Messages in this thread | | | From | Vitaly Kuznetsov <> | Subject | Re: [PATCH v3 1/4] lib/string_helpers: change blk_size to u32 for string_get_size() interface | Date | Tue, 03 Nov 2015 14:13:08 +0100 |
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James Bottomley <jbottomley@odin.com> writes:
> On Mon, 2015-11-02 at 16:58 +0100, Vitaly Kuznetsov wrote: >> James Bottomley <jbottomley@odin.com> writes: >> >> > On Fri, 2015-10-30 at 11:46 +0100, Vitaly Kuznetsov wrote: >> >> James Bottomley <jbottomley@odin.com> writes: >> >> >> >> > On Thu, 2015-10-29 at 17:30 +0100, Vitaly Kuznetsov wrote: >> >> >> string_get_size() can't really handle huge block sizes, especially >> >> >> blk_size > U32_MAX but string_get_size() interface states the opposite. >> >> >> Change blk_size from u64 to u32 to reflect the reality. >> >> > >> >> > What is the actual evidence for this? The calculation is designed to be >> >> > a symmetric 128 bit multiply. When I wrote and tested it, it worked >> >> > fine for huge block sizes. >> >> >> >> We have 'u32 remainder' and then we do: >> >> >> >> exp = divisor[units] / (u32)blk_size; >> >> ... >> >> remainder = do_div(size, divisor[units]); >> >> remainder *= blk_size; >> >> >> >> I'm pretty sure it will overflow for some inputs. >> > >> > It shouldn't; the full code snippet does this: >> > >> > while (blk_size >= divisor[units]) { >> > remainder = do_div(blk_size, divisor[units]); >> > i++; >> > } >> > >> > exp = divisor[units] / (u32)blk_size; >> > >> > So by the time it reaches the statement you complain about, blk_size is >> > already less than or equal to the divisor (which is 1000 or 1024) so >> > truncating to 32 bits is always correct. >> > >> >> I overlooked, sorry! >> >> > I'm sort of getting the impression you don't quite understand the >> > mathematics: i is the logarithm to the base divisor[units]. We reduce >> > both operands to exponents of the logarithm base (adding the two bases >> > together in i), which means they are by definition in a range between >> > zero and the base and then multiply the remaining exponents correcting >> > the result for a base overflow (so the result is always a correct >> > exponent and i is the logarithm to the base). It's actually simply >> > Napier's algorithm. >> > >> > The reason we're getting the up to 2.5% rounding errors you complain >> > about is because at each logarithm until the last one, we throw away the >> > remainder (it's legitimate because it's always 1000x smaller than the >> > exponent), but in the case of a large remainder it provides a small >> > correction to the final operation which we don't account for. If you >> > want to make a true correction, you save the penultimate residue in each >> > case, multiply each by the *other* exponent add them together, divide by >> > the base and increment the final result by the remainder. >> >> My assumption was that we don't really need to support blk_sizes > >> U32_MAX and we can simplify string_get_size() instead of adding >> additional complexity. Apparently, the assumption was wrong. >> >> > >> > However, for 2.5% the physicist in me says the above is way overkill. >> > >> >> It is getting was over 2.5% if blk_size is not a power of 2. While it is >> probably never the case for block subsystem the function is in lib and >> pretends to be general-enough. I'll try to make proper correction and >> let's see if it's worth the effort. > > OK, this is the full calculation. It also includes an arithmetic > rounding to the final figure print. I suppose it's not that much more > complexity than the original, and it does make the algorithm easier to > understand. > > We could do with running the comments by some other non-mathematician, > now I've explained it in detail to you two, to see if they actually give > an understanding of the algorithm.
Thanks, to me they look great! One nitpick below ...
> > James > > --- > > diff --git a/lib/string_helpers.c b/lib/string_helpers.c > index 5939f63..1ec7e77a 100644 > --- a/lib/string_helpers.c > +++ b/lib/string_helpers.c > @@ -44,7 +44,7 @@ void string_get_size(u64 size, u64 blk_size, const enum string_size_units units, > [STRING_UNITS_2] = 1024, > }; > int i, j; > - u32 remainder = 0, sf_cap, exp; > + u32 remainder = 0, sf_cap, r1 = 0, r2 = 0, round; > char tmp[8]; > const char *unit; > > @@ -53,27 +53,46 @@ void string_get_size(u64 size, u64 blk_size, const enum string_size_units units, > if (!size) > goto out; > > + /* This is napier's algorithm. Reduce the original block size to > + * > + * co * divisor[units]^i > + * > + * where co = blk_size + r1/divisor[units]; > + * > + * and the same for size. We simply add to the exponent i, because > + * the final calculation we're looking for is > + * > + * (co1 * co2) * divisor[units]^i > + */ > + > + > while (blk_size >= divisor[units]) { > - remainder = do_div(blk_size, divisor[units]); > + r1 = do_div(blk_size, divisor[units]); > i++; > } > > - exp = divisor[units] / (u32)blk_size; > - /* > - * size must be strictly greater than exp here to ensure that remainder > - * is greater than divisor[units] coming out of the if below. > - */ > - if (size > exp) { > - remainder = do_div(size, divisor[units]); > - remainder *= blk_size; > + while (size >= divisor[units]) { > + r2 = do_div(size, divisor[units]); > i++; > - } else { > - remainder *= size; > } > > - size *= blk_size; > - size += remainder / divisor[units]; > - remainder %= divisor[units]; > + /* here's the magic. co1 * co2 may be > divisor[i], so correct for > + * that in the exponent and make sure that the additional corrections > + * from the remainders is added in. > + * > + * co1 *co2 = (blk_size + r1/divisor[units])*(size + r2/divisor[units]) > + * > + * therefore > + * > + * co1*co2*divisor[units] = blk_size*size*divisor[units] + > + * r1*size + r2*size + r1*r2/divisor[units] > + * > + * drop the last term because it's too small and perform the > + * calculation cleverly by decremeting i to be automatically dealing > + * with everything multiplied by divisor[units] */ > + > + --i; > + size = size * blk_size * divisor[units] + r1 * size + r2 * > blk_size;
The last term is actually not that small. Here is an example:
size = 8192 blk_size = 1024
'As is' the algorithm gives us '8.38 MB', and if we add "+ r1 * r1 / divisor[units]" we get '8.39 MB' (the correct answer is 8192 * 1024 = 8388608 which is 8.39).
Both r1 and r2 are < divisor[units] here so r1 * r2 won't overflow u32, I suggest we add this term.
> > while (size >= divisor[units]) { > remainder = do_div(size, divisor[units]); > @@ -81,8 +100,15 @@ void string_get_size(u64 size, u64 blk_size, const enum string_size_units units, > } > > sf_cap = size; > - for (j = 0; sf_cap*10 < 1000; j++) > + round = 500; > + for (j = 0; sf_cap*10 < 1000; j++) { > sf_cap *= 10; > + round /= 10; > + } > + > + /* add a 5 to the digit below what will be printed to ensure > + * an arithmetical round up */ > + remainder += round; > > if (j) { > remainder *= 1000;
Can I post this solution with your Suggested-by or do you plan to do it yourself?
Thanks,
-- Vitaly
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