint a, const Wint &b) { return a += b; }
Wint &operator++() { return *this += 1; } //前置版本，懒
Wint operator++(int)					  //后置版本
{
Wint b(*this);
return ++*this, b;
}
Wint &operator-=(const Wint &b) //a<b会使a变为0
{
if (size() < b.size())
resize(b.size()); //保证有足够的位数
for (int i = 0; i < b.size(); ++i)
(*this)[i] -= b[i];
return trim(); //单独进位防自运算
}
friend Wint operator-(Wint a, const Wint &b) { return a -= b; }
Wint &operator--() { return *this -= 1; } //前置版本，懒
Wint operator--(int)					  //后置版本
{
Wint b(*this);
return --*this, b;
}
Wint &operator*=(const Wint &b) //高精度乘法，常规写法
{
Wint c;
c.assign(size() + b.size(), 0);
for (int j = 0, k, l; j < b.size(); ++j)
if (b[j]) //稀疏优化，特殊情况很有效
for (int i = 0; i < size(); ++i)
{
unsigned long long n = (*this)[i];
for (n *= b[j], k = i + j; n; n /= base)
c[k++] += n % base;
for (l = i + j; c[l] >= base || l + 1 < k; c[l++] %= base)
c[l + 1] += c[l] / base;
}
return swap(c), trim(0);
}
/*
Wint &operator*=(const Wint &b) //一种效率略高但对位宽有限制的写法
{
vector<unsigned long long> n(size() + b.size(), 0); //防爆int
//乘法算完后统一进位效率高，防止乘法溢出（unsigned long long范围0~1.8e19）
//位宽为9时size()不能超过18（十进制162位），位宽为8时size()不能超过1800（十进制14400位）等等。
for (int j = 0; j != b.size(); ++j)
if (b[j]) //稀疏优化，特殊情况很有效
for (int i = 0; i != size(); ++i)
n[i + j] += (unsigned long long)(*this)[i] * b[j];
for (int i = 1; i < n.size(); ++i) //这里用<防止位数0，单独进位防自运算
n[i] += n[i - 1] / base, n[i - 1] %= base;
return assign(n.begin(), n.end()), trim(0);
}
Wint &operator*=(const Wint &b) //fft优化乘法，注意double仅15位有效数字，调小Wint::width不超过2，计算自2*log2(base)+2*log2(len)<53
{
vector<ll> ax(begin(), end()), bx(b.begin(), b.end());
ax = FFT(size() + b.size()).ask(ax, bx);
for (int i = 1; i < ax.size(); ++i)
ax[i] += ax[i - 1] / base, ax[i - 1] %= base;
return assign(ax.begin(), ax.end()), trim(0);
}
Wint &operator*=(const Wint &b) //ntt优化，Wint::width不超过2
{
vector<ll> ax(begin(), end()), bx(b.begin(), b.end());
ax = FNTT(size() + b.size(), (7 << 26) + 1, 3).ask(ax, bx);
for (int i = 1; i < ax.size(); ++i)
ax[i] += ax[i - 1] / base, ax[i - 1] %= base;
return assign(ax.begin(), ax.end()), trim(0);
}
*/
friend Wint operator*(Wint a, const Wint &b) { return a *= b; }
Wint &operator/=(Wint b)
{
Wint r, c, d = b.base / (b.back() + 1);
*this *= d, b *= d, c.assign(size(), 0);
for (int i = size() - 1; ~i; --i)
{
r.insert(r.begin(), (*this)[i]);
unsigned long long s = 0;
for (int j = b.size(); j + 1 >= b.size(); --j) //b.size()==0肯定第一行就出问题的
s = s * b.base + (j < r.size() ? r[j] : 0);
for (d = c[i] = s / b.back(), d *= b; r < d; r += b)
--c[i];
r -= d;
}
return swap(c), trim(0); //r为加倍后的余数，可通过高精度除低精度得到真正余数，此处略
}
friend Wint operator/(Wint a, const Wint &b) { return a /= b; }
Wint &operator%=(const Wint &b) { return *this -= *this / b * b; }
friend Wint operator%(Wint a, const Wint &b) { return a %= b; }
//开平方，改自ZJU模板
bool cmp(long long c, int d, const Wint &b) const
{
if ((int)b.size() - (int)size() < d + 1 && c)
return 1;
long long t = 0;
for (int i = b.size() - 1, lo = -(base << 1); lo <= t && t <= 0 && ~i; --i)
if (t = t * base - b[i], 0 <= i - d - 1 && i - d - 1 < size())
t += (*this)[i - d - 1] * c;
return t > 0;
}
Wint &sub(const Wint &b, long long k, int d)
{
int l = b.size() + d;
for (int i = d + 1; i <= l; ++i)
{
long long tmp = (*this)[i] - k * b[i - d - 1];
if (tmp < 0)
{
(*this)[i + 1] += (tmp - base + 1) / base;
(*this)[i] = tmp - (tmp - base + 1) / base * base;
}
else
(*this)[i] = tmp;
}
for (int i = l + 1; i < size() && (*this)[i] < 0; ++i)
{
(*this)[i + 1] += ((*this)[i] - base + 1) / base;
(*this)[i] -= ((*this)[i] - base + 1) / base * base;
}
return trim(0);
}
friend Wint sqrt(Wint a)
{
Wint n;
n.assign(a.size() + 1 >> 1, 0);
for (int i = n.size() - 1, l, r; ~i; --i)
{
for (l = 0, r = a.base, n[i] = l + r >> 1; r - l > 1; n[i] = l + r >> 1)
{
if (n.cmp(n[i], i - 1, a))
r = n[i];
else
l = n[i];
}
a.sub(n, n[i], i - 1), n[i] += l + r >> 1;
}
for (int i = 0; i < n.size(); ++i)
n[i] >>= 1;
return n.trim(0);
}
/*
friend Wint sqrt(const Wint &a) //常规牛顿迭代实现的开平方算法，慢但是好敲
{
Wint b = a, c = (b + 1) / 2;
while (b != c)
swap(b, c), c = (b + a / b) / 2;
return c;
}
friend Wint sqrt(const Wint &a)
{
Wint ret, t;
ret.assign((a.size() + 1) >> 1, 0);
for (int i = ret.size() - 1, l, r; ~i; --i)
{
for (l = 0, r = a.base; r - l > 1;)
{
ret[i] = l + (r - l) / 2;
t = ret * ret;
if (a < t)
r = ret[i];
else
l = ret[i];
}
if (!l && i == ret.size() - 1)
ret.pop_back();
else
ret[i] = l;
}
return ret;
}
*/
};
``````

## 分数

``````struct Fraction
{
ll num, den;
Fraction(ll n = 0, ll d = 1) : num(n), den(d)
{
d = __gcd(num, den), num /= d, den /= d;
if (den < 0)
num = -num, den = -den;
}
friend Fraction operator+(const Fraction &A, const Fraction &B)
{
ll d = __gcd(A.den, B.den);
return Fraction(B.den / d * A.num + A.den / d * B.num, A.den / d * B.den);
}
Fraction &operator+=(const Fraction &c) { return *this = *this + c; }
Fraction operator-() const
{
Fraction r(*this);
return r.num = -r.num, r;
}
friend Fraction operator-(const Fraction &a, const Fraction &c) { return -c + a; }
Fraction &operator-=(const Fraction &c) { return *this = *this - c; }
friend Fraction operator*(const Fraction &A, const Fraction &B) { return Fraction(A.num * B.num, A.den * B.den); }
Fraction &operator*=(const Fraction &c) { return *this = *this * c; }
friend Fraction operator/(const Fraction &A, const Fraction &B) { return Fraction(A.num * B.den, A.den * B.num); }
Fraction &operator/=(const Fraction &c) { return *this = *this / c; }
friend Fraction operator%(const Fraction &a, const Fraction &c) { return Fraction(a.num * c.den % (c.num * a.den), a.den * c.den); }
Fraction &operator%=(const Fraction &c) { return *this = *this % c; }
friend bool operator==(const Fraction &a, const Fraction &b) { return a.num * b.den == a.den * b.num; }
friend bool operator!=(const Fraction &a, const Fraction &b) { return !(a == b); }
friend bool operator<(const Fraction &a, const Fraction &b) { return a.num * b.den < a.den * b.num; }
friend bool operator>(const Fraction &a, const Fraction &b) { return b < a; }
friend bool operator<=(const Fraction &a, const Fraction &b) { return !(a > b); }
friend bool operator>=(const Fraction &a, const Fraction &b) { return !(a < b); }
friend Fraction abs(Fraction f)
{
if (f.num < 0)
f.num = -f.num;
return f;
}
friend ostream &operator<<(ostream &os, const Fraction &f) { return !f.num ? os << 0 : f.den == 1 ? os << f.num : os << f.num << '/' << f.den; }
};
``````

## bitset 高精度

``````typedef bitset<127> Bint;
/*
Bint b(unsigned long long u=0);//用u的低N位初始化b，N是一个可转成ULL类型的常量表达式，高位补0
Bint bs(string s,int pos,int m=string::npos,char zero='0',char one='1');//用s从pos位开始的m位初始化b，s中只含zero和one

b.size();//b的大小，即N
b.count();//b中1的个数

b[pos];//b中pos位的引用

b.set();//b全赋1
b.reset();//b全赋0
b.flip();//b全反转

b.to_ull();//b转成unsigned long long，b.size()>64时出错
b.to_string(char zero='0',char one='1');//按参数输出字符串

os<<b;//按'0'和'1'打印b
is>>b;//按'0'和'1'读入b，当下一个字符不是'0'或'1'或读到b.size()个数后停止

==、!=、<<、>>、&、|、^//保持它们在整型运算中的含义
*/
//大小比较，其他运算符类似
bool operator<(const Bint &a, const Bint &b)
{
for (int i = a.size() - 1; ~i; --i)
if (a[i] != b[i])
return a[i] < b[i];
return 0;
}
bool operator!=(const Bint &a, const Bint &b)
{
for (int i = a.size() - 1; ~i; --i)
if (a[i] != b[i])
return 1;
return 0;
}
//加法
Bint operator+(const Bint &a, const Bint &b) { return b.any() ? (a ^ b) + ((a & b) << 1) : a; }
Bint &operator+=(Bint &a, const Bint &b) { return a = a + b; }
//减法
Bint operator-(const Bint &a) { return Bint(1) + ~a; }
Bint &operator-=(Bint &a, const Bint &b) { return a += -b; }
Bint operator-(Bint a, const Bint &b) { return a -= b; }
//乘法
Bint operator*(Bint a, Bint b)
{
Bint r(0);
for (; b.any(); b >>= 1, a += a)
if (b[0])
r += a;
return r;
}
Bint &operator*=(Bint &a, const Bint &b) { return a = a * b; }
//整除，取模
Bint operator%=(Bint &dividend, const Bint &divisor)
{
if (dividend < divisor || divisor.none())
return dividend;
Bint c, res = 0;
for (int k; divisor < dividend;)
{
for (k = 0, c = divisor; !(dividend < c); c <<= 1, ++k)
if (dividend < divisor + c)
{
res += 1 << k;
break;
}
if (dividend < divisor + c)
break;
res += 1 << (k - 1);
dividend -= c >> 1;
}
return dividend; //res是商
}
//输入输出，bitset已经原生重载了输入输出运算符，避免歧义。
istream &getb(istream &is, Bint &val)
{
int sign = 1, ch = is.get();
for (; !isdigit(ch) && ch != EOF; ch = is.get())
if (ch == '-')
sign = -sign;
for (val = 0; isdigit(ch); ch = is.get())
val = (val << 3) + (val << 1) + (ch ^ '0');
if (sign == -1)
val = -val;
return is.putback(ch);
}
ostream &putb(ostream &os, const Bint &val)
{
if (Bint(9) < val)
putb(os, val / 10);
return os.put(val.to_ulong() % 10 + '0');
}
``````