Let’s make integer multiplication fast ● Crypto means “multiplying really big numbers together” ● 64 bits x 64 bits will produce 128-bit output ● We don't have 128-bit registers on 64-bit machines ● Results have to be stored in multiple registers (ignoring vector insns) ● Processors provide "widening multiply" instructions that do this ● Some languages provide uint128 and handle this for you ● Go does not ○ https://github.com/golang/go/issues/9455
A multiplier Basically, splits large numbers into smaller components that we can handle more easily result = hi << 64 + lo func multiply1(a, b uint64) [2]uint64 { al := a & 0xFFFFFFFF ah := a >> 32 bl := b & 0xFFFFFFFF bh := b >> 32 c0 := (al * bl) >> 32 t1 := ah*bl + c0 t1_lo := t1 & 0xFFFFFFFF c1 := t1 >> 32 t2 := al*bh + t1_lo c2 := t2 >> 32 hi := ah*bh + c1 + c2 lo := (a * b) return [2]uint64{lo, hi} }
I heard assembly is fast The amd64 instruction we want is called mulq mulq X Unsigned full multiply of %rax by X Result stored in %rdx:%rax The -q suffix means "quadword" - an eight-byte (64-bit) value.
Multiplier in asm Uses MULQ Nothing clever How much do we love Plan9? // +build amd64 // func mulq(x, y uint64) (lo uint64, hi uint64) TEXT ·mulq(SB),4,$0 MOVQ x+0(FP), AX MOVQ y+8(FP), CX MULQ CX MOVQ AX, ret+16(FP) // result low bits MOVQ DX, ret+24(FP) // result high bits RET
Benchmarks Take my word for it, this is still REALLY slow. // Arbitrary value. aLargeNumber^2 is 110 bits const aLargeNumber uint64 = 3*(1<<52) + 7*(1<<51) func BenchmarkMultiplyAsm(b *testing.B) { for i := 0; i < b.N; i++ { _ = mulq(aLargeNumber, aLargeNumber) } } $ go test -bench MultiplyAsm goos: linux goarch: amd64 pkg: github.com/gtank/gophercon2017-examples BenchmarkMultiplyAsm-2 300000000 5.59 ns/op PASS
Function call overhead A function call takes multiple nanoseconds, which completely dominates the runtime of small arithmetic functions like this multiplier. pprof again! this mode is called “weblist” https://github.com/google/pprof
Inlining To avoid this problem, compilers try to move the code of small/simple functions directly into the caller. This is called inlining. The Go inliner lives here: https://github.com/golang/go/blob/master/src/cmd/compile/internal/gc/inl.go It walks the instruction tree of each function calculating a “cost” that roughly reflects the complexity of the function. If the cost exceeds the hard-coded max budget, the function will not be inlined.
Inlining Functions accrue +1 cost for each node in the instruction tree Some instructions are more equal than others: ● Slice ops are +2 ● 3-arg slice ops are +3 ● Direct function calls (OCALLFUNC/OCALLMETH) only OK if target is inlinable and we have budget for it Some things are hard stops: ● Most other types of calls. Interface calls, type conversions ● Panic/recover ● Certain runtime funcs and all non-intrinsic assembly [#17373]
Multiplier #3 Inlines! Propagated some assignments Shifts cost less than shift + assign if you only use them once But how do we know? func multiply3(a, b uint64) (uint64, uint64) { al := a & 0xFFFFFFFF ah := a >> 32 bl := b & 0xFFFFFFFF bh := b >> 32 t1 := ah*bl + ((al * bl) >> 32) t2 := al*bh + (t1 & 0xFFFFFFFF) hi := ah*bh + t1>>32 + t2>>32 lo := (a * b) return lo, hi }
Compiler Hack Prints the inliner cost of each function during builds The magic number is 80 $ go test -bench . -gcflags -m # github.com/gtank/gophercon2017-examples [inl] func multiply1 costs 99 [inl] func multiply2 costs 97 [inl] func multiply3 costs 77 ./multiply.go:50:6: can inline multiply3
Benchmarks Computers *are* fast $ go test -bench Multiply3 goos: linux goarch: amd64 pkg: github.com/gtank/gophercon2017-examples BenchmarkMultiply3-2 2000000000 1.18 ns/op PASS That’s ⅕ of the time the assembly function took.
gtank
khmer495
takesinoda
nsuz
track3jyo
ivargrimstad
octu0
skume
akatsuki174
danilop
k0kubun
munetoshi
dalinaum
yeseniaperezcruz
mojombo
nonsquared
jonyablonski
deanohume
wjessup
bermonpainter
mza
rocio
cherdarchuk
sstephenson
destraynor