6.2.2440 WITHIN CORE EXT

( 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.

A.6.2.2440 WITHIN.

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: 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: WITHIN must work for both signed and unsigned arguments. One obvious implementation does not work: 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:

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 - ROT ROT - U> ;