1-9 CODE OPTIMIZATION - USER TECHNIQUES
***************************************
(Thanks to Craig Burley for the excellent comments.
Thanks to Timothy Prince for the important comments)
FORTRAN is used for heavy number crunching, and often decreasing the
running time of a program is desirable, however, it shouldn't be set
as a primary goal except on special cases.
While doing manual optimizations, it's a good idea to check the code
also from a numerical point of view. At least, avoid optimizations
that may lead to accuracy deterioration.
The need for profiling
----------------------
Since it's common that programs spend most of the execution time in
a few small parts of the code (usually some loops), it is recommended
to perform first "profiling", i.e. analysis of the relative execution
time spent in different parts of the program.
Profiling is also important when you apply optimizations, compilers
and computer hardware are complicated systems, and the results of
applying an optimization may be hard to guess.
Optimizations that can't be done automatically
----------------------------------------------
There are some optimizations you can do, which a compiler can't,
for example:
1) Using a fast algorithm
2) Using unformatted I/O
3) Using higher optimizations levels
4) Performing "aggressive" manual optimization
5) Writing in assembly language
Using a fast algorithm
----------------------
A good optimization is choosing a fast algorithm.
Other types of optimizations probably can't make linear sort (a N**2
algorithm) compete with heapsort (N*logN) for long sequences, or make
a Runge-Kutta integrator compete with a Rosenbruck integrator for
very stiff problems.
Make yourself familiar with basic good algorithms. See the chapter on
literature for references to some classic books on algorithmics.
Using unformatted I/O
---------------------
Unformatted I/O is often better because:
1) The CPU intensive translation from internal and external
representation is avoided.
2) The radix conversion and roundoff errors associated
with the translation are avoided, so the accuracy
of computations involving numbers written to, and
then read back from a file increases.
3) It is more efficient because unformatted data takes
less space, hence more data can be passed in each
I/O operation, and less (slow) operations are used.
4) The resulting data files are smaller.
The disadvantages of unformatted I/O are that data files are not
portable to other systems, and cannot be edited and studied directly
by a text editor such as EVE, Emacs or vi.
Classification of optimizations types
-------------------------------------
Optimizations that are performed automatically by a compiler or manually
by the programmer, can be classified by various characteristics.
The "scope" of the optimization:
1) Local optimizations - Performed in a part of one procedure.
1) Common sub-expression elimination (e.g. those
occurring when translating array indices to memory
addresses.
2) Using registers for temporary results, and if
possible for variables.
3) Replacing multiplication and division by shift
and add operations.
2) Global optimizations - Performed with the help of data flow
analysis (see below) and split-lifetime analysis.
1) Code motion (hoisting) outside of loops
2) Value propagation
3) Strength reductions
3) Inter-procedural optimizations -
What is improved in the optimization:
1) Space optimizations - Reduces the size of the executable/object.
1) Constant pooling
2) Dead-code elimination.
2) Speed optimizations - Most optimizations belong to this category
There are important optimizations not covered above, e.g. the various
loop transformations:
1) Loop unrolling - Full or partial transformation of
a loop into straight code
2) Loop blocking (tiling) - Minimizes cache misses by
replacing each array processing loop into two loops,
dividing the "iteration space" into smaller "blocks"
3) Loop interchange - Change the nesting order of loops,
may make it possible to perform other transformations
4) Loop distribution - Replace a loop by two (or more)
equivalent loops
5) Loop fusion - Make one loop out of two (or more)
Automatic/Manual Optimizations
------------------------------
A good compiler can automatically perform many code optimizations,
for example:
1) Alignment of data
2) Proper loop nesting
3) Procedure inlining
4) Loop unrolling
5) Loop blocking (tiling)
6) Local optimizations
Manually performing this optimizations is recommended, as long as
program clarity is not affected adversely.
Alignment of data
-----------------
See the chapter on data alignment for information on the influence
of data alignment on performance.
Proper loop nesting
-------------------
Operating systems (except DOS) don't have to load all of your program
into memory during execution, the memory your program needs to store
code or data is partitioned into pages, and the pages are read and
written from and to the disk as necessary.
Computers use memory caches to reduce memory references. The cache is
a small and fast memory unit, when you reference a memory location,
the cache automatically saves the memory locations nearby. Later, if
you reference a memory location that happens to be in the cache it can
be brought much faster.
The way memory management and caches work has important implications
on array processing, it determines the nesting order of loops that
process multi-dimensional arrays.
FORTRAN arrays with more than one index are stored in memory in a
'reverse' order (relative to C). For example a two-dimensional array
is stored column after column. Such storage scheme is called a
'column major' or 'column first' scheme.
When you have large matrices that you process element after element,
you may save a lot of CPU consuming paging activity, and a lot of
'cache misses', if you nest your loops properly:
DO 100 J=1,50
DO 200 I=1,50
A(I,J) = 0.5 * A(I,J)
200 CONTINUE
100 CONTINUE
The inner loop runs over the left matrix index. so when you run the
program, memory references to the matrix don't jump back and forth,
they progress a step at a time over the matrix. In this way page faults
are minimized and the cache is utilized efficiently.
Similar considerations apply to IMPLIED-DO loops in I/O statements,
for similar reasons.
Procedure inlining
------------------
This is an efficient language-independent optimization technique.
Done manually, it makes your program look horrible, but many compilers
can perform it automatically. Note that this technique enlarges the
size of the executable.
Procedure inlining eliminates the overhead of procedure calls,
instead of keeping a single copy of a procedure's code and inserting
repeatedly the code required to call it, the compiler inserts the
procedure's code itself every time it is needed. In this way the
execution of the CALLING SEQUENCEs is eliminated at the small price
of enlarging the executable file. These transformation is even more
effective on highly pipelined CPUs.
Loop unrolling
--------------
This is another efficient language-independent optimization technique.
Note that it also enlarges the size of the executable.
There is overhead inherent in every loop, upon initializing the loop,
and on every iteration (see the chapter on DO loops), the overhead
may be larger if the loop does little, e.g. initialization loops.
Eliminating the loop and writing code separately for each loop index
significantly increases speed .
On every iteration of a DO loop some operations must be performed:
1) Decrementing the loop counter
2) Checking if further iterations are needed
3) If #2 is positive, modify the loop variable
and jump to the start of the loop.
If #2 is negative exit the loop
Saving loop iterations saves all this "book keeping" operations.
Loop unrolling is even more effective on highly pipelined CPUs,
without it the instruction cache may be trashed every iteration.
You may use partial loop unrolling, in that method you process
several loop indexes in one loop iteration (see example below).
The following example program tests partial loop unrolling on VMS:
PROGRAM CPUCLK
C ------------------------------------------------------------------
INCLUDE '($JPIDEF)'
C ------------------------------------------------------------------
INTEGER
* TIME1, TIME2,
* I
C ------------------------------------------------------------------
REAL
* TMP
C ------------------------------------------------------------------
CALL LIB$GETJPI(JPI$_CPUTIM,,,TIME1)
C ------------------------------------------------------------------
TMP = 0.0
DO I = 1, 1000000
TMP = TMP + 1.0 / REAL(I)
ENDDO
WRITE(*,*) ' Sum of original loop: ', TMP
C ------------------------------------------------------------------
CALL LIB$GETJPI(JPI$_CPUTIM,,,TIME2)
WRITE(*,*) ' Time of original loop: ', TIME2 - TIME1
CALL LIB$GETJPI(JPI$_CPUTIM,,,TIME1)
C ------------------------------------------------------------------
TMP = 0.0
DO I = 1, 1000000, 4
TMP = TMP + 1.0 / REAL(I) + 1.0 / REAL(I+1)
& + 1.0 / REAL(I+2) + 1.0 / REAL(I+3)
ENDDO
WRITE(*,*) ' Sum of unrolled loop: ', TMP
C ------------------------------------------------------------------
CALL LIB$GETJPI(JPI$_CPUTIM,,,TIME2)
WRITE(*,*) ' Time of unrolled loop: ', TIME2 - TIME1
C ------------------------------------------------------------------
END
This example program tests partial loop unrolling (by a factor of 4).
The only result-related I/O is a single WRITE statement (added so the
compiler wouldn't optimize away all computations).
Note that the partial unrolling of the loop, made it possible to save
some operations, we don't have to add every term of the series to the
partial-sum accumulator TMP, instead of:
TMP = TMP + 1.0 / REAL(I)
TMP = TMP + 1.0 / REAL(I+1)
TMP = TMP + 1.0 / REAL(I+2)
TMP = TMP + 1.0 / REAL(I+3)
with 4 assignments, we have:
TMP = TMP + 1.0 / REAL(I) + 1.0 / REAL(I+1)
& + 1.0 / REAL(I+2) + 1.0 / REAL(I+3)
with the same number of arithmetic operations, but only one assignment.
If TMP is not kept in a register this could be significant since the
main memory is about one order of magnitude slower than CPU registers.
The same saving of "memory references" can occur in a nest of loops,
let's generalize a bit the previous example:
INTEGER I, J
REAL A(M)
DO J = 1, 2*N
DO I = 1, M
A(I) = A(I) + 1.0 / REAL(I+J)
END DO
END DO
For every array element A(I), 2*N stores are performed, one for each
iteration of the outer loop. Let's unroll the outer loop two times,
and perform "two iterations in one go":
DO J = 1, 2*N, 2
DO I = 1, M
A(I) = A(I) + 1.0 / REAL(I+J)
END DO
DO I = 1, M
A(I) = A(I) + 1.0 / REAL(I+J+1)
END DO
END DO
We can perform "loop fusion" on the two new inner loops:
DO J = 1, 2*N, 2
DO I = 1, M
A(I) = A(I) + 1.0 / REAL(I+J)
A(I) = A(I) + 1.0 / REAL(I+J+1)
END DO
END DO
"Fusing" the two inner statements we get:
DO J = 1, 2*N, 2
DO I = 1, M
A(I) = A(I) + 1.0 / REAL(I+J) + 1.0 / REAL(I+J+1)
END DO
END DO
We can interchange the two loops to minimize paging activity
and cache misses:
DO I = 1, M
DO J = 1, 2*N, 2
A(I) = A(I) + 1.0 / REAL(I+J) + 1.0 / REAL(I+J+1)
END DO
END DO
Here only N stores are performed to each array element.
Two remarks:
1) Interchanging the two loops from the beginning, and
unrolling the J-loop would be a shorter procedure.
However, loop interchanging is not always possible.
2) In any case we will pay the price of making the loop
nest more complicated.
Loop interchange
----------------
Changing the nesting order of loops is a powerful technique.
In a previous section we saw that a proper nesting order can improve
the localization of memory references. In other cases it may enable
further loop transformations.
Let's examine if it's possible to interchange simple loop nests of
the form:
INTEGER M, N, IOFF, JOFF, I, J
PARAMETER (M = ???, N = ???, IOFF = ???, JOFF = ???)
REAL A(M,N)
DO I = 1, M
DO J = 1, N
A(I,J) = some-function-of(A(I+IOFF, J+JOFF))
END DO
END DO
We have here a "self-modifying array", the loop nest scans the array in
the "wrong" way, i.e. row after row. Of course, the ranges of the loops
should be modified so array reference doesn't get out of bounds.
Will the interchanged nest:
DO J = 1, N
DO I = 1, M
A(I,J) = some-function-of(A(I+IOFF, J+JOFF))
END DO
END DO
give the same results?
Possible values of (IOFF, JOFF) parameters can be partitioned into 4
regions, by the lines IOFF = 0, and JOFF = 0:
IOFF = 0
|
Absolute | Reversing
past | zone
-----------+----------- JOFF = 0
Reversing | Absolute
zone | future
|
Having IOFF and JOFF in the "absolute" regions,
.................................................
Loop blocking (tiling)
----------------------
Cache misses and paging activity can be minimized by partitioning
large matrices to small enough rectangular blocks. This can be done
by partitioning the "iteration space" into blocks.
A good example is matrix multiplication of square matrices:
INTEGER I, J, K
REAL A(N,N), B(N,N), C(N,N)
DO J = 1, N
DO I = 1, N
A(I,J) = 0.0
END DO
END DO
DO I = 1, N
DO J = 1, N
DO K = 1, N
A(I,J) = A(I,J) + B(I,K) * C(K,J)
END DO
END DO
END DO
This loop nest implements A = B X C. The inner K-loop performs a dot
product of the I-th row of matrix B, with the J-th column of matrix C.
On every execution of the K-loop we get references all over B and C.
We can block an ordinary loop:
DO I = 1, N
.........
END DO
If we replace it with:
DO IT = 1, N, IS
DO I = IT , MIN(N, IT+IS-1)
.........................
END DO
END DO
The outer loop goes over the "blocks", the inner traverses each block
in its turn. The block size IS, should be choosed to fit in the cache
or a page (whichever is smaller).
Performing blocking on all three loops:
DO IT = 1, N, IS
DO I = IT, MIN(N, IT+IS-1)
DO JT = 1, N, JS
DO J = JT, MIN(N, JT+JS-1)
DO KT = 1, N, KS
DO K = KT, MIN(N, KT+KS-1)
A(I,J) = A(I,J) + B(I,K) * C(K,J)
END DO
END DO
END DO
END DO
END DO
END DO
Here loop interchanging can be performed to yield:
DO IT = 1, N, IS
DO JT = 1, N, JS
DO KT = 1, N, KS
DO I = IT, MIN(N, IT+IS-1)
DO J = JT, MIN(N, JT+JS-1)
DO K = KT, MIN(N, KT+KS-1)
A(I,J) = A(I,J) + B(I,K) * C(K,J)
END DO
END DO
END DO
END DO
END DO
END DO
The size of each 3D block is IS * JS * KS, it should fit in the cache
or a memory page.
This horrible loop nest has more overhead in loop initializing and
control, but paging activity is minimized and reusing data already
in cache is maximized.
Data Flow Analysis (DFA)
------------------------
Compilers can 'follow' the computations in a program to some degree,
and detect the points at which the variables get modified. This is
called DATA-FLOW ANALYSIS. The information gathered in this way is
used to identify safe code optimizations automatically.
Theoretical considerations may limit data-flow analysis, DFA seems
more difficult than the famous halting problem, a problem known to
be undecidable, i.e. a general algorithm doesn't exist. So it's
probably impossible to have a general DFA analyzer - a program that
can perform complete data-flow analysis on every possible program.
Real DFA has additional limitations that theoretical DFA may ignore,
there are some things the compiler cannot know, e.g. what value a user
will supply interactively at run time. Even more stringent limits are
usually imposed in practice to keep compilation times reasonably short,
so compilers doesn't even try to perform complete DFAs.
In the following sections you will see that good programming practises
usually help the compiler optimize your code. Efficient code doesn't
have to be ugly, on the contrary, ugly code doesn't optimize well.
Programming practises that may help compiler optimization
---------------------------------------------------------
1) Using VARIABLE FORMATS instead of RUN-TIME FORMATS.
Variable formats are non-standard in all Fortrans, and not
widely supported either, but are a pretty cool extension.
Variable format is compiled to some degree, run-time
format is done by the format control interpreter at run-time.
See the chapter on formats for more information.
2) Using small simple statement functions (compiler dependant)
3) Avoid implied do loops in I/O statements. You can use
the array name (if not passed in the assumed-size method).
Many compilers do detect cases where the implied-DO construct
clearly identifies an ordered "chunk" of memory just as if an
EQUIVALENCEd array is involved, and thus optimize that accordingly.
Some (like DEC ones, I believe) can even handle those that are
"chunked" with a constant "stride" between elements
(e.g. "WRITE (10) (A(I),I=1,10000,2)").
4) Using the system low-level I/O routines (non-standard!)
5) Using larger I/O blocks, pre-allocation of disk area for
a file (instead of repeatedly extending it) (non-standard!)
Programming practises that may hinder compiler optimizations
------------------------------------------------------------
1) Partitioning the program to procedures, the compiler may not
perform the data-flow analysis on variables and arrays that
are passed between compilation units.
2) Declaring too many variables and arrays, the data-flow analysis
may be limited to some predetermined number of variables.
Such a problem may happen in routines or main programs that are
very large.
3) Using unnecessary common blocks, the compiler may not perform
data-flow analysis on variables belonging to a common block,
especially if there is a call to another routine in which the
same common block is declared.
4) Variables that are used in Variable Format Expressions may be
excluded from analysis.
5) Equivalenced variables are too much bother, so the compiler
may not analyze them individually.
6) Using control constructs like an ASSIGNED GOTO may make some
compilers give up further analysis.
Programming practises that inhibit compiler optimizations
---------------------------------------------------------
1) Partitioning the program to several files, the compiler can't
perform the data-flow analysis on variables and arrays that
are passed to/from such external compilation units.
2) Declaring variables as volatile, makes the compiler exclude
them from analysis. Volatile variables are used in procedures
that are executed when an interrupt or exception seizes control
(e.g. floating-point underflow handlers).
A performance detrimental FORTRAN statement
-------------------------------------------
On delimited-variable-size files, and on counted-variable-length files
that don't have a suffix count field (VMS), backspacing may be very
inefficient (see the files and records chapter).
On such files BACKSPACE goes to the beginning of the file, and reads all
previous records.
A much more important factor in the social movement than those already mentioned was the ever-increasing influence of women. This probably stood at the lowest point to which it has ever fallen, during the classic age of Greek life and thought. In the history of Thucydides, so far as it forms a connected series of events, four times only during a period of nearly seventy years does a woman cross the scene. In each instance her apparition only lasts for a moment. In three of the four instances she is a queen or a princess, and belongs either to the half-barbarous kingdoms of northern Hellas or to wholly barbarous Thrace. In the one remaining instance208— that of the woman who helps some of the trapped Thebans to make their escape from Plataea—while her deed of mercy will live for ever, her name is for ever lost.319 But no sooner did philosophy abandon physics for ethics and religion than the importance of those subjects to women was perceived, first by Socrates, and after him by Xenophon and Plato. Women are said to have attended Plato’s lectures disguised as men. Women formed part of the circle which gathered round Epicurus in his suburban retreat. Others aspired not only to learn but to teach. Arêtê, the daughter of Aristippus, handed on the Cyrenaic doctrine to her son, the younger Aristippus. Hipparchia, the wife of Crates the Cynic, earned a place among the representatives of his school. But all these were exceptions; some of them belonged to the class of Hetaerae; and philosophy, although it might address itself to them, remained unaffected by their influence. The case was widely different in Rome, where women were far more highly honoured than in Greece;320 and even if the prominent part assigned to them in the legendary history of the city be a proof, among others, of its untrustworthiness, still that such stories should be thought worth inventing and preserving is an indirect proof of the extent to which feminine influence prevailed. With the loss of political liberty, their importance, as always happens at such a conjuncture, was considerably increased. Under a personal government there is far more scope for intrigue than where law is king; and as intriguers women are at least the209 equals of men. Moreover, they profited fully by the levelling tendencies of the age. One great service of the imperial jurisconsults was to remove some of the disabilities under which women formerly suffered. According to the old law, they were placed under male guardianship through their whole life, but this restraint was first reduced to a legal fiction by compelling the guardian to do what they wished, and at last it was entirely abolished. Their powers both of inheritance and bequest were extended; they frequently possessed immense wealth; and their wealth was sometimes expended for purposes of public munificence. Their social freedom seems to have been unlimited, and they formed combinations among themselves which probably served to increase their general influence.321 The old religions of Greece and Italy were essentially oracular. While inculcating the existence of supernatural beings, and prescribing the modes according to which such beings were to be worshipped, they paid most attention to the interpretation of the signs by which either future events in general, or the consequences of particular actions, were supposed to be divinely revealed. Of these intimations, some were given to the whole world, so that he who ran might read, others were reserved for certain favoured localities, and only communicated through the appointed ministers of the god. The Delphic oracle in particular enjoyed an enormous reputation both among Greeks and barbarians for guidance afforded under the latter conditions; and during a considerable period it may even be said to have directed the course of Hellenic civilisation. It was also under this form that supernatural religion suffered most injury from the great intellectual movement which followed the Persian wars. Men who had learned to study the constant sequences of Nature for themselves, and to shape their conduct according to fixed principles of prudence or of justice, either thought it irreverent to trouble the god about questions on which they were competent to form an opinion for themselves, or did not choose to place a well-considered scheme at the mercy of his possibly interested responses. That such a revolution occurred about the middle of the fifth century B.C., seems proved by the great change of tone in reference to this subject which one perceives on passing from Aeschylus to Sophocles. That anyone should question the veracity of an oracle is a supposition which never crosses the mind of the elder dramatist. A knowledge of augury counts among the greatest benefits222 conferred by Prometheus on mankind, and the Titan brings Zeus himself to terms by his acquaintance with the secrets of destiny. Sophocles, on the other hand, evidently has to deal with a sceptical generation, despising prophecies and needing to be warned of the fearful consequences brought about by neglecting their injunctions. The stranger had a pleasant, round face, with eyes that twinkled in spite of the creases around them that showed worry. No wonder he was worried, Sandy thought: having deserted the craft they had foiled in its attempt to get the gems, the man had returned from some short foray to discover his craft replaced by another. “Thanks,” Dick retorted, without smiling. When they reached him, in the dying glow of the flashlight Dick trained on a body lying in a heap, they identified the man who had been warned by his gypsy fortune teller to “look out for a hidden enemy.” He was lying at full length in the mould and leaves. "But that is sport," she answered carelessly. On the retirement of Townshend, Walpole reigned supreme and without a rival in the Cabinet. Henry Pelham was made Secretary at War; Compton Earl of Wilmington Privy Seal. He left foreign affairs chiefly to Stanhope, now Lord Harrington, and to the Duke of Newcastle, impressing on them by all means to avoid quarrels with foreign Powers, and maintain the blessings of peace. With all the faults of Walpole, this was the praise of his political system, which system, on the meeting of Parliament in the spring of 1731, was violently attacked by Wyndham and Pulteney, on the plea that we were making ruinous treaties, and sacrificing British interests, in order to benefit Hanover, the eternal millstone round the neck of England. Pulteney and Bolingbroke carried the same attack into the pages of The Craftsman, but they failed to move Walpole, or to shake his power. The English Government, instead of treating Wilkes with a dignified indifference, was weak enough to show how deeply it was touched by him, dismissed him from his commission of Colonel of the Buckinghamshire Militia, and treated Lord Temple as an abettor of his, by depriving him of the Lord-Lieutenancy of the same county, and striking his name from the list of Privy Councillors, giving the Lord-Lieutenancy to Dashwood, now Lord Le Despencer. "I tell you what I'll do," said the Deacon, after a little consideration. "I feel as if both Si and you kin stand a little more'n you had yesterday. I'll cook two to-day. We'll send a big cupful over to Capt. McGillicuddy. That'll leave us two for to-morrer. After that we'll have to trust to Providence." "Indeed you won't," said the Surgeon decisively. "You'll go straight home, and stay there until you are well. You won't be fit for duty for at least a month yet, if then. If you went out into camp now you would have a relapse, and be dead inside of a week. The country between here and Chattanooga is dotted with the graves of men who have been sent back to the front too soon." "Adone do wud that—though you sound more as if you wur in a black temper wud me than as if you pitied me." "Wot about this gal he's married?" "Don't come any further." "Davy, it 'ud be cruel of us to go and leave him." "Insolent priest!" interrupted De Boteler, "do you dare to justify what you have done? Now, by my faith, if you had with proper humility acknowledged your fault and sued for pardon—pardon you should have had. But now, you leave this castle instantly. I will teach you that De Boteler will yet be master of his own house, and his own vassals. And here I swear (and the baron of Sudley uttered an imprecation) that, for your meddling knavery, no priest or monk shall ever again abide here. If the varlets want to shrieve, they can go to the Abbey; and if they want to hear mass, a priest can come from Winchcombe. But never shall another of your meddling fraternity abide at Sudley while Roland de Boteler is its lord." "My lord," said Edith, in her defence, "this woman has sworn falsely. The medicine I gave was a sovereign remedy, if given as I ordered. Ten drops would have saved the child's life; but the contents of the phial destroyed it. The words I uttered were prayers for the life of the child. My children, and all who know me, can bear witness that I have a custom of asking His blessing upon all I take in hand. I raised my eyes towards heaven, and muttered words; but, my lord, they were words of prayer—and I looked up as I prayed, to the footstool of the Lord. But it is in vain to contend: the malice of the wicked will triumph, and Edith Holgrave, who even in thought never harmed one of God's creatures, must be sacrificed to cover the guilt, or hide the thoughtlessness of another." "Aye, Sir Treasurer, thou hast reason to sink thy head! Thy odious poll-tax has mingled vengeance—nay, blood—with the cry of the bond." HoME古一级毛片免费观看
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