Re: [PATCH v3 1/4] lib/string_helpers: change blk_size to u32 for string_get_size() interface

[James Bottomley <jbottomley@odin.com>]

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On Tue, 2015-11-03 at 14:13 +0100, Vitaly Kuznetsov wrote:
> 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:

It's always < 1 in the final equation because it's divided by the square
of divisor[units].  However, that can make a contribution to the round
up, I suppose leading to the truncation you see

> 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?

It was a suggestion when I explained what the missing sources of
precision were, I don't think it's really a suggestion when it comes
with an exemplary patch.  However, the whole thing needs polishing
because all the divisions need to be eliminated, since they're a huge
source of problems for most 32 bit CPUs, so I can fix it all the way and
repost.

James

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