D5.7: Take advantage of multiple cores in the matrix Fourier Algorithm component of the FFT for integer and polynomial arithmetic,and include assembly primitives for SIMD processor instructions (e.g. AVX, etc.), especially in the FFT butterflies #120

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minrk opened this Issue Sep 8, 2015 · 18 comments

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minrk commented Sep 8, 2015

Report on parallelising the FFT

Problem statement

Given two polynomials of length n, the time to multiply them using classical schoolboy multiplication is O(n^2). But there are numerous algorithms which can do better. The Karatsuba method already takes time O(n^log_2(3)). There are other methods, including Toom-Cook which slightly improve the exponent.

The Fast Fourier technique allows multiplication of such polynomials in O(n log(n)) operations. This is a technique that goes back as far as Gauß, but has seen extensive development since then, with over 800 papers on the method and related techniques, with applications from signal processing to string search or polynomial and integer arithmetic.

The version of the FFT that is used in Flint and MPIR is the Schoenhage-Strassen method. Instead of doing a convolution over the complex numbers, which would make use of imprecise floating point numbers, which would be subject to rounding error, it makes use of an exact ring, namely Z/pZ where p = 2^(2^n) + 1. This technique allows exact multiplication of polynomials and integers with nearly linear complexity.

In summary, the existing FFT in Flint is used for:

  • Large integer multiplication
  • Schoenhage-Strassen univariate polynomial multiplication
  • Kronecker-Segmentation univariate polynomial multiplication

The purpose of this task was to parallelise the FFT in Flint.

Typically, parallelising the FFT algorithm is difficult. However, Flint makes use of a cache-friendly implementation of the FFT which uses the Matrix Fourier Algorithm. This breaks one very large FFT convolution up into many smaller FFT's.

The existing FFT implementation in Flint (and MPIR) is world class and includes:

  • truncated fourier transform
  • use of low level GMP/MPIR assembly optimised functions
  • square root of 2 trick
  • Matrix Fourier Algorithm
  • Nussbaumer convolution
  • Chinese remainder with naive convolution

The method

In order to thread the FFT in Flint, we used OpenMP. The level at which we threaded it was at the level of the Matrix Fourier Algorithm. This involved separating temporary storage that is used throughout the algorithm, on a per thread level, and then adding OpenMP primitives to the part of the Matrix Fourier Algorithm that breaks the FFT into lots of smaller FFTs.

We also threaded the code which splits large integers into FFT coefficients. Unfortunately it is difficult, or even impossible to fully parallelise the recombination that happens after the FFT convolution has run, so this wasn't attempted. However, it is a negligible portion of the run time.

Fortunately, once the Matrix Fourier Algorithm becomes more efficient than a single large FFT (due to its cache aware properties), the threaded version also becomes more efficient than the single-threaded version. In fact, the tuning crossover was found to be at exactly the same point! This is an interesting coincidence and made tuning very easy.

To maximise the benefit of threads, we combine parts of the small inward FFTs, the relevant pointwise multiplications and parts of the outward inverse FFTs into combined blocks that each run on a single thread without interruption. The whole FFT convolution consists of many of these smaller blocks. This was by design rather than accident!

The algorithm in Flint also combines the truncated Fourier transform and Matrix Fourier algorithm in such a way that the entire large FFT breaks down exactly into the smaller threaded blocks discussed above, with no additional bits that have to be dealt with serially. This is due to an innovation in the Flint FFT which isn't available elsewhere. Again, this was a design feature, not an accident. The scope of this method is exceptionally technical and well beyond the scope of this report to describe.

In fact, we were able to preserve every single one of the technical tricks mentioned above in our parallel implementation of the FFT in Flint.

Results

The new code for the threaded Matrix Fourier algorithm has been implemented as part of this deliverable and merged into the main Flint repository.

Here are timings of the new code in Flint on a single core, versus four and eight cores for various sized integer multiplications on a 64 bit machine.

limbs 1 core 4 core 8 core
114525 0.066s 0.049s 0.049s
229725 0.14s 0.11s 0.11s
360237 0.32s 0.12s 0.09s
721709 0.65s 0.25s 0.19s
1245101 1.14s 0.39s 0.27s
2492333 2.33s 0.81s 0.55s
4587132 4.45s 1.52s 1.02s
9178748 9.07s 3.02s 2.06s
25947772 28.1s 9.35s 6.25s
51908220 57.9s 24.0s 13.8s
118997068 143s 48.4s 33.2s
238026828 309s 105s 65.7s
506425420 801s 241s 146s

Testing the parallel FFT

The Flint repository is available here.

To build and test the code mentioned above, you must have GMP/MPIR and MPFR installed on your machine (refer to your system documentation for how to do this). Then do:

git clone https://github.com/wbhart/flint2
cd flint
./configure --with-mpir=/path/to/mpir --with-mpfr=/path/to/mpfr --enable-openmp
export OMP_NUM_THREADS=4
make
make check MOD=fft

Full instructions on how to build Flint are available in the Flint documentation, available at the Flint website.

The description of the FFT interface is well beyond the scope of this documentation, but can be found in the Flint documentation (625 pp.) There is also additional information specific to the FFT in the Flint FFT README

Report on writing assembly primitives for the FFT butterflies

Problem statement

For this deliverable, our task was to improve existing functions or write new ones to use features of recent microprocessors (esp. AVX2) to speed up the Schönhage-Strassen FFT butterflies. Such assembly primitives are provided by the MPIR library.

The main operations used in the FFT butterflies are:

  • Compute a+b, a-b for given a,b
  • Compute -(a+b), a-b for given a,b
  • Bit-wise shifts by varying bit-counts
  • Subtraction, and to a lesser extent addition and negation

Some of these operations already had assembly primitives available as part of the MPIR library. However, these were not optimised for recent architectures using AVX, for example. In this task, we also added a new assembly primitive, as described below, which is used directly in the FFT butterflies (where most of the FFT work is actually done).

Each year or two, Intel and AMD release new CPU microarchitectures. The ones we focused on for this deliverable were Intel Haswell and Skylake and AMD Bulldozer. These are not the most recent architectures, but they are coming into widespread use at the present time.

Results

The microarchitectures for which we optimized the code are mainly Intel Haswell and Intel Skylake, and to a lesser extent AMD Bulldozer. For Bulldozer (and Piledriver) it should be noted that the opportunities
for optimization are rather limited: the microarchitecture generally performs poorly, especially in hyper-threading mode, and the AVX instructions in particular are so slow as to be practically useless. The newer AMD Steamroller fares better, but we did not have access to one.

For Haswell and Skylake, the mpn_lshift1, mpn_rshift1, mpn_lshift, and mpn_rshift have been written anew, using AVX2 instructions which gave a large speed-up over the previous code. The mpn_add_n/mpn_sub_n functions (which are identical, performance-wise) have been modified from existing code and optimized according to the respective micro-architecture. An mpn_sumdiff_n (computes a+b, a-b) has been introduced into MPIR; this function existed for older processors but not for recent x86_64.

We are very grateful to Jens Nurmann who contributed significant amounts of code and expertise on AVX2 programming.

Haswell microarchitecture

Timings in cycles per limb:

Function Old New
mpn_lshift1 1.11 0.564
mpn_rshift1 1.39 0.589
mpn_lshift 1.85 0.568
mpn_rshift 1.40 0.578
mpn_add_n 1.32 1.11
mpn_sumdiff_n 2.62(1) 2.42
mpn_nsumdiff_n 3.23(2) 2.64

(1) The sum of the times of mpn_add_n, mpn_sub_n.
(2) The sum of the times of mpn_add_n, mpn_sub_n, mpn_neg_n.

Timings for the full Schönhage-Strassen large integer multiplication (mpn_mul_n) in seconds:

Limbs Old New Ratio
10000 0.002399728 0.002171788 0.91
100000 0.026374851 0.022960783 0.87
1000000 0.357847841 0.302023203 0.84

Note that these timings include the effect of code improvements made for D5.5 (#118), in particular, better mpn_mul_basecase and Karatsuba code.

Skylake microarchitecture

Timings in cycles per limb:

Function Old New
mpn_lshift1 1.01 0.601
mpn_rshift1 1.52 0.601
mpn_lshift 2.01 0.608
mpn_rshift 1.52 0.606
mpn_add_n 1.22 1.04
mpn_sumdiff_n 2.44(1) 2.04
mpn_nsumdiff_n 3.06(2) 2.32

Of note here is the speed of mpn_add_n/mpn_sub_n, at essentially 1c/l for the core loop, optimal both in terms of the data dependency chain and memory accesses, as Skylake can in theory execute 2 read and 1 write per clock cycle. In practice, presumably the instruction scheduler falls into a bad pattern after running at 1c/l for a while, and from then on runs the loop only at ~1.2c/l. Jens Nurmann found that inserting a meaningless AVX2 instruction into the core loop (which does not otherwise use AVX2)
breaks up this bad scheduling pattern, allowing these critically important core functions to run at the optimal speed reliably.

Timings for mpn_mul_n in seconds:

Limbs Old New Ratio
10000 0.002125143 0.001711500 0.81
100000 0.025231292 0.020712453 0.82
1000000 0.304166761 0.258099884 0.85

Bulldozer microarchitecture

Much less optimization effort was made for Bulldozer than for Haswell and Skylake, owing to the age and poor performance of this processor. No code was written from scratch, but among all the existing implementations for a given function, the one that ran fastest on Bulldozer was chosen.

Among those functions that were replaced by faster versions, these three are relevant to the FFT butterflies:

Function Old New
com_n 1.28 0.723
rshift 2 1.11
lshift 2.42 1.24

Timings for mpn_mul_n in seconds:

Limbs Old New Ratio
10000 0.004771156 0.004764643 1.0
100000 0.054624774 0.053038739 0.97
1000000 0.651062127 0.652278285 1.0

Unfortunately, the improvements to the mpn_[lr]shift functions are barely visible in the integer multiplication benchmark on Bulldozer.

All code written for this deliverable has been committed to Alex Kruppa's fork of the MPIR repository at https://github.com/akruppa/mpir and merged into the main MPIR repository at https://github.com/wbhart/mpir and will be available in the MPIR-3.0.0 release, available at the MPIR website.

Build instructions for MPIR are as follows:

Download MPIR-3.0.0 from: http://mpir.org/

Note that you also need to have the latest Yasm assembler to build MPIR: http://yasm.tortall.net/

To build Yasm, download the tarball:

./configure
make

To test MPIR, download the tarball:

./configure --enable-gmpcompat --with-yasm=/path_to_yasm/yasm
make
make check

A Haswell, Skylake, or Bulldozer CPU is required to test the changes referred to above.

@minrk minrk added this to the D5.7 milestone Sep 8, 2015

@nthiery nthiery modified the milestones: Month 18: 2017-02-28, D5.7 Mar 22, 2016

@ClementPernet ClementPernet assigned wbhart and unassigned ClementPernet Sep 9, 2016

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@ClementPernet (WP leader) and @wbhart (Lead beneficiary)
This deliverable is due for February 2017

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bpilorget commented Nov 21, 2016

@ClementPernet (WP leader) and @wbhart (Lead beneficiary)
This deliverable is due for February 2017

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This is not a massive project and should be no problem to deliver on time.
Thanks for reminding me it is one of the deliverables for February!

On 21 November 2016 at 17:25, bpilorget notifications@github.com wrote:

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https://github.com/wbhart (Lead beneficiary)
This deliverable is due for February 2017


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wbhart commented Nov 21, 2016

This is not a massive project and should be no problem to deliver on time.
Thanks for reminding me it is one of the deliverables for February!

On 21 November 2016 at 17:25, bpilorget notifications@github.com wrote:

@ClementPernet https://github.com/ClementPernet (WP leader) and @wbhart
https://github.com/wbhart (Lead beneficiary)
This deliverable is due for February 2017


You are receiving this because you were mentioned.
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#120 (comment),
or mute the thread
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@wbhart wbhart changed the title from D5.7: Take advantage of multiple cores in the matrix Fourier Algorithm component of the FFT for integer and polynomial arithmetic,and include assembly primitives for SIMD processor instructions (AVX, Knight's Bridge, etc.), especially in the FFT butterflies to D5.7: Take advantage of multiple cores in the matrix Fourier Algorithm component of the FFT for integer and polynomial arithmetic,and include assembly primitives for SIMD processor instructions (e.g. AVX, etc.), especially in the FFT butterflies Dec 6, 2016

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I have removed the non-existent "Knight's Bridge" from the description of this deliverable. We now have more of an idea of what is required here, and we think SIMD support such as SSE and AVX is what is required.

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wbhart commented Dec 6, 2016

I have removed the non-existent "Knight's Bridge" from the description of this deliverable. We now have more of an idea of what is required here, and we think SIMD support such as SSE and AVX is what is required.

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The multiple core version of the FFT is now implemented.

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wbhart commented Jan 24, 2017

The multiple core version of the FFT is now implemented.

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Dear M18 deliverable leaders,

Just a reminder that reports are due for mid-february, to buy us some time for proofreading, feedback, and final submission before February 28th. See our README for details on the process.

In practice, I'll be offline February 12-19, and the week right after will be pretty busy. Therefore, it would be helpful if a first draft could be available sometime this week, so that I can have a head start reviewing it.

Thanks in advance!

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nthiery commented Feb 6, 2017

Dear M18 deliverable leaders,

Just a reminder that reports are due for mid-february, to buy us some time for proofreading, feedback, and final submission before February 28th. See our README for details on the process.

In practice, I'll be offline February 12-19, and the week right after will be pretty busy. Therefore, it would be helpful if a first draft could be available sometime this week, so that I can have a head start reviewing it.

Thanks in advance!

@ClementPernet ClementPernet added the FLINT label Feb 16, 2017

@minrk minrk referenced this issue Feb 16, 2017

Open

T5.5: MPIR #103

@wbhart wbhart added the MPIR label Feb 21, 2017

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Hi @wbhart,
What's the status of the current report? Ready for final proofreading and submission?
Same question for your other deliverable reports?

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nthiery commented Feb 23, 2017

Hi @wbhart,
What's the status of the current report? Ready for final proofreading and submission?
Same question for your other deliverable reports?

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nthiery commented Feb 23, 2017

@minrk minrk referenced this issue Feb 24, 2017

Open

T5.4: Singular #102

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Hi @wbhart: I probably won't get to proofread the issue description this afternoon. Could you add a little section of context at the top of the issue description, in the vein of that of D5.5 and D5.6? Please also implement the other little cosmetic changes I did in D5.5 and D5.6 (reminder of what Bulldozer/... are, code blocks for shell instructions, software names in backticks, paragraphs on a single line, ...). Thanks!

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nthiery commented Feb 24, 2017

Hi @wbhart: I probably won't get to proofread the issue description this afternoon. Could you add a little section of context at the top of the issue description, in the vein of that of D5.5 and D5.6? Please also implement the other little cosmetic changes I did in D5.5 and D5.6 (reminder of what Bulldozer/... are, code blocks for shell instructions, software names in backticks, paragraphs on a single line, ...). Thanks!

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Thanks; ping me when you are done!

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nthiery commented Feb 27, 2017

Thanks; ping me when you are done!

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@nthiery All done here too I think, with the exception of a blog post, which should appear today or tomorrow.

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wbhart commented Feb 27, 2017

@nthiery All done here too I think, with the exception of a blog post, which should appear today or tomorrow.

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Ok. It's ready to submit on my side too. Ping me when the blog post is online.

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nthiery commented Feb 27, 2017

Ok. It's ready to submit on my side too. Ping me when the blog post is online.

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@nthiery I have added a link to the now complete blog post

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wbhart commented Feb 28, 2017

@nthiery I have added a link to the now complete blog post

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Submitted!
And one, and two, and three deliverables for Bill!
Congratulations for all the hard work you put in with your teammates.

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nthiery commented Feb 28, 2017

Submitted!
And one, and two, and three deliverables for Bill!
Congratulations for all the hard work you put in with your teammates.

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