This appendix is in two parts:
- A list of all functions which can cause overflow.*
- A note on the accuracy of the division process.
A. Functions that can cause overflow *
||Circumstances causing overflow
||01, 11, 21, 31
||In the case where the number to be negated is -1.
||02, 12, 22, 32
||When C(N) is 1 - 2-38
||04, 14, 24, 34
||Whenever the sum is outside the range -1 to 1 - 2-38
||05, 15, 25, 35
||\Whenever the difference is outside the range -1 to
1 - 2-38 inclusive
|Negate and Add
||07, 17, 27, 37
||In the case of -1 x -1 only.
||Whenever the correct result lies out the range
-1 to 1 - 2-38 inclusive.
||Whenever the modulus of the numerator (dividend) is greater
than that of the denominator (divisor) and in the three
cases shown in B(c) below.
* This does not include "Floating-Point Overflow":
See Appendix 5
The Division Process. This note concerns the accuracy
of the result produced by the computer when the true quotient lies within
the range -1 to +1 inclusive, in function 56.
- Since the result is not rounded, when the true quotient is not
exactly expressible as a 39-digit computer number, the result may
be 2-38 less than that which would be obtained by using
a process containing a round-off stage.
- If a and b are both positive and b > a, and if the quotient
is exactly expressible as a 39-digit computer word,
|a/b, -a/b, 0/b
||the computer results are correct,
|a/-b, -a/-b, 0/-b
||the computer results are 2-38
less than the arithmetically correct results.
- For the division ±a/±a, a > 0, the effects
of (b), combined with the fact that there is no representation
of +1, cause the results below to be produced. Note also the
result obtained for 0/0, and that the overflow indicator is
set for this case, and for both cases in which the result
of ±a/±a errs by more than 2-38.