In the following sections, we briefly go through a few techniques that can help make your Julia code run as fast as possible.
A global variable might have its value, and therefore its type, change at any point. This makes it difficult for the compiler to optimize code using global variables. Variables should be local, or passed as arguments to functions, whenever possible.
Any code that is performance critical or being benchmarked should be inside a function.
We find that global names are frequently constants, and declaring them as such greatly improves performance:
const DEFAULT_VAL = 0
Uses of non-constant globals can be optimized by annotating their types at the point of use:
global x y = f(x::Int + 1)
Writing functions is better style. It leads to more reusable code and clarifies what steps are being done, and what their inputs and outputs are.
NOTE: All code in the REPL is evaluated in global scope, so a variable defined and assigned at toplevel will be a global variable.
In the following REPL session:
julia> x = 1.0
is equivalent to:
julia> global x = 1.0
so all the performance issues discussed previously apply.
The most useful tool for measuring performance is the @time macro. The following example illustrates good working style:
julia> function f(n)
s = 0
for i = 1:n
s += i/2
end
s
end
f (generic function with 1 method)
julia> @time f(1)
elapsed time: 0.004710563 seconds (93504 bytes allocated)
0.5
julia> @time f(10^6)
elapsed time: 0.04123202 seconds (32002136 bytes allocated)
2.5000025e11
On the first call (@time f(1)), f gets compiled. (If you’ve not yet used @time in this session, it will also compile functions needed for timing.) You should not take the results of this run seriously. For the second run, note that in addition to reporting the time, it also indicated that a large amount of memory was allocated. This is the single biggest advantage of @time vs. functions like tic() and toc(), which only report time.
Unexpected memory allocation is almost always a sign of some problem with your code, usually a problem with type-stability. Consequently, in addition to the allocation itself, it’s very likely that the code generated for your function is far from optimal. Take such indications seriously and follow the advice below.
As a teaser, note that an improved version of this function allocates no memory (except to pass back the result back to the REPL) and has an order of magnitude faster execution after the first call:
julia> @time f_improved(1) # first call elapsed time: 0.003702172 seconds (78944 bytes allocated) 0.5 julia> @time f_improved(10^6) elapsed time: 0.004313644 seconds (112 bytes allocated) 2.5000025e11
Below you’ll learn how to spot the problem with f and how to fix it.
In some situations, your function may need to allocate memory as part of its operation, and this can complicate the simple picture above. In such cases, consider using one of the tools below to diagnose problems, or write a version of your function that separates allocation from its algorithmic aspects (see Pre-allocating outputs).
Julia and its package ecosystem includes tools that may help you diagnose problems and improve the performance of your code:
@time, @allocated, or the profiler (through calls to the garbage-collection routines)—hint that there might be issues with your code. If you don’t see another reason for the allocations, suspect a type problem. You can also start Julia with the --track-allocation=user option and examine the resulting *.mem files to see information about where those allocations occur. See Memory allocation analysis.@code_warntype generates a representation of your code that can be helpful in finding expressions that result in type uncertainty. See @code_warntype below.When working with parameterized types, including arrays, it is best to avoid parameterizing with abstract types where possible.
Consider the following:
a = Real[] # typeof(a) = Array{Real,1}
if (f = rand()) < .8
push!(a, f)
end
Because a is a an array of abstract type Real, it must be able to hold any Real value. Since Real objects can be of arbitrary size and structure, a must be represented as an array of pointers to individually allocated Real objects. Because f will always be a Float64, we should instead, use:
a = Float64[] # typeof(a) = Array{Float64,1}
which will create a contiguous block of 64-bit floating-point values that can be manipulated efficiently.
See also the discussion under Parametric Types.
In many languages with optional type declarations, adding declarations is the principal way to make code run faster. This is not the case in Julia. In Julia, the compiler generally knows the types of all function arguments, local variables, and expressions. However, there are a few specific instances where declarations are helpful.
Types can be declared without specifying the types of their fields:
julia> type MyAmbiguousType
a
end
This allows a to be of any type. This can often be useful, but it does have a downside: for objects of type MyAmbiguousType, the compiler will not be able to generate high-performance code. The reason is that the compiler uses the types of objects, not their values, to determine how to build code. Unfortunately, very little can be inferred about an object of type MyAmbiguousType:
julia> b = MyAmbiguousType("Hello")
MyAmbiguousType("Hello")
julia> c = MyAmbiguousType(17)
MyAmbiguousType(17)
julia> typeof(b)
MyAmbiguousType
julia> typeof(c)
MyAmbiguousType
b and c have the same type, yet their underlying representation of data in memory is very different. Even if you stored just numeric values in field a, the fact that the memory representation of a UInt8 differs from a Float64 also means that the CPU needs to handle them using two different kinds of instructions. Since the required information is not available in the type, such decisions have to be made at run-time. This slows performance.
You can do better by declaring the type of a. Here, we are focused on the case where a might be any one of several types, in which case the natural solution is to use parameters. For example:
julia> type MyType{T<:AbstractFloat}
a::T
end
This is a better choice than
julia> type MyStillAmbiguousType
a::AbstractFloat
end
because the first version specifies the type of a from the type of the wrapper object. For example:
julia> m = MyType(3.2)
MyType{Float64}(3.2)
julia> t = MyStillAmbiguousType(3.2)
MyStillAmbiguousType(3.2)
julia> typeof(m)
MyType{Float64}
julia> typeof(t)
MyStillAmbiguousType
The type of field a can be readily determined from the type of m, but not from the type of t. Indeed, in t it’s possible to change the type of field a:
julia> typeof(t.a) Float64 julia> t.a = 4.5f0 4.5f0 julia> typeof(t.a) Float32
In contrast, once m is constructed, the type of m.a cannot change:
julia> m.a = 4.5f0 4.5f0 julia> typeof(m.a) Float64
The fact that the type of m.a is known from m‘s type—coupled with the fact that its type cannot change mid-function—allows the compiler to generate highly-optimized code for objects like m but not for objects like t.
Of course, all of this is true only if we construct m with a concrete type. We can break this by explicitly constructing it with an abstract type:
julia> m = MyType{AbstractFloat}(3.2)
MyType{AbstractFloat}(3.2)
julia> typeof(m.a)
Float64
julia> m.a = 4.5f0
4.5f0
julia> typeof(m.a)
Float32
For all practical purposes, such objects behave identically to those of MyStillAmbiguousType.
It’s quite instructive to compare the sheer amount code generated for a simple function
func(m::MyType) = m.a+1
using
code_llvm(func,(MyType{Float64},))
code_llvm(func,(MyType{AbstractFloat},))
code_llvm(func,(MyType,))
For reasons of length the results are not shown here, but you may wish to try this yourself. Because the type is fully-specified in the first case, the compiler doesn’t need to generate any code to resolve the type at run-time. This results in shorter and faster code.
The same best practices also work for container types:
julia> type MySimpleContainer{A<:AbstractVector}
a::A
end
julia> type MyAmbiguousContainer{T}
a::AbstractVector{T}
end
For example:
julia> c = MySimpleContainer(1:3);
julia> typeof(c)
MySimpleContainer{UnitRange{Int64}}
julia> c = MySimpleContainer([1:3;]);
julia> typeof(c)
MySimpleContainer{Array{Int64,1}}
julia> b = MyAmbiguousContainer(1:3);
julia> typeof(b)
MyAmbiguousContainer{Int64}
julia> b = MyAmbiguousContainer([1:3;]);
julia> typeof(b)
MyAmbiguousContainer{Int64}
For MySimpleContainer, the object is fully-specified by its type and parameters, so the compiler can generate optimized functions. In most instances, this will probably suffice.
While the compiler can now do its job perfectly well, there are cases where you might wish that your code could do different things depending on the element type of a. Usually the best way to achieve this is to wrap your specific operation (here, foo) in a separate function:
function sumfoo(c::MySimpleContainer)
s = 0
for x in c.a
s += foo(x)
end
s
end
foo(x::Integer) = x
foo(x::AbstractFloat) = round(x)
This keeps things simple, while allowing the compiler to generate optimized code in all cases.
However, there are cases where you may need to declare different versions of the outer function for different element types of a. You could do it like this:
function myfun{T<:AbstractFloat}(c::MySimpleContainer{Vector{T}})
...
end
function myfun{T<:Integer}(c::MySimpleContainer{Vector{T}})
...
end
This works fine for Vector{T}, but we’d also have to write explicit versions for UnitRange{T} or other abstract types. To prevent such tedium, you can use two parameters in the declaration of MyContainer:
type MyContainer{T, A<:AbstractVector}
a::A
end
MyContainer(v::AbstractVector) = MyContainer{eltype(v), typeof(v)}(v)
julia> b = MyContainer(1.3:5);
julia> typeof(b)
MyContainer{Float64,UnitRange{Float64}}
Note the somewhat surprising fact that T doesn’t appear in the declaration of field a, a point that we’ll return to in a moment. With this approach, one can write functions such as:
function myfunc{T<:Integer, A<:AbstractArray}(c::MyContainer{T,A})
return c.a[1]+1
end
# Note: because we can only define MyContainer for
# A<:AbstractArray, and any unspecified parameters are arbitrary,
# the previous could have been written more succinctly as
# function myfunc{T<:Integer}(c::MyContainer{T})
function myfunc{T<:AbstractFloat}(c::MyContainer{T})
return c.a[1]+2
end
function myfunc{T<:Integer}(c::MyContainer{T,Vector{T}})
return c.a[1]+3
end
julia> myfunc(MyContainer(1:3))
2
julia> myfunc(MyContainer(1.0:3))
3.0
julia> myfunc(MyContainer([1:3]))
4
As you can see, with this approach it’s possible to specialize on both the element type T and the array type A.
However, there’s one remaining hole: we haven’t enforced that A has element type T, so it’s perfectly possible to construct an object like this:
julia> b = MyContainer{Int64, UnitRange{Float64}}(1.3:5);
julia> typeof(b)
MyContainer{Int64,UnitRange{Float64}}
To prevent this, we can add an inner constructor:
type MyBetterContainer{T<:Real, A<:AbstractVector}
a::A
MyBetterContainer(v::AbstractVector{T}) = new(v)
end
MyBetterContainer(v::AbstractVector) = MyBetterContainer{eltype(v),typeof(v)}(v)
julia> b = MyBetterContainer(1.3:5);
julia> typeof(b)
MyBetterContainer{Float64,UnitRange{Float64}}
julia> b = MyBetterContainer{Int64, UnitRange{Float64}}(1.3:5);
ERROR: no method MyBetterContainer(UnitRange{Float64},)
The inner constructor requires that the element type of A be T.
It is often convenient to work with data structures that may contain values of any type (arrays of type Array{Any}). But, if you’re using one of these structures and happen to know the type of an element, it helps to share this knowledge with the compiler:
function foo(a::Array{Any,1})
x = a[1]::Int32
b = x+1
...
end
Here, we happened to know that the first element of a would be an Int32. Making an annotation like this has the added benefit that it will raise a run-time error if the value is not of the expected type, potentially catching certain bugs earlier.
Keyword arguments can have declared types:
function with_keyword(x; name::Int = 1)
...
end
Functions are specialized on the types of keyword arguments, so these declarations will not affect performance of code inside the function. However, they will reduce the overhead of calls to the function that include keyword arguments.
Functions with keyword arguments have near-zero overhead for call sites that pass only positional arguments.
Passing dynamic lists of keyword arguments, as in f(x; keywords...), can be slow and should be avoided in performance-sensitive code.
Writing a function as many small definitions allows the compiler to directly call the most applicable code, or even inline it.
Here is an example of a “compound function” that should really be written as multiple definitions:
function norm(A)
if isa(A, Vector)
return sqrt(real(dot(A,A)))
elseif isa(A, Matrix)
return max(svd(A)[2])
else
error("norm: invalid argument")
end
end
This can be written more concisely and efficiently as:
norm(x::Vector) = sqrt(real(dot(x,x))) norm(A::Matrix) = max(svd(A)[2])
When possible, it helps to ensure that a function always returns a value of the same type. Consider the following definition:
pos(x) = x < 0 ? 0 : x
Although this seems innocent enough, the problem is that 0 is an integer (of type Int) and x might be of any type. Thus, depending on the value of x, this function might return a value of either of two types. This behavior is allowed, and may be desirable in some cases. But it can easily be fixed as follows:
pos(x) = x < 0 ? zero(x) : x
There is also a one() function, and a more general oftype(x,y) function, which returns y converted to the type of x.
An analogous “type-stability” problem exists for variables used repeatedly within a function:
function foo()
x = 1
for i = 1:10
x = x/bar()
end
return x
end
Local variable x starts as an integer, and after one loop iteration becomes a floating-point number (the result of / operator). This makes it more difficult for the compiler to optimize the body of the loop. There are several possible fixes:
x with x = 1.0
x: x::Float64 = 1
x = one(T)
Many functions follow a pattern of performing some set-up work, and then running many iterations to perform a core computation. Where possible, it is a good idea to put these core computations in separate functions. For example, the following contrived function returns an array of a randomly-chosen type:
function strange_twos(n)
a = Array(rand(Bool) ? Int64 : Float64, n)
for i = 1:n
a[i] = 2
end
return a
end
This should be written as:
function fill_twos!(a)
for i=1:length(a)
a[i] = 2
end
end
function strange_twos(n)
a = Array(rand(Bool) ? Int64 : Float64, n)
fill_twos!(a)
return a
end
Julia’s compiler specializes code for argument types at function boundaries, so in the original implementation it does not know the type of a during the loop (since it is chosen randomly). Therefore the second version is generally faster since the inner loop can be recompiled as part of fill_twos! for different types of a.
The second form is also often better style and can lead to more code reuse.
This pattern is used in several places in the standard library. For example, see hvcat_fill in abstractarray.jl, or the fill! function, which we could have used instead of writing our own fill_twos!.
Functions like strange_twos occur when dealing with data of uncertain type, for example data loaded from an input file that might contain either integers, floats, strings, or something else.
Let’s say you want to create an N-dimensional array that has size 3 along each axis. Such arrays can be created like this:
A = fill(5.0, (3, 3))
This approach works very well: the compiler can figure out that A is an Array{Float64,2} because it knows the type of the fill value (5.0::Float64) and the dimensionality ((3, 3)::NTuple{2,Int}). This implies that the compiler can generate very efficient code for any future usage of A in the same function.
But now let’s say you want to write a function that creates a 3×3×... array in arbitrary dimensions; you might be tempted to write a function
function array3(fillval, N)
fill(fillval, ntuple(d->3, N))
end
This works, but (as you can verify for yourself using @code_warntype
array3(5.0, 2)) the problem is that the output type cannot be inferred: the argument N is a value of type Int, and type-inference does not (and cannot) predict its value in advance. This means that code using the output of this function has to be conservative, checking the type on each access of A; such code will be very slow.
Now, one very good way to solve such problems is by using the function-barrier technique. However, in some cases you might want to eliminate the type-instability altogether. In such cases, one approach is to pass the dimensionality as a parameter, for example through Val{T} (see “Value types”):
function array3{N}(fillval, ::Type{Val{N}})
fill(fillval, ntuple(d->3, Val{N}))
end
Julia has a specialized version of ntuple that accepts a Val{::Int} as the second parameter; by passing N as a type-parameter, you make its “value” known to the compiler. Consequently, this version of array3 allows the compiler to predict the return type.
However, making use of such techniques can be surprisingly subtle. For example, it would be of no help if you called array3 from a function like this:
function call_array3(fillval, n)
A = array3(fillval, Val{n})
end
Here, you’ve created the same problem all over again: the compiler can’t guess the type of n, so it doesn’t know the type of Val{n}. Attempting to use Val, but doing so incorrectly, can easily make performance worse in many situations. (Only in situations where you’re effectively combining Val with the function-barrier trick, to make the kernel function more efficient, should code like the above be used.)
An example of correct usage of Val would be:
function filter3{T,N}(A::AbstractArray{T,N})
kernel = array3(1, Val{N})
filter(A, kernel)
end
In this example, N is passed as a parameter, so its “value” is known to the compiler. Essentially, Val{T} works only when T is either hard-coded (Val{3}) or already specified in the type-domain.
Once one learns to appreciate multiple dispatch, there’s an understandable tendency to go crazy and try to use it for everything. For example, you might imagine using it to store information, e.g.
immutable Car{Make,Model}
year::Int
...more fields...
end
and then dispatch on objects like Car{:Honda,:Accord}(year, args...).
This might be worthwhile when the following are true:
Car, and it becomes vastly more efficient if you know the Make and Model at compile time.Car to process, so that you can store them all in an Array{Car{:Honda,:Accord},N}.When the latter holds, a function processing such a homogenous array can be productively specialized: Julia knows the type of each element in advance (all objects in the container have the same concrete type), so Julia can “look up” the correct method calls when the function is being compiled (obviating the need to check at run-time) and thereby emit efficient code for processing the whole list.
When these do not hold, then it’s likely that you’ll get no benefit; worse, the resulting “combinatorial explosion of types” will be counterproductive. If items[i+1] has a different type than item[i], Julia has to look up the type at run-time, search for the appropriate method in method tables, decide (via type intersection) which one matches, determine whether it has been JIT-compiled yet (and do so if not), and then make the call. In essence, you’re asking the full type- system and JIT-compilation machinery to basically execute the equivalent of a switch statement or dictionary lookup in your own code.
Some run-time benchmarks comparing (1) type dispatch, (2) dictionary lookup, and (3) a “switch” statement can be found on the mailing list.
Perhaps even worse than the run-time impact is the compile-time impact: Julia will compile specialized functions for each different Car{Make, Model}; if you have hundreds or thousands of such types, then every function that accepts such an object as a parameter (from a custom get_year function you might write yourself, to the generic push! function in the standard library) will have hundreds or thousands of variants compiled for it. Each of these increases the size of the cache of compiled code, the length of internal lists of methods, etc. Excess enthusiasm for values-as-parameters can easily waste enormous resources.
Multidimensional arrays in Julia are stored in column-major order. This means that arrays are stacked one column at a time. This can be verified using the vec function or the syntax [:] as shown below (notice that the array is ordered [1 3 2 4], not [1 2 3 4]):
julia> x = [1 2; 3 4]
2×2 Array{Int64,2}:
1 2
3 4
julia> x[:]
4-element Array{Int64,1}:
1
3
2
4
This convention for ordering arrays is common in many languages like Fortran, Matlab, and R (to name a few). The alternative to column-major ordering is row-major ordering, which is the convention adopted by C and Python (numpy) among other languages. Remembering the ordering of arrays can have significant performance effects when looping over arrays. A rule of thumb to keep in mind is that with column-major arrays, the first index changes most rapidly. Essentially this means that looping will be faster if the inner-most loop index is the first to appear in a slice expression.
Consider the following contrived example. Imagine we wanted to write a function that accepts a Vector and returns a square Matrix with either the rows or the columns filled with copies of the input vector. Assume that it is not important whether rows or columns are filled with these copies (perhaps the rest of the code can be easily adapted accordingly). We could conceivably do this in at least four ways (in addition to the recommended call to the built-in repmat()):
function copy_cols{T}(x::Vector{T})
n = size(x, 1)
out = Array{T}(n, n)
for i=1:n
out[:, i] = x
end
out
end
function copy_rows{T}(x::Vector{T})
n = size(x, 1)
out = Array{T}(n, n)
for i=1:n
out[i, :] = x
end
out
end
function copy_col_row{T}(x::Vector{T})
n = size(x, 1)
out = Array{T}(n, n)
for col=1:n, row=1:n
out[row, col] = x[row]
end
out
end
function copy_row_col{T}(x::Vector{T})
n = size(x, 1)
out = Array{T}(n, n)
for row=1:n, col=1:n
out[row, col] = x[col]
end
out
end
Now we will time each of these functions using the same random 10000 by 1 input vector:
julia> x = randn(10000); julia> fmt(f) = println(rpad(string(f)*": ", 14, ' '), @elapsed f(x)) julia> map(fmt, Any[copy_cols, copy_rows, copy_col_row, copy_row_col]); copy_cols: 0.331706323 copy_rows: 1.799009911 copy_col_row: 0.415630047 copy_row_col: 1.721531501
Notice that copy_cols is much faster than copy_rows. This is expected because copy_cols respects the column-based memory layout of the Matrix and fills it one column at a time. Additionally, copy_col_row is much faster than copy_row_col because it follows our rule of thumb that the first element to appear in a slice expression should be coupled with the inner-most loop.
If your function returns an Array or some other complex type, it may have to allocate memory. Unfortunately, oftentimes allocation and its converse, garbage collection, are substantial bottlenecks.
Sometimes you can circumvent the need to allocate memory on each function call by preallocating the output. As a trivial example, compare
function xinc(x)
return [x, x+1, x+2]
end
function loopinc()
y = 0
for i = 1:10^7
ret = xinc(i)
y += ret[2]
end
y
end
with
function xinc!{T}(ret::AbstractVector{T}, x::T)
ret[1] = x
ret[2] = x+1
ret[3] = x+2
nothing
end
function loopinc_prealloc()
ret = Array{Int}(3)
y = 0
for i = 1:10^7
xinc!(ret, i)
y += ret[2]
end
y
end
Timing results:
julia> @time loopinc() elapsed time: 1.955026528 seconds (1279975584 bytes allocated) 50000015000000 julia> @time loopinc_prealloc() elapsed time: 0.078639163 seconds (144 bytes allocated) 50000015000000
Preallocation has other advantages, for example by allowing the caller to control the “output” type from an algorithm. In the example above, we could have passed a SubArray rather than an Array, had we so desired.
Taken to its extreme, pre-allocation can make your code uglier, so performance measurements and some judgment may be required. However, for “vectorized” (element-wise) functions, the convenient syntax x .= f.(y) can be used for in-place operations with fused loops and no temporary arrays (Dot Syntax for Vectorizing Functions).
When writing data to a file (or other I/O device), forming extra intermediate strings is a source of overhead. Instead of:
println(file, "$a $b")
use:
println(file, a, " ", b)
The first version of the code forms a string, then writes it to the file, while the second version writes values directly to the file. Also notice that in some cases string interpolation can be harder to read. Consider:
println(file, "$(f(a))$(f(b))")
versus:
println(file, f(a), f(b))
When executing a remote function in parallel:
responses = Vector{Any}(nworkers())
@sync begin
for (idx, pid) in enumerate(workers())
@async responses[idx] = remotecall_fetch(pid, foo, args...)
end
end
is faster than:
refs = Vector{Any}(nworkers())
for (idx, pid) in enumerate(workers())
refs[idx] = @spawnat pid foo(args...)
end
responses = [fetch(r) for r in refs]
The former results in a single network round-trip to every worker, while the latter results in two network calls - first by the @spawnat and the second due to the fetch (or even a wait). The fetch/wait is also being executed serially resulting in an overall poorer performance.
A deprecated function internally performs a lookup in order to print a relevant warning only once. This extra lookup can cause a significant slowdown, so all uses of deprecated functions should be modified as suggested by the warnings.
These are some minor points that might help in tight inner loops.
sum([x,y,z]) use x+y+z.abs2(z) instead of abs(z)^2 for complex z. In general, try to rewrite code to use abs2() instead of abs() for complex arguments.div(x,y) for truncating division of integers instead of trunc(x/y), fld(x,y) instead of floor(x/y), and cld(x,y) instead of ceil(x/y).Sometimes you can enable better optimization by promising certain program properties.
@inbounds to eliminate array bounds checking within expressions. Be certain before doing this. If the subscripts are ever out of bounds, you may suffer crashes or silent corruption.@fastmath to allow floating point optimizations that are correct for real numbers, but lead to differences for IEEE numbers. Be careful when doing this, as this may change numerical results. This corresponds to the -ffast-math option of clang.@simd in front of for loops that are amenable to vectorization. This feature is experimental and could change or disappear in future versions of Julia.Note: While @simd needs to be placed directly in front of a loop, both @inbounds and @fastmath can be applied to several statements at once, e.g. using begin ... end, or even to a whole function.
Here is an example with both @inbounds and @simd markup:
function inner( x, y )
s = zero(eltype(x))
for i=1:length(x)
@inbounds s += x[i]*y[i]
end
s
end
function innersimd( x, y )
s = zero(eltype(x))
@simd for i=1:length(x)
@inbounds s += x[i]*y[i]
end
s
end
function timeit( n, reps )
x = rand(Float32,n)
y = rand(Float32,n)
s = zero(Float64)
time = @elapsed for j in 1:reps
s+=inner(x,y)
end
println("GFlop/sec = ",2.0*n*reps/time*1E-9)
time = @elapsed for j in 1:reps
s+=innersimd(x,y)
end
println("GFlop/sec (SIMD) = ",2.0*n*reps/time*1E-9)
end
timeit(1000,1000)
On a computer with a 2.4GHz Intel Core i5 processor, this produces:
GFlop/sec = 1.9467069505224963 GFlop/sec (SIMD) = 17.578554163920018
(GFlop/sec measures the performance, and larger numbers are better.) The range for a @simd for loop should be a one-dimensional range. A variable used for accumulating, such as s in the example, is called a reduction variable. By using @simd, you are asserting several properties of the loop:
@simd.A loop containing break, continue, or @goto will cause a compile-time error.
Using @simd merely gives the compiler license to vectorize. Whether it actually does so depends on the compiler. To actually benefit from the current implementation, your loop should have the following additional properties:
@inbounds is currently needed for all array accesses. The compiler can sometimes turn short &&, ||, and ?: expressions into straight-line code, if it is safe to evaluate all operands unconditionally. Consider using ifelse() instead of ?: in the loop if it is safe to do so.@simd.Here is an example with all three kinds of markup. This program first calculates the finite difference of a one-dimensional array, and then evaluates the L2-norm of the result:
function init!(u)
n = length(u)
dx = 1.0 / (n-1)
@fastmath @inbounds @simd for i in 1:n
u[i] = sin(2pi*dx*i)
end
end
function deriv!(u, du)
n = length(u)
dx = 1.0 / (n-1)
@fastmath @inbounds du[1] = (u[2] - u[1]) / dx
@fastmath @inbounds @simd for i in 2:n-1
du[i] = (u[i+1] - u[i-1]) / (2*dx)
end
@fastmath @inbounds du[n] = (u[n] - u[n-1]) / dx
end
function norm(u)
n = length(u)
T = eltype(u)
s = zero(T)
@fastmath @inbounds @simd for i in 1:n
s += u[i]^2
end
@fastmath @inbounds return sqrt(s/n)
end
function main()
n = 2000
u = Array{Float64}(n)
init!(u)
du = similar(u)
deriv!(u, du)
nu = norm(du)
@time for i in 1:10^6
deriv!(u, du)
nu = norm(du)
end
println(nu)
end
main()
On a computer with a 2.7 GHz Intel Core i7 processor, this produces:
$ julia wave.jl; elapsed time: 1.207814709 seconds (0 bytes allocated) $ julia --math-mode=ieee wave.jl; elapsed time: 4.487083643 seconds (0 bytes allocated)
Here, the option --math-mode=ieee disables the @fastmath macro, so that we can compare results.
In this case, the speedup due to @fastmath is a factor of about 3.7. This is unusually large – in general, the speedup will be smaller. (In this particular example, the working set of the benchmark is small enough to fit into the L1 cache of the processor, so that memory access latency does not play a role, and computing time is dominated by CPU usage. In many real world programs this is not the case.) Also, in this case this optimization does not change the result – in general, the result will be slightly different. In some cases, especially for numerically unstable algorithms, the result can be very different.
The annotation @fastmath re-arranges floating point expressions, e.g. changing the order of evaluation, or assuming that certain special cases (inf, nan) cannot occur. In this case (and on this particular computer), the main difference is that the expression 1 / (2*dx) in the function deriv is hoisted out of the loop (i.e. calculated outside the loop), as if one had written idx = 1 /
(2*dx). In the loop, the expression ... / (2*dx) then becomes ... * idx, which is much faster to evaluate. Of course, both the actual optimization that is applied by the compiler as well as the resulting speedup depend very much on the hardware. You can examine the change in generated code by using Julia’s code_native() function.
Subnormal numbers, formerly called denormal numbers, are useful in many contexts, but incur a performance penalty on some hardware. A call set_zero_subnormals(true) grants permission for floating-point operations to treat subnormal inputs or outputs as zeros, which may improve performance on some hardware. A call set_zero_subnormals(false) enforces strict IEEE behavior for subnormal numbers.
Below is an example where subnormals noticeably impact performance on some hardware:
function timestep{T}( b::Vector{T}, a::Vector{T}, Δt::T )
@assert length(a)==length(b)
n = length(b)
b[1] = 1 # Boundary condition
for i=2:n-1
b[i] = a[i] + (a[i-1] - T(2)*a[i] + a[i+1]) * Δt
end
b[n] = 0 # Boundary condition
end
function heatflow{T}( a::Vector{T}, nstep::Integer )
b = similar(a)
for t=1:div(nstep,2) # Assume nstep is even
timestep(b,a,T(0.1))
timestep(a,b,T(0.1))
end
end
heatflow(zeros(Float32,10),2) # Force compilation
for trial=1:6
a = zeros(Float32,1000)
set_zero_subnormals(iseven(trial)) # Odd trials use strict IEEE arithmetic
@time heatflow(a,1000)
end
This example generates many subnormal numbers because the values in a become an exponentially decreasing curve, which slowly flattens out over time.
Treating subnormals as zeros should be used with caution, because doing so breaks some identities, such as x-y==0 implies x==y:
julia> x=3f-38; y=2f-38; julia> set_zero_subnormals(false); (x-y,x==y) (1.0000001f-38,false) julia> set_zero_subnormals(true); (x-y,x==y) (0.0f0,false)
In some applications, an alternative to zeroing subnormal numbers is to inject a tiny bit of noise. For example, instead of initializing a with zeros, initialize it with:
a = rand(Float32,1000) * 1.f-9
The macro @code_warntype (or its function variant code_warntype()) can sometimes be helpful in diagnosing type-related problems. Here’s an example:
pos(x) = x < 0 ? 0 : x
function f(x)
y = pos(x)
sin(y*x+1)
end
julia> @code_warntype f(3.2)
Variables:
x::Float64
y::UNION(INT64,FLOAT64)
_var0::Float64
_var3::Tuple{Int64}
_var4::UNION(INT64,FLOAT64)
_var1::Float64
_var2::Float64
Body:
begin # none, line 2:
_var0 = (top(box))(Float64,(top(sitofp))(Float64,0))
unless (top(box))(Bool,(top(or_int))((top(lt_float))(x::Float64,_var0::Float64)::Bool,(top(box))(Bool,(top(and_int))((top(box))(Bool,(top(and_int))((top(eq_float))(x::Float64,_var0::Float64)::Bool,(top(lt_float))(_var0::Float64,9.223372036854776e18)::Bool)),(top(slt_int))((top(box))(Int64,(top(fptosi))(Int64,_var0::Float64)),0)::Bool)))) goto 1
_var4 = 0
goto 2
1:
_var4 = x::Float64
2:
y = _var4::UNION(INT64,FLOAT64) # line 3:
_var1 = y::UNION(INT64,FLOAT64) * x::Float64::Float64
_var2 = (top(box))(Float64,(top(add_float))(_var1::Float64,(top(box))(Float64,(top(sitofp))(Float64,1))))
return (GlobalRef(Base.Math,:nan_dom_err))((top(ccall))($(Expr(:call1, :(top(tuple)), "sin", GlobalRef(Base.Math,:libm))),Float64,$(Expr(:call1, :(top(tuple)), :Float64)),_var2::Float64,0)::Float64,_var2::Float64)::Float64
end::Float64
Interpreting the output of @code_warntype, like that of its cousins @code_lowered, @code_typed, @code_llvm, and @code_native, takes a little practice. Your code is being presented in form that has been partially digested on its way to generating compiled machine code. Most of the expressions are annotated by a type, indicated by the ::T (where T might be Float64, for example). The most important characteristic of @code_warntype is that non-concrete types are displayed in red; in the above example, such output is shown in all-caps.
The top part of the output summarizes the type information for the different variables internal to the function. You can see that y, one of the variables you created, is a Union{Int64,Float64}, due to the type-instability of pos. There is another variable, _var4, which you can see also has the same type.
The next lines represent the body of f. The lines starting with a number followed by a colon (1:, 2:) are labels, and represent targets for jumps (via goto) in your code. Looking at the body, you can see that pos has been inlined into f—everything before 2: comes from code defined in pos.
Starting at 2:, the variable y is defined, and again annotated as a Union type. Next, we see that the compiler created the temporary variable _var1 to hold the result of y*x. Because a Float64 times either an Int64 or Float64 yields a Float64, all type-instability ends here. The net result is that f(x::Float64) will not be type-unstable in its output, even if some of the intermediate computations are type-unstable.
How you use this information is up to you. Obviously, it would be far and away best to fix pos to be type-stable: if you did so, all of the variables in f would be concrete, and its performance would be optimal. However, there are circumstances where this kind of ephemeral type instability might not matter too much: for example, if pos is never used in isolation, the fact that f‘s output is type-stable (for Float64 inputs) will shield later code from the propagating effects of type instability. This is particularly relevant in cases where fixing the type instability is difficult or impossible: for example, currently it’s not possible to infer the return type of an anonymous function. In such cases, the tips above (e.g., adding type annotations and/or breaking up functions) are your best tools to contain the “damage” from type instability.
The following examples may help you interpret expressions marked as containing non-leaf types:
end::Union{T1,T2})f(x::T)::Union{T1,T2}(top(arrayref))(A::Array{Any,1},1)::Any(top(getfield))(A::ArrayContainer{Float64},:data)::Array{Float64,N}ArrayContainer had a field data::Array{T}. But Array needs the dimension N, too, to be a concrete type.Array{T,3} or Array{T,N}, where N is now a parameter of ArrayContainer
© 2009–2016 Jeff Bezanson, Stefan Karpinski, Viral B. Shah, and other contributors
Licensed under the MIT License.
http://docs.julialang.org/en/release-0.5/manual/performance-tips/