( n1 | u1 n2 | u2 n3 | u3 -- flag )

Perform a comparison of a test value n1 | u1 with a lower limit n2 | u2 and an upper limit n3 | u3, returning true if either (n2 | u2 < n3 | u3 and (n2 | u2 <= n1 | u1 and n1 | u1 < n3 | u3)) or (n2 | u2 > n3 | u3 and (n2 | u2 <= n1 | u1 or n1 | u1 < n3 | u3)) is true, returning false otherwise. An ambiguous condition exists n1 | u1, n2 | u2, and n3 | u3 are not all the same type.



We describe WITHIN without mentioning circular number spaces (an undefined term) or providing the code. Here is a number line with the overflow point (o) at the far right and the underflow point (u) at the far left:
There are two cases to consider: either the n2 | u2... n3 | u3 range straddles the overflow/underflow points or it does not. Lets examine the non-straddle case first:
The [ denotes n2 | u2, the ) denotes n3 | u3, and the dots and [ are numbers WITHIN the range. n3 | u3 is greater than n2 | u2, so the following tests will determine if n1 | u1 is WITHIN n2 | u2 and n3 | u3:
n2 | u2 <= n1 | u1 and n1 | u1 < n3 | u3.
In the case where the comparison range straddles the overflow/underflow points:
n3 | u3 is less than n2 | u2 and the following tests will determine if n1 | u1 is WITHIN n2 | u2 and n3 | u3:
n2 | u2 <= n1 | u1 or n1 | u1 < n3 | u3.
WITHIN must work for both signed and unsigned arguments. One obvious implementation does not work:
: WITHIN ( test low high -- flag )
   >R OVER < 0= ( test flag1 ) SWAP R> < ( flag1 flag2 ) AND
Assume two's-complement arithmetic on a 16-bit machine, and consider the following test:

   33000 32000 34000 WITHIN

The above implementation returns false for that test, even though the unsigned number 33000 is clearly within the range {{32000 ... 34000}}.

The problem is that, in the incorrect implementation, the signed comparison < gives the wrong answer when 32000 is compared to 33000, because when those numbers are treated as signed numbers, 33000 is treated as negative 32536, while 32000 remains positive.

Replacing < with U< in the above implementation makes it work with unsigned numbers, but causes problems with certain signed number ranges; in particular, the test:

1 -5 5 WITHIN
would give an incorrect answer.

For two's-complement machines that ignore arithmetic overflow (most machines), the following implementation works in all cases:

: WITHIN ( test low high -- flag ) OVER - >R - R> U< ;