题意
给定长度为 $n$ 的数组 $a$ 和 $b$，有 $q$ 次询问，每次询问给定整数 $l, r, x$，求
$$\sum\limits_{i = l}^r\frac{a_i}{b_i + x}$$
$1 \leq 3\times 10^5, 1 \leq q, a_i, b_i \leq 10^6$
精度 $10^{-6}$ 时限 4 s
题解
好题，可以先思考一下
查看题解
$$\sum\limits_{i = l}^r\frac{a_i}{b_i + x}$$
对 $f\left(x\right) = \dfrac a{b + x}$ 进行 麦克劳林展开。展开具体过程
根据 麦克劳林级数公式：
$$f\left(x\right) = \sum\limits_{n = 0}^\infin \frac{f^{\left(n\right)}\left(0\right)}{n!}x^n$$
首先，$f\left(x\right) = \dfrac a{b + x}$
然后，我们可以先对 $f\left(x\right) = \dfrac a{b + x}$ 求导，得到 $f'\left(x\right) = -\dfrac a{\left(b + x\right)^2}$，根据公式，取 $x = 0$，得到 $f'\left(0\right) = -\dfrac a{b^2}$。
接下来，对 $f'\left(x\right) = -\dfrac a{\left(b + x\right)^2}$ 求导，得到 $f''\left(x\right) = \dfrac{2a}{\left(b + x\right)^3}$，根据公式，取 $x = 0$，得到 $f''\left(0\right) = -\dfrac{2a}{b^3}$。
之后在求一项，$f'''\left(x\right) = -\dfrac{6a}{\left(b + x\right)^4}$，就可以发现规律了。
$$f^{\left(n\right)} = \left(-1\right)^n \cdot \frac{a\cdot n!}{\left(b + x\right)^{n + 1}}$$
带入 麦克劳林级数公式 得到：
$$f\left(x\right) = \sum\limits_{n = 0}^\infin \left(-1\right)^n \cdot \frac{a \cdot x^n}{b^{n+1}}$$
写成不用 $\Sigma$ 的形式：
$$\frac ab - \frac{a x}{b^2} + \frac{a x^2}{b^3} - \frac {a x^3}{b^4} + \frac {a x^4}{b^5} - \frac {a x^5}{b^6} + \frac {a x^6}{b^7} - \frac{a x^7}{b^8} + \cdots$$
精度不需要太高，算十几项足以。接着做前缀和，这样对于一个固定的 $x$，查一段区间的 $\dfrac{a_i}{b_i + x}$ 和就只需要常数时间了，但要线性时间预处理。
在 $\dfrac a{b + x}$ 中，$x$ 越大，对答案的影响越小。我们可以让 $h$ 从 $1$ 开始，每次增加 $25\%$，对于每个 $x$ 在 $\left[h, 1.25h\right)$ 之间的询问，把所有 $x$ 都当做 $h$ 算答案即可。
代码
// Please submit with C++17! It's best to use C++20 or higher version.
// By Koicy (https://koicy.ly)
// rbtree (i@koicy.ly)
// This is my kingdom code.
// #define EFILE ""
#include <algorithm>
#include <array>
#include <cmath>
#include <cstring>
#include <iomanip>
#include <list>
#include <map>
#include <numeric>
#include <queue>
#include <random>
#include <set>
#include <stack>
#include <string>
#include <tuple>
#include <unordered_map>
#include <utility>
#include <vector>
#define _CONSOLE 0
#define _MULTI_TESTS 0
using namespace std;
using tp = long long;
constexpr tp ZERO = 0, ONE = 1, INF32 = -1u >> 2, INF = -1ull >> 2;
// :\
namespace _IO {
constexpr size_t BUF = 217217;
unsigned long long h = 5;
char ibuf[BUF], obuf[BUF];
char *li, *ri, *lo = obuf, *ro = obuf + BUF, st[100], *tp = st;
FILE *Istream = stdin, *Ostream = stdout;
char GC() {
#if _CONSOLE
return getchar();
#endif
if (li == ri) {
li = ibuf; ri = li + fread(li, 1, BUF, Istream);
if (li == ri) return '\n';
}
return *li++;
}
void _flush() {
fwrite(obuf, 1, lo - obuf, Ostream);
lo = obuf;
}
void PC(char ch) {
#if _CONSOLE
return (void)putchar(ch);
#endif
if (lo == ro) _flush();
*lo++ = ch;
}
template <typename Type>
void r_int(Type& x) {
bool neg = 0;
char ch = GC();
while (ch < 48 || ch > 57) { neg ^= ch == 45; ch = GC(); }
for (x = 0; ch >= '0' && ch <= '9'; ch = GC()) x = x * 10 + (ch & 15);
if (neg) x = -x;
}
template <typename Type>
void r_uint(Type& x) {
char ch = GC();
while (ch < 48 || ch > 57) ch = GC();
for (x = 0; ch >= '0' && ch <= '9'; ch = GC()) x = x * 10 + (ch & 15);
}
template <typename Type>
void r_db(Type& x) {
bool neg = 0;
char ch = GC();
while (ch < 48 || ch > 57) { neg ^= ch == 45; ch = GC(); }
for (x = 0; ch >= '0' && ch <= '9'; ch = GC()) x = x * 10 + (ch & 15);
if (ch == '.') {
double p = 1;
for (ch = GC(); ch >= '0' && ch <= '9'; ch = GC()) x = x + (p /= 10) * (ch & 15);
}
if (neg) x = -x;
}
template <typename Type>
void w_uint(Type x) {
if (!x) { PC('0'); return; }
while (x) { *tp++ = x % 10; x /= 10; }
while (tp > st) PC(*--tp ^ 48);
}
template <typename Type>
void w_int(Type x) {
if (x < 0) { PC('-'); w_uint(-x); }
else w_uint(x);
}
template <typename Type>
void w_db(Type x) {
double l = floor(abs(x));
string s;
if (x < 0) { PC('-'); x = -x; }
x -= l;
if (!l) PC(48);
while (l) { s.push_back(l - floor(l / 10) * 10); l = floor(l / 10); }
for (auto i = s.rbegin(); i != s.rend(); ++i) PC(*i ^ 48);
PC(46);
for (unsigned long long f = 0; f < h; ++f) {
x *= 10;
PC(floor(x - floor(x / 10) * 10) + 48);
}
}
struct IO {
IO& operator>>(char& x) { do { x = GC(); } while (x == 32 || x == 10 || x == 13); return *this; }
IO& operator>>(string& x) {
char c;
operator>>(c);
x = c;
for (c = GC(); c != 32 && c != 10 && c != 13; c = GC()) x.push_back(c);
x.shrink_to_fit();
return *this;
}
IO& operator>>(short& x) { r_int(x); return *this; }
IO& operator>>(int& x) { r_int(x); return *this; }
IO& operator>>(long& x) { r_int(x); return *this; }
IO& operator>>(long long& x) { r_int(x); return *this; }
IO& operator>>(unsigned short& x) { r_uint(x); return *this; }
IO& operator>>(unsigned int& x) { r_uint(x); return *this; }
IO& operator>>(unsigned long& x) { r_uint(x); return *this; }
IO& operator>>(unsigned long long& x) { r_uint(x); return *this; }
IO& operator>>(float& x) { r_db(x); return *this; }
IO& operator>>(double& x) { r_db(x); return *this; }
IO& operator<<(short x) { w_int(x); return *this; }
IO& operator<<(int x) { w_int(x); return *this; }
IO& operator<<(long x) { w_int(x); return *this; }
IO& operator<<(long long x) { w_int(x); return *this; }
IO& operator<<(unsigned short x) { w_uint(x); return *this; }
IO& operator<<(unsigned int x) { w_uint(x); return *this; }
IO& operator<<(unsigned long x) { w_uint(x); return *this; }
IO& operator<<(unsigned long long x) { w_uint(x); return *this; }
IO& operator<<(char x) { PC(x); return *this; }
IO& operator<<(const string& x) { for (auto& i : x) PC(i); return *this; }
IO& operator<<(float& x) { w_db(x); return *this; }
IO& operator<<(double& x) { w_db(x); return *this; }
void flush() { _flush(); }
void precision(unsigned long long p) { h = p; }
IO() {
#ifdef EFILE
Istream = fopen(EFILE ".in", "r");
Ostream = fopen(EFILE ".out", "w");
#else
#ifdef _LOCAL
Istream = fopen("input.txt", "r");
#endif
#endif
}
~IO() { _flush(); }
} bin;
}
using _IO::bin;
// :/
struct Query {
tp l, r, x, id;
};
tp n, m, l;
void STRUGGLING([[maybe_unused]] tp TEST_NUMBER) {
bin >> n >> m;
vector<tp> a(n), b(n);
vector<Query> q(m);
vector<double> tar(m);
vector<vector<double>> f(15, vector<double>(n));
bin.precision(12);
for (tp i = 0; i < n; ++i) bin >> a[i];
for (tp i = 0; i < n; ++i) bin >> b[i];
for (tp i = 0; i < m; ++i) bin >> q[i].l >> q[i].r >> q[i].x;
for (tp i = 0; i < m; ++i) { --q[i].l; --q[i].r; q[i].id = i; }
stable_sort(q.begin(), q.end(), [](Query a, Query b) { return a.x < b.x; });
for (double h = 1; h <= 1000000; h *= 1.25) {
for (tp i = 0; i < n; ++i) {
double den = 1;
for (tp j = 0; j < 15; ++j) {
den *= b[i] + h;
if (i) f[j][i] = f[j][i - 1] + (j & 1 ? -a[i] / den : a[i] / den);
else f[j][i] = j & 1 ? -a[i] / den : a[i] / den;
}
}
while (l < m && q[l].x >= h && q[l].x < h * 1.25) {
double res = 0, px = 1;
for (int i = 0; i < 15; ++i) {
if (!q[l].l) res += px * f[i][q[l].r];
else res += px * (f[i][q[l].r] - f[i][q[l].l - 1]);
px *= q[l].x - h;
}
tar[q[l++].id] = res;
}
}
for (auto& i : tar) bin << i << '\n';
}
void MIST() {
}
#include <fstream>
signed main() {
tp t = 0, _t = 1;
MIST();
#if _MULTI_TESTS
bin >> _t;
#endif
while (t < _t) STRUGGLING(++t);
return 0;
}
//*/