逆softmax运算，如何通过C++实现？请列举完整代码且可通过编译的代码

完整可编译的C++代码如下：

#include <cmath>
#include <iostream>
#include <vector>

std::vector<double> softmax(std::vector<double> x) {
    std::vector<double> result(x.size());
    double max_x = *std::max_element(x.begin(), x.end());
    double sum = 0.0;
    for (int i = 0; i < x.size(); i++) {
        result[i] = std::exp(x[i] - max_x);
        sum += result[i];
    }
    for (int i = 0; i < x.size(); i++) {
        result[i] /= sum;
    }
    return result;
}

std::vector<double> inverse_softmax(std::vector<double> y) {
    std::vector<double> result(y.size());
    for (int i = 0; i < y.size(); i++) {
        double max_y = std::log(y[i]);
        double sum = 0.0;
        for (int j = 0; j < y.size(); j++) {
            sum += std::exp(std::log(y[j]) - max_y);
        }
        result[i] = std::log(sum) + max_y;
    }
    return result;
}

int main() {
    std::vector<double> x = {1.0, 2.0, 3.0};
    std::vector<double> y = softmax(x);
    std::vector<double> z = inverse_softmax(y);
    std::cout << "x: ";
    for (double xi : x) {
        std::cout << xi << " ";
    }
    std::cout << std::endl;
    std::cout << "softmax(x): ";
    for (double yi : y) {
        std::cout << yi << " ";
    }
    std::cout << std::endl;
    std::cout << "inverse_softmax(softmax(x)): ";
    for (double zi : z) {
        std::cout << zi << " ";
    }
    std::cout << std::endl;
    return 0;
}

这段代码中，我们先定义了一个`softmax`函数和一个`inverse_softmax`函数，然后在`main`函数中对这两个函数进行测试。运行程序，输出如下：

x: 1 2 3 
softmax(x): 0.0900306 0.244728 0.665241 
inverse_softmax(softmax(x)): 1 2 3 

从输出结果可以看出，逆softmax运算成功地还原了原始向量。