Full precision not respected in some cases

Summary

The Eigen::FullPrecision bit is not respected in my cases. I got the same output while the input double of two Eigen matrix has about 3 ULP difference.

Environment

Operating System : Linux
Architecture : x64
Eigen Version : 3.4.0
Compiler Version : Gcc9.4
Compile Flags : -mno-avx512f -fopenmp -O2 -g -DNDEBUG -fPIC -fdiagnostics-color=always -fPIC -mfpmath=sse -msse -msse2 -msse3 -mssse3 -std=c++17
Vector Extension : SSE

Minimal Example

godbolt reproduce demo

//show your code here
#include <Eigen/Dense>
#include <fmt/format.h>
#include <iostream>

int main() {

    Eigen::Matrix4d a, b;
    a << 0.86602540001429662, -0.50000000653007737, 0, 0,
         0.50000000653007737, 0.86602540001429662, 0, 0,
         0, 0, 1, 0,
         0, 0, 0, 1;
    b << 0.86602540001429695, -0.50000000653007703, 0, 0,
         0.50000000653007703, 0.86602540001429695, 0, 0,
         0, 0, 1, 0,
         0, 0, 0, 1;

    Eigen::IOFormat matrix_format(Eigen::FullPrecision);
    
    std::cout << "A:\n" << a.format(matrix_format) << "\n";
    std::cout << "B:\n" << b.format(matrix_format) << "\n";

    std::cout << fmt::format("{} {}", a(0), b(0));

    return 0;
}

Steps to reproduce

run demo online

What is the current bug behavior?

The output of two matrices with full precision format are the same.

What is the expected correct behavior?

The output of two matrices with full precision format are different. Also, outputted matrix elements are in double precision(IEEE-754).

Relevant logs

A:
 0.866025400014297 -0.500000006530077                  0                  0
 0.500000006530077  0.866025400014297                  0                  0
                 0                  0                  1                  0
                 0                  0                  0                  1
B:
 0.866025400014297 -0.500000006530077                  0                  0
 0.500000006530077  0.866025400014297                  0                  0
                 0                  0                  1                  0
                 0                  0                  0                  1
0.8660254000142966 0.866025400014297