Loop unrolling

Loop unrolling, also known as loop unwinding, is a loop transformation technique that attempts to optimize a program's execution speed at the expense of its binary size, which is an approach known as space–time tradeoff. The transformation can be undertaken manually by the programmer or by an optimizing compiler. On modern processors, loop unrolling is often counterproductive, as the increased code size can cause more cache misses; "cf." Duff's device.&lt;ref name="lkml-0008.2/0171"&gt;&lt;/ref&gt;
The goal of loop unwinding is to increase a program's speed by reducing or eliminating instructions that control the loop, such as pointer arithmetic and "end of loop" tests on each iteration; reducing branch penalties; as well as hiding latencies, including the delay in reading data from memory. To eliminate this computational overhead, loops can be re-written as a repeated sequence of similar independent statements.
Loop unrolling is also part of certain formal verification techniques, in particular bounded model checking.
Advantages.
The overhead in "tight" loops often consists of instructions to increment a pointer or index to the next element in an array (pointer arithmetic), as well as "end of loop" tests. If an optimizing compiler or assembler is able to pre-calculate offsets to each "individually referenced" array variable, these can be built into the machine code instructions directly, therefore requiring no additional arithmetic operations at run time.
Optimizing compilers will sometimes perform the unrolling automatically, or upon request.
Static/manual loop unrolling.
Manual (or static) loop unrolling involves the programmer analyzing the loop and interpreting the iterations into a sequence of instructions which will reduce the loop overhead. This is in contrast to dynamic unrolling which is accomplished by the compiler.
Simple manual example in C.
A procedure in a computer program is to delete 100 items from a collection. This is normally accomplished by means of a codice_1-loop which calls the function "delete(item_number)". If this part of the program is to be optimized, and the overhead of the loop requires significant resources compared to those for the "delete(x)" function, unwinding can be used to speed it up.
As a result of this modification, the new program has to make only 20 iterations, instead of 100. Afterwards, only 20% of the jumps and conditional branches need to be taken, and represents, over many iterations, a potentially significant decrease in the loop administration overhead. To produce the optimal benefit, no variables should be specified in the unrolled code that require pointer arithmetic. This usually requires "base plus offset" addressing, rather than indexed referencing.
On the other hand, this "manual" loop unrolling expands the source code size from 3 lines to 7, that have to be produced, checked, and debugged, and the compiler may have to allocate more registers to store variables in the expanded loop iteration . In addition, the loop control variables and number of operations inside the unrolled loop structure have to be chosen carefully so that the result is indeed the same as in the original code (assuming this is a later optimization on already working code). For example, consider the implications if the iteration count were not divisible by 5. The manual amendments required also become somewhat more complicated if the test conditions are variables. See also Duff's device.
Early complexity.
In the simple case, the loop control is merely an administrative overhead that arranges the productive statements. The loop itself contributes nothing to the results desired, merely saving the programmer the tedium of replicating the code a hundred times which could have been done by a pre-processor generating the replications, or a text editor. Similarly, codice_2-statements and other flow control statements could be replaced by code replication, except that code bloat can be the result. Computer programs easily track the combinations, but programmers find this repetition boring and make mistakes. 
Consider:
But of course, the code performed need not be the invocation of a procedure, and this next example involves the index variable in computation:
which, if compiled, might produce a lot of code ("print" statements being notorious) but further optimization is possible. This example makes reference only to "x(i)" and "x(i - 1)" in the loop (the latter only to develop the new value "x(i)") therefore, given that there is no later reference to the array "x" developed here, its usages could be replaced by a simple variable. Such a change would however mean a simple variable "whose value is changed" whereas if staying with the array, the compiler's analysis might note that the array's values are constant, each derived from a previous constant, and therefore carries forward the constant values so that the code becomes 
 print 2, 2;
 print 3, 6;
 print 4, 24;
 ...etc.
In general, the content of a loop might be large, involving intricate array indexing. These cases are probably best left to optimizing compilers to unroll. Replicating innermost loops might allow many possible optimisations yet yield only a small gain unless n is large.
Unrolling WHILE loops.
Consider a pseudocode WHILE loop similar to the following:
In this case, unrolling is faster because the ENDWHILE (a jump to the start of the loop) will be executed 66% less often.
Even better, the "tweaked" pseudocode example, that may be performed automatically by some optimizing compilers, eliminating unconditional jumps altogether.
Dynamic unrolling.
Since the benefits of loop unrolling are frequently dependent on the size of an array—which may often not be known until run time—JIT compilers (for example) can determine whether to invoke a "standard" loop sequence or instead generate a (relatively short) sequence of individual instructions for each element. This flexibility is one of the advantages of just-in-time techniques versus static or manual optimization in the context of loop unrolling. In this situation, it is often with relatively small values of "n" where the savings are still useful—requiring quite small (if any) overall increase in program size (that might be included just once, as part of a standard library).
Assembly language programmers (including optimizing compiler writers) are also able to benefit from the technique of dynamic loop unrolling, using a method similar to that used for efficient branch tables. Here, the advantage is greatest where the maximum offset of any referenced field in a particular array is less than the maximum offset that can be specified in a machine instruction (which will be flagged by the assembler if exceeded).
Assembler example (IBM/360 or Z/Architecture).
This example is for IBM/360 or Z/Architecture assemblers and assumes a field of 100 bytes (at offset zero) is to be copied from array "FROM" to array "TO"—both having 50 entries with element lengths of 256 bytes each.
 LM R15,R2,INIT Set R15 = maximum number of MVC
LOOP EQU * Define LOOP label.
 SR R15,R0 Subtract the remaining number of 
 BNP ALL If R15 is not positive, meaning we
 MH R15,=AL2(ILEN) Multiply R15 by the length of one
 B ALL(R15) Jump to ALL+R15, the address of the
ALL MVC 15*256(100,R2),15*256(R1) Move 100 bytes of 16th entry from 
ILEN EQU *-ALL Set ILEN to the length of the previous
 MVC 14*256(100,R2),14*256(R1) Move 100 bytes of 15th entry.
 MVC 13*256(100,R2),13*256(R1) Move 100 bytes of 14th entry.
 MVC 12*256(100,R2),12*256(R1) Move 100 bytes of 13th entry.
 MVC 11*256(100,R2),11*256(R1) Move 100 bytes of 12th entry.
 MVC 10*256(100,R2),10*256(R1) Move 100 bytes of 11th entry.
 MVC 09*256(100,R2),09*256(R1) Move 100 bytes of 10th entry.
 MVC 08*256(100,R2),08*256(R1) Move 100 bytes of 9th entry.
 MVC 07*256(100,R2),07*256(R1) Move 100 bytes of 8th entry.
 MVC 06*256(100,R2),06*256(R1) Move 100 bytes of 7th entry.
 MVC 05*256(100,R2),05*256(R1) Move 100 bytes of 6th entry.
 MVC 04*256(100,R2),04*256(R1) Move 100 bytes of 5th entry.
 MVC 03*256(100,R2),03*256(R1) Move 100 bytes of 4th entry.
 MVC 02*256(100,R2),02*256(R1) Move 100 bytes of 3rd entry.
 MVC 01*256(100,R2),01*256(R1) Move 100 bytes of 2nd entry.
 MVC 00*256(100,R2),00*256(R1) Move 100 bytes of 1st entry.
 S R0,MAXM1 Reduce the number of remaining entries
 BNPR R14 If no more entries to process, return
 AH R1,=AL2(16*256) Increment 'FROM' array pointer beyond
 AH R2,=AL2(16*256) Increment 'TO' array pointer beyond
 L R15,MAXM1 Reload the maximum number of MVC 
 B LOOP Execute loop again.
INIT DS 0A 4 addresses (pointers) to be 
MAXM1 DC A(16) Maximum number of MVC instructions
N DC A(50) Number of actual entries in array (a 
 DC A(FROM) Address of start of array 1 
 DC A(TO) Address of start of array 2 
FROM DS 50CL256 Array of 50 entries of 256 bytes each.
TO DS 50CL256 Array of 50 entries of 256 bytes each.
In this example, approximately 202 instructions would be required with a "conventional" loop (50 iterations), whereas the above dynamic code would require only about 89 instructions (or a saving of approximately 56%). If the array had consisted of only two entries, it would still execute in approximately the same time as the original unwound loop. The increase in code size is only about 108 bytes even if there are thousands of entries in the array.
Similar techniques can of course be used where multiple instructions are involved, as long as the combined instruction length is adjusted accordingly. For example, in this same example, if it is required to clear the rest of each array entry to nulls immediately after the 100 byte field copied, an additional clear instruction, codice_3, can be added immediately after every MVC in the sequence (where codice_4 matches the value in the MVC above it).
It is, of course, perfectly possible to generate the above code "inline" using a single assembler macro statement, specifying just four or five operands (or alternatively, make it into a library subroutine, accessed by a simple call, passing a list of parameters), making the optimization readily accessible.
C example.
The following example demonstrates dynamic loop unrolling for a simple program written in C. Unlike the assembler example above, pointer/index arithmetic is still generated by the compiler in this example because a variable (i) is still used to address the array element. Full optimization is only possible if absolute indexes are used in the replacement statements.
/* The number of entries processed per loop iteration. */
/* Note that this number is a 'constant constant' reflecting the code below. */
int main(void)
 int i = 0; /* counter */
 int entries = 50; /* total number to process */
 int repeat; /* number of while repetitions*/
 int left = 0; /* remainder (process later) */ 
 /* If the number of elements is not be divisible by BUNCHSIZE, */ 
 /* get repeat times required to do most processing in the while loop */
 repeat = (entries / BUNCHSIZE); /* number of times to repeat */
 left = (entries % BUNCHSIZE); /* calculate remainder */
 /* Unroll the loop in 'bunches' of 8 */ 
 while (repeat--) 
 printf("process(%d)\n", i );
 printf("process(%d)\n", i + 1); 
 printf("process(%d)\n", i + 2); 
 printf("process(%d)\n", i + 3); 
 printf("process(%d)\n", i + 4); 
 printf("process(%d)\n", i + 5); 
 printf("process(%d)\n", i + 6); 
 printf("process(%d)\n", i + 7);
 /* update the index by amount processed in one go */ 
 i += BUNCHSIZE;
 /* Use a switch statement to process remaining by jumping to the case label */ 
 /* at the label that will then drop through to complete the set */ 
 switch (left) 
 case 7 : printf("process(%d)\n", i + 6); /* process and rely on drop 
 through */
 case 6 : printf("process(%d)\n", i + 5); 
 case 5 : printf("process(%d)\n", i + 4); 
 case 4 : printf("process(%d)\n", i + 3); 
 case 3 : printf("process(%d)\n", i + 2); 
 case 2 : printf("process(%d)\n", i + 1); /* two left */
 case 1 : printf("process(%d)\n", i); /* just one left to process */ 
 case 0 : ; /* none left */
Code duplication could be avoided by writing the two parts together as in Duff's device.
C to MIPS assembly language loop unrolling example.
The following example will compute a dot product of two 100-entry vectors A and B of type codice_5. Here is the code in C:
double dotProduct = 0;
for (int i = 0; i &lt; 100; i++) {
 dotProduct += A[i]*B[i];
Converting to MIPS assembly language.
The following is MIPS assembly code that will compute the dot product of two 100-entry vectors, A and B, before implementing loop unrolling. The code below omits the loop initializations:
Note that the size of one element of the arrays (a codice_5) is 8 bytes.
 loop3:
 l.d $f10, 0($5) ; $f10 ← A[i]
 l.d $f12, 0($6) ; $f12 ← B[i]
 mul.d $f10, $f10, $f12 ; $f10 ← A[i]*B[i]
 add.d $f8, $f8, $f10 ; $f8 ← $f8 + A[i]*B[i]
 addi $5, $5, 8 ; increment pointer for A[i] by the size
 ; of a double.
 addi $6, $6, 8 ; increment pointer for B[i] by the size
 ; of a double.
 addi $7, $7, -1 ; decrement loop count
 test:
 bgtz $7, loop3 ; Continue if loop count &gt; 0
Unrolling the Loop in MIPS.
The following is the same as above, but with loop unrolling implemented at a factor of 4. Note again that the size of one element of the arrays (a codice_5) is 8 bytes; thus the 0, 8, 16, 24 displacements and the 32 displacement on each loop.
 loop3:
 l.d $f10, 0($5) ; iteration with displacement 0
 l.d $f12, 0($6)
 mul.d $f10, $f10, $f12
 add.d $f8, $f8, $f10
 l.d $f10, 8($5) ; iteration with displacement 8
 l.d $f12, 8($6)
 mul.d $f10, $f10, $f12
 add.d $f8, $f8, $f10
 l.d $f10, 16($5) ; iteration with displacement 16
 l.d $f12, 16($6)
 mul.d $f10, $f10, $f12
 add.d $f8, $f8, $f10
 l.d $f10, 24($5) ; iteration with displacement 24
 l.d $f12, 24($6)
 mul.d $f10, $f10, $f12
 add.d $f8, $f8, $f10
 addi $5, $5, 32
 addi $6, $6, 32
 addi $7, $7, -4
 test:
 bgtz $7, loop3 ; Continue loop if $7 &gt; 0