1 Operations Within Expressions
2 Incremental Translation
3 Addressing Array Elements
4 Translation of Array References
5 Exercises for Section 6.4

**Translation of Expressions**

*1 Operations Within Expressions*

* 2 Incremental Translation*

* 3 Addressing Array Elements*

* 4 Translation of Array References*

* 5 Exercises for Section 6.4*

The rest of this chapter explores
issues that arise during the translation of ex-pressions and statements. We
begin in this section with the translation of ex-pressions into three-address
code. An expression with more than one operator, like *a + b* c,* will translate into instructions with at most one
operator per in-struction. An array reference *A[i][j]* will expand into a sequence of three-address instructions
that calculate an address for the reference. We shall consider type checking of
expressions in Section 6.5 and the use of boolean expressions to direct the
flow of control through a program in Section 6.6.

**1. Operations Within Expressions**

The syntax-directed definition in Fig. 6.19 builds up the three-address
code for an assignment statement *S*
using attribute *code* for *S* and attributes *addr* and *code *for an
expression* E. *Attributes* S.code *and* E.code *denote the three-address*
*code for *S* and *E,* respectively. Attribute *E.addr* denotes the address that will hold
the value of E. Recall from Section 6.2.1 that an address can be a name, a
constant, or a compiler-generated temporary.

Consider the last production, E ->• id, in the syntax-directed
definition in Fig. 6.19. When an expression is a single identifier, say x, then
x itself holds the value of the expression. The semantic rules for this
production define E.addr to point to the symbol-table entry for this instance
of id. Let top denote the current symbol table. Function top. get retrieves the
entry when it is applied to the string representation id.lexeme of this
instance of id. E.code is set to the empty string.

When E —> (Ei), the translation of E is the same as that of the
subex-pression Ei. Hence, E.addr equals Ei.addr, and E.code equals Ei.code.

The operators + and unary - in Fig. 6.19 are representative of the
operators in a typical language. The semantic rules for E —> E1 + E2,
generate code to compute the value of E from the values of E1 and E2. Values
are computed into newly generated temporary names. If E1 is computed into
Ei.addr and E2 into E2. addr, then E1 + E2 translates into t = E1. addr + E2.
addr, where t is a new temporary name. E.addr is set to t. A sequence of
distinct temporary names ti,t2,... is created by successively executing n e w
Tempi).

For convenience, we use the notation gen(x '=' y '+' z) to represent the
three-address instruction x = y + z. Expressions appearing in place of
variables like x, y, and z are evaluated when passed to gen, and quoted strings
like are taken literally.5 Other three-address instructions will be built up
similarly 5 In syntax-directed definitions, gen builds an instruction and
returns it.

In translation schemes, gen builds an instruction and incrementally
emits it by putting it into the stream by applying gen to a combination of
expressions and strings.

When we translate the production E -> Ei+E2, the semantic rules in
Fig. 6.19 build up E.code by concatenating Ei.code, E2.code, and an instruction
that adds the values of E1 and E2. The instruction puts the result of the
addition into a new temporary name for E, denoted by E.addr.

The translation of E -» - E1 is similar. The rules create a new
temporary for E and generate an instruction to perform the unary minus
operation.

Finally, the production S id = E; generates instructions that assign the
value of expression E to the identifier id. The semantic rule for this
production uses function top.get to determine the address of the identifier
represented by id, as in the rules for E —v id. S.code consists of the
instructions to compute the value of E into an address given by E.addr,
followed by an assignment to the address top.get(id.lexeme) for this instance
of id.

Example 6.11 : The syntax-directed definition in Fig. 6.19 translates
the as-signment statement a = b + - c; into the three-address code sequence

**2. Incremental Translation**

Code attributes can be long strings, so they are usually generated
incremen-tally, as discussed in Section 5.5.2. Thus, instead of building up *E.code* as in Fig. 6.19, we can arrange
to generate only the new three-address instructions, as in the translation
scheme of Fig. 6.20. In the incremental approach, *gen* not only constructs a three-address instruction, it appends the
instruction to the sequence of instructions generated so far. The sequence may
either be retained in memory for further processing, or it may be output
incrementally.

The translation scheme in Fig. 6.20 generates the same code as the
syntax-directed definition in Fig. 6.19. With the incremental approach, the *code* at-tribute is not used, since there
is a single sequence of instructions that is created by successive calls to *gen.* For example, the semantic rule for *E ->• E1 + E** _{2}* in Fig. 6.20 simply calls

The approach of Fig. 6.20 can also be used to build a syntax tree. The new semantic action for E -- > E1 + E2
creates a node by using a constructor, as
in of generated instructions.

E -- > Ei + E2 {E.addr = new Node(''+', E1.addr, E2.addr); }

Here, attribute addr represents the address of a node rather than a
variable or constant.

3. **Addressing Array Elements**

Array elements can be accessed quickly if they are stored in a block of
consecu-tive locations. In C and Java, array elements are numbered 0 , 1 , . .
. , ** n
—** 1, for an array with

where base is the relative address of the storage allocated for the
array. That is, base is the relative address of A[0].

The formula (6.2) generalizes to two or more dimensions. In two dimensions,
we write A[ii][i2] in C and Java for element i2 in row ii. Let w1 be the width
of a row and let w2 be the width of an element in a row. The relative address
of .A[zi][z2] can then be calculated by the formula

where Wj, for 1 =< j= < k, is the generalization of Wi and w2 in
(6.3).

Alternatively, the relative
address of an array reference can be calculated in terms of the numbers of
elements rij along dimension j of the array and the width w = Wk of a single
element of the array. In two dimensions (i.e., k = 2 and w = w2), the location
for A[n][i2 ] is given by

More generally, array elements need not be numbered starting at 0. In a
one-dimensional array, the array elements are numbered *low, low* + 1 , . . . , *high*
and *base* is the relative address of *A[low}.* Formula (6.2) for the address of
*A[i]* is replaced by:

The expressions (6.2) and (6.7) can be both be rewritten as i x w + c,
where the subexpression c = base — low x w can be precalculated at compile time.

Note that c = base when low is 0. We assume that c is saved in the
symbol table entry for A, so the relative address of A[i] is obtained by simply
adding i x w to c.

Compile-time precalculation can
also be applied to address calculations for elements of multidimensional
arrays; see Exercise 6.4.5. However, there is one situation where we cannot use
compile-time precalculation: when the array's size is dynamic. If we do not
know the values of *low* and *high* (or their gen-eralizations in many
dimensions) at compile time, then we cannot compute constants such as c. Then,
formulas like (6.7) must be evaluated as they are written, when the program
executes.

The above address calculations are based on row-major layout for arrays,
which is used in C and Java. A two-dimensional array is normally stored in one
of two forms, either *row-major*
(row-by-row) or *column-major*
(column-by-column). Figure 6.21 shows the layout of a 2 x 3 array *A* in (a) row-major form and (b)
column-major form. Column-major form is used in the Fortran family of
languages.

We can generalize row- or column-major form to many dimensions. The
generalization of row-major form is to store the elements in such a way that,
as we scan down a block of storage, the rightmost subscripts appear to vary
fastest, like the numbers on an odometer. Column-major form generalizes to the
opposite arrangement, with the leftmost subscripts varying fastest.

**4. Translation of Array
References**

The chief problem in generating code for array references is to relate
the address-calculation formulas in Section 6.4.3 to a grammar for array
references. Let nonterminal *L*
generate an array name followed by a sequence of index expressions:

As in C and Java, assume that the lowest-numbered array element is 0.

Let us calculate addresses based on widths, using the formula (6.4),
rather than on numbers of elements, as in (6.6). The translation scheme in Fig.
6.22 generates three-address code for expressions with array references. It
consists of the productions and semantic actions from Fig. 6.20, together with
productions involving nonterminal L.

Nonterminal *L*
has three synthesized attributes:

1. *L.addr* denotes a temporary
that is used while computing the offset for the array reference by summing the
terms ** ij** x

*L.array *is a pointer to the symbol-table
entry for the array name. The* *base
address of the array, say, *L.array.base*
is used to determine the actual

/-value of an array reference after all the index
expressions are analyzed.

*L.type *is the type of the subarray
generated by* L. *For any type* t, *we* *assume that its width is given by *t.width.* We use types as attributes, rather than widths, since
types are needed anyway for type checking. For any array type *t,* suppose that *t.elem* gives the element type.

The production *S*
**-» id =** *E;* represents an assignment to a nonarray vari-able, which is
handled as usual. The semantic action for *S
—?> L = E;* generates an indexed copy instruction to assign the value
denoted by expression *E* to the
location denoted by the array reference *L.*
Recall that attribute *L.array* gives
the symbol-table entry for the array. The array's base address — the address of
its Oth element — is given by *L.array.base.*
Attribute *L.addr* denotes the
temporary that holds the offset for the array reference generated by *L.* The location for the array reference
is therefore *L.array.base[L.addr].*
The generated instruction copies the r-value from address *E.addr* into the location for *L.*

Productions *E -- > Ei+E** _{2}* and

An annotated parse tree for the expression c + a [ i ] [ j ] is shown in
Fig. 6.23.

The expression is translated into the sequence of three-address
instructions in Fig. 6.24. As usual, we have used the name of each identifier
to refer to its symbol-table entry. •

**5. Exercises for Section 6.4**

**Exercise **6** . 4 **. 1 : Add to
the translation of Fig. 6.19 rules for
the following

productions:

a) *E* *-+* *Ei* *** *E*_{2}*.*

b) *E* *->•* *+ Ei *(unary plus).

**Exercise **6** . 4 **. 2 : Repeat Exercise 6.4.1 for the incremental
translation of Fig. 6.20.

**E x e r c i se 6 . 4 . 3:** Use
the translation of Fig. 6.22
to translate the following

assignments:

a) x
= a [ i ] + b [ j ] .

b) x
= a [ i ] [ j ] + b [ i ] [ j ] .

! c) x = a [
b [ i ] [ j ] ] [ c [ k ] ] .

**Exercise 6 . 4 . 4 **: Revise the translation of Fig.
6.22 for array references of the

Fortran style,
that is, *1d[Ei,E*_{2}*,...* *, E*_{n}*]* for an
n-dimensional array.

**Exercise 6 . 4 . 5 **: Generalize formula (6.7) to
multidimensional arrays, and in-dicate what values can be stored in the symbol
table and used to compute offsets. Consider the following cases:

a) An array *A* of two
dimensions, in row-major form. The first dimension has indexes running from *l1* to *hi,* and the second dimension has indexes from *l** _{2}* to

**Symbolic
Type Widths**

The
intermediate code should be relatively independent of the target ma-chine, so
the optimizer does not have to change much if the code generator is replaced by
one for a different machine. However, as we have described the calculation of
type widths, an assumption regarding how basic types is built into the
translation scheme. For instance, Example 6.12 assumes that each element of an
integer array takes four bytes. Some intermediate codes, e.g., P-code for
Pascal, leave it to the code generator to fill in the size of array elements,
so the intermediate code is independent of the size of a machine word. We could
have done the same in our translation scheme if we replaced 4 (as the width of
an integer) by a symbolic constant.

b) The same
as (a), but with the array stored in column-major form.

c) An array *A* of *k* dimensions, stored
in row-major form, with elements of size *w.*
The *jth.* dimension has indexes running
from *lj* to *hj.*

d) The same
as (c) but with the array stored in column-major form.

**E x e r c
i se 6 . 4 . 6 **: An integer array** ***A[i,
j]*** **has index** ***i*** **ranging from 1 to 10 and** **index
*j* ranging from 1 to 20. Integers take
4 bytes each. Suppose array *A* is
stored starting at byte 0. Find the location of:

a) A[4,5] b)A[10,8] c)
A[3,17].

**Exercise
6 . 4 . 7: **Repeat Exercise 6.4.6 if** ***A***
**is stored in column-major order.

**Exercise 6 . 4 . 8: **A real array A[i,j, k] has index i ranging from 1 to 4, index j ranging
from 0 to 4, and index k ranging from 5 to 10. Reals take 8 bytes each. Suppose
array A is stored starting at byte 0. Find the location of:

a) A[3,4,5] b)A[l,2,7] c) A[4,3,9].

**Exercise
6 . 4 . 9 **:
Repeat Exercise 6.4.8 if** ***A***
**is stored in column-major order.

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