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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