Most efficient way to find prime factorization of 2 for a positive integer

Okay, the most efficient way you say? How about (almost) a single assembly instruction?

From the GCC doc (and also available in Clang):

Built-in Function: int __builtin_ctz (unsigned int x)

Returns the number of trailing 0-bits in x, starting at the least significant bit position. If x is 0, the result is undefined.

unsigned Prime_Factor_Two(unsigned x) {
    return x ? __builtin_ctz(x) : 0;
}

No function calls, no loops, only one branch. If you know that the number is positive you can even remove that and just use __builtin_ctz(x).

The __builtin_ctz() built-in:

• On x86 should compile to a single assembly instruction: TZCNT (if supported) or BSF.
• On ARM should compile to two instructions: RBIT + CLZ.
• On PowerPC should compile to 31 - CNTLZ(x & -x) (assuming 32bit unsigned).
• On other platforms, maybe a handful of instructions.

To also support negative integers you can leverage the fact that the two's complement of a number preserves the least significant zeroes, and just change type from unsigned to int:

unsigned Prime_Factor_Two(int x) {
    return x ? __builtin_ctz(x) : 0;
}

Assuming only positive numbers and that your system uses 2's Complement notation, you can first isolate the least significant set bit using the seemingly bizarre x = x & -x operation; then, you can convert that to the set bit's position using the log2(x) function.

Here's a test program:

#include <stdio.h>
#include <math.h>

int main()
{
    int num, ans;
    do {
        printf("Enter a number: ");
        if (scanf("%d", &num) != 1 || num == 0) break;
        ans = (int)(log2(num & -num) + 0.5);
        printf("Answer is: %d\n", ans);
    } while (num > 0);
    return 0;
}

Alternatively, to avoid using floating-point stuffs and the math(s) library, you can use a bit-shift loop (this will also work for negative and zero values):

int main()
{
    int num, ans;
    do {
        printf("Enter a number: ");
        if (scanf("%d", &num) != 1) break;
        num &= -num;
        for (ans = 0; num > 1; ans++) num >>= 1;
        printf("Answer is: %d\n", ans);
    } while (num > 0);
    return 0;
}

EDIT: Of course, both of the above methods are contrived and unnecessary; a simple loop with a shifting, single-bit mask will do the trick - except for a value of zero, which is, anyway, divisible by 2 (with no remainder) infinite times:

#include <stdio.h>

int main()
{
    int num, ans, bit;
    do {
        printf("Enter a number: ");
        if (scanf("%d", &num) != 1 || num == 0) break;
        for (ans = 0, bit = 1; !(num & bit); ans++) bit <<= 1;
        printf("Answer is: %d\n", ans);
    } while (1);
    return 0;
}

An interesting way to do this is with what amounts to a binary search for the least-significant 1 bit. You can even code it as explicitly branchless, though the example below does not quite do so. This approach does, however, require you to know the number of value bits in the argument type.

Example:

/*
 * Returns the number of factors of 2 in the prime factorization of the argument, or
 * returns -1 if the argument is 0.
 */
int factor_of_two_count(uint64_t in) {
    int result = -1;
    uint64_t bottom;
    
    bottom = (in & 0xffffffffu);
    in = bottom ? bottom : (in >> 32);
    result += !bottom * 32;

    bottom = (in & 0xffffu);
    in = bottom ? bottom : (in >> 16);
    result += !bottom * 16;

    bottom = (in & 0xffu);
    in = bottom ? bottom : (in >> 8);
    result += !bottom * 8;

    bottom = (in & 0xfu);
    in = bottom ? bottom : (in >> 4);
    result += !bottom * 4;

    bottom = (in & 0x3u);
    in = bottom ? bottom : (in >> 2);
    result += !bottom * 2;

    bottom = (in & 0x1u);
    result += !bottom;

    return result;
}

However, your bit-by bit loop will likely outperform that on random data, for that is roughly analogous to six passes through such a loop, and fewer than 2% of all random 64-bit inputs would require that many. Only if branch misprediction issues weighed heavily on the bitwise loop or if the distribution of inputs were skewed towards those with many factors of 2 would this be likely to be a winner.

is there anything computational or analytical I can do to optimize this code?

See Count the consecutive zero bits (trailing) on the right with modulus division and lookup.

unsigned int v;  // find the number of trailing zeros in v
int r;           // put the result in r
static const int Mod37BitPosition[] = // map a bit value mod 37 to its position
{
  32, 0, 1, 26, 2, 23, 27, 0, 3, 16, 24, 30, 28, 11, 0, 13, 4,
  7, 17, 0, 25, 22, 31, 15, 29, 10, 12, 6, 0, 21, 14, 9, 5,
  20, 8, 19, 18
};
r = Mod37BitPosition[(-v & v) % 37];

Author explanation:

The code above finds the number of zeros that are trailing on the right, so binary 0100 would produce 2. It makes use of the fact that the first 32 bit position values are relatively prime with 37, so performing a modulus division with 37 gives a unique number from 0 to 36 for each. These numbers may then be mapped to the number of zeros using a small lookup table. It uses only 4 operations, however indexing into a table and performing modulus division may make it unsuitable for some situations.