It is often possible to analyse the range of a finite portion of given
stream, and use this information to determine output digits for an
algorithm computing some operation applied to this stream. For
example, suppose we are computing (1-*x*), and we know that *x* is of
the form (1::*x*'), we can say that , and hence
. We can represent all reals in this range
using a stream starting with the digit one, and so we can output
this digit without examining further digits of *x*. It was thus
possible to generate one output digit by examining only one input
digit. The example given is a specific case, but this general
principle may be applied in numerous ways.

In some cases we can extend this technique as follows. Suppose we know
by range analysis that , but that it need not
necessarily start with the digit zero. If we wish to compute 2*x*, a
legal operation given the range of *x*, the simplest way is to perform
a left shift by removing a zero from the head of the stream
representing *x*. Obviously, however, this method will not work if the
first digit of *x* is not a zero. Fortunately we can define a function
which will re-represent the stream starting with a specified digit, in
this case zero, which we can then remove. This function for signed
binary streams is described in section 4.1.4, and is referred to
in this work as the `p' function.