There are several issues:
The exception you got is caused by infinite recursion at this call:
kara_mul(a + b, c + d)
As these variables are strings, the + is a string concatenation. This means these arguments evaluate to
n and m, which were the arguments to the current execution of the function.
The correct algorithm would perform a numerical addition here, for which you need to provide an implementation (adding two string representations of potentially very long integers)
if (len_n == 1 && len_m == 1) detects the base case, but the base case should kick in when either of these sizes is 1, not necessary both. So this should be an || operator, or should be written as two separate if statements.
The input strings should be split such that b and d are equal in size. This is not what your code does. Note how the Wikipedia article stresses this point:
The second argument of the split_at function specifies the number of digits to extract from the right
stol should never be called on strings that could potentially be too long for conversion to long. So for example, stol(p1) is not safe, as p1 could have 20 or more digits.
As a consequence of the previous point, you'll need to implement functions that add or subtract two string representations of numbers, and also one that can multiply a string representation with a single digit (the base case).
Here is an implementation that corrects these issues:
#include <iostream>
#include <algorithm>
int digit(std::string n, int i) {
return i >= n.size() ? 0 : n[n.size() - i - 1] - '0';
}
std::string add(std::string n, std::string m) {
int len = std::max(n.size(), m.size());
std::string result;
int carry = 0;
for (int i = 0; i < len; i++) {
int sum = digit(n, i) + digit(m, i) + carry;
result += (char) (sum % 10 + '0');
carry = sum >= 10;
}
if (carry) result += '1';
reverse(result.begin(), result.end());
return result;
}
std::string subtract(std::string n, std::string m) {
int len = n.size();
if (m.size() > len) throw std::invalid_argument("subtraction overflow");
if (n == m) return "0";
std::string result;
int carry = 0;
for (int i = 0; i < len; i++) {
int diff = digit(n, i) - digit(m, i) - carry;
carry = diff < 0;
result += (char) (diff + carry * 10 + '0');
}
if (carry) throw std::invalid_argument("subtraction overflow");
result.erase(result.find_last_not_of('0') + 1);
reverse(result.begin(), result.end());
return result;
}
std::string simple_mul(std::string n, int coefficient) {
if (coefficient < 2) return coefficient ? n : "0";
std::string result = simple_mul(add(n, n), coefficient / 2);
return coefficient % 2 ? add(result, n) : result;
}
std::string kara_mul(std::string n, std::string m) {
int len_n = n.size();
int len_m = m.size();
if (len_n == 1) return simple_mul(m, digit(n, 0));
if (len_m == 1) return simple_mul(n, digit(m, 0));
int len_min2 = std::min(len_n, len_m) / 2;
std::string a = n.substr(0, len_n - len_min2);
std::string b = n.substr(len_n - len_min2);
std::string c = m.substr(0, len_m - len_min2);
std::string d = m.substr(len_m - len_min2);
std::string p1 = kara_mul(a, c);
std::string p2 = kara_mul(b, d);
std::string p3 = subtract(kara_mul(add(a, b), add(c, d)), add(p1, p2));
return add(add(p1 + std::string(len_min2*2, '0'), p2), p3 + std::string(len_min2, '0'));
}
