### 16.14 Canonicalization of Instructions

There are often cases where multiple RTL expressions could represent an operation performed by a single machine instruction. This situation is most commonly encountered with logical, branch, and multiply-accumulate instructions. In such cases, the compiler attempts to convert these multiple RTL expressions into a single canonical form to reduce the number of insn patterns required.

In addition to algebraic simplifications, following canonicalizations are performed:

- For commutative and comparison operators, a constant is always made the second operand. If a machine only supports a constant as the second operand, only patterns that match a constant in the second operand need be supplied.
- For associative operators, a sequence of operators will always chain to the left; for instance, only the left operand of an integer `plus` can itself be a `plus`. `and`, `ior`, `xor`, `plus`, `mult`, `smin`, `smax`, `umin`, and `umax` are associative when applied to integers, and sometimes to floating-point.
- For these operators, if only one operand is a `neg`, `not`, `mult`, `plus`, or `minus` expression, it will be the first operand.
- In combinations of `neg`, `mult`, `plus`, and `minus`, the `neg` operations (if any) will be moved inside the operations as far as possible. For instance, `(neg (mult A B))` is canonicalized as `(mult (neg A) B)`, but `(plus (mult (neg B) C) A)` is canonicalized as `(minus A (mult B C))`.

- For the `compare` operator, a constant is always the second operand if the first argument is a condition code register or `(cc0)`.
- An operand of `neg`, `not`, `mult`, `plus`, or `minus` is made the first operand under the same conditions as above.
- `(ltu (plus` a b`)` b`)` is converted to `(ltu (plus` a b`)` a`)`. Likewise with `geu` instead of `ltu`.
- `(minus` x`(const_int` n`))` is converted to `(plus` x`(const_int` -n`))`.
- Within address computations (i.e., inside `mem`), a left shift is converted into the appropriate multiplication by a power of two.

- De Morgan's Law is used to move bitwise negation inside a bitwise logical-and or logical-or operation. If this results in only one operand being a `not` expression, it will be the first one.

  A machine that has an instruction that performs a bitwise logical-and of one operand with the bitwise negation of the other should specify the pattern for that instruction as

                (define_insn ""
                  [(set (match_operand:m 0 ...)
                        (and:m (not:m (match_operand:m 1 ...))
                                     (match_operand:m 2 ...)))]
                  "..."
                  "...")

  Similarly, a pattern for a “NAND” instruction should be written

                (define_insn ""
                  [(set (match_operand:m 0 ...)
                        (ior:m (not:m (match_operand:m 1 ...))
                                     (not:m (match_operand:m 2 ...))))]
                  "..."
                  "...")

  In both cases, it is not necessary to include patterns for the many logically equivalent RTL expressions.

- The only possible RTL expressions involving both bitwise exclusive-or and bitwise negation are `(xor:`m x y`)` and `(not:`m`(xor:`m x y`))`.
- The sum of three items, one of which is a constant, will only appear in the form

                (plus:m (plus:m x y) constant)

- Equality comparisons of a group of bits (usually a single bit) with zero will be written using `zero_extract` rather than the equivalent `and` or `sign_extract` operations.

- `(sign_extend:`m1`(mult:`m2`(sign_extend:`m2 x`) (sign_extend:`m2 y`)))` is converted to `(mult:`m1`(sign_extend:`m1 x`) (sign_extend:`m1 y`))`, and likewise for `zero_extend`.
- `(sign_extend:`m1`(mult:`m2`(ashiftrt:`m2 x s`) (sign_extend:`m2 y`)))` is converted to `(mult:`m1`(sign_extend:`m1`(ashiftrt:`m2 x s`)) (sign_extend:`m1 y`))`, and likewise for patterns using `zero_extend` and `lshiftrt`. If the second operand of `mult` is also a shift, then that is extended also. This transformation is only applied when it can be proven that the original operation had sufficient precision to prevent overflow.

Further canonicalization rules are defined in the function `commutative_operand_precedence` in `gcc/rtlanal.c`.