Anti-derivative

// Impossible is not for the faint of heart.

#include <fstream>
#include <iostream>
#include <vector>
#include <string>
#include <cmath>

int main() {
    std::ifstream input_file("comp", std::ios::binary);
    std::string input((std::istreambuf_iterator<char>(input_file)), std::istreambuf_iterator<char>());

    const int N = input.length();
    std::vector<double> x(N + 1);
    // input = "We the People We the People";  // First 5 characters of the US Constitution
    // input.resize(N);
    x[0] = 1;
    std::cout << "Original string: " << input << std::endl;

    std::cout << "Forward x_n sequence: ";
    for (int n = 0; n < N; ++n) {
        unsigned char y = input[n];
        x[n + 1] = x[n] + (static_cast<double>(y) / (n + 1));
        std::cout << x[n] << " ";
    }
    std::cout << x[N] << std::endl;

    double compressed_value = x[N];
    std::cout << "Compressed value (x_N): " << compressed_value << std::endl;

    std::string decompressed;
    for (int n = N - 1; n >= 0; --n) {
        double y_calc = (x[n + 1] - x[n]) * (n + 1);
        int y = std::round(y_calc);
        y = std::max(0, std::min(255, y));
        decompressed = static_cast<char>(y) + decompressed;
        x[n] = x[n + 1] - (static_cast<double>(y) / (n + 1));
    }

    std::cout << "Decompressed string: " << decompressed << std::endl;

    if (input == decompressed) {
        std::cout << "Compression is fully reversible." << std::endl;
    } else {
        std::cout << "Reversibility failed." << std::endl;
    }

    return 0;
}