Fastest way to calculate a 128-bit integer modulo a 64-bit integer

Question

I have a 128-bit unsigned integer A and a 64-bit unsigned integer B. What\'s the fastest way to calculate A % B - that is the (64-bit) remainder from dividing A

Answer 1

I know the question specified 32-bit code, but the answer for 64-bit may be useful or interesting to others.

And yes, 64b/32b => 32b division does make a useful building-block for 128b % 64b => 64b. libgcc's __umoddi3 (source linked below) gives an idea of how to do that sort of thing, but it only implements 2N % 2N => 2N on top of a 2N / N => N division, not 4N % 2N => 2N.

Wider multi-precision libraries are available, e.g. https://gmplib.org/manual/Integer-Division.html#Integer-Division.

---

GNU C on 64-bit machines does provide an __int128 type, and libgcc functions to multiply and divide as efficiently as possible on the target architecture.

x86-64's div r/m64 instruction does 128b/64b => 64b division (also producing remainder as a second output), but it faults if the quotient overflows. So you can't directly use it if A/B > 2^64-1, but you can get gcc to use it for you (or even inline the same code that libgcc uses).

---

This compiles (Godbolt compiler explorer) to one or two div instructions (which happen inside a libgcc function call). If there was a faster way, libgcc would probably use that instead.

#include 
uint64_t AmodB(unsigned __int128 A, uint64_t B) {
  return A % B;
}

The __umodti3 function it calls calculates a full 128b/128b modulo, but the implementation of that function does check for the special case where the divisor's high half is 0, as you can see in the libgcc source. (libgcc builds the si/di/ti version of the function from that code, as appropriate for the target architecture. udiv_qrnnd is an inline asm macro that does unsigned 2N/N => N division for the target architecture.

For x86-64 (and other architectures with a hardware divide instruction), the fast-path (when high_half(A) < B; guaranteeing div won't fault) is just two not-taken branches, some fluff for out-of-order CPUs to chew through, and a single div r64 instruction, which takes about 50-100 cycles¹ on modern x86 CPUs, according to Agner Fog's insn tables. Some other work can be happening in parallel with div, but the integer divide unit is not very pipelined and div decodes to a lot of uops (unlike FP division).

The fallback path still only uses two 64-bit div instructions for the case where B is only 64-bit, but A/B doesn't fit in 64 bits so A/B directly would fault.

Note that libgcc's __umodti3 just inlines __udivmoddi4 into a wrapper that only returns the remainder.

Footnote 1: 32-bit div is over 2x faster on Intel CPUs. On AMD CPUs, performance only depends on the size of the actual input values, even if they're small values in a 64-bit register. If small values are common, it might be worth benchmarking a branch to a simple 32-bit division version before doing 64-bit or 128-bit division.

---

For repeated modulo by the same B

It might be worth considering calculating a fixed-point multiplicative inverse for B, if one exists. For example, with compile-time constants, gcc does the optimization for types narrower than 128b.

uint64_t modulo_by_constant64(uint64_t A) { return A % 0x12345678ABULL; }

    movabs  rdx, -2233785418547900415
    mov     rax, rdi
    mul     rdx
    mov     rax, rdx             # wasted instruction, could have kept using RDX.
    movabs  rdx, 78187493547
    shr     rax, 36            # division result
    imul    rax, rdx           # multiply and subtract to get the modulo
    sub     rdi, rax
    mov     rax, rdi
    ret

x86's mul r64 instruction does 64b*64b => 128b (rdx:rax) multiplication, and can be used as a building block to construct a 128b * 128b => 256b multiply to implement the same algorithm. Since we only need the high half of the full 256b result, that saves a few multiplies.

Modern Intel CPUs have very high performance mul: 3c latency, one per clock throughput. However, the exact combination of shifts and adds required varies with the constant, so the general case of calculating a multiplicative inverse at run-time isn't quite as efficient each time its used as a JIT-compiled or statically-compiled version (even on top of the pre-computation overhead).

IDK where the break-even point would be. For JIT-compiling, it will be higher than ~200 reuses, unless you cache generated code for commonly-used B values. For the "normal" way, it might possibly be in the range of 200 reuses, but IDK how expensive it would be to find a modular multiplicative inverse for 128-bit / 64-bit division.

libdivide can do this for you, but only for 32 and 64-bit types. Still, it's probably a good starting point.