C++, 93ms

Compile args g++ -O2, input from stdin as float128 binary, output as float128

#include <stdio.h>
#include <math.h>
#include <time.h>
#include <algorithm>
typedef long double ld;
typedef unsigned uint;
void setId(ld& x, uint y) { 3[(int*)&x] = y; }
uint getId(ld& x) { return 3[(int*)&x]; }
typedef struct pz{
    unsigned int a[4];
    bool operator< (const pz& o) const {
        return *(unsigned long*)&a[1] >
             *(unsigned long*)&o.a[1]; }
} *pp;
void mult(uint N, ld* in, ld* out) {
    uint M = pow(N,.7); fprintf(stderr, "IN %ld\n", clock());
    static ld tmp[9999999];
    for (uint i=0; i<N; ++i) {
        tmp[i] = in[i];
        setId(tmp[i], i);
    }
    for (uint i=0; i<N+N-1; ++i) {
        out[i] = 0;
    }
    std::nth_element(pp(tmp), pp(tmp+M), pp(tmp+N));
    std::sort(pp(tmp), pp(tmp+M)); fprintf(stderr, "PRE %ld %d\n", clock(), M);
    for (uint i=0; i<M; ++i)
    for (uint j=0; j<M; ++j) {//fprintf(stderr, "[%d %d]", M, getId(tmp[0]));
        out[getId(tmp[i])+getId(tmp[j])] += tmp[i] * tmp[j];
    }
    ld thro = tmp[M] * tmp[M] * 1e-4;fprintf(stderr, "IRO %ld\n", clock());
    int kk = 0;
    for (uint i=0; i<N+N-1; ++i) if (out[i]<thro){ ++kk;
        ld ret = 0;
        if (i < N) {
            for (uint j=0; j<=i; ++j)
                ret += in[j] * in[i-j];
        } else { // i-j=N-1
            for (uint j=i-N+1; j<N; ++j)
                ret += in[j] * in[i-j];
        }
        out[i] = ret;
    }fprintf(stderr, "IIRO %ld %d\n", clock(), kk);
}

int main() {
    const int N = 100000;
    static ld Input[N];
    static ld Output[N*2];
    //for (uint i=0; i<N; ++i) scanf("%Le", Input+i);
    fread(Input, sizeof Input, 1, stdin);
    mult(N, Input, Output);
    fread(Output, sizeof Output, 1, stdout);
    //for (uint i=0; i<N+N-1; ++i) printf("%Le\n", Output[i]);
    
    /*
    mult(10, Input, Output);
    for (int i=0; i<20; ++i) printf("%Le\n", Output[i]);*/
}

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