Comparison by subtraction

Also called: compare by subtracting, subtract-and-compare, compare instruction

Comparing two numbers by computing A − B and reading the flags: Z for equal, the carry flag for unsigned order, and N ⊕ V for signed order.

A processor doesn't need a separate comparator to compare numbers. It subtracts, throws the difference away, and reads the status flags. A compare instruction does exactly this.

After computing A − B as A + + 1:

Equal: Z = 1, because the difference is zero.
Unsigned A < B: C = 0, because a borrow happened. Unsigned A ≥ B: C = 1.
Signed A < B: = 1. Signed A ≥ B: = 0.

Why N ⊕ V and not just N? If the subtraction didn't overflow, the sign of the difference is right, so N alone answers "is A − B negative?". If it did overflow (V = 1), the sign is flipped, so N is wrong and must be inverted. XOR with V does exactly that.

Greater-than and less-or-equal add Z: unsigned A > B is C = 1 and Z = 0; signed A ≤ B is N ⊕ V = 1 or Z = 1.

One subtraction answers the signed and the unsigned question at the same time. The program picks the flags that match its data type.


0	0	0
0	1	1
1	0	1
1	1	0

Worked examples

Example

Signed: when N gives the wrong answer

A = 0110 (+6), B = 1101 (−3). Compute A − B = 0110 + 0010 + 1.

1.
Bit 0: 0 + 0 + 1 → 1, carry 0. Bit 1: 1 + 1 → 0, carry 1.
2.
Bit 2: 1 + 0 + 1 → 0, carry 1, so C3 = 1. Bit 3: 0 + 0 + 1 → 1, carry 0, so C4 = 0.
3.
Result 1001: N = 1, so the difference looks negative.
4.
V = C4 ⊕ C3 = 1: 6 − (−3) = 9 didn't fit, so the sign is flipped.
5.
N ⊕ V = 0, so A is not less than B. Correct: 6 > −3.

Example

Unsigned: the same bits, the other question

Read the same patterns as unsigned: A = 6, B = 13. The subtraction above had no carry-out, so C = 0: a borrow, meaning A < B. Correct, since 6 < 13. One subtraction, two different but correct verdicts.

Common mistakes

Using N alone for a signed comparison.
Using C for signed numbers, or N ⊕ V for unsigned ones.
Reading C = 1 as A < B. After A − B, C = 1 means A ≥ B.

Practice Comparison by subtraction

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