From 25d18f985d258a34bc9b89fb41151174ab8f729b Mon Sep 17 00:00:00 2001 From: doinachiroiu Date: Thu, 20 Aug 2020 11:19:49 +0000 Subject: [PATCH] Updated README and change names of main files --- oss-internship-2020/pffft/README.md | 67 +++ oss-internship-2020/pffft/main_pffft.c | 419 ++++++++++++++++++ .../pffft/main_pffft_sandboxed.cc | 208 +++++++++ .../pffft/pffft_library_notes.txt | 416 +++++++++++++++++ 4 files changed, 1110 insertions(+) create mode 100644 oss-internship-2020/pffft/README.md create mode 100644 oss-internship-2020/pffft/main_pffft.c create mode 100644 oss-internship-2020/pffft/main_pffft_sandboxed.cc create mode 100644 oss-internship-2020/pffft/pffft_library_notes.txt diff --git a/oss-internship-2020/pffft/README.md b/oss-internship-2020/pffft/README.md new file mode 100644 index 0000000..52facf3 --- /dev/null +++ b/oss-internship-2020/pffft/README.md @@ -0,0 +1,67 @@ +# Sandboxing PFFFT library + +Builder: CMake +OS: Linux + +### For testing: +`cd build`, then `./pffft_sandboxed` + +### For debug: +`SAPI_VLOG_LEVEL=1 ./pffft_sandboxed --v=100 +--sandbox2_danger_danger_permit_all_and_log ` + +## ***About the project*** +*PFFFT library is concerned with 1D Fast-Fourier Transformations finding a +compromise between accuracy and speed. It deals with real and complex +vectors, both cases being illustrated in the testing part (`main_pffft.c` +for initially and original version, `main_pffft_sandboxed.cc` for our +currently implemented sandboxed version). +The original files can be found at: https://bitbucket.org/jpommier/pffft/src.* + +*The purpose of sandboxing is to limit the permissions and capabilities of +library’s methods, in order to secure the usage of them. +After obtaining the sandbox, the functions will be called through an +Sandbox API (being called `api` in the current test) and so, the +operations, system calls or namspaces access may be controlled. +From both `pffft.h` and `fftpack.h` headers, useful methods are added to +sapi library builded with CMake. There is also a need to link math library +as the transformations made require mathematical operators. +Regarding the testing of the methods, one main is doing this job by +iterating through a set of values, that represents the accuracy of +transformations and print the speed for each value and type of +transformation. More specifically, the input length is the target for +accuracy (named as `N`) and it stands for the number of data points from +the series that calculate the result of transformation. It is also +important to mention that the `cplx` variable stands for a boolean value +that tells the type of transformation (0 for REAL and 1 for COMPLEX) and +it is taken into account while testing. +In the end, the performance of PFFFT library it is outlined by the output.* + +#### CMake observations resume: + * linking pffft and fftpack (which contains necessary functions for pffft) + * set math library + +#### Sandboxed main observations resume: + * containing two testing parts (fft / pffft benchmarks) + * showing the performance of the transformations implies + testing them through various FFT dimenstions. + Variable N, the input length, will take specific values + meaning the number of points to which it is set the calculus + (more details of mathematical purpose of N - https://en.wikipedia.org/wiki/Cooley%E2%80%93Tukey_FFT_algorithm). + * output shows speed depending on the input length + + + +### Bugs history + - [Solved] pffft benchmark bug: "Sandbox not active" + N = 64, status OK, pffft_transform generates error + N > 64, status not OK + Problem on initialising sapi::StatusOr s; the memory that stays + for s is not the same with the address passed in pffft_transform function. + (sapi :: v :: GenericPtr - to be changed) + + Temporary solution: change the generated files to accept + uintptr_t instead of PFFFT_Setup + + Solution: using "sapi::v::RemotePtr" instead of "sapi::v::GenericPtr" + to access the memory of object s \ No newline at end of file diff --git a/oss-internship-2020/pffft/main_pffft.c b/oss-internship-2020/pffft/main_pffft.c new file mode 100644 index 0000000..a5d20c2 --- /dev/null +++ b/oss-internship-2020/pffft/main_pffft.c @@ -0,0 +1,419 @@ +/* + Copyright (c) 2013 Julien Pommier. + + Small test & bench for PFFFT, comparing its performance with the scalar FFTPACK, FFTW, and Apple vDSP + + How to build: + + on linux, with fftw3: + gcc -o test_pffft -DHAVE_FFTW -msse -mfpmath=sse -O3 -Wall -W pffft.c test_pffft.c fftpack.c -L/usr/local/lib -I/usr/local/include/ -lfftw3f -lm + + on macos, without fftw3: + clang -o test_pffft -DHAVE_VECLIB -O3 -Wall -W pffft.c test_pffft.c fftpack.c -L/usr/local/lib -I/usr/local/include/ -framework Accelerate + + on macos, with fftw3: + clang -o test_pffft -DHAVE_FFTW -DHAVE_VECLIB -O3 -Wall -W pffft.c test_pffft.c fftpack.c -L/usr/local/lib -I/usr/local/include/ -lfftw3f -framework Accelerate + + on windows, with visual c++: + cl /Ox -D_USE_MATH_DEFINES /arch:SSE test_pffft.c pffft.c fftpack.c + + build without SIMD instructions: + gcc -o test_pffft -DPFFFT_SIMD_DISABLE -O3 -Wall -W pffft.c test_pffft.c fftpack.c -lm + + */ + +#include "pffft.h" +#include "fftpack.h" + +#include +#include +#include +#include +#include +#include + +#ifdef HAVE_SYS_TIMES +# include +# include +#endif + +#ifdef HAVE_VECLIB +# include +#endif + +#ifdef HAVE_FFTW +# include +#endif + +#define MAX(x,y) ((x)>(y)?(x):(y)) + +double frand() { + return rand()/(double)RAND_MAX; +} + +#if defined(HAVE_SYS_TIMES) + inline double uclock_sec(void) { + static double ttclk = 0.; + if (ttclk == 0.) ttclk = sysconf(_SC_CLK_TCK); + struct tms t; return ((double)times(&t)) / ttclk; + } +# else + double uclock_sec(void) +{ return (double)clock()/(double)CLOCKS_PER_SEC; } +#endif + + +/* compare results with the regular fftpack */ +void pffft_validate_N(int N, int cplx) { + int Nfloat = N*(cplx?2:1); + int Nbytes = Nfloat * sizeof(float); + float *ref, *in, *out, *tmp, *tmp2; + PFFFT_Setup *s = pffft_new_setup(N, cplx ? PFFFT_COMPLEX : PFFFT_REAL); + int pass; + + if (!s) { printf("Skipping N=%d, not supported\n", N); return; } + ref = pffft_aligned_malloc(Nbytes); + in = pffft_aligned_malloc(Nbytes); + out = pffft_aligned_malloc(Nbytes); + tmp = pffft_aligned_malloc(Nbytes); + tmp2 = pffft_aligned_malloc(Nbytes); + + for (pass=0; pass < 2; ++pass) { + float ref_max = 0; + int k; + //printf("N=%d pass=%d cplx=%d\n", N, pass, cplx); + // compute reference solution with FFTPACK + if (pass == 0) { + float *wrk = malloc(2*Nbytes+15*sizeof(float)); + for (k=0; k < Nfloat; ++k) { + ref[k] = in[k] = frand()*2-1; + out[k] = 1e30; + } + if (!cplx) { + rffti(N, wrk); + rfftf(N, ref, wrk); + // use our ordering for real ffts instead of the one of fftpack + { + float refN=ref[N-1]; + for (k=N-2; k >= 1; --k) ref[k+1] = ref[k]; + ref[1] = refN; + } + } else { + cffti(N, wrk); + cfftf(N, ref, wrk); + } + free(wrk); + } + + for (k = 0; k < Nfloat; ++k) ref_max = MAX(ref_max, fabs(ref[k])); + + + // pass 0 : non canonical ordering of transform coefficients + if (pass == 0) { + // test forward transform, with different input / output + pffft_transform(s, in, tmp, 0, PFFFT_FORWARD); + memcpy(tmp2, tmp, Nbytes); + memcpy(tmp, in, Nbytes); + pffft_transform(s, tmp, tmp, 0, PFFFT_FORWARD); + for (k = 0; k < Nfloat; ++k) { + assert(tmp2[k] == tmp[k]); + } + + // test reordering + pffft_zreorder(s, tmp, out, PFFFT_FORWARD); + pffft_zreorder(s, out, tmp, PFFFT_BACKWARD); + for (k = 0; k < Nfloat; ++k) { + assert(tmp2[k] == tmp[k]); + } + pffft_zreorder(s, tmp, out, PFFFT_FORWARD); + } else { + // pass 1 : canonical ordering of transform coeffs. + pffft_transform_ordered(s, in, tmp, 0, PFFFT_FORWARD); + memcpy(tmp2, tmp, Nbytes); + memcpy(tmp, in, Nbytes); + pffft_transform_ordered(s, tmp, tmp, 0, PFFFT_FORWARD); + for (k = 0; k < Nfloat; ++k) { + assert(tmp2[k] == tmp[k]); + } + memcpy(out, tmp, Nbytes); + } + + { + for (k=0; k < Nfloat; ++k) { + if (!(fabs(ref[k] - out[k]) < 1e-3*ref_max)) { + printf("%s forward PFFFT mismatch found for N=%d\n", (cplx?"CPLX":"REAL"), N); + exit(1); + } + } + + if (pass == 0) pffft_transform(s, tmp, out, 0, PFFFT_BACKWARD); + else pffft_transform_ordered(s, tmp, out, 0, PFFFT_BACKWARD); + memcpy(tmp2, out, Nbytes); + memcpy(out, tmp, Nbytes); + if (pass == 0) pffft_transform(s, out, out, 0, PFFFT_BACKWARD); + else pffft_transform_ordered(s, out, out, 0, PFFFT_BACKWARD); + for (k = 0; k < Nfloat; ++k) { + assert(tmp2[k] == out[k]); + out[k] *= 1.f/N; + } + for (k = 0; k < Nfloat; ++k) { + if (fabs(in[k] - out[k]) > 1e-3 * ref_max) { + printf("pass=%d, %s IFFFT does not match for N=%d\n", pass, (cplx?"CPLX":"REAL"), N); break; + exit(1); + } + } + } + + // quick test of the circular convolution in fft domain + { + float conv_err = 0, conv_max = 0; + + pffft_zreorder(s, ref, tmp, PFFFT_FORWARD); + memset(out, 0, Nbytes); + pffft_zconvolve_accumulate(s, ref, ref, out, 1.0); + pffft_zreorder(s, out, tmp2, PFFFT_FORWARD); + + for (k=0; k < Nfloat; k += 2) { + float ar = tmp[k], ai=tmp[k+1]; + if (cplx || k > 0) { + tmp[k] = ar*ar - ai*ai; + tmp[k+1] = 2*ar*ai; + } else { + tmp[0] = ar*ar; + tmp[1] = ai*ai; + } + } + + for (k=0; k < Nfloat; ++k) { + float d = fabs(tmp[k] - tmp2[k]), e = fabs(tmp[k]); + if (d > conv_err) conv_err = d; + if (e > conv_max) conv_max = e; + } + if (conv_err > 1e-5*conv_max) { + printf("zconvolve error ? %g %g\n", conv_err, conv_max); exit(1); + } + } + + } + + printf("%s PFFFT is OK for N=%d\n", (cplx?"CPLX":"REAL"), N); fflush(stdout); + + pffft_destroy_setup(s); + pffft_aligned_free(ref); + pffft_aligned_free(in); + pffft_aligned_free(out); + pffft_aligned_free(tmp); + pffft_aligned_free(tmp2); +} + +void pffft_validate(int cplx) { + static int Ntest[] = { 16, 32, 64, 96, 128, 160, 192, 256, 288, 384, 5*96, 512, 576, 5*128, 800, 864, 1024, 2048, 2592, 4000, 4096, 12000, 36864, 0}; + int k; + for (k = 0; Ntest[k]; ++k) { + int N = Ntest[k]; + if (N == 16 && !cplx) continue; + pffft_validate_N(N, cplx); + } +} + +int array_output_format = 0; + +void show_output(const char *name, int N, int cplx, float flops, float t0, float t1, int max_iter) { + float mflops = flops/1e6/(t1 - t0 + 1e-16); + if (array_output_format) { + if (flops != -1) { + printf("|%9.0f ", mflops); + } else printf("| n/a "); + } else { + if (flops != -1) { + printf("N=%5d, %s %16s : %6.0f MFlops [t=%6.0f ns, %d runs]\n", N, (cplx?"CPLX":"REAL"), name, mflops, (t1-t0)/2/max_iter * 1e9, max_iter); + } + } + fflush(stdout); +} + +void benchmark_ffts(int N, int cplx) { + int Nfloat = (cplx ? N*2 : N); + int Nbytes = Nfloat * sizeof(float); + float *X = pffft_aligned_malloc(Nbytes), *Y = pffft_aligned_malloc(Nbytes), *Z = pffft_aligned_malloc(Nbytes); + + double t0, t1, flops; + + int k; + int max_iter = 5120000/N*4; +#ifdef __arm__ + max_iter /= 4; +#endif + int iter; + + for (k = 0; k < Nfloat; ++k) { + X[k] = 0; //sqrtf(k+1); + } + + // FFTPack benchmark + { + float *wrk = malloc(2*Nbytes + 15*sizeof(float)); + int max_iter_ = max_iter/pffft_simd_size(); if (max_iter_ == 0) max_iter_ = 1; + if (cplx) cffti(N, wrk); + else rffti(N, wrk); + t0 = uclock_sec(); + + for (iter = 0; iter < max_iter_; ++iter) { + if (cplx) { + cfftf(N, X, wrk); + cfftb(N, X, wrk); + } else { + rfftf(N, X, wrk); + rfftb(N, X, wrk); + } + } + t1 = uclock_sec(); + free(wrk); + + flops = (max_iter_*2) * ((cplx ? 5 : 2.5)*N*log((double)N)/M_LN2); // see http://www.fftw.org/speed/method.html + show_output("FFTPack", N, cplx, flops, t0, t1, max_iter_); + } + +#ifdef HAVE_VECLIB + int log2N = (int)(log(N)/log(2) + 0.5f); + if (N == (1< 1 && strcmp(argv[1], "--array-format") == 0) { + array_output_format = 1; + } + +#ifndef PFFFT_SIMD_DISABLE + validate_pffft_simd(); +#endif + pffft_validate(1); + pffft_validate(0); + if (!array_output_format) { + for (i=0; Nvalues[i] > 0; ++i) { + benchmark_ffts(Nvalues[i], 0 /* real fft */); + } + for (i=0; Nvalues[i] > 0; ++i) { + benchmark_ffts(Nvalues[i], 1 /* cplx fft */); + } + } else { + printf("| input len "); + printf("|real FFTPack"); +#ifdef HAVE_VECLIB + printf("| real vDSP "); +#endif +#ifdef HAVE_FFTW + printf("| real FFTW "); +#endif + printf("| real PFFFT | "); + + printf("|cplx FFTPack"); +#ifdef HAVE_VECLIB + printf("| cplx vDSP "); +#endif +#ifdef HAVE_FFTW + printf("| cplx FFTW "); +#endif + printf("| cplx PFFFT |\n"); + for (i=0; Nvalues[i] > 0; ++i) { + printf("|%9d ", Nvalues[i]); + benchmark_ffts(Nvalues[i], 0); + printf("| "); + benchmark_ffts(Nvalues[i], 1); + printf("|\n"); + } + printf(" (numbers are given in MFlops)\n"); + } + + + return 0; +} diff --git a/oss-internship-2020/pffft/main_pffft_sandboxed.cc b/oss-internship-2020/pffft/main_pffft_sandboxed.cc new file mode 100644 index 0000000..662b949 --- /dev/null +++ b/oss-internship-2020/pffft/main_pffft_sandboxed.cc @@ -0,0 +1,208 @@ +#define GOOGLE_STRIP_LOG 1 + +#include +#include +#include +#include +#include +#include +#include +#include +#include + +#include "fftpack.h" +#include "pffft_sapi.sapi.h" +#include "sandboxed_api/util/flag.h" +#include "sandboxed_api/vars.h" + +ABSL_DECLARE_FLAG(string, sandbox2_danger_danger_permit_all); +ABSL_DECLARE_FLAG(string, sandbox2_danger_danger_permit_all_and_log); + +class pffftSapiSandbox : public pffftSandbox { + public: + std::unique_ptr ModifyPolicy( + sandbox2::PolicyBuilder*) override { + return sandbox2::PolicyBuilder() + .AllowStaticStartup() + .AllowOpen() + .AllowRead() + .AllowWrite() + .AllowSystemMalloc() + .AllowExit() + .AllowSyscalls({ + __NR_futex, + __NR_close, + __NR_getrusage, + }) + .DisableNamespaces() + .BuildOrDie(); + } +}; + +double frand() { return rand() / (double)RAND_MAX; } + +double uclock_sec(void) { return (double)clock() / (double)CLOCKS_PER_SEC; } + +int array_output_format = 0; + +void show_output(const char* name, int N, int cplx, float flops, float t0, + float t1, int max_iter) { + float mflops = flops / 1e6 / (t1 - t0 + 1e-16); + if (array_output_format) { + if (flops != -1) { + printf("|%9.0f ", mflops); + } else + printf("| n/a "); + } else { + if (flops != -1) { + printf("N=%5d, %s %16s : %6.0f MFlops [t=%6.0f ns, %d runs]\n", N, + (cplx ? "CPLX" : "REAL"), name, mflops, + (t1 - t0) / 2 / max_iter * 1e9, max_iter); + } + } + fflush(stdout); +} + +int main(int argc, char* argv[]) { + /* + * Initialize Google's logging library. + */ + google::InitGoogleLogging(argv[0]); + + gflags::ParseCommandLineFlags(&argc, &argv, true); + /* + * Nvalues is a vector keeping the values by which iterates N, its value + * representing the input length. More concrete, N is the number of + * data points the caclulus is up to (determinating its accuracy). + * To show the performance of Fast-Fourier Transformations the program is + * testing for various values of N. + */ + int Nvalues[] = {64, 96, 128, 160, 192, 256, + 384, 5 * 96, 512, 5 * 128, 3 * 256, 800, + 1024, 2048, 2400, 4096, 8192, 9 * 1024, + 16384, 32768, 256 * 1024, 1024 * 1024, -1}; + int i; + + VLOG(1) << "Initializing sandbox...\n"; + + pffftSapiSandbox sandbox; + sandbox.Init().IgnoreError(); + + VLOG(1) << "Initialization: " << sandbox.Init().ToString().c_str() << "\n"; + + pffftApi api(&sandbox); + + int N, cplx; + + cplx = 0; + + do { + for (i = 0; i < 23; i++) { + N = Nvalues[i]; + + int Nfloat = N * (cplx ? 2 : 1); + int Nbytes = Nfloat * sizeof(float); + int pass; + + float ref[Nbytes], in[Nbytes], out[Nbytes], tmp[Nbytes], tmp2[Nbytes]; + + sapi::v::Array ref_(ref, Nbytes); + sapi::v::Array in_(in, Nbytes); + sapi::v::Array out_(out, Nbytes); + sapi::v::Array tmp_(tmp, Nbytes); + sapi::v::Array tmp2_(tmp2, Nbytes); + + float wrk[2 * Nbytes + 15 * sizeof(float)]; + sapi::v::Array wrk_(wrk, 2 * Nbytes + 15 * sizeof(float)); + + float ref_max = 0; + int k; + + Nfloat = (cplx ? N * 2 : N); + float X[Nbytes], Y[Nbytes], Z[Nbytes]; + sapi::v::Array X_(X, Nbytes), Y_(Y, Nbytes), Z_(Z, Nbytes); + + double t0, t1, flops; + + int max_iter = 5120000 / N * 4; +#ifdef __arm__ + max_iter /= 4; +#endif + int iter; + + for (k = 0; k < Nfloat; ++k) { + X[k] = 0; + } + + /* + * FFTPack benchmark + */ + { + /* + * SIMD_SZ == 4 (returning value of pffft_simd_size()) + */ + int max_iter_ = max_iter / 4; + + if (max_iter_ == 0) max_iter_ = 1; + if (cplx) { + api.cffti(N, wrk_.PtrBoth()).IgnoreError(); + } else { + api.rffti(N, wrk_.PtrBoth()).IgnoreError(); + } + t0 = uclock_sec(); + + for (iter = 0; iter < max_iter_; ++iter) { + if (cplx) { + api.cfftf(N, X_.PtrBoth(), wrk_.PtrBoth()).IgnoreError(); + api.cfftb(N, X_.PtrBoth(), wrk_.PtrBoth()).IgnoreError(); + } else { + api.rfftf(N, X_.PtrBoth(), wrk_.PtrBoth()).IgnoreError(); + api.rfftb(N, X_.PtrBoth(), wrk_.PtrBoth()).IgnoreError(); + } + } + t1 = uclock_sec(); + + flops = + (max_iter_ * 2) * ((cplx ? 5 : 2.5) * N * log((double)N) / M_LN2); + show_output("FFTPack", N, cplx, flops, t0, t1, max_iter_); + } + + /* + * PFFFT benchmark + */ + { + sapi::StatusOr s = + api.pffft_new_setup(N, cplx ? PFFFT_COMPLEX : PFFFT_REAL); + + VLOG(1) << "Setup status is: " << s.status().ToString().c_str() << "\n"; + + if (s.ok()) { + sapi::v::RemotePtr s_reg(s.value()); + + t0 = uclock_sec(); + for (iter = 0; iter < max_iter; ++iter) { + api.pffft_transform(&s_reg, X_.PtrBoth(), Z_.PtrBoth(), + Y_.PtrBoth(), PFFFT_FORWARD) + .IgnoreError(); + api.pffft_transform(&s_reg, X_.PtrBoth(), Z_.PtrBoth(), + Y_.PtrBoth(), PFFFT_FORWARD) + .IgnoreError(); + } + + t1 = uclock_sec(); + api.pffft_destroy_setup(&s_reg).IgnoreError(); + + flops = + (max_iter * 2) * ((cplx ? 5 : 2.5) * N * log((double)N) / M_LN2); + show_output("PFFFT", N, cplx, flops, t0, t1, max_iter); + } + + VLOG(1) << "N = " << N << " SUCCESSFULLY\n\n"; + } + } + + cplx = !cplx; + } while (cplx); + + return 0; +} \ No newline at end of file diff --git a/oss-internship-2020/pffft/pffft_library_notes.txt b/oss-internship-2020/pffft/pffft_library_notes.txt new file mode 100644 index 0000000..ee20b42 --- /dev/null +++ b/oss-internship-2020/pffft/pffft_library_notes.txt @@ -0,0 +1,416 @@ +PFFFT: a pretty fast FFT. + +TL;DR +-- + +PFFFT does 1D Fast Fourier Transforms, of single precision real and +complex vectors. It tries do it fast, it tries to be correct, and it +tries to be small. Computations do take advantage of SSE1 instructions +on x86 cpus, Altivec on powerpc cpus, and NEON on ARM cpus. The +license is BSD-like. + + +Why does it exist: +-- + +I was in search of a good performing FFT library , preferably very +small and with a very liberal license. + +When one says "fft library", FFTW ("Fastest Fourier Transform in the +West") is probably the first name that comes to mind -- I guess that +99% of open-source projects that need a FFT do use FFTW, and are happy +with it. However, it is quite a large library , which does everything +fft related (2d transforms, 3d transforms, other transformations such +as discrete cosine , or fast hartley). And it is licensed under the +GNU GPL , which means that it cannot be used in non open-source +products. + +An alternative to FFTW that is really small, is the venerable FFTPACK +v4, which is available on NETLIB. A more recent version (v5) exists, +but it is larger as it deals with multi-dimensional transforms. This +is a library that is written in FORTRAN 77, a language that is now +considered as a bit antiquated by many. FFTPACKv4 was written in 1985, +by Dr Paul Swarztrauber of NCAR, more than 25 years ago ! And despite +its age, benchmarks show it that it still a very good performing FFT +library, see for example the 1d single precision benchmarks here: +http://www.fftw.org/speed/opteron-2.2GHz-32bit/ . It is however not +competitive with the fastest ones, such as FFTW, Intel MKL, AMD ACML, +Apple vDSP. The reason for that is that those libraries do take +advantage of the SSE SIMD instructions available on Intel CPUs, +available since the days of the Pentium III. These instructions deal +with small vectors of 4 floats at a time, instead of a single float +for a traditionnal FPU, so when using these instructions one may expect +a 4-fold performance improvement. + +The idea was to take this fortran fftpack v4 code, translate to C, +modify it to deal with those SSE instructions, and check that the +final performance is not completely ridiculous when compared to other +SIMD FFT libraries. Translation to C was performed with f2c ( +http://www.netlib.org/f2c/ ). The resulting file was a bit edited in +order to remove the thousands of gotos that were introduced by +f2c. You will find the fftpack.h and fftpack.c sources in the +repository, this a complete translation of +http://www.netlib.org/fftpack/ , with the discrete cosine transform +and the test program. There is no license information in the netlib +repository, but it was confirmed to me by the fftpack v5 curators that +the same terms do apply to fftpack v4: +http://www.cisl.ucar.edu/css/software/fftpack5/ftpk.html . This is a +"BSD-like" license, it is compatible with proprietary projects. + +Adapting fftpack to deal with the SIMD 4-element vectors instead of +scalar single precision numbers was more complex than I originally +thought, especially with the real transforms, and I ended up writing +more code than I planned.. + + +The code: +-- + +Only two files, in good old C, pffft.c and pffft.h . The API is very +very simple, just make sure that you read the comments in pffft.h. + + +Comparison with other FFTs: +-- + +The idea was not to break speed records, but to get a decently fast +fft that is at least 50% as fast as the fastest FFT -- especially on +slowest computers . I'm more focused on getting the best performance +on slow cpus (Atom, Intel Core 1, old Athlons, ARM Cortex-A9...), than +on getting top performance on today fastest cpus. + +It can be used in a real-time context as the fft functions do not +perform any memory allocation -- that is why they accept a 'work' +array in their arguments. + +It is also a bit focused on performing 1D convolutions, that is why it +provides "unordered" FFTs , and a fourier domain convolution +operation. + + +Benchmark results (cpu tested: core i7 2600, core 2 quad, core 1 duo, atom N270, cortex-A9, cortex-A15, A8X) +-- + +The benchmark shows the performance of various fft implementations measured in +MFlops, with the number of floating point operations being defined as 5Nlog2(N) +for a length N complex fft, and 2.5*Nlog2(N) for a real fft. +See http://www.fftw.org/speed/method.html for an explanation of these formulas. + +MacOS Lion, gcc 4.2, 64-bit, fftw 3.3 on a 3.4 GHz core i7 2600 + +Built with: + + gcc-4.2 -o test_pffft -arch x86_64 -O3 -Wall -W pffft.c test_pffft.c fftpack.c -L/usr/local/lib -I/usr/local/include/ -DHAVE_VECLIB -framework veclib -DHAVE_FFTW -lfftw3f + +| input len |real FFTPack| real vDSP | real FFTW | real PFFFT | |cplx FFTPack| cplx vDSP | cplx FFTW | cplx PFFFT | +|-----------+------------+------------+------------+------------| |------------+------------+------------+------------| +| 64 | 2816 | 8596 | 7329 | 8187 | | 2887 | 14898 | 14668 | 11108 | +| 96 | 3298 | n/a | 8378 | 7727 | | 3953 | n/a | 15680 | 10878 | +| 128 | 3507 | 11575 | 9266 | 10108 | | 4233 | 17598 | 16427 | 12000 | +| 160 | 3391 | n/a | 9838 | 10711 | | 4220 | n/a | 16653 | 11187 | +| 192 | 3919 | n/a | 9868 | 10956 | | 4297 | n/a | 15770 | 12540 | +| 256 | 4283 | 13179 | 10694 | 13128 | | 4545 | 19550 | 16350 | 13822 | +| 384 | 3136 | n/a | 10810 | 12061 | | 3600 | n/a | 16103 | 13240 | +| 480 | 3477 | n/a | 10632 | 12074 | | 3536 | n/a | 11630 | 12522 | +| 512 | 3783 | 15141 | 11267 | 13838 | | 3649 | 20002 | 16560 | 13580 | +| 640 | 3639 | n/a | 11164 | 13946 | | 3695 | n/a | 15416 | 13890 | +| 768 | 3800 | n/a | 11245 | 13495 | | 3590 | n/a | 15802 | 14552 | +| 800 | 3440 | n/a | 10499 | 13301 | | 3659 | n/a | 12056 | 13268 | +| 1024 | 3924 | 15605 | 11450 | 15339 | | 3769 | 20963 | 13941 | 15467 | +| 2048 | 4518 | 16195 | 11551 | 15532 | | 4258 | 20413 | 13723 | 15042 | +| 2400 | 4294 | n/a | 10685 | 13078 | | 4093 | n/a | 12777 | 13119 | +| 4096 | 4750 | 16596 | 11672 | 15817 | | 4157 | 19662 | 14316 | 14336 | +| 8192 | 3820 | 16227 | 11084 | 12555 | | 3691 | 18132 | 12102 | 13813 | +| 9216 | 3864 | n/a | 10254 | 12870 | | 3586 | n/a | 12119 | 13994 | +| 16384 | 3822 | 15123 | 10454 | 12822 | | 3613 | 16874 | 12370 | 13881 | +| 32768 | 4175 | 14512 | 10662 | 11095 | | 3881 | 14702 | 11619 | 11524 | +| 262144 | 3317 | 11429 | 6269 | 9517 | | 2810 | 11729 | 7757 | 10179 | +| 1048576 | 2913 | 10551 | 4730 | 5867 | | 2661 | 7881 | 3520 | 5350 | +|-----------+------------+------------+------------+------------| |------------+------------+------------+------------| + + +Debian 6, gcc 4.4.5, 64-bit, fftw 3.3.1 on a 3.4 GHz core i7 2600 + +Built with: +gcc -o test_pffft -DHAVE_FFTW -msse2 -O3 -Wall -W pffft.c test_pffft.c fftpack.c -L$HOME/local/lib -I$HOME/local/include/ -lfftw3f -lm + +| N (input length) | real FFTPack | real FFTW | real PFFFT | | cplx FFTPack | cplx FFTW | cplx PFFFT | +|------------------+--------------+--------------+--------------| |--------------+--------------+--------------| +| 64 | 3840 | 7680 | 8777 | | 4389 | 20480 | 11171 | +| 96 | 4214 | 9633 | 8429 | | 4816 | 22477 | 11238 | +| 128 | 3584 | 10240 | 10240 | | 5120 | 23893 | 11947 | +| 192 | 4854 | 11095 | 12945 | | 4854 | 22191 | 14121 | +| 256 | 4096 | 11703 | 16384 | | 5120 | 23406 | 13653 | +| 384 | 4395 | 14651 | 12558 | | 4884 | 19535 | 14651 | +| 512 | 5760 | 13166 | 15360 | | 4608 | 23040 | 15360 | +| 768 | 4907 | 14020 | 16357 | | 4461 | 19628 | 14020 | +| 1024 | 5120 | 14629 | 14629 | | 5120 | 20480 | 15754 | +| 2048 | 5632 | 14080 | 18773 | | 4693 | 12516 | 16091 | +| 4096 | 5120 | 13653 | 17554 | | 4726 | 7680 | 14456 | +| 8192 | 4160 | 7396 | 13312 | | 4437 | 14791 | 13312 | +| 9216 | 4210 | 6124 | 13473 | | 4491 | 7282 | 14970 | +| 16384 | 3976 | 11010 | 14313 | | 4210 | 11450 | 13631 | +| 32768 | 4260 | 10224 | 10954 | | 4260 | 6816 | 11797 | +| 262144 | 3736 | 6896 | 9961 | | 2359 | 8965 | 9437 | +| 1048576 | 2796 | 4534 | 6453 | | 1864 | 3078 | 5592 | +|------------------+--------------+--------------+--------------| |--------------+--------------+--------------| + + + +MacOS Snow Leopard, gcc 4.0, 32-bit, fftw 3.3 on a 1.83 GHz core 1 duo + +Built with: + + gcc -o test_pffft -DHAVE_FFTW -DHAVE_VECLIB -O3 -Wall -W pffft.c test_pffft.c fftpack.c -L/usr/local/lib -I/usr/local/include/ -lfftw3f -framework veclib + +| input len |real FFTPack| real vDSP | real FFTW | real PFFFT | |cplx FFTPack| cplx vDSP | cplx FFTW | cplx PFFFT | +|-----------+------------+------------+------------+------------| |------------+------------+------------+------------| +| 64 | 745 | 2145 | 1706 | 2028 | | 961 | 3356 | 3313 | 2300 | +| 96 | 877 | n/a | 1976 | 1978 | | 1059 | n/a | 3333 | 2233 | +| 128 | 951 | 2808 | 2213 | 2279 | | 1202 | 3803 | 3739 | 2494 | +| 192 | 1002 | n/a | 2456 | 2429 | | 1186 | n/a | 3701 | 2508 | +| 256 | 1065 | 3205 | 2641 | 2793 | | 1302 | 4013 | 3912 | 2663 | +| 384 | 845 | n/a | 2759 | 2499 | | 948 | n/a | 3729 | 2504 | +| 512 | 900 | 3476 | 2956 | 2759 | | 974 | 4057 | 3954 | 2645 | +| 768 | 910 | n/a | 2912 | 2737 | | 975 | n/a | 3837 | 2614 | +| 1024 | 936 | 3583 | 3107 | 3009 | | 1006 | 4124 | 3821 | 2697 | +| 2048 | 1057 | 3585 | 3091 | 2837 | | 1089 | 3889 | 3701 | 2513 | +| 4096 | 1083 | 3524 | 3092 | 2733 | | 1039 | 3617 | 3462 | 2364 | +| 8192 | 874 | 3252 | 2967 | 2363 | | 911 | 3106 | 2789 | 2302 | +| 9216 | 898 | n/a | 2420 | 2290 | | 865 | n/a | 2676 | 2204 | +| 16384 | 903 | 2892 | 2506 | 2421 | | 899 | 3026 | 2797 | 2289 | +| 32768 | 965 | 2837 | 2550 | 2358 | | 920 | 2922 | 2763 | 2240 | +| 262144 | 738 | 2422 | 1589 | 1708 | | 610 | 2038 | 1436 | 1091 | +| 1048576 | 528 | 1207 | 845 | 880 | | 606 | 1020 | 669 | 1036 | +|-----------+------------+------------+------------+------------| |------------+------------+------------+------------| + + + +Ubuntu 11.04, gcc 4.5, 32-bit, fftw 3.2 on a 2.66 core 2 quad + +Built with: +gcc -o test_pffft -DHAVE_FFTW -msse -mfpmath=sse -O3 -Wall -W pffft.c test_pffft.c fftpack.c -L/usr/local/lib -I/usr/local/include/ -lfftw3f -lm + +| input len |real FFTPack| real FFTW | real PFFFT | |cplx FFTPack| cplx FFTW | cplx PFFFT | +|-----------+------------+------------+------------| |------------+------------+------------| +| 64 | 1920 | 3614 | 5120 | | 2194 | 7680 | 6467 | +| 96 | 1873 | 3549 | 5187 | | 2107 | 8429 | 5863 | +| 128 | 2240 | 3773 | 5514 | | 2560 | 7964 | 6827 | +| 192 | 1765 | 4569 | 7767 | | 2284 | 9137 | 7061 | +| 256 | 2048 | 5461 | 7447 | | 2731 | 9638 | 7802 | +| 384 | 1998 | 5861 | 6762 | | 2313 | 9253 | 7644 | +| 512 | 2095 | 6144 | 7680 | | 2194 | 10240 | 7089 | +| 768 | 2230 | 5773 | 7549 | | 2045 | 10331 | 7010 | +| 1024 | 2133 | 6400 | 8533 | | 2133 | 10779 | 7877 | +| 2048 | 2011 | 7040 | 8665 | | 1942 | 10240 | 7768 | +| 4096 | 2194 | 6827 | 8777 | | 1755 | 9452 | 6827 | +| 8192 | 1849 | 6656 | 6656 | | 1752 | 7831 | 6827 | +| 9216 | 1871 | 5858 | 6416 | | 1643 | 6909 | 6266 | +| 16384 | 1883 | 6223 | 6506 | | 1664 | 7340 | 6982 | +| 32768 | 1826 | 6390 | 6667 | | 1631 | 7481 | 6971 | +| 262144 | 1546 | 4075 | 5977 | | 1299 | 3415 | 3551 | +| 1048576 | 1104 | 2071 | 1730 | | 1104 | 1149 | 1834 | +|-----------+------------+------------+------------| |------------+------------+------------| + + + +Ubuntu 11.04, gcc 4.5, 32-bit, fftw 3.3 on a 1.6 GHz Atom N270 + +Built with: +gcc -o test_pffft -DHAVE_FFTW -msse -mfpmath=sse -O3 -Wall -W pffft.c test_pffft.c fftpack.c -L/usr/local/lib -I/usr/local/include/ -lfftw3f -lm + +| N (input length) | real FFTPack | real FFTW | real PFFFT | | cplx FFTPack | cplx FFTW | cplx PFFFT | +|------------------+--------------+--------------+--------------| |--------------+--------------+--------------| +| 64 | 452 | 1041 | 1336 | | 549 | 2318 | 1781 | +| 96 | 444 | 1297 | 1297 | | 503 | 2408 | 1686 | +| 128 | 527 | 1525 | 1707 | | 543 | 2655 | 1886 | +| 192 | 498 | 1653 | 1849 | | 539 | 2678 | 1942 | +| 256 | 585 | 1862 | 2156 | | 594 | 2777 | 2244 | +| 384 | 499 | 1870 | 1998 | | 511 | 2586 | 1890 | +| 512 | 562 | 2095 | 2194 | | 542 | 2973 | 2194 | +| 768 | 545 | 2045 | 2133 | | 545 | 2365 | 2133 | +| 1024 | 595 | 2133 | 2438 | | 569 | 2695 | 2179 | +| 2048 | 587 | 2125 | 2347 | | 521 | 2230 | 1707 | +| 4096 | 495 | 1890 | 1834 | | 492 | 1876 | 1672 | +| 8192 | 469 | 1548 | 1729 | | 438 | 1740 | 1664 | +| 9216 | 468 | 1663 | 1663 | | 446 | 1585 | 1531 | +| 16384 | 453 | 1608 | 1767 | | 398 | 1476 | 1664 | +| 32768 | 456 | 1420 | 1503 | | 387 | 1388 | 1345 | +| 262144 | 309 | 385 | 726 | | 262 | 415 | 840 | +| 1048576 | 280 | 351 | 739 | | 261 | 313 | 797 | +|------------------+--------------+--------------+--------------| |--------------+--------------+--------------| + + + +Windows 7, visual c++ 2010 on a 1.6 GHz Atom N270 + +Built with: +cl /Ox -D_USE_MATH_DEFINES /arch:SSE test_pffft.c pffft.c fftpack.c + +(visual c++ is definitively not very good with SSE intrinsics...) + +| N (input length) | real FFTPack | real PFFFT | | cplx FFTPack | cplx PFFFT | +|------------------+--------------+--------------| |--------------+--------------| +| 64 | 173 | 1009 | | 174 | 1159 | +| 96 | 169 | 1029 | | 188 | 1201 | +| 128 | 195 | 1242 | | 191 | 1275 | +| 192 | 178 | 1312 | | 184 | 1276 | +| 256 | 196 | 1591 | | 186 | 1281 | +| 384 | 172 | 1409 | | 181 | 1281 | +| 512 | 187 | 1640 | | 181 | 1313 | +| 768 | 171 | 1614 | | 176 | 1258 | +| 1024 | 186 | 1812 | | 178 | 1223 | +| 2048 | 190 | 1707 | | 186 | 1099 | +| 4096 | 182 | 1446 | | 177 | 975 | +| 8192 | 175 | 1345 | | 169 | 1034 | +| 9216 | 165 | 1271 | | 168 | 1023 | +| 16384 | 166 | 1396 | | 165 | 949 | +| 32768 | 172 | 1311 | | 161 | 881 | +| 262144 | 136 | 632 | | 134 | 629 | +| 1048576 | 134 | 698 | | 127 | 623 | +|------------------+--------------+--------------| |--------------+--------------| + + + +Ubuntu 12.04, gcc-4.7.3, 32-bit, with fftw 3.3.3 (built with --enable-neon), on a 1.2GHz ARM Cortex A9 (Tegra 3) + +Built with: +gcc-4.7 -O3 -DHAVE_FFTW -march=armv7-a -mtune=cortex-a9 -mfloat-abi=hard -mfpu=neon -ffast-math test_pffft.c pffft.c -o test_pffft_arm fftpack.c -lm -I/usr/local/include/ -L/usr/local/lib/ -lfftw3f + +| input len |real FFTPack| real FFTW | real PFFFT | |cplx FFTPack| cplx FFTW | cplx PFFFT | +|-----------+------------+------------+------------| |------------+------------+------------| +| 64 | 549 | 452 | 731 | | 512 | 602 | 640 | +| 96 | 421 | 272 | 702 | | 496 | 571 | 602 | +| 128 | 498 | 512 | 815 | | 597 | 618 | 652 | +| 160 | 521 | 536 | 815 | | 586 | 669 | 625 | +| 192 | 539 | 571 | 883 | | 485 | 597 | 626 | +| 256 | 640 | 539 | 975 | | 569 | 611 | 671 | +| 384 | 499 | 610 | 879 | | 499 | 602 | 637 | +| 480 | 518 | 507 | 877 | | 496 | 661 | 616 | +| 512 | 524 | 591 | 1002 | | 549 | 678 | 668 | +| 640 | 542 | 612 | 955 | | 568 | 663 | 645 | +| 768 | 557 | 613 | 981 | | 491 | 663 | 598 | +| 800 | 514 | 353 | 882 | | 514 | 360 | 574 | +| 1024 | 640 | 640 | 1067 | | 492 | 683 | 602 | +| 2048 | 587 | 640 | 908 | | 486 | 640 | 552 | +| 2400 | 479 | 368 | 777 | | 422 | 376 | 518 | +| 4096 | 511 | 614 | 853 | | 426 | 640 | 534 | +| 8192 | 415 | 584 | 708 | | 386 | 622 | 516 | +| 9216 | 419 | 571 | 687 | | 364 | 586 | 506 | +| 16384 | 426 | 577 | 716 | | 398 | 606 | 530 | +| 32768 | 417 | 572 | 673 | | 399 | 572 | 468 | +| 262144 | 219 | 380 | 293 | | 255 | 431 | 343 | +| 1048576 | 202 | 274 | 237 | | 265 | 282 | 355 | +|-----------+------------+------------+------------| |------------+------------+------------| + +Same platform as above, but this time pffft and fftpack are built with clang 3.2: + +clang -O3 -DHAVE_FFTW -march=armv7-a -mtune=cortex-a9 -mfloat-abi=hard -mfpu=neon -ffast-math test_pffft.c pffft.c -o test_pffft_arm fftpack.c -lm -I/usr/local/include/ -L/usr/local/lib/ -lfftw3f + +| input len |real FFTPack| real FFTW | real PFFFT | |cplx FFTPack| cplx FFTW | cplx PFFFT | +|-----------+------------+------------+------------| |------------+------------+------------| +| 64 | 427 | 452 | 853 | | 427 | 602 | 1024 | +| 96 | 351 | 276 | 843 | | 337 | 571 | 963 | +| 128 | 373 | 512 | 996 | | 390 | 618 | 1054 | +| 160 | 426 | 536 | 987 | | 375 | 669 | 914 | +| 192 | 404 | 571 | 1079 | | 388 | 588 | 1079 | +| 256 | 465 | 539 | 1205 | | 445 | 602 | 1170 | +| 384 | 366 | 610 | 1099 | | 343 | 594 | 1099 | +| 480 | 356 | 507 | 1140 | | 335 | 651 | 931 | +| 512 | 411 | 591 | 1213 | | 384 | 649 | 1124 | +| 640 | 398 | 612 | 1193 | | 373 | 654 | 901 | +| 768 | 409 | 613 | 1227 | | 383 | 663 | 1044 | +| 800 | 411 | 348 | 1073 | | 353 | 358 | 809 | +| 1024 | 427 | 640 | 1280 | | 413 | 692 | 1004 | +| 2048 | 414 | 626 | 1126 | | 371 | 640 | 853 | +| 2400 | 399 | 373 | 898 | | 319 | 368 | 653 | +| 4096 | 404 | 602 | 1059 | | 357 | 633 | 778 | +| 8192 | 332 | 584 | 792 | | 308 | 616 | 716 | +| 9216 | 322 | 561 | 783 | | 299 | 586 | 687 | +| 16384 | 344 | 568 | 778 | | 314 | 617 | 745 | +| 32768 | 342 | 564 | 737 | | 314 | 552 | 629 | +| 262144 | 201 | 383 | 313 | | 227 | 435 | 413 | +| 1048576 | 187 | 262 | 251 | | 228 | 281 | 409 | +|-----------+------------+------------+------------| |------------+------------+------------| + +So it looks like, on ARM, gcc 4.7 is the best at scalar floating point +(the fftpack performance numbers are better with gcc), while clang is +the best with neon intrinsics (see how pffft perf has improved with +clang 3.2). + + +NVIDIA Jetson TK1 board, gcc-4.8.2. The cpu is a 2.3GHz cortex A15 (Tegra K1). + +Built with: +gcc -O3 -march=armv7-a -mtune=native -mfloat-abi=hard -mfpu=neon -ffast-math test_pffft.c pffft.c -o test_pffft_arm fftpack.c -lm + +| input len |real FFTPack| real PFFFT | |cplx FFTPack| cplx PFFFT | +|-----------+------------+------------| |------------+------------| +| 64 | 1735 | 3308 | | 1994 | 3744 | +| 96 | 1596 | 3448 | | 1987 | 3572 | +| 128 | 1807 | 4076 | | 2255 | 3960 | +| 160 | 1769 | 4083 | | 2071 | 3845 | +| 192 | 1990 | 4233 | | 2017 | 3939 | +| 256 | 2191 | 4882 | | 2254 | 4346 | +| 384 | 1878 | 4492 | | 2073 | 4012 | +| 480 | 1748 | 4398 | | 1923 | 3951 | +| 512 | 2030 | 5064 | | 2267 | 4195 | +| 640 | 1918 | 4756 | | 2094 | 4184 | +| 768 | 2099 | 4907 | | 2048 | 4297 | +| 800 | 1822 | 4555 | | 1880 | 4063 | +| 1024 | 2232 | 5355 | | 2187 | 4420 | +| 2048 | 2176 | 4983 | | 2027 | 3602 | +| 2400 | 1741 | 4256 | | 1710 | 3344 | +| 4096 | 1816 | 3914 | | 1851 | 3349 | +| 8192 | 1716 | 3481 | | 1700 | 3255 | +| 9216 | 1735 | 3589 | | 1653 | 3094 | +| 16384 | 1567 | 3483 | | 1637 | 3244 | +| 32768 | 1624 | 3240 | | 1655 | 3156 | +| 262144 | 1012 | 1898 | | 983 | 1503 | +| 1048576 | 876 | 1154 | | 868 | 1341 | +|-----------+------------+------------| |------------+------------| + +The performance on the tegra K1 is pretty impressive. I'm not +including the FFTW numbers as they as slightly below the scalar +fftpack numbers, so something must be wrong (however it seems to be +correctly configured and is using neon simd instructions). + +When using clang 3.4 the pffft version is even a bit faster, reaching +5.7 GFlops for real ffts of size 1024. + + +iPad Air 2 with iOS9, xcode 8.0, arm64. The cpu is an Apple A8X, supposedly running at 1.5GHz. + +| input len |real FFTPack| real vDSP | real PFFFT | |cplx FFTPack| cplx vDSP | cplx PFFFT | +|-----------+------------+------------+------------| |------------+------------+------------| +| 64 | 2517 | 7995 | 6086 | | 2725 | 13006 | 8495 | +| 96 | 2442 | n/a | 6691 | | 2256 | n/a | 7991 | +| 128 | 2664 | 10186 | 7877 | | 2575 | 15115 | 9115 | +| 160 | 2638 | n/a | 8283 | | 2682 | n/a | 8806 | +| 192 | 2903 | n/a | 9083 | | 2634 | n/a | 8980 | +| 256 | 3184 | 11452 | 10039 | | 3026 | 15410 | 10199 | +| 384 | 2665 | n/a | 10100 | | 2275 | n/a | 9247 | +| 480 | 2546 | n/a | 9863 | | 2341 | n/a | 8892 | +| 512 | 2832 | 12197 | 10989 | | 2547 | 16768 | 10154 | +| 640 | 2755 | n/a | 10461 | | 2569 | n/a | 9666 | +| 768 | 2998 | n/a | 11355 | | 2585 | n/a | 9813 | +| 800 | 2516 | n/a | 10332 | | 2433 | n/a | 9164 | +| 1024 | 3109 | 12965 | 12114 | | 2869 | 16448 | 10519 | +| 2048 | 3027 | 12996 | 12023 | | 2648 | 17304 | 10307 | +| 2400 | 2515 | n/a | 10372 | | 2355 | n/a | 8443 | +| 4096 | 3204 | 13603 | 12359 | | 2814 | 16570 | 9780 | +| 8192 | 2759 | 13422 | 10824 | | 2153 | 15652 | 7884 | +| 9216 | 2700 | n/a | 9938 | | 2241 | n/a | 7900 | +| 16384 | 2280 | 13057 | 7976 | | 593 | 4272 | 2534 | +| 32768 | 768 | 4269 | 2882 | | 606 | 4405 | 2604 | +| 262144 | 724 | 3527 | 2630 | | 534 | 2418 | 2157 | +| 1048576 | 674 | 1467 | 2135 | | 530 | 1621 | 2055 | +|-----------+------------+------------+------------| |------------+------------+------------| + +I double-checked to make sure I did not make a mistake in the time +measurements, as the numbers are much higher than what I initially +expected. They are in fact higher than the number I get on the 2.8GHz +Xeon of my 2008 mac pro.. (except for FFT lengths >= 32768 where +having a big cache is useful). A good surprise is also that the perf +is not too far from apple's vDSP (at least for the real FFT). +