An efficient way to do basic 128 bit integer calculations in C++?

Question

Some years ago I needed a way to do some basic 128 bit integer math with Cuda: 128 bit integer on cuda?. Now I am having the same problem, but this time I need to run some basic 128 bit arithmetics (sums, bitshifts and multiplications) on a 32 bit embedded system (Intel Edison) that does not support 128 bits of any kind. There are, however, 64 bit integers supported directly (unsigned long long int).

I tried naively to use the asm code that was answered to me last time on the CPU, but I got a bunch of errors. I am really not experienced with asm, so: what is the most efficient way, having 64 bit integers, to implement additions, multiplications and bit shifting in 128 bits?

Answer 1

Update: Since the OP hasn't accepted an answer yet <hint><hint>, I've attached a bit more code.

Using the libraries discussed above is probably a good idea. While you might only need a few functions today, eventually you may find that you need one more. Then one more after that. Until eventually you end up writing, debugging and maintaining your own 128bit math library. Which is a waste of your time and effort.

That said. If you are determined to roll your own:

1) The cuda question you asked previously already has c code for multiplication. Was there some problem with it?

2) The shift probably won't benefit from using asm, so a c solution makes sense to me here as well. Although if performance is really an issue here, I'd see if the Edison supports SHLD/SHRD, which might make this a bit faster. Otherwise, m Maybe an approach like this?

my_uint128_t lshift_uint128 (const my_uint128_t a, int b)
{
   my_uint128_t res;
   if (b < 32) {    
      res.x = a.x << b;
      res.y = (a.y << b) | (a.x >> (32 - b));
      res.z = (a.z << b) | (a.y >> (32 - b));
      res.w = (a.w << b) | (a.z >> (32 - b));
   } elseif (b < 64) {
      ...
   }

   return res;
}

Update: Since it appears that the Edison may support SHLD/SHRD, here's an alternative which might be more performant than the 'c' code above. As with all code purporting to be faster, you should test it.

inline
unsigned int __shld(unsigned int into, unsigned int from, unsigned int c)
{
   unsigned int res;

   if (__builtin_constant_p(into) &&
       __builtin_constant_p(from) &&
       __builtin_constant_p(c))
   {
      res = (into << c) | (from >> (32 - c));
   }
   else
   {
      asm("shld %b3, %2, %0"
          : "=rm" (res)
          : "0" (into), "r" (from), "ic" (c)
          : "cc");
   }

   return res;
}

inline
unsigned int __shrd(unsigned int into, unsigned int from, unsigned int c)
{
   unsigned int res;

   if (__builtin_constant_p(into) && 
       __builtin_constant_p(from) && 
       __builtin_constant_p(c))
   {
      res = (into >> c) | (from << (32 - c));
   }
   else
   {
      asm("shrd %b3, %2, %0"
          : "=rm" (res)
          : "0" (into), "r" (from), "ic" (c)
          : "cc");
   }

   return res;
}

my_uint128_t lshift_uint128 (const my_uint128_t a, unsigned int b)
{
   my_uint128_t res;

   if (b < 32) {
      res.x = a.x << b;
      res.y = __shld(a.y, a.x, b);
      res.z = __shld(a.z, a.y, b);
      res.w = __shld(a.w, a.z, b);
   } else if (b < 64) {
      res.x = 0;
      res.y = a.x << (b - 32);
      res.z = __shld(a.y, a.x, b - 32);
      res.w = __shld(a.z, a.y, b - 32);
   } else if (b < 96) {
      res.x = 0;
      res.y = 0;
      res.z = a.x << (b - 64);
      res.w = __shld(a.y, a.x, b - 64);
   } else if (b < 128) {
      res.x = 0;
      res.y = 0;
      res.z = 0;
      res.w = a.x << (b - 96);
   } else {
      memset(&res, 0, sizeof(res));
   }

   return res;
}

my_uint128_t rshift_uint128 (const my_uint128_t a, unsigned int b)
{
   my_uint128_t res;

   if (b < 32) {
      res.x = __shrd(a.x, a.y, b);
      res.y = __shrd(a.y, a.z, b);
      res.z = __shrd(a.z, a.w, b);
      res.w = a.w >> b;
   } else if (b < 64) {
      res.x = __shrd(a.y, a.z, b - 32);
      res.y = __shrd(a.z, a.w, b - 32);
      res.z = a.w >> (b - 32);
      res.w = 0;
   } else if (b < 96) {
      res.x = __shrd(a.z, a.w, b - 64);
      res.y = a.w >> (b - 64);
      res.z = 0;
      res.w = 0;
   } else if (b < 128) {
      res.x = a.w >> (b - 96);
      res.y = 0;
      res.z = 0;
      res.w = 0;
   } else {
      memset(&res, 0, sizeof(res));
   }

   return res;
}

3) The addition may benefit from asm. You could try this:

struct my_uint128_t
{
   unsigned int x;
   unsigned int y;
   unsigned int z;
   unsigned int w;
};

my_uint128_t add_uint128 (const my_uint128_t a, const my_uint128_t b)
{
   my_uint128_t res;

    asm ("addl %5, %[resx]\n\t"
         "adcl %7, %[resy]\n\t"
         "adcl %9, %[resz]\n\t"
         "adcl %11, %[resw]\n\t"
         : [resx] "=&r" (res.x), [resy] "=&r" (res.y), 
           [resz] "=&r" (res.z), [resw] "=&r" (res.w)
         : "%0"(a.x), "irm"(b.x), 
           "%1"(a.y), "irm"(b.y), 
           "%2"(a.z), "irm"(b.z), 
           "%3"(a.w), "irm"(b.w)
         : "cc");

   return res;
}

I just dashed this off, so use at your own risk. I don't have an Edison, but this works with x86.

Update: If you are just doing accumulation (think to += from instead of the code above which is c = a + b), this code might serve you better:

inline
void addto_uint128 (my_uint128_t *to, const my_uint128_t from)
{
   asm ("addl %[fromx], %[tox]\n\t"
        "adcl %[fromy], %[toy]\n\t"
        "adcl %[fromz], %[toz]\n\t"
        "adcl %[fromw], %[tow]\n\t"
        : [tox] "+&r"(to->x), [toy] "+&r"(to->y), 
          [toz] "+&r"(to->z), [tow] "+&r"(to->w)
        : [fromx] "irm"(from.x), [fromy] "irm"(from.y), 
          [fromz] "irm"(from.z), [fromw] "irm"(from.w)
        : "cc");
}

Answer 2

If using an external library is an option then have a look at this question. You can use TTMath which is a very simple header for big precision math. On 32-bit architectures ttmath:UInt<4> will create a 128-bit int type with four 32-bit limbs. Some other alternatives are (u)int128_t in Boost.Multiprecision or calccrypto/uint128_t

If you must write it your own then there are already a lot of solutions on SO and I'll summarize them here

---

For addition and subtraction, it's very easy and straightforward, simply add/subtract the words (which big int libraries often called limbs) from the lowest significant to higher significant, with carry of course.

typedef struct INT128 {
    uint64_t H, L;
} my_uint128_t;

inline my_uint128_t add(my_uint128_t a, my_uint128_t b)
{
    my_uint128_t c;
    c.L = a.L + b.L;
    c.H = a.H + b.H + (c.L < a.L);  // c = a + b
    return c;
}

The assembly output can be checked with Compiler Explorer

The compilers can already generate efficient code for double-word operations, but many aren't smart enough to use "add with carry" when compiling multi-word operations from high level languages as you can see in the question efficient 128-bit addition using carry flag. Therefore using 2 long longs like above will make it not only more readable but also easier for the compiler to emit a little more efficient code.

If that still doesn't suit your performance requirement, you must use intrinsic or write it in assembly. To add the 128-bit value stored in bignum to the 128-bit value in {eax, ebx, ecx, edx} you can use the following code

add edx, [bignum]
adc ecx, [bignum+4]
adc ebx, [bignum+8]
adc eax, [bignum+12]

The equivalent intrinsic will be like this for Clang

unsigned *x, *y, *z, carryin=0, carryout;
z[0] = __builtin_addc(x[0], y[0], carryin, &carryout);
carryin = carryout;
z[1] = __builtin_addc(x[1], y[1], carryin, &carryout);
carryin = carryout;
z[2] = __builtin_addc(x[2], y[2], carryin, &carryout);
carryin = carryout;
z[3] = __builtin_addc(x[3], y[3], carryin, &carryout);

You need to change the intrinsic to the one supported by your compiler, for example __builtin_uadd_overflow in gcc, or _addcarry_u32 for MSVC and ICC

For more information read these

• Working with Big Numbers Using x86 Instructions
• How can I add and subtract 128 bit integers in C or C++ if my compiler does not support them?
• Producing good add with carry code from clang
• multi-word addition using the carry flag

---

For bit shifts you can find the C solution in the question Bitwise shift operation on a 128-bit number. This is a simple left shift but you can unroll the recursive call for more performance

void shiftl128 (
    unsigned int& a,
    unsigned int& b,
    unsigned int& c,
    unsigned int& d,
    size_t k)
{
    assert (k <= 128);
    if (k >= 32) // shifting a 32-bit integer by more than 31 bits is "undefined"
    {
        a=b;
        b=c;
        c=d;
        d=0;
        shiftl128(a,b,c,d,k-32);
    }
    else
    {
        a = (a << k) | (b >> (32-k));
        b = (b << k) | (c >> (32-k));
        c = (c << k) | (d >> (32-k));
        d = (d << k);
    }
}

The assembly for less-than-32-bit shifts can be found in the question 128-bit shifts using assembly language?

shld    edx, ecx, cl
shld    ecx, ebx, cl
shld    ebx, eax, cl
shl     eax, cl

Right shifts can be implemented similarly, or just copy from the above linked question

---

Multiplication and divisions are a lot more complex and you can reference the solution in the question Efficient Multiply/Divide of two 128-bit Integers on x86 (no 64-bit):

class int128_t
{
    uint32_t dw3, dw2, dw1, dw0;

    // Various constrctors, operators, etc...

    int128_t& operator*=(const int128_t&  rhs) __attribute__((always_inline))
    {
        int128_t Urhs(rhs);
        uint32_t lhs_xor_mask = (int32_t(dw3) >> 31);
        uint32_t rhs_xor_mask = (int32_t(Urhs.dw3) >> 31);
        uint32_t result_xor_mask = (lhs_xor_mask ^ rhs_xor_mask);
        dw0 ^= lhs_xor_mask;
        dw1 ^= lhs_xor_mask;
        dw2 ^= lhs_xor_mask;
        dw3 ^= lhs_xor_mask;
        Urhs.dw0 ^= rhs_xor_mask;
        Urhs.dw1 ^= rhs_xor_mask;
        Urhs.dw2 ^= rhs_xor_mask;
        Urhs.dw3 ^= rhs_xor_mask;
        *this += (lhs_xor_mask & 1);
        Urhs += (rhs_xor_mask & 1);

        struct mul128_t
        {
            int128_t dqw1, dqw0;
            mul128_t(const int128_t& dqw1, const int128_t& dqw0): dqw1(dqw1), dqw0(dqw0){}
        };

        mul128_t data(Urhs,*this);
        asm volatile(
        "push      %%ebp                            \n\
        movl       %%eax,   %%ebp                   \n\
        movl       $0x00,   %%ebx                   \n\
        movl       $0x00,   %%ecx                   \n\
        movl       $0x00,   %%esi                   \n\
        movl       $0x00,   %%edi                   \n\
        movl   28(%%ebp),   %%eax #Calc: (dw0*dw0)  \n\
        mull             12(%%ebp)                  \n\
        addl       %%eax,   %%ebx                   \n\
        adcl       %%edx,   %%ecx                   \n\
        adcl       $0x00,   %%esi                   \n\
        adcl       $0x00,   %%edi                   \n\
        movl   24(%%ebp),   %%eax #Calc: (dw1*dw0)  \n\
        mull             12(%%ebp)                  \n\
        addl       %%eax,   %%ecx                   \n\
        adcl       %%edx,   %%esi                   \n\
        adcl       $0x00,   %%edi                   \n\
        movl   20(%%ebp),   %%eax #Calc: (dw2*dw0)  \n\
        mull             12(%%ebp)                  \n\
        addl       %%eax,   %%esi                   \n\
        adcl       %%edx,   %%edi                   \n\
        movl   16(%%ebp),   %%eax #Calc: (dw3*dw0)  \n\
        mull             12(%%ebp)                  \n\
        addl       %%eax,   %%edi                   \n\
        movl   28(%%ebp),   %%eax #Calc: (dw0*dw1)  \n\
        mull              8(%%ebp)                  \n\
        addl       %%eax,   %%ecx                   \n\
        adcl       %%edx,   %%esi                   \n\
        adcl       $0x00,   %%edi                   \n\
        movl   24(%%ebp),   %%eax #Calc: (dw1*dw1)  \n\
        mull              8(%%ebp)                  \n\
        addl       %%eax,   %%esi                   \n\
        adcl       %%edx,   %%edi                   \n\
        movl   20(%%ebp),   %%eax #Calc: (dw2*dw1)  \n\
        mull              8(%%ebp)                  \n\
        addl       %%eax,   %%edi                   \n\
        movl   28(%%ebp),   %%eax #Calc: (dw0*dw2)  \n\
        mull              4(%%ebp)                  \n\
        addl       %%eax,   %%esi                   \n\
        adcl       %%edx,   %%edi                   \n\
        movl   24(%%ebp),  %%eax #Calc: (dw1*dw2)   \n\
        mull              4(%%ebp)                  \n\
        addl       %%eax,   %%edi                   \n\
        movl   28(%%ebp),   %%eax #Calc: (dw0*dw3)  \n\
        mull               (%%ebp)                  \n\
        addl       %%eax,   %%edi                   \n\
        pop        %%ebp                            \n"
        :"=b"(this->dw0),"=c"(this->dw1),"=S"(this->dw2),"=D"(this->dw3)
        :"a"(&data):"%ebp");

        dw0 ^= result_xor_mask;
        dw1 ^= result_xor_mask;
        dw2 ^= result_xor_mask;
        dw3 ^= result_xor_mask;
        return (*this += (result_xor_mask & 1));
    }
};

You can also find a lot of related questions with the 128bit tag