detecting multiplication of uint64_t integers overflow with C

Question

Is there any efficient and portable way to check when multiplication operations with int64_t or uint64_t operands overflow in C?

For instance, for addition of uint64_t I can do:

if (UINT64_MAX - a < b) overflow_detected();
else sum = a + b;

But I can not get to a similar simple expression for multiplication.

All that occurs to me is breaking the operands into high and low uint32_t parts and performing the multiplication of those parts while checking for overflow, something really ugly and probably inefficient too.

Update 1: Some benchmark code implementing several approaches added

Update 2: Jens Gustedt method added

benchmarking program:

#include <stdio.h>
#include <stdint.h>
#include <stdlib.h>

#define N 100000000

int d = 2;

#define POW_2_64 ((double)(1 << 31) * (double)(1 << 31) * 4)

#define calc_b (a + c)
// #define calc_b (a + d)

int main(int argc, char *argv[]) {
    uint64_t a;
    uint64_t c = 0;
    int o = 0;
    int opt;

    if (argc != 2) exit(1);

    opt = atoi(argv[1]);

    switch (opt) {

    case 1: /* faked check, just for timing */
        for (a = 0; a < N; a++) {
            uint64_t b = a + c;
            if (c > a) o++;
            c += b * a;
        }
        break;

    case 2: /* using division */
        for (a = 0; a < N; a++) {
            uint64_t b = a + c;
            if (b && (a > UINT64_MAX / b)) o++;
            c += b * a;
        }
        break;

    case 3: /* using floating point, unreliable */
        for (a = 0; a < N; a++) {
            uint64_t b = a + c;
            if ((double)UINT64_MAX < (double)a * (double)b) o++;
            c += b * a;
        }
        break;

    case 4: /* using floating point and division for difficult cases */
        for (a = 0; a < N; a++) {
            uint64_t b = a + c;
            double m = (double)a * (double)b;
            if ( ((double)(~(uint64_t)(0xffffffff)) < m ) &&
                 ( (POW_2_64 < m) ||
                   ( b &&
                     (a > UINT64_MAX / b) ) ) ) o++;
            c += b * a;
        }
        break;

    case 5: /* Jens Gustedt method */
        for (a = 0; a < N; a++) {
            uint64_t b = a + c;
            uint64_t a1, b1;
            if (a > b) { a1 = a; b1 = b; }
            else       { a1 = b; b1 = a; }
            if (b1 > 0xffffffff) o++;
            else {
                uint64_t a1l = (a1 & 0xffffffff) * b1;
                uint64_t a1h = (a1 >> 32) * b1 + (a1l >> 32);
                if (a1h >> 32) o++;
            }
            c += b1 * a1;
        }
        break;

    default:
        exit(2);
    }
    printf("c: %lu, o: %u\n", c, o);
}

So far, case 4 that uses floating point to filter most cases is the fastest when it is assumed that overflows are very unusual, at least on my computer where it is only two times slower than the do-nothing case.

Case 5, is 30% slower than 4, but it always performs the same, there isn't any special case numbers that require slower processing as happens with 4.

Answer 1

Actually, the same principle can be used for multiplication:

uint64_t a;
uint64_t b;
...
if (b != 0 && a > UINT64_MAX / b) { // if you multiply by b, you get: a * b > UINT64_MAX
    < error >
}
uint64_t c = a * b;

For signed integers similar can be done, you'd probably need a case for each combination of signs.

Answer 2

If you want to avoid division as in Ambroz' answer:

First you have to see that the smaller of the two numbers, say a, is less than 2³², otherwise the result will overflow anyhow. Let b be decomposed into the two 32 bit words that is b = c 2³² + d.

The computation then is not so difficult, I find:

uint64_t mult_with_overflow_check(uint64_t a, uint64_t b) {
  if (a > b) return mult_with_overflow_check(b, a);
  if (a > UINT32_MAX) overflow();
  uint32_t c = b >> 32;
  uint32_t d = UINT32_MAX & b;
  uint64_t r = a * c;
  uint64_t s = a * d;
  if (r > UINT32_MAX) overflow();
  r <<= 32;
  return addition_with_overflow_check(s, r);
}

so this are two multiplications, two shifts, some additions and condition checks. This could be more efficient than the division because e.g the two multiplications can be pipelined in paralle. You'd have to benchmark to see what works better for you.