My tests showed this answer gave the best performance from offered (not surprising it has the most rate). 
I've implemented same algorithm for my project (I really want good performance & memory usage) - my tests showed my implementation has ~2.6-3.25 better speed performace. Also I don't like previous best offered algorithm bcs of bad memory usage - you will have extra memory usage as when apply 1.1 multiplier 'heuristic', as when .append() lead to resize.
So, leave my code here - maybe somebody find it useful.
HtmlPreprocess.h:
#ifndef _HTML_PREPROCESS_H_
#define _HTML_PREPROCESS_H_
#include <string>
class HtmlPreprocess
{
public:
    HtmlPreprocess();
    ~HtmlPreprocess();
    static void htmlspecialchars(
        const std::string & in,
        std::string & out
        );
};
#endif // _HTML_PREPROCESS_H_
HtmlPreprocess.cpp:
#include "HtmlPreprocess.h"
HtmlPreprocess::HtmlPreprocess()
{
}
HtmlPreprocess::~HtmlPreprocess()
{
}
const unsigned char map_char_to_final_size[] = 
{
   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,
   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,
   1,   1,   6,   1,   1,   1,   5,   6,   1,   1,   1,   1,   1,   1,   1,   1,
   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   4,   1,   4,   1,
   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,
   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,
   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,
   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,
   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,
   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,
   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,
   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,
   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,
   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,
   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,
   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1
};
const unsigned char map_char_to_index[] = 
{
   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,
   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,
   0xFF,   0xFF,   2,      0xFF,   0xFF,   0xFF,   0,      1,      0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,
   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   4,      0xFF,   3,      0xFF,
   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,
   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,
   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,
   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,
   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,
   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,
   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,
   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,
   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,
   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,
   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,
   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF,   0xFF
};
void HtmlPreprocess::htmlspecialchars(
    const std::string & in,
    std::string & out
    )
{
    const char * lp_in_stored = &in[0];
    size_t in_size = in.size();
    const char * lp_in = lp_in_stored;
    size_t final_size = 0;
    for (size_t i = 0; i < in_size; i++)
        final_size += map_char_to_final_size[*lp_in++];
    out.resize(final_size);
    lp_in = lp_in_stored;
    char * lp_out = &out[0];
    for (size_t i = 0; i < in_size; i++)
    {
        char current_char = *lp_in++;
        unsigned char next_action = map_char_to_index[current_char];
        switch (next_action){
        case 0:
            *lp_out++ = '&';
            *lp_out++ = 'a';
            *lp_out++ = 'm';
            *lp_out++ = 'p';
            *lp_out++ = ';';
            break;
        case 1:
            *lp_out++ = '&';
            *lp_out++ = 'a';
            *lp_out++ = 'p';
            *lp_out++ = 'o';
            *lp_out++ = 's';
            *lp_out++ = ';';
            break;
        case 2:
            *lp_out++ = '&';
            *lp_out++ = 'q';
            *lp_out++ = 'u';
            *lp_out++ = 'o';
            *lp_out++ = 't';
            *lp_out++ = ';';
            break;
        case 3:
            *lp_out++ = '&';
            *lp_out++ = 'g';
            *lp_out++ = 't';
            *lp_out++ = ';';
            break;
        case 4:
            *lp_out++ = '&';
            *lp_out++ = 'l';
            *lp_out++ = 't';
            *lp_out++ = ';';
            break;
        default:
            *lp_out++ = current_char;
        }
    }
}