An arithmetic coder encodes a sequence of symbols as an infinite precision number. You divide the range of that number
(I will call this the "arithmetic domain") into successively smaller intervals; each interval corresponds to a symbol.
After encoding a symbol, your current coding interval shrinks to the low/high range of that symbol, and you encode further
symbols within that range. The minimum number of bits required to distinguish the desired sequence from another is the
code length of the sequence.

If you make each symbol's interval proportional to the probability of that symbol, the arithmetic coder can produce a
code length equal to the entropy. (we're still assuming infinite precision encoding). If your estimated probabilities do
not match the true symbol probabilities (they never do) you have some loss due to modeling. The difference between the
arithmetic coder's output length and the sum of -log2(P) for all model probabilities is the coding loss.

In order to do arithmetic coding in finite precision, we track the low/high interval of the arithmetic coder. As the top bits
(or bytes) of low & high become equal, we stream them out. This is like "zooming in" on a portion of the infinite number line
where the top & bottom of the interval are in the same binary power of two buckets.

We must then confront the "underflow problem". That is, sometimes (high - low) ("range") can get very small, but the top bits of
high and low never match, eg. if they happen to straddle a binary power of 2. They can be something like

high = 1000000000aaaaa..
low = 0111111111bbbbb..

There are several solutions to the underflow problem. For example CACM87 "bit plus follow" or the MACM / VirtQ approach which you can read about
elsewhere, also the "just force a shrink" method (see links at end).

The method I use in "recip_arith" rather hides the underflow problem. Rather than checking for the top bits (bytes) of low & high
being equal, we simply zoom in regardless. The renormalization step is :

while ( ac->range < (1<<24) )
{
*(ac->ptr)++ = (uint8_t)(ac->low>>24);
ac->low <<= 8;
ac->range <<= 8;
}

low and range are 32 bit here, when range is less than 2^24 the top byte of low & high is the same and can be shifted out, *except* in
the underflow case. In the underflow case, we could have the top byte of low is = FF , and high is actually 100 with an implicit bit above
the 32nd bit (eg. low + range exceeds 2^32). What we do is go ahead and output the FF, then if we later find that we made a mistake
we correct it by propagating a carry into the already sent bits.

(note that you could do range += FF here for slightly less coding loss, but the difference is small; the actual "high" of our arithmetic interval is 1 above range, range can approach that infinitely
closely from below but never touch it; the coding interval is [low,high) inclusive on the bottom & exclusive on the top. Coders that don't
quite get this right have a lot of +1/-1 adjustments around low & high)

Renormalization means we can send an infinite length number while only working on a finite precision portion of that number down in the
active range of bits at the bottom. Renormalization also means that "range" is kept large enough to be able to distinguish symbols with
only integer subdivision of the range, which we shall now address. Renormalization in and of itself does not introduce any coding loss; it is
perfectly accurate (though failing to add FF is coding loss, and schemes like the "force shrink" method or "just dont renormalize" method of
fpaq0p do contribute to coding loss).

The other way we must adapt to finite precision is the division of the interval into ranges proportional to symbol probabilities.
The infinite precision refinement would be (real numbers!) :

arithmetic_low += CDF_low * arithmetic_range / CDF_sum;
arithmetic_range *= CDF_freq / CDF_sum;
(real numbers, no floor divide)
(CDF = cumulative distribution function, aka cumulative probability, sum of previous symbol frequencies)
CDF_freq = CDF_high - CDF_low for the current symbol ; CDFs in [0,CDF_sum]

We don't want to do real numbers, so we just approximate them with integer math. But how exactly?

** The crucial distinguishing aspect of an arithmetic coder is how you map the CDF domain to the arithmetic domain **

The CDF domain is controlled by you; you have modeled probabilities somehow. The CDF domain always starts at 0 and goes to CDF_sum,
which is under your control. In the decoder, you must search in the CDF domain to find what symbol is specified. Working in the CDF
domain is easy. In contrast, the arithmetic interval is always changing; "low/range" is being shrunk by coding, and then zoomed in again by
renormalization.

The forward map takes you from CDF domain to arithmetic domain. Adding on the arithmetic "low" is trivial and we will not include it in the
map. The crucial thing is just scaling by (arithmetic_range / CDF_sum).

We can now write a very general arithmetic encoder :

arithmetic_low += forward(CDF_low,arithmetic_range,CDF_sum);
arithmetic_range = forward(CDF_high,arithmetic_range,CDF_sum) - forward(CDF_low,arithmetic_range,CDF_sum);

our "forward" map will be working on integers. Some properties forward must have :

forward(x) should be monotonic
forward(x+1) > forward(x) strictly (so that range can never shrink to zero)
this may only be true for arithmetic_range > CDF_sum or some similar constraint
forward(0) >= 0
forward(CDF_sum) <= arithmetic_range
forward map of the CDF end points does not need to hit the end points of range, but it must be within them
(failure to use all of range does contribute to coding loss)

The coding loss of our approximation is caused by the difference in forward(high) - forward(low) and the ideal scaling
(which should be proportional to range & symbol probability).

The integer forward map with lowest coding loss is the "CACM87 map" :

forward(cdf,range,cdf_sum) = ( cdf * range ) / cdf_sum;
this is now integers (eg. floor division)
CACM87 has
forward(cdf_sum) = range ; eg. it uses the full range

coding loss is just due to the floor division not exactly matching the real number divide. (note that you might be tempted to
say, hey add (cdf_sum/2) to get a rounded integer division instead of floor; the exact form here is needed to be able to construct
an inverse map with the right properties, which we will get to later).

** Sketch of full arithmetic coding process **

A quick overview of what the decoder has to do. Most of the decoder just replicates the same work as the encoder; it narrows the
arithmetic interval in exactly the same way. Rather than streaming out bytes in renormalization, the decoder streams them in. The
decoder sees the arithmetic code value that the encoder sent, to some precision ahead. It needs enough bits fetched to be able to
resolve the correct symbol (to tell which CDF bin is selected).

In implementation, rather than track the low/high arithmetic interval and the arithmetic number within that interval, we instead just track
(arithmetic - low), the offset inside the interval. I call this the "code" in the decoder.

The decoder needs an extra step that the encoder doesn't do : given the current "code" , figure out what symbol that specifies.
To do that, we have to take the "code" (in the arithmetic interval domain), map it back to CDF domain, then scan the CDF intervals to
find which symbol's bin it falls in.

To do so requires an "inverse" map (arithmetic domain -> CDF domain), the opposite of the "forward" map (CDF -> arithmetic) we just
introduced.

A full general purpose (multi-symbol) arithmetic coder is :

(in integers now)
Encode :
look up CDF of the symbol you want to encode
map CDF interval to range inverval :
lo = forward(cdf_low,range,cdf_sum);
hi = forward(cdf_high,range,cdf_sum);
arithmetic_low += lo;
arithmetic_range = hi - lo;
propagate carry in "arithmetic_low" if necessary
renormalize if necessary
Decode :
take current arithmetic "code"
map it back to CDF domain :
target = inverse(arithmetic_code,range,cdf_sum);
find symbol from CDF target such that :
CDF_low <= target < CDF_high
rest proceeds like encoder:
lo = forward(cdf_low,range,cdf_sum);
hi = forward(cdf_high,range,cdf_sum);
arithmetic_code -= lo;
arithmetic_range = hi - lo;
renormalize if necessary

The encoder needs "forward" twice, the decoder needs "forward" twice plus "inverse" once.

Naive implementation of forward & inverse both need division, which would mean 2 and 3 divides for encode & decode, respectively.

** The inverse map and when you don't need it **

First of all, why do you need the inverse map, and when do you not need it?

One common case where you don't need the inverse map at all is binary arithmetic coding. In that case it is common to just do
the forward map and resolve the symbol in arithmetic domain, rather than CDF domain.

That is :

binary decoder :
arithetmetic_code is known
map the threshold between bits 0 & 1 to arithmetic domain :
arihmetic_mid = forward(cdf_min,range,cdf_sum);
find bin in arithmetic domain :
symbol = arithetmetic_code >= arithmetic_mid;
lo/hi = { 0 , arithmetic_mid , range }

(in the binary case we also only need one forward map, not two, since one of the end points is always 0 or range).

Now, you can do the same technique for small alphabet multi-symbol, for 4 or 8 or 16 symbols (in SIMD vectors); rather than make a CDF target
to look up the symbol, instead take all the symbol CDF's and scale them into arithmetic domain. In practice this means a bunch of calls to
"forward" (ie. a bunch of multiplies) rather than one call to "inverse" (a divide).

But for full alphabet (ie 8 bit, 256 symbol), you don't want to scale all the CDF's. (exception: Fenwick Tree style descent). Typically
you want a static (or semi-static, defsum) probability model, then you can do the CDF -> symbol lookup using just a table. In that case
we can construct the symbol lookup in CDF domain, we need the map from arithmetic domain back to CDF domain.

The inverse map must have properties :

assume range >= cdf_sum
so the forward map is a stretch , inverse map is a contraction
should invert exactly at the CDF end points :
y = forward(x);
inverse(y) == x
the CDF buckets should map to the lower CDF :
lo = forward(x)
hi = forward(x+1)
(hi > lo , it can be much greater than 1 apart)
inverse( anything in [lo,hi) ) = x

in hand wavey terms, you need inverse to act like floor division. eg :

CDF domain [012] -> arithmetic domain [00011122]

For example, for the CACM87 forward map we used above, the inverse is :

CACM87
forward(cdf,range,cdf_sum) = ( cdf * range ) / cdf_sum;
inverse(code,range,cdf_sum) = ( code * cdf_sum + cdf_sum-1 ) / range;
(integers, floor division)

In most general form, both forward & inverse map require division. The forward map is easy to make divide-free, but the inverse map not so.

** Getting rid of division and cdf_sum power of 2 **

We'll now start getting rid of pesky division.

The first thing we can do, which we will adopt henceforth, is to choose cdf_sum to be a power of 2.
We can choose our static model to normalize the cdf sum to a power of 2, or with adaptive modeling use a scheme
that maintains power of 2 sums. (defsum, constant sum shift, sliding window, etc.)

cdf_sum = 1<`<`

cdf_shift;
CACM87 forward(cdf,range,cdf_sum) = ( cdf * range ) >> cdf_shift;

So we have eliminated division from the forward map, but it remains in the inverse map. (the problem is that the CDF domain
is our "home" which is under our control, while the arithmetic domain is constantly moving around, stretching and shrinking,
as the "range" interval is modified).

We're fundamentally stuck needing something like "1 / range" , which is the whole crux of the problem that recip_arith is
trying to attack.

I think we'll come back to that next time, as we've come far enough.

While I'm going into these forward/inverse maps lets go ahead and mention the standard "range coder" :

range coder :
forward(cdf,range,cdf_sum) = cdf * ( range / cdf_sum );
inverse(code,range,cdf_sum) = code / ( range / cdf_sum );
(integer, floor division)
or with power of 2 cdf_sum and in a more C way :
r_norm = range >> cdf_shift;
forward(cdf,range,cdf_sum) = cdf * r_norm;
inverse(code,range,cdf_sum) = code / r_norm;

where I'm introducing "r_norm" , which is just the ratio between "range" and "cdf_sum" , or the scaling from CDF to arithmetic domain.

Historically, the "range coder" map was attractive (vs CACM87) because it allowed 32 bit arithmetic state. In the olden days we needed
to do the multiply and stay in 32 bits. In CACM87 you have to do (cdf * range) in the numerator, so each of those was limited to 16 bits.
Because the arithmetic state (code/range) was only 16 bits, you had to do bit renormalization (in order to keep range large enough to do
large CDF sums (actually byte renormalization was done as far back as 1984, in which case CDF sum was capped at 8 bits and only binary
arithmetic coding could be done)). By adopting the "range coder" map, you
could put the arithmetic state in 32 bits and still use just 32 bit registers. That meant byte renormalization was possible.

So, with modern processors with 64-bit registers there's actually very little advantage to the "range coder" map over the CACM87 map.

The range coder map has some coding loss. The fundamental reason is that the forward() map scaling is not exactly (Probability * range).
Another way to think of that is that not all of range is used. In the "range coder" :

forward(cdf_sum) = cdf_sum * r_norm = r_norm << cdf_shift = (range >> cdf_shift) << cdf_shift
forward(cdf_sum) = range with bottom cdf_shift bits turned off
unused region is (range % cdf_sum)

The coding loss is small in practice (because we ensure that range is much larger than cdf_sum). In typical use, range is 24-31 bits
and cdf_shift is in 11-14 , then the coding loss is on the order of 0.001 bpb. You can make the coding loss of the range coder
arbitrarily small by using larger range (eg. 56-63 bits) with small cdf_sum.

The "range coder" map is actually much simpler than the CACM87 map. It simply takes each integer step in the CDF domain, and turns
that into a step of "r_norm" in the arithmetic domain. The inverse map then just does the opposite, each run of "r_norm" steps in
the arithmetic domain maps to a single integer step in the CDF domain.

.. and that's enough background for now.

The entry page for recip_arith :
cbloom rants A Multi-Symbol Division Free Arithmetic Coder with Low Coding Loss using Reciprocal Multiplication

Links :

cbloom rants 10-05-08 Rant on New Arithmetic Coders

cbloom rants 10-06-08 Followup on the Russian Range Coder

cbloom rants 10-07-08 A little more on arithmetic coding ...

cbloom rants 10-10-08 On the Art of Good Arithmetic Coder Use

cbloom rants 08-16-10 - Range Coder Revisited .. oh wait, nevermind

cbloom rants 09-16-10 - Modern Arithmetic Coding from 1987

cbloom rants 10-08-08 Arithmetic coders throw away accuracy in lots of little places.

cbloom rants 08-11-10 - Huffman - Arithmetic Equivalence

On the Overhead of Range Coders