Appendix 4
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 *
| Title
| Function
| Circumstances causing overflow
|
| Negate
| 01, 11, 21, 31
| In the case where the number to be negated is -1.
|
| Count
| 02, 12, 22, 32
| When C(N) is 1 - 2-38
|
| Add
| 04, 14, 24, 34
| Whenever the sum is outside the range -1 to 1 - 2-38
inclusive.
|
| Subtract
| 05, 15, 25, 35
| \Whenever the difference is outside the range -1 to
1 - 2-38 inclusive
|
| Negate and Add
| 07, 17, 27, 37
| /
|
| Multiply
| 52, 53
| In the case of -1 x -1 only.
|
| Double
| 54, 55
| Whenever the correct result lies out the range
-1 to 1 - 2-38 inclusive.
|
| Divide
| 56
| 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
B.
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,
|
|
| then for
|
| a/b, -a/b, 0/b
| :
| the computer results are correct,
|
| | but for
|
| 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.
| Division
| Computer result
| Overflow Indicator
|
| a/a
| -1
| Set
|
| -a/a
| -1
| Not Set
|
| a/-a
| 1-2-38
| Set
|
| -a/-a
| 1-2-38
| Not Set
|
| 0/0
| -2-38
| Set
|