Luogu4546 THUWC2017 在美妙的数学王国中畅游 LCT、泰勒展开

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题意：反正就是一堆操作

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LCT总是和玄学东西放在一起
我们不妨令$x_0=0.5$（其实取什么都是一样的，但是最好取在$[0,1]$的范围内），将其代入给出的式子，我们得到的$f(x)$的式子就是一个多项式了。
然后复习一下导数：
$(Cf(x))'=Cf'(x)$（$C$为常数）
$sin'(x)=cos(x),cos'(x)=-sin(x),(e^x)'=e^x,C'=0 , (ax+b)'=a$
令$g(x)=u$，则$f[g(x)]' = f'(u) \times g'(x)$
有了这些式子就可以得到给出的三种函数的任意阶导数了。
但是显然我们不能把所有项的系数都算出来。因为在比较靠后的项中，阶乘的值很大，对答案造成的贡献就会小到忽略不计，所以我们可以取前面若干项，这里我取的是前$12$项。
然后用$LCT$维护这些项的系数和，每一次询问把链拿出来直接算就行了，难题变成裸题了qwq

 #include<bits/stdc++.h>
 #define ld long double
 //This code is written by Itst
 using namespace std;

 inline int read(){
     ;
     ;
     char c = getchar();
     while(c != EOF && !isdigit(c)){
         if(c == '-')
             f = ;
         c = getchar();
     }
     while(c != EOF && isdigit(c)){
         a = (a << ) + (a << ) + (c ^ ');
         c = getchar();
     }
     return f ? -a : a;
 }

 ;
 struct node{
     ld point[] , pre[] , a , b;
     ] , fa , type;
     bool mark;
 }Tree[MAXN];
 int N;
 ];

 inline bool nroot(int x){
     ] == x || Tree[Tree[x].fa].ch[] == x;
 }

 inline bool son(int x){
     ] == x;
 }

 inline ld calc(int type , int n , ld a , ld b){
     ld sum = ;
     switch(type){
     :
         ){
         :
             sum = sin(0.5 * a + b);
             break;
         :
             sum = cos(0.5 * a + b);
             break;
         :
             sum = -sin(0.5 * a + b);
             break;
         :
             sum = -cos(0.5 * a + b);
             break;
         }
         return sum * pow(a , n);
     :
         return pow(a , n) * exp(a * 0.5 + b);
     :
         switch(n){
         :
             return a * 0.5 + b;
         :
             return a;
         default:
             ;
         }
     }
 }

 inline void pushup(int x){
      ; i <=  ; ++i)
         Tree[x].point[i] = Tree[x].pre[i] + Tree[Tree[x].ch[]].point[i] + Tree[Tree[x].ch[]].point[i];
 }

 inline void getpre(int x){
      ; i <=  ; ++i)
         Tree[x].pre[i] = calc(Tree[x].type , i , Tree[x].a , Tree[x].b);
 }

 inline void ZigZag(int x){
     bool f = son(x);
     ];
     if(nroot(y))
         Tree[z].ch[son(y)] = x;
     Tree[x].fa = z;
     Tree[x].ch[f ^ ] = y;
     Tree[y].fa = x;
     Tree[y].ch[f] = w;
     if(w)
         Tree[w].fa = y;
     pushup(y);
     pushup(x);
 }

 inline void pushdown(int x){
     if(Tree[x].mark){
         Tree[Tree[x].ch[]].mark ^= ;
         Tree[Tree[x].ch[]].mark ^= ;
         Tree[x].mark = ;
         swap(Tree[x].ch[] , Tree[x].ch[]);
     }
 }

 void pushdown_all(int x){
     if(nroot(x))
         pushdown_all(Tree[x].fa);
     pushdown(x);
 }

 inline void Splay(int x){
     pushdown_all(x);
     while(nroot(x)){
         if(nroot(Tree[x].fa))
             ZigZag(son(x) == son(Tree[x].fa) ? Tree[x].fa : x);
         ZigZag(x);
     }
 }

 inline void access(int x){
      ; x ; y = x , x = Tree[x].fa){
         Splay(x);
         Tree[x].ch[] = y;
         pushup(x);
     }
 }

 inline int findroot(int x){
     access(x);
     Splay(x);
     pushdown(x);
     ])
         pushdown(x = Tree[x].ch[]);
     Splay(x);
     return x;
 }

 inline void makeroot(int x){
     access(x);
     Splay(x);
     Tree[x].mark ^= ;
 }

 inline void split(int x , int y){
     makeroot(x);
     access(y);
     Splay(y);
 }

 inline void link(int x , int y){
     makeroot(x);
     Tree[x].fa = y;
 }

 inline void cut(int x , int y){
     split(x , y);
     Tree[y].ch[] = Tree[x].fa = ;
     pushup(y);
 }

 inline void change(int x , int type , ld a , ld b){
     access(x);
     Splay(x);
     Tree[x].type = type;
     Tree[x].a = a;
     Tree[x].b = b;
     getpre(x);
     pushup(x);
 }

 int main(){
     freopen("4546.in" , "r" , stdin);
     freopen("4546.out" , "w" , stdout);
     N = read();
     int M = read();
     read();
      ; i <= N ; ++i){
         Tree[i].type = read();
         scanf("%Lf %Lf" , &Tree[i].a , &Tree[i].b);
         getpre(i);
     }
     while(M--){
         ld a , b , times , jc;
         int d , e;
         if(scanf("%s" , s) == EOF)
             ;
         ]){
         case 'a':
             d = read() + ;
             e = read() + ;
             link(d , e);
             break;
         case 'd':
             d = read() + ;
             e = read() + ;
             cut(d , e);
             break;
         case 'm':
             d = read() + ;
             e = read();
             scanf("%Lf %Lf" , &a , &b);
             change(d , e , a , b);
             break;
         case 't':
             d = read() + ;
             e = read() + ;
             scanf("%Lf" , &a);
             if(findroot(d) != findroot(e))
                 puts("unreachable");
             else{
                 split(d , e);
                 b = ;
                 times = jc = ;
                 a -= 0.5;
                  ; i <=  ; ++i){
                     b += times * Tree[e].point[i] / jc;
                     times *= a;
                     jc *= (i + );
                 }
                 printf("%.9Lf\n" , b);
             }
         }
     }
     ;
 }