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 > <mailto: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 >>> <mailto: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 >>> <mailto: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 >>>> <mailto: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 >>>>> <mailto: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 >>>>>> <mailto: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 >>>>>>> <mailto: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 >>> <mailto: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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