# Optimized division of uint64_t by a constant on 32-bit microcontrollers

#### Written 2022-07-21

#### Tags:Multiplication Division Optimization ARM

## Problem Statement

Recently, I ran into a bottleneck and my profiler highlighted that a large fraction of program time was being spent in __aeabi_uldivmod(), an ARM math support function for division with the following prototype:

struct result { u64 quot, rem; }; struct result __aeabi_uldivmod(u64 numerator, u64 denominator) { struct result res; //compute res here return res; }

There's several differeny ways to implement __aeabi_uldivmod(), including 1-bit-at-a-time long division, the same with speedup from a count-leading-zeros instrucion like ARM's CLZ, or 32-bit-at-a-time long-division using the udiv instruction present on some ARM cores.

## A partial solution

But most of my program's calls to __aeabi_uldivmod() were with a fixed denominator - this common special-case is often optimized by compilers by replacing division with multiplication by the inverse - since we're dealing with integers, they actually use multiplication by a scaled inverse, something like replacing x/N with x * (2^32/N), then shifting the result right to drop off the remainder. Here's what we need to optimize(and some related functions, like ns_to_us, ns_to_ms, which can be done similarly):

uint64_t ns_to_s( uint64_t ns64 ) { return ns64 / 1000000000ULL; }

And it turns out GCC does know this trick, at least aarch64 GCC10.2.1 does. When we compile this we get:

mov x1, 23123 #Load part of a contant into x1 lsr x0, x0, 9 #x0 is ns64, shift it right 9 bits(divide by 512) movk x1, 0xa09b, lsl 16 #load more bits into x1 movk x1, 0xb82f, lsl 32 #... movk x1, 0x44, lsl 48 #... umulh x0, x0, x1 #multiply x0 by x1, keeping only upper 64 bits(discarding 64lsbs) lsr x0, x0, 11 #unsigned shift right 11 more bits( ret

But when we compile with 32-bit arm GCC10.2.1, we get a call to uldivmod.

ns_to_s: @ args = 0, pretend = 0, frame = 0 @ frame_needed = 0, uses_anonymous_args = 0 push {r3, lr} adr r3, .L14 ldrd r2, [r3] bl __aeabi_uldivmod(PLT) pop {r3, pc} .L14: .word 1000000000 .word 0

At first I thought this made sense, since 32-bit arm doesn't have a umulh instruction that takes two 64bit registers and multiplies them together to compute a 128bit result then truncates it. But 32bit by 32bit multiplication with 64bit result is relatively quick on 32-bit arm cores, and add and subtract instructions can be chained together to do a 128-bit add in only 4 instructions. If only I had a umulh function, the aarch64 division trick above could be made to work...

## The hack

...why not implement one?

At first I started with a slightly naive approach to 128-bit addition by adding after each multiplication using a uint64_t. However, these additions can be overlapped - several uint32_ts can be added together with a uint64_t result without overflowing. By pipelining the 128-bit accumulator to operate LSBs to MSBs, 32 bits at a time, several instructions can be optimized out, and since 64 LSBs are dropped in the result, only their carry-bits are needed. Here's what I came up with:

uint64_t umulh( uint64_t a, uint64_t b ) { const uint32_t a_lo = a; const uint32_t a_hi = a >> 32; const uint32_t b_lo = b; const uint32_t b_hi = b >> 32; /* FOIL method of multiplication See https://en.wikipedia.org/wiki/FOIL_method, but instead of binomials with constants a,b,c,d variables x,y: (ax+b) * (cy + d), consider it with variables a,b,c,d, constants x,y = 1<<32 */ const uint64_t acc0 = (uint64_t)a_lo * b_lo; const uint64_t acc1 = (uint64_t)a_hi * b_lo; const uint64_t acc2 = (uint64_t)a_lo * b_hi; const uint64_t acc3 = (uint64_t)a_hi * b_hi; /* Break up into 32-bit values */ const uint32_t lo0 = acc0; const uint32_t hi0 = acc0 >> 32; const uint32_t lo1 = acc1; const uint32_t hi1 = acc1 >> 32; const uint32_t lo2 = acc2; const uint32_t hi2 = acc2 >> 32; const uint32_t lo3 = acc3; const uint32_t hi3 = acc3 >> 32; /* The first 32 bits aren't used in the result, no need to store them, so no need to compute them. In fact, don't even worry about res0, start computing res1*/ uint64_t acc = 0; const uint32_t res0 = lo0; acc += hi0; acc += lo1; acc += lo2; const uint32_t res1 = acc; acc >>= 32; acc += hi1; acc += hi2; acc += lo3; const uint32_t res2 = (uint32_t)acc; acc >>= 32; acc += hi3; const uint32_t res3 = (uint32_t)acc; return (((uint64_t)res3) << 32 ) + res2; } uint64_t ns_to_s_inv( uint64_t ns64 ) { //constants and shifts from aarch-64 GCC return umulh( 0x44b82fa09b5A53ULL, ns64 >> 9 ) >> 11; }

## Results

I timed it on an nRF52, and it beats all __aeabi_uldivmod implementations I've seen so far, though udiv based approachs are quite close.

Algorithm\InstructionCache | Disabled | Enabled |
---|---|---|

ns_to_s()/__aeabi_uldivmod - traditional | 50.4-64us | 30.8-40.8us |

ns_to_s()/__aeabi_uldivmod - udiv based | 2.64-3.36us | 2.16-4.64us |

ns_to_s_inv()/umulh | 1.26us | 1.09us |

This comes roughly to a 25x to 50x runtime improvement from where I started. The reason for some of this variability is that __aeabi_uldivmod() often takes a different number of cycles based on the values of the inputs. When I saw this in the timing with my oscilloscope, I set the oscilloscope to trigger on the start of each computation, then average 256x resulting triggers together.

## Oscilloscope captures

__aeabi_uldivmod common approach:

Now here's GCC11 for Cortex-M3/M4's udiv-based approach:

And here's my umulh-emulation-based approach - averaging is still on, but since the umulh-based approach contains no loops or other data-dependent control-flow the calculation time is far more repeatable when given different inputs. Also note this is 20x zoomed in time compared to the first plot:

## Applicability

Generally, the umulh-based approach above may apply to any machine with a 32 bit x 32 bit multiply instruction with 64-bit result. This instruction is called umull for ARM cores. When a 32-bit udiv instruction is available, __aeabi_uldivmod is competitive on: Cortex-M3, M4, M33, M35P, M55, and newer Cortex-A cores including Cortex-A7, Cortex-A15, Cortex-A53, Cortex-A57. Cores with umull but without udiv is where the umulh approach really shines: ARM7TDMI, ARM9, ARM10, ARM11, Cortex-A8, and Cortex-A9. Possibly also XScale and StrongARM.