5020: [THUWC 2017]在美妙的数学王国中畅游

传送门

当年听llj讲的时候觉得这简直是个不可做的神题．

现在看来并不是很神，可能是我已经被剧透了的缘故．．．

一开始以为是函数套函数，懵蔽了好久，结果只是求和

被剧透了泰勒展开就比较水了．．只要你不像我一样蠢的最最简单的求导都求错．．．

还有不像我一样蠢展开了看到有常数项不暴力二项式定理展开转而展开f(a*x+b)发现不会求e^b．．．

那么直接泰勒展开然后二项式定理暴力展开后用lct合并即可，维护个17项就差不多了

 //Achen
 #include<algorithm>
 #include<iostream>
 #include<cstring>
 #include<cstdlib>
 #include<cstdio>
 #include<cmath>
 #include<queue>
 #include<set>
 #include<map>
 #define For(i,a,b) for(int i=(a);i<=(b);i++)
 #define Rep(i,a,b) for(int i=(a);i>=(b);i--)
 const int N=+;
 typedef long long LL;
 typedef double db;
 using namespace std;
 int n,m;
 char o[];
 db C[][],inv[],a_i[],b_i[];

 template<typename T>void read(T &x) {
     char ch=getchar(); x=; T f=;
     while(ch!='-'&&(ch<''||ch>'')) ch=getchar();
     if(ch=='-') f=-,ch=getchar();
     for(;ch>=''&&ch<='';ch=getchar()) x=x*+ch-''; x*=f;
 }

 int ch[N][],p[N],flip[N];
 #define lc ch[x][0]
 #define rc ch[x][1]
 struct data {
     db a[];
     friend data operator +(const data&A,const data&B) {
         data rs;
         For(i,,) rs.a[i]=A.a[i]+B.a[i];
         return rs;
     }
 }dt[N],sum[N];

 int isroot(int x) {return (ch[p[x]][]!=x&&ch[p[x]][]!=x);}

 void update(int x) { sum[x]=dt[x]+sum[lc]+sum[rc]; }

 void down(int x) {
     if(!flip[x]) return;
     swap(lc,rc);
     flip[x]^=;
     flip[lc]^=;
     flip[rc]^=;
 } 

 void rotate(int x) {
     int y=p[x],z=p[y],l=(x==ch[y][]),r=l^;
     if(!isroot(y)) ch[z][y==ch[z][]]=x; p[x]=z;
     ch[y][l]=ch[x][r]; p[ch[x][r]]=y;
     ch[x][r]=y; p[y]=x;
     update(y); update(x);
 } 

 void splay(int x) {
     static int g[N],top=,tp;
     for(tp=x;!isroot(tp);tp=p[tp]) g[++top]=tp;
     g[++top]=tp;
     while(top) {down(g[top--]);}
     for(;!isroot(x);rotate(x)) {
         int y=p[x],z=p[y];
         if(!isroot(y))
             ((x==ch[y][])^(y==ch[z][]))?rotate(x):rotate(y);
     }
 }

 void access(int x) {
     for(int t=;x;x=p[t=x]) {
         splay(x);
         rc=t;
         update(x);
     }
 }

 int find_root(int x) {
     access(x);
     splay(x);
     while(lc) x=lc;
     return x;
 }

 void newroot(int x) {
     access(x);
     splay(x);
     flip[x]^=;
 }

 void lik(int x,int y) {
     if(find_root(x)==find_root(y)) return;
     newroot(x);
     splay(x);
     p[x]=y;
 }

 void cut(int x,int y) {
     newroot(x);
     access(y);
     splay(y);
     if(ch[y][]==x) ch[y][]=p[x]=; update(y);
 }

 void get_it(data &tp,int f,db a,db b) {
     For(i,,) tp.a[i]=;
     if(f==) { //sin(a*x+b);
         a_i[]=b_i[]=;
         For(i,,) a_i[i]=a_i[i-]*a,b_i[i]=b_i[i-]*b;
         db f=1.0;
         for(int i=;i<=;i+=,f=-f)
             For(j,,i) tp.a[j]+=f*inv[i]*a_i[j]*b_i[i-j]*C[i][j];
     }
     else if(f==) { //e^(a*x+b);
         a_i[]=b_i[]=;
         For(i,,) a_i[i]=a_i[i-]*a,b_i[i]=b_i[i-]*b;
         For(i,,) For(j,,i)
             tp.a[j]+=inv[i]*a_i[j]*b_i[i-j]*C[i][j];
     }
     else tp.a[]=b,tp.a[]=a;
 }

 db calc(data tp,db x) {
     db rs=,now=;
     For(i,,) {
         rs=rs+tp.a[i]*now;
         now*=x;
     }
     return rs;
 }

 void change(int x,int f,db a,db b) {
     splay(x);
     get_it(dt[x],f,a,b);
     update(x);
 }

 void qry(int x,int y,db z) {
     if(find_root(x)!=find_root(y)) {
         puts("unreachable");
         return;
     }
     newroot(x);
     access(y);
     splay(y);
     printf("%.8le\n",calc(sum[y],z));
 }

 //#define DEBUG
 int main() {
 #ifdef DEBUG
     freopen("1.in","r",stdin);
     //freopen("1.out","w",stdout);
 #endif
     read(n); read(m); scanf("%s",o);
     For(i,,) C[i][]=;
     For(i,,) For(j,,i) C[i][j]=C[i-][j]+C[i-][j-];
     inv[]=inv[]=;
     For(i,,) inv[i]=inv[i-]/(1.0*i);
     for(int i=;i<=n;i++) {
         int f; db a,b;
         scanf("%d %lf %lf",&f,&a,&b);
         get_it(dt[i],f,a,b);
     }
     while(m--) {
         scanf("%s",o);
         if(o[]=='a') {
             int x,y; read(x); read(y);
             lik(x+,y+);
         }
         else if(o[]=='d') {
             int x,y; read(x); read(y); cut(x+,y+);
         }
         else if(o[]=='m') {
             int x,f; db a,b;
             scanf("%d %d %lf %lf",&x,&f,&a,&b);
             change(x+,f,a,b);
         }
         else if(o[]=='t') {
             int x,y; db z;
             scanf("%d %d %lf",&x,&y,&z);
             qry(x+,y+,z);
         }
     }
     return ;
 }