x[i] = math.sqrt(y[i] * z.imag)a shocking host of dictionary look-ups, allocations, and all-around wasteful computations kick into gear.
The trick, then, to getting good numerical performance from Python is to avoid really doing your work in Python. Instead, you should use Python's remarkable capacity as a glue language to coordinate calls between highly optimized lower-level numerical libraries. This is why a certain handsome astrophysicist called Python "the engine of modern science". NumPy plays an extremely important role in enabling this sticky style of programming by providing a high-level Pythonic interface to an unboxed array that can be passed easily into precompiled C and Fortran libraries.
In order to benefit from NumPy and its vast ecosystem, your algorithm must spend most of its time performing some common operation for which someone has already written an efficient library. Want to multiply some matrices? Great news, calling BLAS from Python isn't really much slower than doing it from C. Need to perform a large convolution? No problem, just hop on over to the frequency domain with a call to the always zippy FFTW.
But disaster and tribulation: What if no one has yet written a library that does the heavy lifting that I need? The standard solutions all boil down to "implement the bottleneck in C" (or if you're feeling enlightened, Cython).
Is a different way possible? Must we sacrifice all our abstractions to get performance? Even if we give up all the niceties of Python, we'll still probably churn out some fairly naive native code that woefully underutilizes our computers' capabilities. Think of all those pitifully empty vector registers, despondently idle extra cores, and a swarm of GPU shader units which haven't seen a general purpose computation in weeks. Harnessing all that parallelism from a low-level language requires, for most tasks, a heroic effort.
A worthy challenge is then issued in two parts:
- Find a way to accelerate a meaningfully expressive subset of Python, such that it's possible to still use convenient abstractions without a large runtime cost. This generally implies a just-in-time compiler of some sort (though a few notable exceptions do compile Python statically).
- As long as we're dynamically translating high level abstractions into low-level executables, is there any chance that the "high level"-ness could be useful for parallelization? It sure would be nice to use those other cores...
To be clear, I am not talking about speeding up all of Python, though some very smart and praiseworthy folks have been working on that for a while. Rather, the great thing about many numerically intensive algorithms is that they are remarkably simple. You might get away with using some subset of Python for implementing the core of your computation, and still feel like you are coding at a high level of abstraction (so long as the boundary between the numerical language subset and the rest of Python is mostly seamless).
A surprisingly large number of projects have already risen to meet this challenge. They (roughly) fall onto a spectrum which trades off between the freedom of the compiler to dramatically rewrite/optimize your code and the expressiveness of the sub-language that is exposed to the programmer.
- NumPyPy - an attempt to reimplement all of NumPy in Python and then let PyPy do its meta-tracing magic. It seems to me that the all-or-nothing nature of PyPy's uncooperativeness with existing NumPy libraries makes this a Utopian misadventure in code duplication, requiring the reimplementation of a huge scientific computing code base with faint hope that a largely opaque general-purpose JIT can play the role of an optimizing compiler for scientific programs. Hopefully fijal will prove us detractors wrong.
- Numba - one the several cool projects Travis Oliphant has been cooking up since he started Continuum Analytics. For the most part, Numba's main purpose is to unbox numeric values and make looping fast in Python. It's still a work in progress and seems to be going in multiple directions at once. They're adding support for general-purpose Python constructs, but relying on the traditional Python runtime to implement anything non-numeric, which sequentializes their runtime due to the Global Interpreter Lock. To enable parallelism you can disavow using any constructs that rely on things Numba doesn't compile directly...but that requires that you know what those constructs are. Like I said, it's still evolving. The commercial version of Numba even touts some capacity for targeting GPUs, but I haven't used it and don't know what can actually get parallelized.
- Blaze - another Travis Oliphant creation, though this one is even more ambitious than Numba. Whereas NumPy is a good abstraction for dense in-memory arrays with varying layouts, Blaze is intended to work with more complex data types and "is designed to handle out-of-core computations on large datasets that exceed the system memory capacity, as well as on distributed and streaming data". Travis is billing Blaze as the successor to NumPy. The underlying abstractions are to a large degree inspired by the Haskell library Repa 3, which is very cool and worth reading about. One key difference between Blaze and NumPy (aside from the much richer array type) is that Blaze delays array computations and then compiles them on-demand. I get the sense that Blaze is pretty far off from being ready for the masses, but I'm sure it will be Awesome Upon Arrival.
- Copperhead - Copperhead takes the direct route to parallelism by forcing you to write your code using data parallel operators which have clear compilation schemes onto multicore and GPU targets. To further simplify the compiler's job, Copperhead forces your code to be purely functional, which goes far against the grain of idiomatic Python. In exchange for these semantics handcuffs, you get some pretty speedy parallel programs. Unfortuantely, the author Bryan Catanzaro has disappeared from github, so I'm not sure if Copperhead is still being developed.
- Theano - Theano is both more cumbersome and more honest than projects like Numba or Copperhead, which take code that looks like Python but then execute it under different assumptions/semantics. With Theano, on the other hand, you have to explicitly piece together symbolic expressions representing your computation. You're always aware that you're constructing Theano syntax explicitly. In exchange for your effort though, Theano can work small feats of magic. For example, Theano can group and reorganize matrix multiplications, reorder floating point operations for stability, and compute gradients using automatic differentiation. Their backend has some preliminary support for CUDA and should eventually add in multi-core and SIMD code generation.
To add another compiler-critter into the fray, I've written Parakeet, a just-in-time compiler for numerical Python which specializes functions for given input types. Parakeet makes extensive use of the data parallel operators such as map, reduce, and (prefix) scan. It's not essential to use these operators when programming with Parakeet, but they do enable parallelism and more aggressive optimizations. Luckily, it's quite easy to end up using these operators by accident, since our library functions are implemented on top of them.
(edited to present Parakeet less sheepishly)
On the spectrum described above, Parakeet sits somewhere between Numba and Copperhead. Like Copperhead, Parakeet's subset of Python is limited to using a small set of data types, library functions and data parallel operators. On the other hand, unlike Copperhead, you don't have to program in a purely function style: if you write loop-heavy numerical code you'll miss out on parallelization but will still see good single-core performance. The main difference from Numba is the absence of any sort of "object layer" which uses the Python C API. Parakeet will (in the long run) support a smaller more numerically-focused subset of Python for the purpose of giving the programmer a clear sense of what will run fast (and if a feature is slow, then Parakeet simply doesn't support it). Additionally, Parakeet's implementation of Python and NumPy library functions leans heavily on data parallel operators, which gives me hope for making pervasive use of GPUs and multi-core hardware.
If you want to learn more about Parakeet check out some of the following presentations:
- HotPar 2012: We submitted a paper describing an old version of our compiler (written in OCaml with a fragile GPU backend).
- SciPy 2013 Lightning Talk: A 5-minute overview of the rewritten Parakeet with an LLVM backend.
- PyData Boston 2013: A longer presentation with more extensive comparison to Numba.
pip install parakeet) or just clone the github repo. Let me know how it goes!