This is getting interesting but maybe only for me, so I'll stop here. I did some profiling and the majority of the time is spent in math/big.addMulVVW. Thinking I might be doing something stupid, I compared it with the Go implementation at http://www.luschny.de/math/factorial/scala/FactorialScalaCsharp.htm, and found that my version (although "straightforward"), is about the same amount of code but without special casing and support tables, yet about 25% faster - and yes, they get the same answer.
So I believe the factor of 60 comes from comparing an implementation in a language and libraries designed for numerical computation against a much less specialized and optimized world. Or perhaps from Julia using a different and dramatically more efficient algorithm. It does seem like a big gap, but I am no expert in this area. Maybe worth investigating further but not by me. -rob On Sun, Jan 14, 2024 at 9:31 AM Rob Pike <r...@golang.org> wrote: > Oh, I did say my implementation was straightforward. It's free of any > clever multiplication algorithms or mathematical delights. It could easily > be giving up 10x or more for that reason alone. And I haven't even profiled > it yet. > > -rob > > > On Sat, Jan 13, 2024 at 7:04 PM Bakul Shah <ba...@iitbombay.org> wrote: > >> FYI Julia (on M1 MBP) seems much faster: >> >> julia> @which factorial(big(100000000)) >> factorial(x::BigInt) in Base.GMP at gmp.jl:645 >> >> julia> @time begin; factorial(big(100000000)); 1; end >> 27.849116 seconds (1.39 M allocations: 11.963 GiB, 0.22% gc time) >> >> >> Probably they use Schönhage-Strassen multiplication algorithm for very >> large numbers as the 1E8! result will have over a 3/4 billion digits. I >> should try this in Gambit-Scheme (which has an excellent multiply >> implementation). >> >> On Jan 12, 2024, at 9:32 PM, Rob Pike <r...@golang.org> wrote: >> >> Thanks for the tip. A fairly straightforward implementation of this >> algorithm gives me about a factor of two speedup for pretty much any value. >> I went up to 1e8!, which took about half an hour compared to nearly an hour >> for MulRange. >> >> I'll probably stick in ivy after a little more tuning. I may even try >> parallelization. >> >> -rob >> >> >> On Tue, Jan 9, 2024 at 4:54 PM Bakul Shah <ba...@iitbombay.org> wrote: >> >>> For that you may wish to explore Peter Luschny's "prime swing" factorial >>> algorithm and variations! >>> https://oeis.org/A000142/a000142.pdf >>> >>> And implementations in various languages including go: >>> https://github.com/PeterLuschny/Fast-Factorial-Functions >>> >>> On Jan 8, 2024, at 9:22 PM, Rob Pike <r...@golang.org> wrote: >>> >>> Here's an example where it's the bottleneck: ivy factorial >>> >>> >>> !1e7 >>> 1.20242340052e+65657059 >>> >>> )cpu >>> 1m10s (1m10s user, 167.330ms sys) >>> >>> >>> -rob >>> >>> >>> On Tue, Jan 9, 2024 at 2:21 PM Bakul Shah <ba...@iitbombay.org> wrote: >>> >>>> Perhaps you were thinking of this? >>>> >>>> At iteration number k, the value xk contains O(klog(k)) digits, thus >>>> the computation of xk+1 = kxk has cost O(klog(k)). Finally, the total >>>> cost with this basic approach is O(2log(2)+¼+n log(n)) = O(n2log(n)). >>>> >>>> A better approach is the *binary splitting* : it just consists in >>>> recursively cutting the product of m consecutive integers in half. It leads >>>> to better results when products on large integers are performed with a fast >>>> method. >>>> >>>> http://numbers.computation.free.fr/Constants/Algorithms/splitting.html >>>> >>>> >>>> I think you can do recursive splitting without using function recursion >>>> by allocating N/2 array (where b = a+N-1) and iterating over it; each time >>>> the array "shrinks" by half. A "cleverer" algorithm would allocate an array >>>> of *words* of a bignum, as you know that the upper limit on size is N*64 >>>> (for 64 bit numbers) so you can just reuse the same space for each outer >>>> iteration (N/2 multiplie, N/4 ...) and apply Karatsuba 2nd outer iteration >>>> onwards. Not sure if this is easy in Go. >>>> >>>> On Jan 8, 2024, at 11:47 AM, Robert Griesemer <g...@golang.org> wrote: >>>> >>>> Hello John; >>>> >>>> Thanks for your interest in this code. >>>> >>>> In a (long past) implementation of the factorial function, I noticed >>>> that computing a * (a+1) * (a+2) * ... (b-1) * b was much faster when >>>> computed in a recursive fashion than when computed iteratively: the reason >>>> (I believed) was that the iterative approach seemed to produce a lot more >>>> "internal fragmentation", that is medium-size intermediate results where >>>> the most significant word (or "limb" as is the term in other >>>> implementations) is only marginally used, resulting in more work than >>>> necessary if those words were fully used. >>>> >>>> I never fully investigated, it was enough at the time that the >>>> recursive approach was much faster. In retrospect, I don't quite believe my >>>> own theory. Also, that implementation didn't have Karatsuba multiplication, >>>> it just used grade-school multiplication. >>>> >>>> Since a, b are uint64 values (words), this could probably be >>>> implemented in terms of mulAddVWW directly, with a suitable initial >>>> allocation for the result - ideally this should just need one allocation >>>> (not sure how close we can get to the right size). That would cut down the >>>> allocations massively. >>>> >>>> In a next step, one should benchmark the implementation again. >>>> >>>> But at the very least, the overflow bug should be fixed, thanks for >>>> finding it! I will send out a CL to fix that today. >>>> >>>> Thanks, >>>> - gri >>>> >>>> >>>> >>>> On Sun, Jan 7, 2024 at 4:47 AM John Jannotti <janno...@gmail.com> >>>> wrote: >>>> >>>>> Actually, both implementations have bugs! >>>>> >>>>> The recursive implementation ends with: >>>>> ``` >>>>> m := (a + b) / 2 >>>>> return z.mul(nat(nil).mulRange(a, m), nat(nil).mulRange(m+1, b)) >>>>> ``` >>>>> >>>>> That's a bug whenever `(a+b)` overflows, making `m` small. >>>>> FIX: `m := a + (b-a)/2` >>>>> >>>>> My iterative implementation went into an infinite loop here: >>>>> `for m := a + 1; m <= b; m++ {` >>>>> if b is `math.MaxUint64` >>>>> FIX: add `&& m > a` to the exit condition is an easy fix, but pays a >>>>> small penalty for the vast majority of calls that don't have b=MaxUint64 >>>>> >>>>> I would add these to `mulRangesN` of the unit test: >>>>> ``` >>>>> {math.MaxUint64 - 3, math.MaxUint64 - 1, >>>>> "6277101735386680760773248120919220245411599323494568951784"}, >>>>> {math.MaxUint64 - 3, math.MaxUint64, >>>>> "115792089237316195360799967654821100226821973275796746098729803619699194331160"} >>>>> ``` >>>>> >>>>> On Sun, Jan 7, 2024 at 6:34 AM John Jannotti <janno...@gmail.com> >>>>> wrote: >>>>> >>>>>> I'm equally curious. >>>>>> >>>>>> FWIW, I realized the loop should perhaps be >>>>>> ``` >>>>>> mb := nat(nil).setUint64(b) // ensure mb starts big enough for b, >>>>>> even on 32-bit arch >>>>>> for m := a + 1; m <= b; m++ { >>>>>> mb.setUint64(m) >>>>>> z = z.mul(z, mb) >>>>>> } >>>>>> ``` >>>>>> to avoid allocating repeatedly for `m`, which yields: >>>>>> BenchmarkIterativeMulRangeN-10 354685 3032 ns/op 2129 >>>>>> B/op 48 allocs/op >>>>>> >>>>>> On Sun, Jan 7, 2024 at 2:41 AM Rob Pike <r...@golang.org> wrote: >>>>>> >>>>>>> It seems reasonable but first I'd like to understand why the >>>>>>> recursive method is used. I can't deduce why, but the CL that adds it, >>>>>>> by >>>>>>> gri, does Karatsuba multiplication, which implies something deep is >>>>>>> going >>>>>>> on. I'll add him to the conversation. >>>>>>> >>>>>>> -rob >>>>>>> >>>>>>> >>>>>>> >>>>>>> >>>>>>> On Sun, Jan 7, 2024 at 5:46 PM John Jannotti <janno...@gmail.com> >>>>>>> wrote: >>>>>>> >>>>>>>> I enjoy bignum implementations, so I was looking through nat.go and >>>>>>>> saw that `mulRange` is implemented in a surprising, recursive way,. >>>>>>>> In the >>>>>>>> non-base case, `mulRange(a, b)` returns `mulrange(a, (a+b)/2) * >>>>>>>> mulRange(1+(a+b)/2, b)` (lots of big.Int ceremony elided). >>>>>>>> >>>>>>>> That's fine, but I didn't see any advantage over the >>>>>>>> straightforward (and simpler?) for loop. >>>>>>>> >>>>>>>> ``` >>>>>>>> z = z.setUint64(a) >>>>>>>> for m := a + 1; m <= b; m++ { >>>>>>>> z = z.mul(z, nat(nil).setUint64(m)) >>>>>>>> } >>>>>>>> return z >>>>>>>> ``` >>>>>>>> >>>>>>>> In fact, I suspected the existing code was slower, and allocated a >>>>>>>> lot more. That seems true. A quick benchmark, using the existing unit >>>>>>>> test >>>>>>>> as the benchmark, yields >>>>>>>> BenchmarkRecusiveMulRangeN-10 169417 6856 ns/op >>>>>>>> 9452 B/op 338 allocs/op >>>>>>>> BenchmarkIterativeMulRangeN-10 265354 4269 ns/op >>>>>>>> 2505 B/op 196 allocs/op >>>>>>>> >>>>>>>> I doubt `mulRange` is a performance bottleneck in anyone's code! >>>>>>>> But it is exported as `int.MulRange` so I guess it's viewed with some >>>>>>>> value. And seeing as how the for-loop seems even easier to understand >>>>>>>> that >>>>>>>> the recursive version, maybe it's worth submitting a PR? (If so, >>>>>>>> should I >>>>>>>> create an issue first?) >>>>>>>> >>>>>>>> >>>>>>>> >>>>>>>> >>>>>>>> -- >>>>>>>> You received this message because you are subscribed to the Google >>>>>>>> Groups "golang-nuts" group. >>>>>>>> To unsubscribe from this group and stop receiving emails from it, >>>>>>>> send an email to golang-nuts+unsubscr...@googlegroups.com. >>>>>>>> To view this discussion on the web visit >>>>>>>> https://groups.google.com/d/msgid/golang-nuts/e6ceb75a-f8b7-4f77-97dc-9445fb750782n%40googlegroups.com >>>>>>>> <https://groups.google.com/d/msgid/golang-nuts/e6ceb75a-f8b7-4f77-97dc-9445fb750782n%40googlegroups.com?utm_medium=email&utm_source=footer> >>>>>>>> . >>>>>>>> >>>>>>> >>>> -- >>>> You received this message because you are subscribed to the Google >>>> Groups "golang-nuts" group. >>>> To unsubscribe from this group and stop receiving emails from it, send >>>> an email to golang-nuts+unsubscr...@googlegroups.com. >>>> To view this discussion on the web visit >>>> https://groups.google.com/d/msgid/golang-nuts/CAKy0tf7Lcd8hiF2Qv3NFfjGcfvXDn%2BA%2BxJ1bfKta1w9P-OAs%3Dw%40mail.gmail.com >>>> <https://groups.google.com/d/msgid/golang-nuts/CAKy0tf7Lcd8hiF2Qv3NFfjGcfvXDn%2BA%2BxJ1bfKta1w9P-OAs%3Dw%40mail.gmail.com?utm_medium=email&utm_source=footer> >>>> . >>>> >>>> >>>> >>> >> -- You received this message because you are subscribed to the Google Groups "golang-nuts" group. 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