自己写的高精度加法函数没法正常运行

string work(string a,string b)
{
    int l1=a.size(),l2=b.size();
    for(int i=0;i<l1/2;i++){char ch=a[i];a[i]=a[l1-1-i];a[l1-i-1]=ch;}
    for(int i=0;i<l2/2;i++){char ch=b[i];b[i]=b[l2-1-i];b[l2-i-1]=ch;}
    string c;
    if(l1>=l2) {
        c=a;
        for(int i=0;i<l2;i++) {
            c[i]+=b[i]-'0';
            if(c[i]>'9')
            {
                if(i+1==c.size()) { c[i+1]='1';}
                else {c[i+1]+=1;}
            }
        }  
    }
    else {
        c=b;
        for(int i=0;i<l1;i++) {
            c[i]+=a[i]-'0';
            if(c[i]>'9')
            {
                if(i+1==c.size()) { c[i+1]='1';}
                else {c[i+1]+=1;}
            }
        }
    }
    int l3=c.size();
    for(int i=0;i<l3/2;i++){char ch=c[i];c[i]=c[l3-i-1];c[l3-i-1]=ch;}
    return c;
}

请各位指点一下,写了好多次都写不对

#include <iostream>
#include <string>
#include <iterator>

class BigInt
{
public:
    BigInt(unsigned int n = 0)
    {
        if (n == 0)
            _data.push_back('0');
        else
        {
            while (n)
            {
                _data.push_back(n % 10 + '0');
                n /= 10;
            }
        }
    }

    BigInt(const std::string &s) : _data(s.rbegin(), s.rend()) {}

    BigInt &operator+=(const BigInt &other);

    BigInt operator+(const BigInt &other) const
    {
        BigInt t = *this;
        t += other;
        return t;
    }

private:
    std::string _data;

    friend std::istream &operator>>(std::istream &is, BigInt &a);
    friend std::ostream &operator<<(std::ostream &os, const BigInt &a);
};

BigInt &BigInt::operator+=(const BigInt &other)
{
    int carry = 0;
    auto itr1 = _data.begin();
    auto itr2 = other._data.begin();
    while (itr1 != _data.end() && itr2 != other._data.end())
    {
        int a = *itr1 - '0';
        int b = *itr2 - '0';
        int c = a + b + carry;
        carry = c / 10;
        c %= 10;
        *itr1 = c + '0';
        ++itr1;
        ++itr2;
    }
    while (itr1 != _data.end())
    {
        int a = *itr1 - '0' + carry;
        carry = a / 10;
        a %= 10;
        *itr1 = a + '0';
        ++itr1;
    }
    while (itr2 != other._data.end())
    {
        int a = *itr2 - '0' + carry;
        carry = a / 10;
        a %= 10;
        _data.push_back(a + '0');
        ++itr2;
    }
    if (carry > 0)
        _data.push_back(carry + '0');
    return *this;
}

std::istream &operator>>(std::istream &is, BigInt &a)
{
    std::string s;
    is >> s;
    a = s;
    return is;
}

std::ostream &operator<<(std::ostream &os, const BigInt &a)
{
    std::copy(a._data.rbegin(), a._data.rend(), std::ostream_iterator<char>(os));
    return os;
}

int main()
{
    BigInt a, b;
    std::cin >> a >> b;
    std::cout << a + b << std::endl;
    return 0;
}

$ g++ -Wall main.cpp
$ ./a.out
1234567890123456789012345678901234567890
9999999999999999999999999999999999999999 
11234567890123456789012345678901234567889

#include <iostream>
#include <string>
#include <iterator>

class BigInt
{
public:
    BigInt(unsigned int n = 0)
    {
        if (n == 0)
            _data.push_back('0');
        else
        {
            while (n)
            {
                _data.push_back(n % 10 + '0');
                n /= 10;
            }
        }
    }

    BigInt(const std::string &s) : _data(s.rbegin(), s.rend()) {}

    BigInt &operator+=(const BigInt &other);

    BigInt operator+(const BigInt &other) const
    {
        BigInt t = *this;
        t += other;
        return t;
    }

private:
    std::string _data;

    friend std::istream &operator>>(std::istream &is, BigInt &a);
    friend std::ostream &operator<<(std::ostream &os, const BigInt &a);
};

BigInt &BigInt::operator+=(const BigInt &other)
{
    int carry = 0;
    auto itr1 = _data.begin();
    auto itr2 = other._data.begin();
    while (itr1 != _data.end() && itr2 != other._data.end())
    {
        int a = *itr1 - '0';
        int b = *itr2 - '0';
        int c = a + b + carry;
        carry = c / 10;
        c %= 10;
        *itr1 = c + '0';
        ++itr1;
        ++itr2;
    }
    while (itr1 != _data.end())
    {
        int a = *itr1 - '0' + carry;
        carry = a / 10;
        a %= 10;
        *itr1 = a + '0';
        ++itr1;
    }
    while (itr2 != other._data.end())
    {
        int a = *itr2 - '0' + carry;
        carry = a / 10;
        a %= 10;
        _data.push_back(a + '0');
        ++itr2;
    }
    if (carry > 0)
        _data.push_back(carry + '0');
    return *this;
}

std::istream &operator>>(std::istream &is, BigInt &a)
{
    std::string s;
    is >> s;
    a = s;
    return is;
}

std::ostream &operator<<(std::ostream &os, const BigInt &a)
{
    std::copy(a._data.rbegin(), a._data.rend(), std::ostream_iterator<char>(os));
    return os;
}

int main()
{
    BigInt a, b;
    std::cin >> a >> b;
    std::cout << a + b << std::endl;
    return 0;
}

$ g++ -Wall main.cpp
$ ./a.out
1234567890123456789012345678901234567890
9999999999999999999999999999999999999999 
11234567890123456789012345678901234567889