Fast divisibility tests (by 2,3,4,5,.., 16)?

Question

What are the fastest divisibility tests? Say, given a little-endian architecture and a 32-bit signed integer: how to calculate very fast that a number is divisible by 2,3,4,

Answer 1

As @James mentioned, let the compiler simplify it for you. If n is a constant, any descent compiler is able to recognize the pattern and change it to a more efficient equivalent.

For example, the code

#include 

int main() {
    size_t x;
    scanf("%u\n", &x);
    __asm__ volatile ("nop;nop;nop;nop;nop;");
    const char* volatile foo = (x%3 == 0) ? "yes" : "no";
    __asm__ volatile ("nop;nop;nop;nop;nop;");
    printf("%s\n", foo);
    return 0;
}

compiled with g++-4.5 -O3, the relevant part of x%3 == 0 will become

mov    rcx,QWORD PTR [rbp-0x8]   # rbp-0x8 = &x
mov    rdx,0xaaaaaaaaaaaaaaab
mov    rax,rcx
mul    rdx
lea    rax,"yes"
shr    rdx,1
lea    rdx,[rdx+rdx*2]
cmp    rcx,rdx
lea    rdx,"no"
cmovne rax,rdx
mov    QWORD PTR [rbp-0x10],rax

which, translated back to C code, means

(hi64bit(x * 0xaaaaaaaaaaaaaaab) / 2) * 3 == x ? "yes" : "no"
// equivalatent to:                 x % 3 == 0 ? "yes" : "no"

no division involved here. (Note that 0xaaaaaaaaaaaaaaab == 0x20000000000000001L/3)

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Edit:

• The magic constant 0xaaaaaaaaaaaaaaab can be computed in http://www.hackersdelight.org/magic.htm
• For divisors of the form 2ⁿ - 1, check http://graphics.stanford.edu/~seander/bithacks.html#ModulusDivision