On Fri, 2010-11-19 at 17:28 -0500, Bob Copeland wrote:
quoted text
> On Fri, Nov 19, 2010 at 06:07:05PM +0900, Bruno Randolf wrote:
> > Hmm, maybe I suck in mathemathics, but I don't see a way to do that given the 
> > formula:
> > 
> > (((internal * (weight - 1)) + (val * factor)) / weight
> 
> I was thinking something along the lines of:
> 
> round = (1 << n) - 1;
> (((internal * (weight - 1) + round) >> n) + val) * ((1 << n) / weight)
> 
> where (1 << n) is the factor and ((1 << n) / weight) can be precomputed.
> If you think about it, this is just reciprocal multiplication in fixed-
> point math with n bits of decimal resolution. 
> 
> The problem is the shift of the older terms introduces roundoff error, but
> there are some tricks you can do to maintain bounded error, e.g. shifting
> by some smaller factor of n and scaling other terms -- in the limit you
> reinvent floating point and then it's slower than division :)

Sure, x/y := x/z * z/y, and by picking z := 2^n, we can pre-compute z/y
and write x/z using a shift. The problem however is always range vs
granularity, you chose to first /z and then *z/y, this avoids some
overflow issues but truncates the lower n bits of x.

If you first *z/y and then /z you keep your low bits but risk loosing
the top bits to an overflow.

I guess the question is do we really need weights outside of 2^n? If
not, you can use the weight := 2^n version. If you do, you get to pick
either of the previously mentioned options.

Sadly gcc doesn't sanely support a u128 type, which would be very useful
to avoid some of these overflow issues (like we used to use u64 mults
for u32 fixed points mults).
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