Suppose we have inputs (*x*,*y*,*z*) where

We examine the first digits of *x* and *z*, and use simple analysis of
the range of possible reals represented, as described in
section 3.6.2 to determine whether or not we can
determine digits of a stream representing using only the
known digits of *x* and *z*.

Having examined a some finite number of digits from *x* and *z*, we
will encounter on of three qualitatively distinct cases; either we can
say that , or we can find a digit or digits
whose range encloses all possible values of , or we cannot
find such a digit but cannot say that . In the
first case it is safe to cease examination of *x* and *z* and use *y*
to return the remainder of the result. In the second and third cases
we must continue to examine *x* and *z*, generating digits where
possible, until we can say for certain that .

Suppose the ranges *r* = [*r*_{min}, *r*_{max}] and *s* =
[*s*_{min},*s*_{max}] contain the lower and upper bounds *x* and *z* of
*y*. There are a number of possible situations that may arise.
Figure 5.1 illustrates with some examples.

**Case 1:**- The lower and upper bounds are too far apart to determine a digit,
and ranges are disjoint. Therefore lower and upper bounds not equal
and we can safely return
*y*.Example 1 illustrates such a situation.

**Case 2:**- We can find a digit such whose range encloses [
*r*_{min},*s*_{max}]. Output this digit and examine the inputs further.In example 2, we can deduce that the range of

*y*is [0,1], so we can generate the digit 1 as the first digit of*y*. Similarly in example 3, the range of*y*must be , so we can generate the digits (or equivalently ).Example 4 is an interesting possibility which would be handled by here. We can see that the least possible value of the lower bound

*r*_{min}is equal to the greatest possible value of the upper bound*s*_{max}. We can deduce that this is the only possible value for*y*and output it directly without examining further inputs or intervals. **Case 3:**- We cannot find a digit such whose range encloses [
*r*_{min},*s*_{max}], but we cannot say for certain that the lower and upper end-points are not equal. Therefore we must examine more digits of the end-points.Example 5 falls into this category. Suppose the numeral and , we can see that . If this were the case, we would have to output the number zero (eg. the numeral ) without examining

*y*or any further intervals by the condition stated in section 5.1. We must therefore examine further digits of*x*and*z*until either we can ascertain that they are both equal, or we can generate a digit as in examples 2, 3, or 4.