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Another SSE Intrinsics Approach

In this section we follow the idea of BLIS micro kernel for i86-64 architectures with SSE. Compared to the original BLIS micro kernel we simplify quite a few things:

Select the demo-naive-sse-with-intrinsics-unrolled Branch

Again, we do a make clean before switching a branch:

$shell> cd ulmBLAS                                                       
$shell> make clean                                                       
for dir in src refblas test bench; do make -C $dir clean; done
rm -f  auxiliary/xerbla.o  level1/dasum.o level1/daxpy.o level1/dcopy.o level1/ddot.o level1/dnrm2.o level1/drot.o level1/drotg.o level1/drotm.o level1/drotmg.o level1/dscal.o level1/dswap.o level1/idamax.o  level3/dgemm.o level3/dgemm_nn.o level3/dsymm.o level3/stubs.o
rm -f  auxiliary/atl_xerbla.o  level1/atl_dasum.o level1/atl_daxpy.o level1/atl_dcopy.o level1/atl_ddot.o level1/atl_dnrm2.o level1/atl_drot.o level1/atl_drotg.o level1/atl_drotm.o level1/atl_drotmg.o level1/atl_dscal.o level1/atl_dswap.o level1/atl_idamax.o  level3/atl_dgemm.o level3/atl_dgemm_nn.o level3/atl_dsymm.o level3/atl_stubs.o
rm -f ../libulmblas.a
rm -f ../libatlulmblas.a
rm -f caxpy.o ccopy.o cdotc.o cdotu.o cgbmv.o cgemm.o cgemv.o cgerc.o cgeru.o chbmv.o chemm.o chemv.o cher.o cher2.o cher2k.o cherk.o chpmv.o chpr.o chpr2.o crotg.o cscal.o csrot.o csscal.o cswap.o csymm.o csyr2k.o csyrk.o ctbmv.o ctbsv.o ctpmv.o ctpsv.o ctrmm.o ctrmv.o ctrsm.o ctrsv.o dasum.o daxpy.o dcabs1.o dcopy.o ddot.o dgbmv.o dgemm.o dgemv.o dger.o dnrm2.o drot.o drotg.o drotm.o drotmg.o dsbmv.o dscal.o dsdot.o dspmv.o dspr.o dspr2.o dswap.o dsymm.o dsymv.o dsyr.o dsyr2.o dsyr2k.o dsyrk.o dtbmv.o dtbsv.o dtpmv.o dtpsv.o dtrmm.o dtrmv.o dtrsm.o dtrsv.o dzasum.o dznrm2.o icamax.o idamax.o isamax.o izamax.o lsame.o sasum.o saxpy.o scabs1.o scasum.o scnrm2.o scopy.o sdot.o sdsdot.o sgbmv.o sgemm.o sgemv.o sger.o snrm2.o srot.o srotg.o srotm.o srotmg.o ssbmv.o sscal.o sspmv.o sspr.o sspr2.o sswap.o ssymm.o ssymv.o ssyr.o ssyr2.o ssyr2k.o ssyrk.o stbmv.o stbsv.o stpmv.o stpsv.o strmm.o strmv.o strsm.o strsv.o xerbla.o xerbla_array.o zaxpy.o zcopy.o zdotc.o zdotu.o zdrot.o zdscal.o zgbmv.o zgemm.o zgemv.o zgerc.o zgeru.o zhbmv.o zhemm.o zhemv.o zher.o zher2.o zher2k.o zherk.o zhpmv.o zhpr.o zhpr2.o zrotg.o zscal.o zswap.o zsymm.o zsyr2k.o zsyrk.o ztbmv.o ztbsv.o ztpmv.o ztpsv.o ztrmm.o ztrmv.o ztrsm.o ztrsv.o
rm -f ../librefblas.a
rm -f  dblat1_ref  dblat3_ref  dblat1_ulm  dblat3_ulm *.SUMM
rm -f xdl1blastst libtstatlas.a l1blastst.o  ATL_cputime.o  ATL_epsilon.o  ATL_f77amax.o  ATL_f77asum.o  ATL_f77axpy.o  ATL_f77copy.o  ATL_f77dot.o  ATL_f77gemm.o  ATL_f77nrm2.o  ATL_f77rot.o  ATL_f77rotg.o  ATL_f77rotm.o  ATL_f77rotmg.o  ATL_f77scal.o  ATL_f77swap.o  ATL_f77symm.o  ATL_f77syr2k.o  ATL_f77syrk.o  ATL_f77trmm.o  ATL_f77trsm.o  ATL_flushcache.o  ATL_gediffnrm1.o  ATL_gegen.o  ATL_genrm1.o  ATL_infnrm.o  ATL_rand.o  ATL_set.o  ATL_synrm.o  ATL_trnrm1.o  ATL_vdiff.o  ATL_zero.o  ATL_df77wrap.o

Then we are checking out the demo-naive-sse-with-intrinsics branch:

$shell> git branch -a                                                    
  demo-naive-sse-with-intrinsics
* demo-naive-sse-with-intrinsics-unrolled
  demo-pure-c
  master
  remotes/origin/HEAD -> origin/master
  remotes/origin/bench-atlas
  remotes/origin/bench-blis
  remotes/origin/bench-eigen
  remotes/origin/bench-mkl
  remotes/origin/blis-avx-microkernel
  remotes/origin/demo-naive-avx-with-intrinsics
  remotes/origin/demo-naive-sse-with-intrinsics
  remotes/origin/demo-naive-sse-with-intrinsics-unrolled
  remotes/origin/demo-pure-c
  remotes/origin/demo-sse-all-asm
  remotes/origin/demo-sse-all-asm-try-prefetching
  remotes/origin/demo-sse-all-asm-try-prefetching-v2
  remotes/origin/demo-sse-all-asm-with-prefetching
  remotes/origin/demo-sse-asm
  remotes/origin/demo-sse-asm-for-AB-loop
  remotes/origin/demo-sse-asm-unrolled
  remotes/origin/demo-sse-asm-unrolled-v2
  remotes/origin/demo-sse-asm-unrolled-v3
  remotes/origin/demo-sse-asm-unrolled-with-prefetch
  remotes/origin/demo-sse-intrinsics
  remotes/origin/demo-sse-intrinsics-for-AB-loop
  remotes/origin/demo-sse-intrinsics-v2
  remotes/origin/demo-sse-intrinsics-v3
  remotes/origin/demo-with-sse-intrinsics
  remotes/origin/master
  remotes/origin/trsm-assignment
  remotes/origin/trsm-pure-c
$shell> git checkout -B demo-sse-intrinsics remotes/origin/demo-sse-intrinsics               
Switched to a new branch 'demo-sse-intrinsics'
Branch demo-sse-intrinsics set up to track remote branch demo-sse-intrinsics from origin.

Then we compile the project

$shell> make                                                             
make -C src
clang -Wall -I. -O2  -mfpmath=sse -fomit-frame-pointer -DULM_BLOCKED -c -o auxiliary/xerbla.o auxiliary/xerbla.c
clang -Wall -I. -O2  -mfpmath=sse -fomit-frame-pointer -DULM_BLOCKED -c -o level1/dasum.o level1/dasum.c
clang -Wall -I. -O2  -mfpmath=sse -fomit-frame-pointer -DULM_BLOCKED -c -o level1/daxpy.o level1/daxpy.c
clang -Wall -I. -O2  -mfpmath=sse -fomit-frame-pointer -DULM_BLOCKED -c -o level1/dcopy.o level1/dcopy.c
clang -Wall -I. -O2  -mfpmath=sse -fomit-frame-pointer -DULM_BLOCKED -c -o level1/ddot.o level1/ddot.c
clang -Wall -I. -O2  -mfpmath=sse -fomit-frame-pointer -DULM_BLOCKED -c -o level1/dnrm2.o level1/dnrm2.c
clang -Wall -I. -O2  -mfpmath=sse -fomit-frame-pointer -DULM_BLOCKED -c -o level1/drot.o level1/drot.c
clang -Wall -I. -O2  -mfpmath=sse -fomit-frame-pointer -DULM_BLOCKED -c -o level1/drotg.o level1/drotg.c
clang -Wall -I. -O2  -mfpmath=sse -fomit-frame-pointer -DULM_BLOCKED -c -o level1/drotm.o level1/drotm.c
clang -Wall -I. -O2  -mfpmath=sse -fomit-frame-pointer -DULM_BLOCKED -c -o level1/drotmg.o level1/drotmg.c
clang -Wall -I. -O2  -mfpmath=sse -fomit-frame-pointer -DULM_BLOCKED -c -o level1/dscal.o level1/dscal.c
clang -Wall -I. -O2  -mfpmath=sse -fomit-frame-pointer -DULM_BLOCKED -c -o level1/dswap.o level1/dswap.c
clang -Wall -I. -O2  -mfpmath=sse -fomit-frame-pointer -DULM_BLOCKED -c -o level1/idamax.o level1/idamax.c
clang -Wall -I. -O2  -mfpmath=sse -fomit-frame-pointer -DULM_BLOCKED -c -o level3/dgemm.o level3/dgemm.c
clang -Wall -I. -O2  -mfpmath=sse -fomit-frame-pointer -DULM_BLOCKED -c -o level3/dgemm_nn.o level3/dgemm_nn.c
clang -Wall -I. -O2  -mfpmath=sse -fomit-frame-pointer -DULM_BLOCKED -c -o level3/dsymm.o level3/dsymm.c
clang -Wall -I. -O2  -mfpmath=sse -fomit-frame-pointer -DULM_BLOCKED -c -o level3/stubs.o level3/stubs.c
ar cru ../libulmblas.a  auxiliary/xerbla.o  level1/dasum.o level1/daxpy.o level1/dcopy.o level1/ddot.o level1/dnrm2.o level1/drot.o level1/drotg.o level1/drotm.o level1/drotmg.o level1/dscal.o level1/dswap.o level1/idamax.o  level3/dgemm.o level3/dgemm_nn.o level3/dsymm.o level3/stubs.o
ranlib ../libulmblas.a
clang -Wall -I. -O2  -mfpmath=sse -fomit-frame-pointer -DULM_BLOCKED -DFAKE_ATLAS -c -o auxiliary/atl_xerbla.o auxiliary/xerbla.c
clang -Wall -I. -O2  -mfpmath=sse -fomit-frame-pointer -DULM_BLOCKED -DFAKE_ATLAS -c -o level1/atl_dasum.o level1/dasum.c
clang -Wall -I. -O2  -mfpmath=sse -fomit-frame-pointer -DULM_BLOCKED -DFAKE_ATLAS -c -o level1/atl_daxpy.o level1/daxpy.c
clang -Wall -I. -O2  -mfpmath=sse -fomit-frame-pointer -DULM_BLOCKED -DFAKE_ATLAS -c -o level1/atl_dcopy.o level1/dcopy.c
clang -Wall -I. -O2  -mfpmath=sse -fomit-frame-pointer -DULM_BLOCKED -DFAKE_ATLAS -c -o level1/atl_ddot.o level1/ddot.c
clang -Wall -I. -O2  -mfpmath=sse -fomit-frame-pointer -DULM_BLOCKED -DFAKE_ATLAS -c -o level1/atl_dnrm2.o level1/dnrm2.c
clang -Wall -I. -O2  -mfpmath=sse -fomit-frame-pointer -DULM_BLOCKED -DFAKE_ATLAS -c -o level1/atl_drot.o level1/drot.c
clang -Wall -I. -O2  -mfpmath=sse -fomit-frame-pointer -DULM_BLOCKED -DFAKE_ATLAS -c -o level1/atl_drotg.o level1/drotg.c
clang -Wall -I. -O2  -mfpmath=sse -fomit-frame-pointer -DULM_BLOCKED -DFAKE_ATLAS -c -o level1/atl_drotm.o level1/drotm.c
clang -Wall -I. -O2  -mfpmath=sse -fomit-frame-pointer -DULM_BLOCKED -DFAKE_ATLAS -c -o level1/atl_drotmg.o level1/drotmg.c
clang -Wall -I. -O2  -mfpmath=sse -fomit-frame-pointer -DULM_BLOCKED -DFAKE_ATLAS -c -o level1/atl_dscal.o level1/dscal.c
clang -Wall -I. -O2  -mfpmath=sse -fomit-frame-pointer -DULM_BLOCKED -DFAKE_ATLAS -c -o level1/atl_dswap.o level1/dswap.c
clang -Wall -I. -O2  -mfpmath=sse -fomit-frame-pointer -DULM_BLOCKED -DFAKE_ATLAS -c -o level1/atl_idamax.o level1/idamax.c
clang -Wall -I. -O2  -mfpmath=sse -fomit-frame-pointer -DULM_BLOCKED -DFAKE_ATLAS -c -o level3/atl_dgemm.o level3/dgemm.c
clang -Wall -I. -O2  -mfpmath=sse -fomit-frame-pointer -DULM_BLOCKED -DFAKE_ATLAS -c -o level3/atl_dgemm_nn.o level3/dgemm_nn.c
clang -Wall -I. -O2  -mfpmath=sse -fomit-frame-pointer -DULM_BLOCKED -DFAKE_ATLAS -c -o level3/atl_dsymm.o level3/dsymm.c
clang -Wall -I. -O2  -mfpmath=sse -fomit-frame-pointer -DULM_BLOCKED -DFAKE_ATLAS -c -o level3/atl_stubs.o level3/stubs.c
ar cru ../libatlulmblas.a  auxiliary/atl_xerbla.o  level1/atl_dasum.o level1/atl_daxpy.o level1/atl_dcopy.o level1/atl_ddot.o level1/atl_dnrm2.o level1/atl_drot.o level1/atl_drotg.o level1/atl_drotm.o level1/atl_drotmg.o level1/atl_dscal.o level1/atl_dswap.o level1/atl_idamax.o  level3/atl_dgemm.o level3/atl_dgemm_nn.o level3/atl_dsymm.o level3/atl_stubs.o
ranlib ../libatlulmblas.a
make -C refblas
gfortran -fimplicit-none -O3 -c -o caxpy.o caxpy.f
gfortran -fimplicit-none -O3 -c -o ccopy.o ccopy.f
gfortran -fimplicit-none -O3 -c -o cdotc.o cdotc.f
gfortran -fimplicit-none -O3 -c -o cdotu.o cdotu.f
gfortran -fimplicit-none -O3 -c -o cgbmv.o cgbmv.f
gfortran -fimplicit-none -O3 -c -o cgemm.o cgemm.f
gfortran -fimplicit-none -O3 -c -o cgemv.o cgemv.f
gfortran -fimplicit-none -O3 -c -o cgerc.o cgerc.f
gfortran -fimplicit-none -O3 -c -o cgeru.o cgeru.f
gfortran -fimplicit-none -O3 -c -o chbmv.o chbmv.f
gfortran -fimplicit-none -O3 -c -o chemm.o chemm.f
gfortran -fimplicit-none -O3 -c -o chemv.o chemv.f
gfortran -fimplicit-none -O3 -c -o cher.o cher.f
gfortran -fimplicit-none -O3 -c -o cher2.o cher2.f
gfortran -fimplicit-none -O3 -c -o cher2k.o cher2k.f
gfortran -fimplicit-none -O3 -c -o cherk.o cherk.f
gfortran -fimplicit-none -O3 -c -o chpmv.o chpmv.f
gfortran -fimplicit-none -O3 -c -o chpr.o chpr.f
gfortran -fimplicit-none -O3 -c -o chpr2.o chpr2.f
gfortran -fimplicit-none -O3 -c -o crotg.o crotg.f
gfortran -fimplicit-none -O3 -c -o cscal.o cscal.f
gfortran -fimplicit-none -O3 -c -o csrot.o csrot.f
gfortran -fimplicit-none -O3 -c -o csscal.o csscal.f
gfortran -fimplicit-none -O3 -c -o cswap.o cswap.f
gfortran -fimplicit-none -O3 -c -o csymm.o csymm.f
gfortran -fimplicit-none -O3 -c -o csyr2k.o csyr2k.f
gfortran -fimplicit-none -O3 -c -o csyrk.o csyrk.f
gfortran -fimplicit-none -O3 -c -o ctbmv.o ctbmv.f
gfortran -fimplicit-none -O3 -c -o ctbsv.o ctbsv.f
gfortran -fimplicit-none -O3 -c -o ctpmv.o ctpmv.f
gfortran -fimplicit-none -O3 -c -o ctpsv.o ctpsv.f
gfortran -fimplicit-none -O3 -c -o ctrmm.o ctrmm.f
gfortran -fimplicit-none -O3 -c -o ctrmv.o ctrmv.f
gfortran -fimplicit-none -O3 -c -o ctrsm.o ctrsm.f
gfortran -fimplicit-none -O3 -c -o ctrsv.o ctrsv.f
gfortran -fimplicit-none -O3 -c -o dasum.o dasum.f
gfortran -fimplicit-none -O3 -c -o daxpy.o daxpy.f
gfortran -fimplicit-none -O3 -c -o dcabs1.o dcabs1.f
gfortran -fimplicit-none -O3 -c -o dcopy.o dcopy.f
gfortran -fimplicit-none -O3 -c -o ddot.o ddot.f
gfortran -fimplicit-none -O3 -c -o dgbmv.o dgbmv.f
gfortran -fimplicit-none -O3 -c -o dgemm.o dgemm.f
gfortran -fimplicit-none -O3 -c -o dgemv.o dgemv.f
gfortran -fimplicit-none -O3 -c -o dger.o dger.f
gfortran -fimplicit-none -O3 -c -o dnrm2.o dnrm2.f
gfortran -fimplicit-none -O3 -c -o drot.o drot.f
gfortran -fimplicit-none -O3 -c -o drotg.o drotg.f
gfortran -fimplicit-none -O3 -c -o drotm.o drotm.f
gfortran -fimplicit-none -O3 -c -o drotmg.o drotmg.f
gfortran -fimplicit-none -O3 -c -o dsbmv.o dsbmv.f
gfortran -fimplicit-none -O3 -c -o dscal.o dscal.f
gfortran -fimplicit-none -O3 -c -o dsdot.o dsdot.f
gfortran -fimplicit-none -O3 -c -o dspmv.o dspmv.f
gfortran -fimplicit-none -O3 -c -o dspr.o dspr.f
gfortran -fimplicit-none -O3 -c -o dspr2.o dspr2.f
gfortran -fimplicit-none -O3 -c -o dswap.o dswap.f
gfortran -fimplicit-none -O3 -c -o dsymm.o dsymm.f
gfortran -fimplicit-none -O3 -c -o dsymv.o dsymv.f
gfortran -fimplicit-none -O3 -c -o dsyr.o dsyr.f
gfortran -fimplicit-none -O3 -c -o dsyr2.o dsyr2.f
gfortran -fimplicit-none -O3 -c -o dsyr2k.o dsyr2k.f
gfortran -fimplicit-none -O3 -c -o dsyrk.o dsyrk.f
gfortran -fimplicit-none -O3 -c -o dtbmv.o dtbmv.f
gfortran -fimplicit-none -O3 -c -o dtbsv.o dtbsv.f
gfortran -fimplicit-none -O3 -c -o dtpmv.o dtpmv.f
gfortran -fimplicit-none -O3 -c -o dtpsv.o dtpsv.f
gfortran -fimplicit-none -O3 -c -o dtrmm.o dtrmm.f
gfortran -fimplicit-none -O3 -c -o dtrmv.o dtrmv.f
gfortran -fimplicit-none -O3 -c -o dtrsm.o dtrsm.f
gfortran -fimplicit-none -O3 -c -o dtrsv.o dtrsv.f
gfortran -fimplicit-none -O3 -c -o dzasum.o dzasum.f
gfortran -fimplicit-none -O3 -c -o dznrm2.o dznrm2.f
gfortran -fimplicit-none -O3 -c -o icamax.o icamax.f
gfortran -fimplicit-none -O3 -c -o idamax.o idamax.f
gfortran -fimplicit-none -O3 -c -o isamax.o isamax.f
gfortran -fimplicit-none -O3 -c -o izamax.o izamax.f
gfortran -fimplicit-none -O3 -c -o lsame.o lsame.f
gfortran -fimplicit-none -O3 -c -o sasum.o sasum.f
gfortran -fimplicit-none -O3 -c -o saxpy.o saxpy.f
gfortran -fimplicit-none -O3 -c -o scabs1.o scabs1.f
gfortran -fimplicit-none -O3 -c -o scasum.o scasum.f
gfortran -fimplicit-none -O3 -c -o scnrm2.o scnrm2.f
gfortran -fimplicit-none -O3 -c -o scopy.o scopy.f
gfortran -fimplicit-none -O3 -c -o sdot.o sdot.f
gfortran -fimplicit-none -O3 -c -o sdsdot.o sdsdot.f
gfortran -fimplicit-none -O3 -c -o sgbmv.o sgbmv.f
gfortran -fimplicit-none -O3 -c -o sgemm.o sgemm.f
gfortran -fimplicit-none -O3 -c -o sgemv.o sgemv.f
gfortran -fimplicit-none -O3 -c -o sger.o sger.f
gfortran -fimplicit-none -O3 -c -o snrm2.o snrm2.f
gfortran -fimplicit-none -O3 -c -o srot.o srot.f
gfortran -fimplicit-none -O3 -c -o srotg.o srotg.f
gfortran -fimplicit-none -O3 -c -o srotm.o srotm.f
gfortran -fimplicit-none -O3 -c -o srotmg.o srotmg.f
gfortran -fimplicit-none -O3 -c -o ssbmv.o ssbmv.f
gfortran -fimplicit-none -O3 -c -o sscal.o sscal.f
gfortran -fimplicit-none -O3 -c -o sspmv.o sspmv.f
gfortran -fimplicit-none -O3 -c -o sspr.o sspr.f
gfortran -fimplicit-none -O3 -c -o sspr2.o sspr2.f
gfortran -fimplicit-none -O3 -c -o sswap.o sswap.f
gfortran -fimplicit-none -O3 -c -o ssymm.o ssymm.f
gfortran -fimplicit-none -O3 -c -o ssymv.o ssymv.f
gfortran -fimplicit-none -O3 -c -o ssyr.o ssyr.f
gfortran -fimplicit-none -O3 -c -o ssyr2.o ssyr2.f
gfortran -fimplicit-none -O3 -c -o ssyr2k.o ssyr2k.f
gfortran -fimplicit-none -O3 -c -o ssyrk.o ssyrk.f
gfortran -fimplicit-none -O3 -c -o stbmv.o stbmv.f
gfortran -fimplicit-none -O3 -c -o stbsv.o stbsv.f
gfortran -fimplicit-none -O3 -c -o stpmv.o stpmv.f
gfortran -fimplicit-none -O3 -c -o stpsv.o stpsv.f
gfortran -fimplicit-none -O3 -c -o strmm.o strmm.f
gfortran -fimplicit-none -O3 -c -o strmv.o strmv.f
gfortran -fimplicit-none -O3 -c -o strsm.o strsm.f
gfortran -fimplicit-none -O3 -c -o strsv.o strsv.f
gfortran -fimplicit-none -O3 -c -o xerbla.o xerbla.f
gfortran -fimplicit-none -O3 -c -o xerbla_array.o xerbla_array.f
gfortran -fimplicit-none -O3 -c -o zaxpy.o zaxpy.f
gfortran -fimplicit-none -O3 -c -o zcopy.o zcopy.f
gfortran -fimplicit-none -O3 -c -o zdotc.o zdotc.f
gfortran -fimplicit-none -O3 -c -o zdotu.o zdotu.f
gfortran -fimplicit-none -O3 -c -o zdrot.o zdrot.f
gfortran -fimplicit-none -O3 -c -o zdscal.o zdscal.f
gfortran -fimplicit-none -O3 -c -o zgbmv.o zgbmv.f
gfortran -fimplicit-none -O3 -c -o zgemm.o zgemm.f
gfortran -fimplicit-none -O3 -c -o zgemv.o zgemv.f
gfortran -fimplicit-none -O3 -c -o zgerc.o zgerc.f
gfortran -fimplicit-none -O3 -c -o zgeru.o zgeru.f
gfortran -fimplicit-none -O3 -c -o zhbmv.o zhbmv.f
gfortran -fimplicit-none -O3 -c -o zhemm.o zhemm.f
gfortran -fimplicit-none -O3 -c -o zhemv.o zhemv.f
gfortran -fimplicit-none -O3 -c -o zher.o zher.f
gfortran -fimplicit-none -O3 -c -o zher2.o zher2.f
gfortran -fimplicit-none -O3 -c -o zher2k.o zher2k.f
gfortran -fimplicit-none -O3 -c -o zherk.o zherk.f
gfortran -fimplicit-none -O3 -c -o zhpmv.o zhpmv.f
gfortran -fimplicit-none -O3 -c -o zhpr.o zhpr.f
gfortran -fimplicit-none -O3 -c -o zhpr2.o zhpr2.f
gfortran -fimplicit-none -O3 -c -o zrotg.o zrotg.f
gfortran -fimplicit-none -O3 -c -o zscal.o zscal.f
gfortran -fimplicit-none -O3 -c -o zswap.o zswap.f
gfortran -fimplicit-none -O3 -c -o zsymm.o zsymm.f
gfortran -fimplicit-none -O3 -c -o zsyr2k.o zsyr2k.f
gfortran -fimplicit-none -O3 -c -o zsyrk.o zsyrk.f
gfortran -fimplicit-none -O3 -c -o ztbmv.o ztbmv.f
gfortran -fimplicit-none -O3 -c -o ztbsv.o ztbsv.f
gfortran -fimplicit-none -O3 -c -o ztpmv.o ztpmv.f
gfortran -fimplicit-none -O3 -c -o ztpsv.o ztpsv.f
gfortran -fimplicit-none -O3 -c -o ztrmm.o ztrmm.f
gfortran -fimplicit-none -O3 -c -o ztrmv.o ztrmv.f
gfortran -fimplicit-none -O3 -c -o ztrsm.o ztrsm.f
gfortran -fimplicit-none -O3 -c -o ztrsv.o ztrsv.f
ar cru ../librefblas.a caxpy.o ccopy.o cdotc.o cdotu.o cgbmv.o cgemm.o cgemv.o cgerc.o cgeru.o chbmv.o chemm.o chemv.o cher.o cher2.o cher2k.o cherk.o chpmv.o chpr.o chpr2.o crotg.o cscal.o csrot.o csscal.o cswap.o csymm.o csyr2k.o csyrk.o ctbmv.o ctbsv.o ctpmv.o ctpsv.o ctrmm.o ctrmv.o ctrsm.o ctrsv.o dasum.o daxpy.o dcabs1.o dcopy.o ddot.o dgbmv.o dgemm.o dgemv.o dger.o dnrm2.o drot.o drotg.o drotm.o drotmg.o dsbmv.o dscal.o dsdot.o dspmv.o dspr.o dspr2.o dswap.o dsymm.o dsymv.o dsyr.o dsyr2.o dsyr2k.o dsyrk.o dtbmv.o dtbsv.o dtpmv.o dtpsv.o dtrmm.o dtrmv.o dtrsm.o dtrsv.o dzasum.o dznrm2.o icamax.o idamax.o isamax.o izamax.o lsame.o sasum.o saxpy.o scabs1.o scasum.o scnrm2.o scopy.o sdot.o sdsdot.o sgbmv.o sgemm.o sgemv.o sger.o snrm2.o srot.o srotg.o srotm.o srotmg.o ssbmv.o sscal.o sspmv.o sspr.o sspr2.o sswap.o ssymm.o ssymv.o ssyr.o ssyr2.o ssyr2k.o ssyrk.o stbmv.o stbsv.o stpmv.o stpsv.o strmm.o strmv.o strsm.o strsv.o xerbla.o xerbla_array.o zaxpy.o zcopy.o zdotc.o zdotu.o zdrot.o zdscal.o zgbmv.o zgemm.o zgemv.o zgerc.o zgeru.o zhbmv.o zhemm.o zhemv.o zher.o zher2.o zher2k.o zherk.o zhpmv.o zhpr.o zhpr2.o zrotg.o zscal.o zswap.o zsymm.o zsyr2k.o zsyrk.o ztbmv.o ztbsv.o ztpmv.o ztpsv.o ztrmm.o ztrmv.o ztrsm.o ztrsv.o
ranlib ../librefblas.a
make -C test
gfortran dblat1.f -L.. -lrefblas -o dblat1_ref
dblat1.f:215.44:
               CALL STEST1(DNRM2(N,SX,INCX),STEMP,STEMP,SFAC)           
                                            1
Warning: Rank mismatch in argument 'strue1' at (1) (scalar and rank-1)
dblat1.f:219.44:
               CALL STEST1(DASUM(N,SX,INCX),STEMP,STEMP,SFAC)           
                                            1
Warning: Rank mismatch in argument 'strue1' at (1) (scalar and rank-1)
gfortran dblat3.f -L.. -lrefblas -o dblat3_ref
gfortran dblat1.f -L.. -lulmblas -o dblat1_ulm
dblat1.f:215.44:
               CALL STEST1(DNRM2(N,SX,INCX),STEMP,STEMP,SFAC)           
                                            1
Warning: Rank mismatch in argument 'strue1' at (1) (scalar and rank-1)
dblat1.f:219.44:
               CALL STEST1(DASUM(N,SX,INCX),STEMP,STEMP,SFAC)           
                                            1
Warning: Rank mismatch in argument 'strue1' at (1) (scalar and rank-1)
gfortran dblat3.f -L.. -lulmblas -o dblat3_ulm
make -C bench
gcc-4.8 -c -DL2SIZE=4194304 -DAdd_ -DF77_INTEGER=int -DStringSunStyle -DATL_SSE2 -DDREAL   -c -o l1blastst.o l1blastst.c
gcc-4.8 -c -DL2SIZE=4194304 -DAdd_ -DF77_INTEGER=int -DStringSunStyle -DATL_SSE2 -DDREAL   -c -o ATL_cputime.o ATL_cputime.c
gcc-4.8 -c -DL2SIZE=4194304 -DAdd_ -DF77_INTEGER=int -DStringSunStyle -DATL_SSE2 -DDREAL   -c -o ATL_epsilon.o ATL_epsilon.c
gcc-4.8 -c -DL2SIZE=4194304 -DAdd_ -DF77_INTEGER=int -DStringSunStyle -DATL_SSE2 -DDREAL   -c -o ATL_f77amax.o ATL_f77amax.c
gcc-4.8 -c -DL2SIZE=4194304 -DAdd_ -DF77_INTEGER=int -DStringSunStyle -DATL_SSE2 -DDREAL   -c -o ATL_f77asum.o ATL_f77asum.c
gcc-4.8 -c -DL2SIZE=4194304 -DAdd_ -DF77_INTEGER=int -DStringSunStyle -DATL_SSE2 -DDREAL   -c -o ATL_f77axpy.o ATL_f77axpy.c
gcc-4.8 -c -DL2SIZE=4194304 -DAdd_ -DF77_INTEGER=int -DStringSunStyle -DATL_SSE2 -DDREAL   -c -o ATL_f77copy.o ATL_f77copy.c
gcc-4.8 -c -DL2SIZE=4194304 -DAdd_ -DF77_INTEGER=int -DStringSunStyle -DATL_SSE2 -DDREAL   -c -o ATL_f77dot.o ATL_f77dot.c
gcc-4.8 -c -DL2SIZE=4194304 -DAdd_ -DF77_INTEGER=int -DStringSunStyle -DATL_SSE2 -DDREAL   -c -o ATL_f77gemm.o ATL_f77gemm.c
gcc-4.8 -c -DL2SIZE=4194304 -DAdd_ -DF77_INTEGER=int -DStringSunStyle -DATL_SSE2 -DDREAL   -c -o ATL_f77nrm2.o ATL_f77nrm2.c
gcc-4.8 -c -DL2SIZE=4194304 -DAdd_ -DF77_INTEGER=int -DStringSunStyle -DATL_SSE2 -DDREAL   -c -o ATL_f77rot.o ATL_f77rot.c
gcc-4.8 -c -DL2SIZE=4194304 -DAdd_ -DF77_INTEGER=int -DStringSunStyle -DATL_SSE2 -DDREAL   -c -o ATL_f77rotg.o ATL_f77rotg.c
gcc-4.8 -c -DL2SIZE=4194304 -DAdd_ -DF77_INTEGER=int -DStringSunStyle -DATL_SSE2 -DDREAL   -c -o ATL_f77rotm.o ATL_f77rotm.c
gcc-4.8 -c -DL2SIZE=4194304 -DAdd_ -DF77_INTEGER=int -DStringSunStyle -DATL_SSE2 -DDREAL   -c -o ATL_f77rotmg.o ATL_f77rotmg.c
gcc-4.8 -c -DL2SIZE=4194304 -DAdd_ -DF77_INTEGER=int -DStringSunStyle -DATL_SSE2 -DDREAL   -c -o ATL_f77scal.o ATL_f77scal.c
gcc-4.8 -c -DL2SIZE=4194304 -DAdd_ -DF77_INTEGER=int -DStringSunStyle -DATL_SSE2 -DDREAL   -c -o ATL_f77swap.o ATL_f77swap.c
gcc-4.8 -c -DL2SIZE=4194304 -DAdd_ -DF77_INTEGER=int -DStringSunStyle -DATL_SSE2 -DDREAL   -c -o ATL_f77symm.o ATL_f77symm.c
gcc-4.8 -c -DL2SIZE=4194304 -DAdd_ -DF77_INTEGER=int -DStringSunStyle -DATL_SSE2 -DDREAL   -c -o ATL_f77syr2k.o ATL_f77syr2k.c
gcc-4.8 -c -DL2SIZE=4194304 -DAdd_ -DF77_INTEGER=int -DStringSunStyle -DATL_SSE2 -DDREAL   -c -o ATL_f77syrk.o ATL_f77syrk.c
gcc-4.8 -c -DL2SIZE=4194304 -DAdd_ -DF77_INTEGER=int -DStringSunStyle -DATL_SSE2 -DDREAL   -c -o ATL_f77trmm.o ATL_f77trmm.c
gcc-4.8 -c -DL2SIZE=4194304 -DAdd_ -DF77_INTEGER=int -DStringSunStyle -DATL_SSE2 -DDREAL   -c -o ATL_f77trsm.o ATL_f77trsm.c
gcc-4.8 -c -DL2SIZE=4194304 -DAdd_ -DF77_INTEGER=int -DStringSunStyle -DATL_SSE2 -DDREAL   -c -o ATL_flushcache.o ATL_flushcache.c
gcc-4.8 -c -DL2SIZE=4194304 -DAdd_ -DF77_INTEGER=int -DStringSunStyle -DATL_SSE2 -DDREAL   -c -o ATL_gediffnrm1.o ATL_gediffnrm1.c
gcc-4.8 -c -DL2SIZE=4194304 -DAdd_ -DF77_INTEGER=int -DStringSunStyle -DATL_SSE2 -DDREAL   -c -o ATL_gegen.o ATL_gegen.c
gcc-4.8 -c -DL2SIZE=4194304 -DAdd_ -DF77_INTEGER=int -DStringSunStyle -DATL_SSE2 -DDREAL   -c -o ATL_genrm1.o ATL_genrm1.c
gcc-4.8 -c -DL2SIZE=4194304 -DAdd_ -DF77_INTEGER=int -DStringSunStyle -DATL_SSE2 -DDREAL   -c -o ATL_infnrm.o ATL_infnrm.c
gcc-4.8 -c -DL2SIZE=4194304 -DAdd_ -DF77_INTEGER=int -DStringSunStyle -DATL_SSE2 -DDREAL   -c -o ATL_rand.o ATL_rand.c
gcc-4.8 -c -DL2SIZE=4194304 -DAdd_ -DF77_INTEGER=int -DStringSunStyle -DATL_SSE2 -DDREAL   -c -o ATL_set.o ATL_set.c
gcc-4.8 -c -DL2SIZE=4194304 -DAdd_ -DF77_INTEGER=int -DStringSunStyle -DATL_SSE2 -DDREAL   -c -o ATL_synrm.o ATL_synrm.c
gcc-4.8 -c -DL2SIZE=4194304 -DAdd_ -DF77_INTEGER=int -DStringSunStyle -DATL_SSE2 -DDREAL   -c -o ATL_trnrm1.o ATL_trnrm1.c
gcc-4.8 -c -DL2SIZE=4194304 -DAdd_ -DF77_INTEGER=int -DStringSunStyle -DATL_SSE2 -DDREAL   -c -o ATL_vdiff.o ATL_vdiff.c
gcc-4.8 -c -DL2SIZE=4194304 -DAdd_ -DF77_INTEGER=int -DStringSunStyle -DATL_SSE2 -DDREAL   -c -o ATL_zero.o ATL_zero.c
gfortran   -c -o ATL_df77wrap.o ATL_df77wrap.f
ar r libtstatlas.a  ATL_cputime.o  ATL_epsilon.o  ATL_f77amax.o  ATL_f77asum.o  ATL_f77axpy.o  ATL_f77copy.o  ATL_f77dot.o  ATL_f77gemm.o  ATL_f77nrm2.o  ATL_f77rot.o  ATL_f77rotg.o  ATL_f77rotm.o  ATL_f77rotmg.o  ATL_f77scal.o  ATL_f77swap.o  ATL_f77symm.o  ATL_f77syr2k.o  ATL_f77syrk.o  ATL_f77trmm.o  ATL_f77trsm.o  ATL_flushcache.o  ATL_gediffnrm1.o  ATL_gegen.o  ATL_genrm1.o  ATL_infnrm.o  ATL_rand.o  ATL_set.o  ATL_synrm.o  ATL_trnrm1.o  ATL_vdiff.o  ATL_zero.o  ATL_df77wrap.o
ar: creating archive libtstatlas.a
ranlib libtstatlas.a
gfortran -o xdl1blastst l1blastst.o libtstatlas.a ../libatlulmblas.a ../librefblas.a
gfortran -o xdl3blastst l3blastst.o libtstatlas.a ../libatlulmblas.a ../librefblas.a

The Micro Kernel Algorithm

Again we consider the update step

\[\mathbf{AB} \leftarrow \mathbf{AB} + \begin{pmatrix} a_{4l} \\ a_{4l+1} \\ a_{4l+2} \\ a_{4l+3}\end{pmatrix} \begin{pmatrix} b_{4l}, & b_{4l+1}, & b_{4l+2}, & b_{4l+3}\end{pmatrix}\]

And again we will keep the complete matrix \(\mathbf{AB}\) in eight SSE registers. However, we will do this with a slight modification that we illustrate later.

Remember that in our so called naive approach the 64-bit operand \(b_{4l}\) was duplicated and stored in a 128-bit SSE register. This time we will store the two 64-bit operands \(b_{4l}\) and \(b_{4l+1}\) in a common 128-bit SSE register. That is basically the main difference. More precise we store the operands in SSE registers \(\mathbb{tmp}_0\) to \(\mathbb{tmp}_3\)

\[\begin{array}{llll}\mathbb{tmp}_{0} \leftarrow \begin{pmatrix} a_{4l } \\ a_{4l+1} \end{pmatrix}, &\mathbb{tmp}_{1} \leftarrow \begin{pmatrix} a_{4l+2} \\ a_{4l+3} \end{pmatrix}, &\mathbb{tmp}_{2} \leftarrow \begin{pmatrix} b_{4l } \\ b_{4l+1} \end{pmatrix}, &\mathbb{tmp}_{3} \leftarrow \begin{pmatrix} b_{4l+2} \\ b_{4l+3} \end{pmatrix}, &\end{array}\]

Now we notice that a component wise SSE multiplication like \(\mathbb{tmp}_{0} \odot \mathbb{tmp}_{2}\) computes \(\begin{pmatrix} a_{4l} b_{4l} \\ a_{4l+1} b_{4l+1} \end{pmatrix}\) which contributes to \(\mathbf{AB}_{0,0}\) and \(\mathbf{AB}_{1,1}\). With this registers further contributions can be computed for \((\mathbf{AB}_{2,0}, \mathbf{AB}_{3,1})\), \((\mathbf{AB}_{0,2}, \mathbf{AB}_{1,3})\) and \((\mathbf{AB}_{2,2}, \mathbf{AB}_{3,3})\).

For the remaining entries of \(\mathbf{AB}\) contributions can be computed after swapping \(\mathbb{tmp}_{2}\) and \(\mathbb{tmp}_{3}\):

\[\begin{array}{llll}\mathbb{tmp}_{4} \leftarrow \begin{pmatrix} b_{4l+1} \\ b_{4l } \end{pmatrix}, &\mathbb{tmp}_{5} \leftarrow \begin{pmatrix} b_{4l+3} \\ b_{4l+2} \end{pmatrix}, &\end{array}\]

Using eight SSE registers for \(\mathbf{AB}\) we have two SSE registers \(\mathbb{tmp}_6\) and \(\mathbb{tmp}_7\) left for intermediate results.

Denoting elements of \(\mathbf{AB}\) with \(\mathbb{ab}_{\cdot,\cdot}\) we update diags and anti-diags in the first two columns with

\[\begin{array}{lll}\mathbb{tmp}_6 &\leftarrow& \mathbb{tmp}_2 \\\mathbb{tmp}_2 &\leftarrow& \mathbb{tmp}_2 \odot \mathbb{tmp}_0 \\\mathbb{tmp}_6 &\leftarrow& \mathbb{tmp}_6 \odot \mathbb{tmp}_0 \\\mathbb{ab}_{00,11} &\leftarrow& \mathbb{ab}_{00,11} + \mathbb{tmp}_2 \\\mathbb{ab}_{20,31} &\leftarrow& \mathbb{ab}_{20,31} + \mathbb{tmp}_6 \\& & \\\mathbb{tmp}_7 &\leftarrow& \mathbb{tmp}_4 \\\mathbb{tmp}_4 &\leftarrow& \mathbb{tmp}_4 \odot \mathbb{tmp}_0 \\\mathbb{tmp}_7 &\leftarrow& \mathbb{tmp}_7 \odot \mathbb{tmp}_0 \\\mathbb{ab}_{01,10} &\leftarrow& \mathbb{ab}_{01,10} + \mathbb{tmp}_4 \\\mathbb{ab}_{21,30} &\leftarrow& \mathbb{ab}_{21,30} + \mathbb{tmp}_7 \\\end{array}\]

Analogously we compute updates for the last two columns of \(\mathbf{AB}\)

\[\begin{array}{lll}\mathbb{tmp}_6 &\leftarrow& \mathbb{tmp}_3 \\\mathbb{tmp}_3 &\leftarrow& \mathbb{tmp}_3 \odot \mathbb{tmp}_0 \\\mathbb{tmp}_6 &\leftarrow& \mathbb{tmp}_6 \odot \mathbb{tmp}_0 \\\mathbb{ab}_{00,11} &\leftarrow& \mathbb{ab}_{00,11} + \mathbb{tmp}_2 \\\mathbb{ab}_{20,31} &\leftarrow& \mathbb{ab}_{20,31} + \mathbb{tmp}_6 \\& & \\\mathbb{tmp}_7 &\leftarrow& \mathbb{tmp}_5 \\\mathbb{tmp}_5 &\leftarrow& \mathbb{tmp}_5 \odot \mathbb{tmp}_0 \\\mathbb{tmp}_7 &\leftarrow& \mathbb{tmp}_7 \odot \mathbb{tmp}_0 \\\mathbb{ab}_{01,10} &\leftarrow& \mathbb{ab}_{01,10} + \mathbb{tmp}_4 \\\mathbb{ab}_{21,30} &\leftarrow& \mathbb{ab}_{21,30} + \mathbb{tmp}_7 \\\end{array}\]

When copying \(\mathbb{ab}_{\cdot,\cdot}\) back two memory we have to move lower and hight double separately. For example the lower double of \(\mathbb{ab}_{00,11}\) gets moved to \(\mathbf{AB}_{0,0}\) and the higher double to \(\mathbf{AB}_{1,1}\).

The dgemm_nn Code

ulmBLAS/src/level3/dgemm_nn.c
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#include <ulmblas.h>
#include <stdio.h>
#include <emmintrin.h>
#include <immintrin.h>

#define MC  384
#define KC  384
#define NC  4096

#define MR  4
#define NR  4

//
//  Local buffers for storing panels from A, B and C
//
static double _A[MC*KC] __attribute__ ((aligned (16)));
static double _B[KC*NC] __attribute__ ((aligned (16)));
static double _C[MR*NR] __attribute__ ((aligned (16)));

//
//  Packing complete panels from A (i.e. without padding)
//
static void
pack_MRxk(int k, const double *A, int incRowA, int incColA,
          double *buffer)
{
    int i, j;

    for (j=0; j<k; ++j) {
        for (i=0; i<MR; ++i) {
            buffer[i] = A[i*incRowA];
        }
        buffer += MR;
        A      += incColA;
    }
}

//
//  Packing panels from A with padding if required
//
static void
pack_A(int mc, int kc, const double *A, int incRowA, int incColA,
       double *buffer)
{
    int mp  = mc / MR;
    int _mr = mc % MR;

    int i, j;

    for (i=0; i<mp; ++i) {
        pack_MRxk(kc, A, incRowA, incColA, buffer);
        buffer += kc*MR;
        A      += MR*incRowA;
    }
    if (_mr>0) {
        for (j=0; j<kc; ++j) {
            for (i=0; i<_mr; ++i) {
                buffer[i] = A[i*incRowA];
            }
            for (i=_mr; i<MR; ++i) {
                buffer[i] = 0.0;
            }
            buffer += MR;
            A      += incColA;
        }
    }
}

//
//  Packing complete panels from B (i.e. without padding)
//
static void
pack_kxNR(int k, const double *B, int incRowB, int incColB,
          double *buffer)
{
    int i, j;

    for (i=0; i<k; ++i) {
        for (j=0; j<NR; ++j) {
            buffer[j] = B[j*incColB];
        }
        buffer += NR;
        B      += incRowB;
    }
}

//
//  Packing panels from B with padding if required
//
static void
pack_B(int kc, int nc, const double *B, int incRowB, int incColB,
       double *buffer)
{
    int np  = nc / NR;
    int _nr = nc % NR;

    int i, j;

    for (j=0; j<np; ++j) {
        pack_kxNR(kc, B, incRowB, incColB, buffer);
        buffer += kc*NR;
        B      += NR*incColB;
    }
    if (_nr>0) {
        for (i=0; i<kc; ++i) {
            for (j=0; j<_nr; ++j) {
                buffer[j] = B[j*incColB];
            }
            for (j=_nr; j<NR; ++j) {
                buffer[j] = 0.0;
            }
            buffer += NR;
            B      += incRowB;
        }
    }
}

//
//  Micro kernel for multiplying panels from A and B.
//
static void
dgemm_micro_kernel(long kc,
                   double alpha, const double *A, const double *B,
                   double beta,
                   double *C, long incRowC, long incColC)
{
    double AB[MR*NR] __attribute__ ((aligned (16)));

    int i, j, l;

//
//  Compute AB = A*B
//
    register __m128d ab_00_11, ab_20_31;
    register __m128d ab_01_10, ab_21_30;
    register __m128d ab_02_13, ab_22_33;
    register __m128d ab_03_12, ab_23_32;

    register __m128d tmp0, tmp1, tmp2, tmp3;
    register __m128d tmp4, tmp5, tmp6, tmp7;

    ab_00_11 = _mm_setzero_pd(); ab_20_31 = _mm_setzero_pd();
    ab_01_10 = _mm_setzero_pd(); ab_21_30 = _mm_setzero_pd();
    ab_02_13 = _mm_setzero_pd(); ab_22_33 = _mm_setzero_pd();
    ab_03_12 = _mm_setzero_pd(); ab_23_32 = _mm_setzero_pd();

    for (l=0; l<kc; ++l) {
        tmp0     = _mm_load_pd(A);
        tmp1     = _mm_load_pd(A+2);

        tmp2     = _mm_load_pd(B);
        tmp3     = _mm_load_pd(B+2);

        tmp4     = _mm_shuffle_pd(tmp2, tmp2, _MM_SHUFFLE2(0, 1));
        tmp5     = _mm_shuffle_pd(tmp3, tmp3, _MM_SHUFFLE2(0, 1));

        tmp6     = tmp2;
        tmp2     = _mm_mul_pd(tmp2, tmp0);
        tmp6     = _mm_mul_pd(tmp6, tmp1);
        ab_00_11 = _mm_add_pd(ab_00_11, tmp2);
        ab_20_31 = _mm_add_pd(ab_20_31, tmp6);

        tmp7     = tmp4;
        tmp4     = _mm_mul_pd(tmp4, tmp0);
        tmp7     = _mm_mul_pd(tmp7, tmp1);
        ab_01_10 = _mm_add_pd(ab_01_10, tmp4);
        ab_21_30 = _mm_add_pd(ab_21_30, tmp7);

        tmp6     = tmp3;
        tmp3     = _mm_mul_pd(tmp3, tmp0);
        tmp6     = _mm_mul_pd(tmp6, tmp1);
        ab_02_13 = _mm_add_pd(ab_02_13, tmp3);
        ab_22_33 = _mm_add_pd(ab_22_33, tmp6);

        tmp7     = tmp5;
        tmp5     = _mm_mul_pd(tmp5, tmp0);
        tmp7     = _mm_mul_pd(tmp7, tmp1);
        ab_03_12 = _mm_add_pd(ab_03_12, tmp5);
        ab_23_32 = _mm_add_pd(ab_23_32, tmp7);

        A += 4;
        B += 4;
    }

    _mm_storel_pd(&AB[0+0*4], ab_00_11);
    _mm_storeh_pd(&AB[1+0*4], ab_01_10);
    _mm_storel_pd(&AB[2+0*4], ab_20_31);
    _mm_storeh_pd(&AB[3+0*4], ab_21_30);

    _mm_storel_pd(&AB[0+1*4], ab_01_10);
    _mm_storeh_pd(&AB[1+1*4], ab_00_11);
    _mm_storel_pd(&AB[2+1*4], ab_21_30);
    _mm_storeh_pd(&AB[3+1*4], ab_20_31);

    _mm_storel_pd(&AB[0+2*4], ab_02_13);
    _mm_storeh_pd(&AB[1+2*4], ab_03_12);
    _mm_storel_pd(&AB[2+2*4], ab_22_33);
    _mm_storeh_pd(&AB[3+2*4], ab_23_32);

    _mm_storel_pd(&AB[0+3*4], ab_03_12);
    _mm_storeh_pd(&AB[1+3*4], ab_02_13);
    _mm_storel_pd(&AB[2+3*4], ab_23_32);
    _mm_storeh_pd(&AB[3+3*4], ab_22_33);

//
//  Update C <- beta*C
//
    if (beta==0.0) {
        for (j=0; j<NR; ++j) {
            for (i=0; i<MR; ++i) {
                C[i*incRowC+j*incColC] = 0.0;
            }
        }
    } else if (beta!=1.0) {
        for (j=0; j<NR; ++j) {
            for (i=0; i<MR; ++i) {
                C[i*incRowC+j*incColC] *= beta;
            }
        }
    }

//
//  Update C <- C + alpha*AB (note: the case alpha==0.0 was already treated in
//                                  the above layer dgemm_nn)
//
    if (alpha==1.0) {
        for (j=0; j<NR; ++j) {
            for (i=0; i<MR; ++i) {
                C[i*incRowC+j*incColC] += AB[i+j*MR];
            }
        }
    } else {
        for (j=0; j<NR; ++j) {
            for (i=0; i<MR; ++i) {
                C[i*incRowC+j*incColC] += alpha*AB[i+j*MR];
            }
        }
    }
}

//
//  Compute Y += alpha*X
//
static void
dgeaxpy(int           m,
        int           n,
        double        alpha,
        const double  *X,
        int           incRowX,
        int           incColX,
        double        *Y,
        int           incRowY,
        int           incColY)
{
    int i, j;


    if (alpha!=1.0) {
        for (j=0; j<n; ++j) {
            for (i=0; i<m; ++i) {
                Y[i*incRowY+j*incColY] += alpha*X[i*incRowX+j*incColX];
            }
        }
    } else {
        for (j=0; j<n; ++j) {
            for (i=0; i<m; ++i) {
                Y[i*incRowY+j*incColY] += X[i*incRowX+j*incColX];
            }
        }
    }
}

//
//  Compute X *= alpha
//
static void
dgescal(int     m,
        int     n,
        double  alpha,
        double  *X,
        int     incRowX,
        int     incColX)
{
    int i, j;

    if (alpha!=0.0) {
        for (j=0; j<n; ++j) {
            for (i=0; i<m; ++i) {
                X[i*incRowX+j*incColX] *= alpha;
            }
        }
    } else {
        for (j=0; j<n; ++j) {
            for (i=0; i<m; ++i) {
                X[i*incRowX+j*incColX] = 0.0;
            }
        }
    }
}

//
//  Macro Kernel for the multiplication of blocks of A and B.  We assume that
//  these blocks were previously packed to buffers _A and _B.
//
static void
dgemm_macro_kernel(int     mc,
                   int     nc,
                   int     kc,
                   double  alpha,
                   double  beta,
                   double  *C,
                   int     incRowC,
                   int     incColC)
{
    int mp = (mc+MR-1) / MR;
    int np = (nc+NR-1) / NR;

    int _mr = mc % MR;
    int _nr = nc % NR;

    int mr, nr;
    int i, j;

    for (j=0; j<np; ++j) {
        nr    = (j!=np-1 || _nr==0) ? NR : _nr;

        for (i=0; i<mp; ++i) {
            mr    = (i!=mp-1 || _mr==0) ? MR : _mr;

            if (mr==MR && nr==NR) {
                dgemm_micro_kernel(kc, alpha, &_A[i*kc*MR], &_B[j*kc*NR],
                                   beta,
                                   &C[i*MR*incRowC+j*NR*incColC],
                                   incRowC, incColC);
            } else {
                dgemm_micro_kernel(kc, alpha, &_A[i*kc*MR], &_B[j*kc*NR],
                                   0.0,
                                   _C, 1, MR);
                dgescal(mr, nr, beta,
                        &C[i*MR*incRowC+j*NR*incColC], incRowC, incColC);
                dgeaxpy(mr, nr, 1.0, _C, 1, MR,
                        &C[i*MR*incRowC+j*NR*incColC], incRowC, incColC);
            }
        }
    }
}

//
//  Compute C <- beta*C + alpha*A*B
//
void
ULMBLAS(dgemm_nn)(int            m,
                  int            n,
                  int            k,
                  double         alpha,
                  const double   *A,
                  int            incRowA,
                  int            incColA,
                  const double   *B,
                  int            incRowB,
                  int            incColB,
                  double         beta,
                  double         *C,
                  int            incRowC,
                  int            incColC)
{
    int mb = (m+MC-1) / MC;
    int nb = (n+NC-1) / NC;
    int kb = (k+KC-1) / KC;

    int _mc = m % MC;
    int _nc = n % NC;
    int _kc = k % KC;

    int mc, nc, kc;
    int i, j, l;

    double _beta;

    if (alpha==0.0 || k==0) {
        dgescal(m, n, beta, C, incRowC, incColC);
        return;
    }

    for (j=0; j<nb; ++j) {
        nc = (j!=nb-1 || _nc==0) ? NC : _nc;

        for (l=0; l<kb; ++l) {
            kc    = (l!=kb-1 || _kc==0) ? KC   : _kc;
            _beta = (l==0) ? beta : 1.0;

            pack_B(kc, nc,
                   &B[l*KC*incRowB+j*NC*incColB], incRowB, incColB,
                   _B);

            for (i=0; i<mb; ++i) {
                mc = (i!=mb-1 || _mc==0) ? MC : _mc;

                pack_A(mc, kc,
                       &A[i*MC*incRowA+l*KC*incColA], incRowA, incColA,
                       _A);

                dgemm_macro_kernel(mc, nc, kc, alpha, _beta,
                                   &C[i*MC*incRowC+j*NC*incColC],
                                   incRowC, incColC);
            }
        }
    }
}

Benchmark Results

We run the benchmarks

$shell> cd bench                                                         
$shell> ./xdl3blastst > report                                           
$shell> cat report                                                       
./xdl3blastst 
--------------------------------- GEMM ----------------------------------
TST# A B    M    N    K ALPHA  LDA  LDB  BETA  LDC  TIME MFLOP SpUp  TEST
==== = = ==== ==== ==== ===== ==== ==== ===== ==== ===== ===== ==== =====
   0 N N  100  100  100   1.0 1000 1000   1.0 1000  0.00 1806.7 1.00 -----
   0 N N  100  100  100   1.0 1000 1000   1.0 1000  0.00 4640.4 2.57 PASS 
   1 N N  200  200  200   1.0 1000 1000   1.0 1000  0.01 1961.3 1.00 -----
   1 N N  200  200  200   1.0 1000 1000   1.0 1000  0.00 5596.4 2.85 PASS 
   2 N N  300  300  300   1.0 1000 1000   1.0 1000  0.03 2018.2 1.00 -----
   2 N N  300  300  300   1.0 1000 1000   1.0 1000  0.01 5856.8 2.90 PASS 
   3 N N  400  400  400   1.0 1000 1000   1.0 1000  0.06 2065.7 1.00 -----
   3 N N  400  400  400   1.0 1000 1000   1.0 1000  0.02 5750.0 2.78 PASS 
   4 N N  500  500  500   1.0 1000 1000   1.0 1000  0.12 2067.0 1.00 -----
   4 N N  500  500  500   1.0 1000 1000   1.0 1000  0.04 5919.1 2.86 PASS 
   5 N N  600  600  600   1.0 1000 1000   1.0 1000  0.33 1292.4 1.00 -----
   5 N N  600  600  600   1.0 1000 1000   1.0 1000  0.07 5952.3 4.61 PASS 
   6 N N  700  700  700   1.0 1000 1000   1.0 1000  0.65 1059.8 1.00 -----
   6 N N  700  700  700   1.0 1000 1000   1.0 1000  0.11 6077.2 5.73 PASS 
   7 N N  800  800  800   1.0 1000 1000   1.0 1000  1.00 1022.0 1.00 -----
   7 N N  800  800  800   1.0 1000 1000   1.0 1000  0.17 6028.7 5.90 PASS 
   8 N N  900  900  900   1.0 1000 1000   1.0 1000  1.38 1058.5 1.00 -----
   8 N N  900  900  900   1.0 1000 1000   1.0 1000  0.24 6076.0 5.74 PASS 
   9 N N 1000 1000 1000   1.0 1000 1000   1.0 1000  1.71 1166.3 1.00 -----
   9 N N 1000 1000 1000   1.0 1000 1000   1.0 1000  0.33 6126.5 5.25 PASS 
10 tests run, 10 passed

and filter out the results for the demo-sse-intrinsics branch:

$shell> grep PASS report > demo-sse-intrinsics

With the gnuplot script

ulmBLAS/bench/bench5.gps
set terminal svg size 1140,480
set output "bench5.svg"
set xlabel "Matrix dimensions N=M=K"
set ylabel "MFLOPS"
set yrange [0:9600]
set title "Compute C + A*B"
set key outside
plot "refBLAS" using 4:13 with linespoints lt 2 title "Netlib RefBLAS", "demo-pure-c" using 4:13 with linespoints lt 4 title "demo-pure-c", "demo-naive-sse-with-intrinsics" using 4:13 with linespoints lt 5 title "demo-naive-sse-with-intrinsics", "demo-naive-sse-with-intrinsics-unrolled" using 4:13 with linespoints lt 6 title "demo-naive-sse-with-intrinsics-unrolled", "demo-sse-intrinsics" using 4:13 with linespoints lt 7 title "demo-sse-intrinsics (clang)"

we feed gnuplot

$shell> gnuplot bench5.gps

and get

Sensitivity to Compilers

Maybe you noticed that this time the clang compiler was used above! This times we get poor results when using gcc 4.8:

$shell> gcc-4.8 --version                                                
gcc-4.8 (Homebrew gcc 4.8.3_1) 4.8.3
Copyright (C) 2013 Free Software Foundation, Inc.
This is free software; see the source for copying conditions.  There is NO
warranty; not even for MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.
$shell> cd src                                                           
$shell> gcc-4.8 -Wall -I. -O3  -msse3 -mfpmath=sse -fomit-frame-pointer -DFAKE_ATLAS -c -o level3/atl_dgemm_nn.o level3/dgemm_nn.c               
$shell> make                                                             
ar cru ../libatlulmblas.a  auxiliary/atl_xerbla.o  level1/atl_dasum.o level1/atl_daxpy.o level1/atl_dcopy.o level1/atl_ddot.o level1/atl_dnrm2.o level1/atl_drot.o level1/atl_drotg.o level1/atl_drotm.o level1/atl_drotmg.o level1/atl_dscal.o level1/atl_dswap.o level1/atl_idamax.o  level3/atl_dgemm.o level3/atl_dgemm_nn.o level3/atl_dsymm.o level3/atl_stubs.o
ranlib ../libatlulmblas.a
$shell> cd ../bench                                                      
$shell> make                                                             
gfortran -o xdl1blastst l1blastst.o libtstatlas.a ../libatlulmblas.a ../librefblas.a
gfortran -o xdl3blastst l3blastst.o libtstatlas.a ../libatlulmblas.a ../librefblas.a
$shell> ./xdl3blastst                                                    
./xdl3blastst 
--------------------------------- GEMM ----------------------------------
TST# A B    M    N    K ALPHA  LDA  LDB  BETA  LDC  TIME MFLOP SpUp  TEST
==== = = ==== ==== ==== ===== ==== ==== ===== ==== ===== ===== ==== =====
   0 N N  100  100  100   1.0 1000 1000   1.0 1000  0.00 1810.0 1.00 -----
   0 N N  100  100  100   1.0 1000 1000   1.0 1000  0.00 4201.7 2.32 PASS 
   1 N N  200  200  200   1.0 1000 1000   1.0 1000  0.01 1963.7 1.00 -----
   1 N N  200  200  200   1.0 1000 1000   1.0 1000  0.00 4906.5 2.50 PASS 
   2 N N  300  300  300   1.0 1000 1000   1.0 1000  0.03 2017.8 1.00 -----
   2 N N  300  300  300   1.0 1000 1000   1.0 1000  0.01 5145.3 2.55 PASS 
   3 N N  400  400  400   1.0 1000 1000   1.0 1000  0.06 2067.3 1.00 -----
   3 N N  400  400  400   1.0 1000 1000   1.0 1000  0.03 5054.5 2.44 PASS 
   4 N N  500  500  500   1.0 1000 1000   1.0 1000  0.12 2042.8 1.00 -----
   4 N N  500  500  500   1.0 1000 1000   1.0 1000  0.05 5103.9 2.50 PASS 
   5 N N  600  600  600   1.0 1000 1000   1.0 1000  0.32 1368.4 1.00 -----
   5 N N  600  600  600   1.0 1000 1000   1.0 1000  0.08 5219.9 3.81 PASS 
   6 N N  700  700  700   1.0 1000 1000   1.0 1000  0.66 1039.3 1.00 -----
   6 N N  700  700  700   1.0 1000 1000   1.0 1000  0.13 5294.4 5.09 PASS 
   7 N N  800  800  800   1.0 1000 1000   1.0 1000  0.98 1049.0 1.00 -----
   7 N N  800  800  800   1.0 1000 1000   1.0 1000  0.20 5201.2 4.96 PASS 
   8 N N  900  900  900   1.0 1000 1000   1.0 1000  1.38 1055.7 1.00 -----
   8 N N  900  900  900   1.0 1000 1000   1.0 1000  0.28 5259.6 4.98 PASS 
   9 N N 1000 1000 1000   1.0 1000 1000   1.0 1000  1.71 1168.5 1.00 -----
   9 N N 1000 1000 1000   1.0 1000 1000   1.0 1000  0.38 5309.8 4.54 PASS 
10 tests run, 10 passed

We can visualize this again

$shell> ./xdl3blastst > report                                           
$shell> grep PASS report > demo-sse-intrinsics-gcc

By adopting the gnuplot script

ulmBLAS/bench/bench6.gps
set terminal svg size 1140,480
set output "bench6.svg"
set xlabel "Matrix dimensions N=M=K"
set ylabel "MFLOPS"
set yrange [0:9600]
set title "Compute C + A*B"
set key outside
plot "refBLAS" using 4:13 with linespoints lt 2 title "Netlib RefBLAS", "demo-pure-c" using 4:13 with linespoints lt 4 title "demo-pure-c", "demo-naive-sse-with-intrinsics" using 4:13 with linespoints lt 5 title "demo-naive-sse-with-intrinsics", "demo-naive-sse-with-intrinsics-unrolled" using 4:13 with linespoints lt 6 title "demo-naive-sse-with-intrinsics-unrolled", "demo-sse-intrinsics" using 4:13 with linespoints lt 7 title "demo-sse-intrinsics (clang)", "demo-sse-intrinsics-gcc" using 4:13 with linespoints lt 8 title "demo-sse-intrinsics (gcc 4.8)

and feeding gnuplot with this script

$shell> gnuplot bench6.gps

We get

Analyzing Assembler Code from gcc 4.8

#  ...
                xorpd           %xmm9, %xmm9
                movapd          %xmm9, %xmm11
                addq            %rdx, %rdi
                movapd          %xmm9, %xmm8
                movapd          %xmm9, %xmm10
                movapd          %xmm9, %xmm13
                movapd          %xmm9, %xmm15
                movapd          %xmm9, %xmm12
                movapd          %xmm9, %xmm14
                .align 4,0x90
L3:
                movapd          (%rdx), %xmm5
                addq            $32, %rdx
                addq            $32, %rsi
                movapd          -16(%rsi), %xmm2
                movapd          %xmm5, %xmm7
                movapd          %xmm5, %xmm1
                movapd          -32(%rsi), %xmm3
                shufpd          $1, %xmm5, %xmm7
                mulpd           %xmm2, %xmm5
                movapd          -16(%rdx), %xmm4
                mulpd           %xmm3, %xmm1
                cmpq            %rdi, %rdx
                movapd          %xmm4, %xmm6
                shufpd          $1, %xmm4, %xmm6
                addpd           %xmm5, %xmm12
                movapd          %xmm7, %xmm5
                mulpd           %xmm3, %xmm5
                addpd           %xmm1, %xmm14
                mulpd           %xmm2, %xmm7
                addpd           %xmm5, %xmm15
                movapd          %xmm4, %xmm5
                mulpd           %xmm3, %xmm5
                addpd           %xmm7, %xmm13
                mulpd           %xmm2, %xmm4
                mulpd           %xmm6, %xmm3
                mulpd           %xmm6, %xmm2
                addpd           %xmm5, %xmm10
                addpd           %xmm4, %xmm8
                addpd           %xmm3, %xmm11
                addpd           %xmm2, %xmm9
                jne             L3
L2:
                movd            %r10, %xmm1
                movapd          %xmm14, %xmm2
                movsd           %xmm14, -120(%rsp)
                ucomisd         LC0(%rip), %xmm1
                movhpd          %xmm15, -112(%rsp)
                movlpd          %xmm12, -104(%rsp)
                movhpd          %xmm13, -96(%rsp)
                movlpd          %xmm15, -88(%rsp)
                movhpd          %xmm14, -80(%rsp)
                movlpd          %xmm13, -72(%rsp)
                movhpd          %xmm12, -64(%rsp)
                movlpd          %xmm10, -56(%rsp)
                movhpd          %xmm11, -48(%rsp)
                movlpd          %xmm8, -40(%rsp)
                movhpd          %xmm9, -32(%rsp)
                movlpd          %xmm11, -24(%rsp)
                movhpd          %xmm10, -16(%rsp)
                movlpd          %xmm9, -8(%rsp)
                movhpd          %xmm8, (%rsp)
#  ...

TODO: re-code this to equivalent SSE intrinsics

Analyzing Assembler Code from clang

#  ...
                xorpd           %xmm8, %xmm8
                xorpd           %xmm9, %xmm9
                xorpd           %xmm14, %xmm14
                xorpd           %xmm15, %xmm15
                xorpd           %xmm10, %xmm10
                xorpd           %xmm11, %xmm11
                xorpd           %xmm12, %xmm12
                xorpd           %xmm13, %xmm13
                jle             LBB1_2
                .align          4, 0x90
LBB1_1:                                 ## %.lr.ph
                                        ## =>This Inner Loop Header: Depth=1
                movapd          (%rsi), %xmm6
                movapd          16(%rsi), %xmm7
                movapd          (%rdx), %xmm2
                movapd          16(%rdx), %xmm3
                movapd          %xmm6, %xmm4
                mulpd           %xmm2, %xmm4
                movapd          %xmm7, %xmm5
                mulpd           %xmm2, %xmm5
                pshufd          $78, %xmm2, %xmm2       ## xmm2 = xmm2[2,3,0,1]
                addpd           %xmm4, %xmm8
                addpd           %xmm5, %xmm9
                movapd          %xmm6, %xmm4
                mulpd           %xmm2, %xmm4
                mulpd           %xmm7, %xmm2
                addpd           %xmm4, %xmm14
                addpd           %xmm2, %xmm15
                movapd          %xmm6, %xmm2
                mulpd           %xmm3, %xmm2
                movapd          %xmm7, %xmm4
                mulpd           %xmm3, %xmm4
                pshufd          $78, %xmm3, %xmm3       ## xmm3 = xmm3[2,3,0,1]
                addpd           %xmm2, %xmm10
                addpd           %xmm4, %xmm11
                mulpd           %xmm3, %xmm6
                mulpd           %xmm3, %xmm7
                addpd           %xmm6, %xmm12
                addpd           %xmm7, %xmm13
                addq            $32, %rsi
                addq            $32, %rdx
                decq            %rdi
                jne             LBB1_1
LBB1_2:                                 ## %._crit_edge
                movlpd          %xmm8, (%rsp)
                movhpd          %xmm14, 8(%rsp)
                movlpd          %xmm9, 16(%rsp)
                movhpd          %xmm15, 24(%rsp)
                movlpd          %xmm14, 32(%rsp)
                movhpd          %xmm8, 40(%rsp)
                movlpd          %xmm15, 48(%rsp)
                movhpd          %xmm9, 56(%rsp)
                movlpd          %xmm10, 64(%rsp)
                movhpd          %xmm12, 72(%rsp)
                movlpd          %xmm11, 80(%rsp)
                movhpd          %xmm13, 88(%rsp)
                movlpd          %xmm12, 96(%rsp)
                movhpd          %xmm10, 104(%rsp)
                movlpd          %xmm13, 112(%rsp)
                movhpd          %xmm11, 120(%rsp)
#  ...

TODO: re-code this to equivalent SSE intrinsics

Conclusion

The performance difference is mainly due to latency issues. We have to take into account that the execution of each SSE instruction also involves some latency time. Assume that instruction inst2 requires that inst1 is already completed. Than we can improve pipelining by doing in between some other useful things.

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