Function type A -> B in some sense is not very good. Though functions are first class values, one often cannot freely operate them due to efficiency problems. You can't apply too many transformations (A -> B) -> (C -> D), at some point you have to compute a value.
Obviously this is due to the non-strict nature of -> .
There are well know tricks to deal with functions of type Double -> Double. One can represent them as vectors given a certain basis, which can consist of trig functions, polynomials etc.
Are there any general tricks to get round the inefficiency of the A -> B type?
Or alternatives to -> ?
Your concern seems to be that given h === f • g, f • g is typically less efficient than h. Given a composition of functions known at compile time, there are two tricks performed by the compiler which can render f • g more efficient than you would suspect -- inlining, and fusion. Inlining avoids the extra indirection of a second function call, and opens up many new opportunities for optimizations. Stream fusion (or build/foldr fusion) can (to give a basic example) turn compositions such as map f . map g into map (f . g) thereby reducing the number of traversals of a structure by a constant factor. Fusion operates not only on lists, but other structures, and provides one reason for the efficient performance of Haskell libraries such as Vector.
Short cut fusion: http://www.haskell.org/haskellwiki/Correctness_of_short_cut_fusion
Stream fusion: What is Haskell's Stream Fusion
I cannot confirm this. As a productive user and implementor of AFRP, I am performing transformations on fully polymorphic functions a lot, deeply nested and for long running applications. Note that Haskell compilers do not use the traditional stack-based function calling paradigm. They use graph reduction algorithms. We don't have the same problems as, say, C.
One of the most general tricks is memoization - storing the value of a function after you computed it. Links: Haskellwiki, SO example, MemoCombinators. As you mentioned, the other trick is when you have a nice type of function (polynomial, vector, Taylor series etc.) - then it can be represented as a list, expression etc.
FWIW: In Felix, which is a whole program analyser relying heavily on inlining for performance, function arguments have three kinds: eager, lazy, or "let the compiler decide".
Eager arguments are evaluated and assigned to variable before the body of the function is evaluated.
Lazy arguments are evaluated by replacing the parameter with the argument expression wherever it occurs.
The default is "let the compiler decide". For a large amount of "ordinary" code (whatever that means) it doesn't make any difference whether you use eager or lazy evaluation.
Generally in Felix lazy evaluation is faster: note carefully this does NOT mean closures. It means inlining. However sometimes the compiler will chose eager evaluation, it reduces code bloat, and too much inlining is counter productive. I make no claim the algorithm is any good .. however Felix can sometimes outperform C and Ocaml (GHC didn't get into the finals).
As a simple example .. type classes. Felix has typeclasses, sort of like Haskell. No or very little performance overhead .. certainly no dictionaries!
In my view, Haskell would be a lot better if you just chucked out the archaic concept of separate compilation: whole program analysers can do so much more, and text is much faster to work with than object code (given the complete freedom to cache compilation results). It's crazy to have a lazy language use a compilation model designed for eager evaluation!
The other thing a Haskell variant might try is to drop the idea all functions are lazy, and instead adopt the idea that the evaluation strategy is irrelevant, unless otherwise specified. That may allow a lot more optimisation opportunities.
Related
I assume it doesn't.
My reason is that Haskell is pure functional programming (without I/O Monad), they could have made every "call by name" use the same evaluated value if the "name"s are the same.
I don't know anything about the implementation details but I'm really interested.
Detailed explanations will be much appreciated :)
BTW, I tried google, it was quite hard to get anything useful.
First of all, Haskell is a specification, not an implementation; the report does not actually require use of call-by-name evaluation, or lazy evaluation for that matter. Haskell implementations are only required to be non-strict, which does rule out call-by-value and similar strategies.
So, strictly (ha, ha) speaking, evaluation strategies can't slow down Haskell. I'm not sure what can slow down Haskell, though clearly something has or else it wouldn't have taken 12 years to get the next version of the Report out after Haskell 98. My guess is that it involves committees somehow.
Anyway, "lazy evaluation" refers to a "call by need" strategy, which is the most common implementation choice for Haskell. This differs from call-by-name in that if a subexpression is used in multiple places, it will be evaluated at most once.
The details of what qualifies as a subexpression that will be shared is a bit subtle and probably somewhat implementation-dependent, but to use an example from GHC Haskell: Consider the function cycle, which repeats an input list infinitely. A naive implementation might be:
cycle xs = xs ++ cycle xs
This ends up being inefficient because there is no single cycle xs expression that can be shared, so the resulting list has to be constructed continually as it's traversed, allocating more memory and doing more computation each time.
In contrast, the actual implementation looks like this:
cycle xs = xs' where xs' = xs ++ xs'
Here the name xs' is defined recursively as itself appended to the end of the input list. This time xs' is shared, and evaluated only once; the resulting infinite list is actually a finite, circular linked list in memory, and once the entire loop has been evaluated no further work is needed.
In general, GHC will not memoize functions for you: given f and x, each use of f x will be re-evaluated unless you give the result a name and use that. The resulting value will be the same in either case, but the performance can differ significantly. This is mostly a matter of avoiding pessimizations--it would be easy for GHC to memoize things for you, but in many cases this would cost large amounts of memory to gain tiny or nonexistent amounts of speed.
The flip side is that shared values are retained; if you have a data structure that's very expensive to compute, naming the result of constructing it and passing that to functions using it ensures that no work is duplicated--even if it's used simultaneously by different threads.
You can also pessimize things yourself this way--if a data structure is cheap to compute and uses lots of memory, you should probably avoid sharing references to the full structure, as that will keep the whole thing alive in memory as long as anything could possibly use it later.
Yes, it does, somewhat. The problem is that Haskell can't, in general, calculate the value too early (e.g. if it would lead to an exception), so it sometimes needs to keep a thunk (code for calculating the value) instead of the value itself, which uses more memory and slows things down. The compiler tries to detect cases where this can be avoided, but it's impossible to detect all of them.
The changelog for version 0.8 of vector lists the following change with a warning:
Functor, Monad, Applicative, Alternative, Foldable and Traversable
instances for boxed vectors (WARNING: they tend to be slow and are
only provided for completeness).
Could someone explain why this is the case? Is it just the normal cost of typeclass specialization, or something more interesting?
Update: Looking at some particular instances, one sees for example:
instance Foldable.Foldable Vector where
{-# INLINE foldr #-}
foldr = foldr
and similarly for the other folds. Does this mean that folding is slow for Vectors in general? If not, what makes a non-specialized fold slow enough to warrant a warning?
I submitted the original set of these instances to Roman a year and a half ago and have maintained vector-instances since then. (I had to remove these instances from vector-instances once they migrated into vector, and now maintain it solely for the really exotic stuff). His concern was that if folks used these instances polymorphically then the RULES that make Vectors fuse away can't fire unless the polymorphic function gets inlined and monomorphized.
They exist because not every bit of code on the planet is Vector-specific and even then it is nice to sometimes use the common names.
Slow here is relative. The worst case is they perform like anybody else's folds, binds, etc. but Roman takes every single boxed value as a personal insult. :)
I've just had a quick look at the source code and the implementations don't look excessively slow. I'd argue the authors added this warning because when you're writing a program in the Vector monad, you're working from such a high-level point of view that it's easy to forget that every >>= is, in fact, a concatMap, which tends to be inherently slow.
Another thing: Vector is particularly fast for unboxed types. So a user might be attracted to use the monad notation (for convenience), while he should actually be using an unboxed type (for speed).
Does anyone know of any papers discussion inlining algorithms? And closely related, the relationship of parent-child graph to call graph.
Background: I have a compiler written in Ocaml which aggressively inlines functions, primarily as a result of this and some other optimisations it generates faster code for my programming language than most others in many circumstances (including even C).
Problem #1: The algorithm has trouble with recursion. For this my rule is only to inline children into parents, to prevent infinite recursion, but this precludes sibling functions inlining once into each other.
Problem #2: I do not know of a simple way to optimise inlining operations. My algorithm is imperative with mutable representation of function bodies because it does not seem even remotely possible to make an efficient functional inlining algorithm. If the call graph is a tree, it is clear that a bottom up inlining is optimal.
Technical information: Inlining consists of a number of inlining steps. The problem is the ordering of the steps.
Each step works as follows:
we make a copy of the function to be inlined and beta-reduce by
replacing both type parameters and value parameters with arguments.
We then replace return statement with an assignment to a new variable
followed by a jump to the end of the function body.
The original call to the function is then replaced by this body.
However we're not finished. We must also clone all the children of
the function, beta-reducting them as well, and reparent the clones to
the calling function.
The cloning operation makes it extremely hard to inline recursive functions. The usual trick of keeping a list of what is already in progress and just checking to see if we're already processing this call does not work in naive form because the recursive call is now moved into the beta-reduced code being stuffed into the calling function, and the recursion target may have changed to a cloned child. However that child, in calling the parent, is still calling the original parent which calls its child, and now the unrolling of the recursion will not stop. As mentioned I broke this regress by only allowing inlining a recursive call to a child, preventing sibling recursions being inlined.
The cost of inlining is further complicated by the need to garbage collect unused functions. Since inlining is potentially exponential, this is essential. If all the calls to a function are inlined, we should get rid of the function if it has not been inlined into yet, otherwise we'll waste time inlining into a function which is no longer used. Actually keeping track of who calls what is extremely difficult, because when inlining we're not working with an actual function representation, but an "unravelled" one: for example, the list of instructions is being processed sequentially and a new list built up, and at any one point in time there may not be a coherent instruction list.
In his ML compiler Steven Weeks chose to use a number of small optimisations applied repeatedly, since this made the optimisations easy to write and easy to control, but unfortunately this misses a lot of optimisation opportunities compared to a recursive algorithm.
Problem #3: when is it safe to inline a function call?
To explain this problem generically: in a lazy functional language, arguments are wrapped in closures and then we can inline an application; this is the standard model for Haskell. However it also explains why Haskell is so slow. The closures are not required if the argument is known, then the parameter can be replaced directly with its argument where is occurs (this is normal order beta-reduction).
However if it is known the argument evaluation is not non-terminating, eager evaluation can be used instead: the parameter is assigned the value of the expression once, and then reused. A hybrid of these two techniques is to use a closure but cache the result inside the closure object. Still, GHC hasn't succeeded in producing very efficient code: it is clearly very difficult, especially if you have separate compilation.
In Felix, I took the opposite approach. Instead of demanding correctness and gradually improving efficiency by proving optimisations preserved semantics, I mandate that the optimisation defines the semantics. This guarantees correct operation of the optimiser at the expense of uncertainty about what how certain code will behave. The idea is to provide ways for the programmer to force the optimiser to conform to intended semantics if the default optimisation strategy is too aggressive.
For example, the default parameter passing mode allows the compiler to chose whether to wrap the argument in a closure, replace the parameter with the argument, or assign the argument to the parameter. If the programmer wants to force a closure, they can just pass in a closure. If the programmer wants to force eager evaluation, they mark the parameter var.
The complexity here is much greater than a functional programming language: Felix is a procedural language with variables and pointers. It also has Haskell style typeclasses. This makes the inlining routine extremely complex, for example, type-class instances replace abstract functions whenever possible (due to type specialisation when calling a polymorphic function, it may be possible to find an instance whilst inlining, so now we have a new function we can inline).
Just to be clear I have to add some more notes.
Inlining and several other optimisations such as user defined term reductions, typeclass instantiations, linear data flow checks for variable elimination, tail rec optimisation, are done all at once on a given function.
The ordering problem isn't the order to apply different optimisations, the problem is to order the functions.
I use a brain dead algorithm to detect recursion: I build up a list of everything used directly by a each function, find the closure, and then check if the function is in the result. Note the usage set is built up many times during optimisation, and this is a serious bottleneck.
Whether a function is recursive or not can change unfortunately. A recursive function might become non-recursive after tail rec optimisation. But there is a much harder case: instantiating a typeclass "virtual" function can make what appeared to be non-recursive recursive.
As to sibling calls, the problem is that given f and g where f calls g and g calls f I actually want to inline f into g, and g into f .. once. My infinite regress stopping rule is to only allow inlining of f into g if they're mutually recursive if f is a child of g, which excludes inlining siblings.
Basically I want to "flatten out" all code "as much as possible".
I realize you probably already know all this, but it seems important to still write it in full, at least for further reference.
In the functional community, there is some litterature mostly from the GHC people. Note that they consider inlining as a transformation in the source language, while you seem to work at a lower level. Working in the source language -- or an intermediate language of reasonably similar semantics -- is, I believe, a big help for simplicity and correctness.
GHC Wiki : Inlining (contains a bibliography)
Secrets of the Glasgow Haskell inliner
For the question of the ordering compiler passes, this is quite arcane. Still in a Haskell setting, there is the Compilation by Transformation in a Non-strict Functional Language PhD Thesis which discusses the ordering of different compiler passes (and also inlining).
There is also the quite recent paper on Compilation by Equality Saturation which propose a novel approach to optimisation passes ordering. I'm not sure it has yet demonstrated applicability at a large scale, but it's certainly an interesting direction to explore.
As for the recursion case, you could use Tarjan algorithm on your call graph to detect circular dependency clusters, and exclude them from inlining. It won't affect sibling calls.
http://en.wikipedia.org/wiki/Tarjan%27s_strongly_connected_components_algorithm
coming from the Ocaml community, I'm trying to learn a bit of Haskell. The transition goes quite well but I'm a bit confused with debugging. I used to put (lots of) "printf" in my ocaml code, to inspect some intermediate values, or as flag to see where the computation exactly failed.
Since printf is an IO action, do I have to lift all my haskell code inside the IO monad to be able to this kind of debugging ? Or is there a better way to do this (I really don't want to do it by hand if it can be avoided)
I also find the trace function :
http://www.haskell.org/haskellwiki/Debugging#Printf_and_friends
which seems exactly what I want, but I don't understand it's type: there is no IO anywhere!
Can someone explain me the behaviour of the trace function ?
trace is the easiest to use method for debugging. It's not in IO exactly for the reason you pointed: no need to lift your code in the IO monad. It's implemented like this
trace :: String -> a -> a
trace string expr = unsafePerformIO $ do
putTraceMsg string
return expr
So there is IO behind the scenes but unsafePerformIO is used to escape out of it. That's a function which potentially breaks referential transparency which you can guess looking at its type IO a -> a and also its name.
trace is simply made impure. The point of the IO monad is to preserve purity (no IO unnoticed by the type system) and define the order of execution of statements, which would otherwise be practically undefined through lazy evaluation.
On own risk however, you can nevertheless hack together some IO a -> a, i.e. perform impure IO. This is a hack and of course "suffers" from lazy evaluation, but that's what trace simply does for the sake of debugging.
Nevertheless though, you should probably go other ways for debugging:
Reducing the need for debugging intermediate values
Write small, reusable, clear, generic functions whose correctness is obvious.
Combine the correct pieces to greater correct pieces.
Write tests or try out pieces interactively.
Use breakpoints etc. (compiler-based debugging)
Use generic monads. If your code is monadic nevertheless, write it independent of a concrete monad. Use type M a = ... instead of plain IO .... You can afterwards easily combine monads through transformers and put a debugging monad on top of it. Even if the need for monads is gone, you could just insert Identity a for pure values.
For what it's worth, there are actually two kinds of "debugging" at issue here:
Logging intermediate values, such as the value a particular subexpression has on each call into a recursive function
Inspecting the runtime behavior of the evaluation of an expression
In a strict imperative language these usually coincide. In Haskell, they often do not:
Recording intermediate values can change the runtime behavior, such as by forcing the evaluation of terms that would otherwise be discarded.
The actual process of computation can dramatically differ from the apparent structure of an expression due to laziness and shared subexpressions.
If you just want to keep a log of intermediate values, there are many ways to do so--for instance, rather than lifting everything into IO, a simple Writer monad will suffice, this being equivalent to making functions return a 2-tuple of their actual result and an accumulator value (some sort of list, typically).
It's also not usually necessary to put everything into the monad, only the functions that need to write to the "log" value--for instance, you can factor out just the subexpressions that might need to do logging, leaving the main logic pure, then reassemble the overall computation by combining pure functions and logging computations in the usual manner with fmaps and whatnot. Keep in mind that Writer is kind of a sorry excuse for a monad: with no way to read from the log, only write to it, each computation is logically independent of its context, which makes it easier to juggle things around.
But in some cases even that's overkill--for many pure functions, just moving subexpressions to the toplevel and trying things out in the REPL works pretty well.
If you want to actually inspect run-time behavior of pure code, however--for instance, to figure out why a subexpression diverges--there is in general no way to do so from other pure code--in fact, this is essentially the definition of purity. So in that case, you have no choice but to use tools that exist "outside" the pure language: either impure functions such as unsafePerformPrintfDebugging--errr, I mean trace--or a modified runtime environment, such as the GHCi debugger.
trace also tends to over-evaluate its argument for printing, losing a lot of the benefits of laziness in the process.
If you can wait until the program is finished before studying the output, then stacking a Writer monad is the classic approach to implementing a logger. I use this here to return a result set from impure HDBC code.
Well, since whole Haskell is built around principle of lazy evaluation (so that order of calculations is in fact non-deterministic), use of printf's make very little sense in it.
If REPL+inspect resulting values is really not enough for your debugging, wrapping everything into IO is the only choice (but it's not THE RIGHT WAY of Haskell programming).
The following two Haskell programs for computing the n'th term of the Fibonacci sequence have greatly different performance characteristics:
fib1 n =
case n of
0 -> 1
1 -> 1
x -> (fib1 (x-1)) + (fib1 (x-2))
fib2 n = fibArr !! n where
fibArr = 1:1:[a + b | (a, b) <- zip fibArr (tail fibArr)]
They are very close to mathematically identical, but fib2 uses the list notation to memoize its intermediate results, while fib1 has explicit recursion. Despite the potential for the intermediate results to be cached in fib1, the execution time gets to be a problem even for fib1 25, suggesting that the recursive steps are always evaluated. Does referential transparency contribute anything to Haskell's performance? How can I know ahead of time if it will or won't?
This is just an example of the sort of thing I'm worried about. I'd like to hear any thoughts about overcoming the difficulty inherent in reasoning about the performance of a lazily-executed, functional programming language.
Summary: I'm accepting 3lectrologos's answer, because the point that you don't reason so much about the language's performance, as about your compiler's optimization, seems to be extremely important in Haskell - more so than in any other language I'm familiar with. I'm inclined to say that the importance of the compiler is the factor that differentiates reasoning about performance in lazy, functional langauges, from reasoning about the performance of any other type.
Addendum: Anyone happening on this question may want to look at the slides from Johan Tibell's talk about high performance Haskell.
In your particular Fibonacci example, it's not very hard to see why the second one should run faster (although you haven't specified what f2 is).
It's mainly an algorithmic issue:
fib1 implements the purely recursive algorithm and (as far as I know) Haskell has no mechanism for "implicit memoization".
fib2 uses explicit memoization (using the fibArr list to store previously computed values.
In general, it's much harder to make performance assumptions for a lazy language like Haskell, than for an eager one. Nevertheless, if you understand the underlying mechanisms (especially for laziness) and gather some experience, you will be able to make some "predictions" about performance.
Referential transparency increases (potentially) performance in (at least) two ways:
First, you (as a programmer) can be sure that two calls to the same function will always return the same, so you can exploit this in various cases to benefit in performance.
Second (and more important), the Haskell compiler can be sure for the above fact and this may enable many optimizations that can't be enabled in impure languages (if you've ever written a compiler or have any experience in compiler optimizations you are probably aware of the importance of this).
If you want to read more about the reasoning behind the design choices (laziness, pureness) of Haskell, I'd suggest reading this.
Reasoning about performance is generally hard in Haskell and lazy languages in general, although not impossible. Some techniques are covered in Chris Okasaki's Purely Function Data Structures (also available online in a previous version).
Another way to ensure performance is to fix the evaluation order, either using annotations or continuation passing style. That way you get to control when things are evaluated.
In your example you might calculate the numbers "bottom up" and pass the previous two numbers along to each iteration:
fib n = fib_iter(1,1,n)
where
fib_iter(a,b,0) = a
fib_iter(a,b,1) = a
fib_iter(a,b,n) = fib_iter(a+b,a,n-1)
This results in a linear time algorithm.
Whenever you have a dynamic programming algorithm where each result relies on the N previous results, you can use this technique. Otherwise you might have to use an array or something completely different.
Your implementation of fib2 uses memoization but each time you call fib2 it rebuild the "whole" result. Turn on ghci time and size profiling:
Prelude> :set +s
If it was doing memoisation "between" calls the subsequent calls would be faster and use no memory. Call fib2 20000 twice and see for yourself.
By comparison a more idiomatic version where you define the exact mathematical identity:
-- the infinite list of all fibs numbers.
fibs = 1 : 1 : zipWith (+) fibs (tail fibs)
memoFib n = fibs !! n
actually do use memoisation, explicit as you see. If you run memoFib 20000 twice you'll see the time and space taken the first time then the second call is instantaneous and take no memory. No magic and no implicit memoization like a comment might have hinted at.
Now about your original question: optimizing and reasoning about performance in Haskell...
I wouldn't call myself an expert in Haskell, I have only been using it for 3 years, 2 of which at my workplace but I did have to optimize and get to understand how to reason somewhat about its performance.
As mentionned in other post laziness is your friend and can help you gain performance however YOU have to be in control of what is lazily evaluated and what is strictly evaluated.
Check this comparison of foldl vs foldr
foldl actually stores "how" to compute the value i.e. it is lazy. In some case you saves time and space beeing lazy, like the "infinite" fibs. The infinite "fibs" doesn't generate all of them but knows how. When you know you will need the value you might as well just get it "strictly" speaking... That's where strictness annotation are usefull, to give you back control.
I recall reading many times that in lisp you have to "minimize" consing.
Understanding what is stricly evaluated and how to force it is important but so is understanding how much "trashing" you do to the memory. Remember Haskell is immutable, that means that updating a "variable" is actually creating a copy with the modification. Prepending with (:) is vastly more efficient than appending with (++) because (:) does not copy memory contrarily to (++). Whenever a big atomic block is updated (even for a single char) the whole block needs to be copied to represent the "updated" version. The way you structure data and update it can have a big impact on performance. The ghc profiler is your friend and will help you spot these. Sure the garbage collector is fast but not having it do anything is faster!
Cheers
Aside from the memoization issue, fib1 also uses non-tailcall recursion. Tailcall recursion can be re-factored automatically into a simple goto and perform very well, but the recursion in fib1 cannot be optimized in this way, because you need the stack frame from each instance of fib1 in order to calculate the result. If you rewrote fib1 to pass a running total as an argument, thus allowing a tail call instead of needing to keep the stack frame for the final addition, the performance would improve immensely. But not as much as the memoized example, of course :)
Since allocation is a major cost in any functional language, an important part of understanding performance is to understand when objects are allocated, how long they live, when they die, and when they are reclaimed. To get this information you need a heap profiler. It's an essential tool, and luckily GHC ships with a good one.
For more information, read Colin Runciman's papers.