What is the full “for” loop syntax in C?

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The comma is not exclusive of for loops; it is the comma operator.

x = (a, b);

will do first a, then b, then set x to the value of b.

The for syntax is:

for (init; condition; increment)
    ...

Which is somewhat (ignoring continue and break for now) equivalent to:

init;
while (condition) {
    ...
    increment;
}

So your for loop example is (again ignoring continue and break) equivalent to

p=0;
while (p+=(a&1)*b,a!=1) {
    ...
    a>>=1,b<<=1;
}

Which acts as if it were (again ignoring continue and break):

p=0; 
while (true) {
    p+=(a&1)*b;
    if (a == 1) break;
    ...
    a>>=1;
    b<<=1;
}

Two extra details of the for loop which were not in the simplified conversion to a while loop above:

  • If the condition is omitted, it is always true (resulting in an infinite loop unless a break, goto, or something else breaks the loop).
  • A continue acts as if it were a goto to a label just before the increment, unlike a continue in the while loop which would skip the increment.

Also, an important detail about the comma operator: it is a sequence point, like && and || (which is why I can split it in separate statements and keep its meaning intact).

Changes in C99

The C99 standard introduces a couple of nuances not mentioned earlier in this explanation (which is very good for C89/C90).

First, all loops are blocks in their own right. Effectively,

for (...) { ... }

is itself wrapped in a pair of braces

{
for (...) { ... }
}

The standard sayeth:

ISO/IEC 9899:1999 §6.8.5 Iteration statements

¶5 An iteration statement is a block whose scope is a strict subset of the scope of its
enclosing block. The loop body is also a block whose scope is a strict subset of the scope
of the iteration statement.

This is also described in the Rationale in terms of the extra set of braces.

Secondly, the init portion in C99 can be a (single) declaration, as in

for (int i = 0; i < sizeof(something); i++) { ... }

Now the ‘block wrapped around the loop’ comes into its own; it explains why the variable i cannot be accessed outside the loop. You can declare more than one variable, but they must all be of the same type:

for (int i = 0, j = sizeof(something); i < j; i++, j--) { ... }

The standard sayeth:

ISO/IEC 9899:1999 §6.8.5.3 The for statement

The statement

for ( clause-1 ; expression-2 ; expression-3 ) statement

behaves as follows: The expression expression-2 is the controlling expression that is
evaluated before each execution of the loop body. The expression expression-3 is
evaluated as a void expression after each execution of the loop body. If clause-1 is a
declaration, the scope of any variables it declares is the remainder of the declaration and
the entire loop, including the other two expressions; it is reached in the order of execution
before the first evaluation of the controlling expression. If clause-1 is an expression, it is
evaluated as a void expression before the first evaluation of the controlling expression.133)

Both clause-1 and expression-3 can be omitted. An omitted expression-2 is replaced by a
nonzero constant.

133) Thus, clause-1 specifies initialization for the loop, possibly declaring one or more variables for use in
the loop; the controlling expression, expression-2, specifies an evaluation made before each iteration,
such that execution of the loop continues until the expression compares equal to 0; and expression-3
specifies an operation (such as incrementing) that is performed after each iteration.

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