Numerical recipes

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20.5 Arithmetic Coding 911
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Figure 20.5.1. Arithmetic coding of the message “IOU...” in the fictitious language Vowellish.
Successive characters give successively finer subdivisions of the initial interval between 0 and 1. The final
value can be output as the digits of a fraction in any desired radix. Note how the subinterval allocated
to a character is proportional to its probability of occurrence.
The routine arcmak constructs the cumulative frequency distribution table used
to partition the interval at each stage. In the principal routine arcode, when an
interval of size jdif is to be partitioned in the proportions of some n to some ntot,
say, then we must compute (n*jdif)/ntot. With integer arithmetic, the numerator
is likely to overflow; and, unfortunately, an expression like jdif/(ntot/n) is not
equivalent. In the implementation below, we resort to double precision floating
arithmetic for this calculation. Not only is this inefficient, but different roundoff
errors can (albeit very rarely) make different machines encode differently, though any
one type of machine will decode exactly what it encoded, since identical roundoff
errors occur in the two processes. For serious use, one needs to replace this floating
calculation with an integer computation in a double register (not available to the
C programmer).
The internally set variable minint, which is the minimum allowed number
of discrete steps between the upper and lower bounds, determines when new lowsignificance
digits are added. minint must be large enough to provide resolution of
all the input characters. That is, we must have pi × minint > 1 for all i. A value
of 100Nch, or1.1/ min pi, whichever is larger, is generally adequate. However, for
safety, the routine below takes minint to be as large as possible, with the product
minint*nradd just smaller than overflow. This results in some time inefficiency,
and in a few unnecessary characters being output at the end of a message. You can
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