476 lines
17 KiB
C
476 lines
17 KiB
C
/*
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* Copyright (c) IBM Corporation 2020.
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* All rights reserved.
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*
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* Redistribution and use in source and binary forms, with or without
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* modification, are permitted provided that the following conditions are
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* met:
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*
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* 1. Redistributions of source code must retain the above copyright
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* notice, this list of conditions and the following disclaimer.
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*
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* 2. Redistributions in binary form must reproduce the above copyright
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* notice, this list of conditions and the following disclaimer in
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* the documentation and/or other materials provided with the
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* distribution.
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* 3. Neither the name of the OpenBLAS project nor the names of
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* its contributors may be used to endorse or promote products
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* derived from this software without specific prior written
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* permission.
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*
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* THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
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* AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
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* IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
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* ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE
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* LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
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* DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR
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* SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER
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* CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY,
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* OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE
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* USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
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*/
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#include "common.h"
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#include <vecintrin.h>
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#include <stdbool.h>
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#include <stdio.h>
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#include <stdlib.h>
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#ifdef COMPLEX
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#error "Handling for complex numbers is not supported in this kernel"
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#endif
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#ifdef DOUBLE
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#define UNROLL_M DGEMM_DEFAULT_UNROLL_M
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#define UNROLL_N DGEMM_DEFAULT_UNROLL_N
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#else
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#define UNROLL_M SGEMM_DEFAULT_UNROLL_M
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#define UNROLL_N SGEMM_DEFAULT_UNROLL_N
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#endif
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static const size_t unroll_m = UNROLL_M;
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static const size_t unroll_n = UNROLL_N;
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/* Handling of triangular matrices */
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#ifdef TRMMKERNEL
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static const bool trmm = true;
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static const bool left =
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#ifdef LEFT
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true;
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#else
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false;
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#endif
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static const bool backwards =
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#if defined(LEFT) != defined(TRANSA)
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true;
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#else
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false;
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#endif
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#else
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static const bool trmm = false;
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static const bool left = false;
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static const bool backwards = false;
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#endif /* TRMMKERNEL */
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/*
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* Background:
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*
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* The algorithm of GotoBLAS / OpenBLAS breaks down the matrix multiplication
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* problem by splitting all matrices into partitions multiple times, so that the
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* submatrices fit into the L1 or L2 caches. As a result, each multiplication of
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* submatrices can stream data fast from L1 and L2 caches. Inbetween, it copies
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* and rearranges the submatrices to enable contiguous memory accesses to
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* improve locality in both caches and TLBs.
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*
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* At the heart of the algorithm is this kernel, which multiplies, a "Block
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* matrix" A (small dimensions) with a "Panel matrix" B (number of rows is
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* small) and adds the result into a "Panel matrix" C; GotoBLAS calls this
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* operation GEBP. This kernel further partitions GEBP twice, such that (1)
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* submatrices of C and B fit into the L1 caches (GEBP_column_block) and (2) a
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* block of C fits into the registers, while multiplying panels from A and B
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* streamed from the L2 and L1 cache, respectively (GEBP_block).
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*
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*
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* Algorithm GEBP(A, B, C, m, n, k, alpha):
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*
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* The problem is calculating C += alpha * (A * B)
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* C is an m x n matrix, A is an m x k matrix, B is an k x n matrix.
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*
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* - C is in column-major-order, with an offset of ldc to the element in the
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* next column (same row).
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* - A is in row-major-order yet stores SGEMM_UNROLL_M elements of each column
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* contiguously while walking along rows.
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* - B is in column-major-order but packs SGEMM_UNROLL_N elements of a row
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* contiguously.
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* If the numbers of rows and columns are not multiples of SGEMM_UNROLL_M or
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* SGEMM_UNROLL_N, the remaining elements are arranged in blocks with power-of-2
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* dimensions (e.g., 5 remaining columns would be in a block-of-4 and a
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* block-of-1).
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*
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* Note that packing A and B into that form is taken care of by the caller in
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* driver/level3/level3.c (actually done by "copy kernels").
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*
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* Steps:
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* - Partition C and B into blocks of n_r (SGEMM_UNROLL_N) columns, C_j and B_j.
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* Now, B_j should fit into the L1 cache.
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* - For each partition, calculate C_j += alpha * (A * B_j) by
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* (1) Calculate C_aux := A * B_j (see below)
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* (2) unpack C_j = C_j + alpha * C_aux
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*
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*
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* Algorithm for Calculating C_aux:
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*
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* - Further partition C_aux and A into groups of m_r (SGEMM_UNROLL_M) rows,
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* such that the m_r x n_r-submatrix of C_aux can be held in registers. Each
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* submatrix of C_aux can be calculated independently, and the registers are
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* added back into C_j.
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*
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* - For each row-block of C_aux:
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* (uses a row block of A and full B_j)
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* - stream over all columns of A, multiply with elements from B and
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* accumulate in registers. (use different inner-kernels to exploit
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* vectorization for varying block sizes)
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* - add alpha * row block of C_aux back into C_j.
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*
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* Note that there are additional mechanics for handling triangular matrices,
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* calculating B := alpha (A * B) where either of the matrices A or B can be
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* triangular. In case of A, the macro "LEFT" is defined. In addition, A can
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* optionally be transposed.
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* The code effectively skips an "offset" number of columns in A and rows of B
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* in each block, to save unnecessary work by exploiting the triangular nature.
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* To handle all cases, the code discerns (1) a "left" mode when A is triangular
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* and (2) "forward" / "backwards" modes where only the first "offset"
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* columns/rows of A/B are used or where the first "offset" columns/rows are
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* skipped, respectively.
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*
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* Reference:
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*
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* The summary above is based on staring at various kernel implementations and:
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* K. Goto and R. A. Van de Geijn, Anatomy of High-Performance Matrix
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* Multiplication, in ACM Transactions of Mathematical Software, Vol. 34, No.
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* 3, May 2008.
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*/
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#define VLEN_BYTES 16
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#define VLEN_FLOATS (VLEN_BYTES / sizeof(FLOAT))
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typedef FLOAT vector_float __attribute__ ((vector_size (16)));
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/**
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* Load a vector into register, and hint on 8-byte alignment to improve
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* performance. gcc-9 and newer will create these hints by itself. For older
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* compiler versions, use inline assembly to explicitly express the hint.
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* Provide explicit hex encoding to cater for binutils versions that do not know
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* about vector-load with alignment hints yet.
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*
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* Note that, for block sizes where we apply vectorization, vectors in A will
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* always be 8-byte aligned.
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*/
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static inline vector_float vec_load_hinted(FLOAT const *restrict a) {
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vector_float const *restrict addr = (vector_float const *restrict)a;
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vector_float y;
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#if __GNUC__ < 9
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// hex-encode vl %[out],%[addr],3
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asm(".insn vrx,0xe70000003006,%[out],%[addr],3"
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: [ out ] "=v"(y)
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: [ addr ] "R"(*addr));
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#else
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y = *addr;
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#endif
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return y;
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}
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/**
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* Calculate for a row-block in C_i of size ROWSxCOLS using vector intrinsics.
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*
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* @param[in] A Pointer current block of input matrix A.
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* @param[in] k Number of columns in A.
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* @param[in] B Pointer current block of input matrix B.
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* @param[inout] C Pointer current block of output matrix C.
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* @param[in] ldc Offset between elements in adjacent columns in C.
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* @param[in] alpha Scalar factor.
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*/
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#define VECTOR_BLOCK(ROWS, COLS) \
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static inline void GEBP_block_##ROWS##_##COLS( \
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FLOAT const *restrict A, BLASLONG bk, FLOAT const *restrict B, \
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FLOAT *restrict C, BLASLONG ldc, FLOAT alpha) { \
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_Static_assert( \
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ROWS % VLEN_FLOATS == 0, \
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"rows in block must be multiples of vector length"); \
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vector_float Caux[ROWS / VLEN_FLOATS][COLS]; \
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\
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for (BLASLONG i = 0; i < ROWS / VLEN_FLOATS; i++) { \
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vector_float A0 = \
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vec_load_hinted(A + i * VLEN_FLOATS); \
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for (BLASLONG j = 0; j < COLS; j++) \
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Caux[i][j] = A0 * B[j]; \
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} \
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\
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/* \
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* Stream over the row-block of A, which is packed \
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* column-by-column, multiply by coefficients in B and add up \
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* into temporaries Caux (which the compiler will hold in \
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* registers). Vectorization: Multiply column vectors from A \
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* with scalars from B and add up in column vectors of Caux. \
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* That equates to unrolling the loop over rows (in i) and \
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* executing each unrolled iteration as a vector element. \
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*/ \
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for (BLASLONG k = 1; k < bk; k++) { \
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for (BLASLONG i = 0; i < ROWS / VLEN_FLOATS; i++) { \
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vector_float Ak = vec_load_hinted( \
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A + i * VLEN_FLOATS + k * ROWS); \
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\
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for (BLASLONG j = 0; j < COLS; j++) \
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Caux[i][j] += Ak * B[j + k * COLS]; \
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} \
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} \
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\
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/* \
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* Unpack row-block of C_aux into outer C_i, multiply by \
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* alpha and add up. \
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*/ \
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for (BLASLONG j = 0; j < COLS; j++) { \
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for (BLASLONG i = 0; i < ROWS / VLEN_FLOATS; i++) { \
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vector_float *C_ij = \
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(vector_float *)(C + i * VLEN_FLOATS + \
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j * ldc); \
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if (trmm) { \
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*C_ij = alpha * Caux[i][j]; \
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} else { \
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*C_ij += alpha * Caux[i][j]; \
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} \
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} \
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} \
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}
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#if UNROLL_M == 16
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VECTOR_BLOCK(16, 4)
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VECTOR_BLOCK(16, 2)
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VECTOR_BLOCK(16, 1)
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#endif
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#if UNROLL_N == 8
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VECTOR_BLOCK(8, 8)
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VECTOR_BLOCK(4, 8)
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#endif
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VECTOR_BLOCK(8, 4)
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VECTOR_BLOCK(8, 2)
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VECTOR_BLOCK(8, 1)
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VECTOR_BLOCK(4, 4)
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VECTOR_BLOCK(4, 2)
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VECTOR_BLOCK(4, 1)
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#ifdef DOUBLE
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VECTOR_BLOCK(2, 4)
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VECTOR_BLOCK(2, 2)
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#endif
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/**
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* Handle calculation for row blocks in C_i of any size by dispatching into
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* macro-defined (inline) functions or by deferring to a simple generic
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* implementation. Note that the compiler can remove this awkward-looking
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* dispatching code while inlineing.
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*
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* @param[in] m Number of rows in block C_i.
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* @param[in] n Number of columns in block C_i.
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* @param[in] first_row Index of first row of the block C_i (relative to C).
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* @param[in] A Pointer to input matrix A (note: all of it).
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* @param[in] k Number of columns in A and rows in B.
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* @param[in] B Pointer to current column block (panel) of input matrix B.
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* @param[inout] C Pointer to current column block (panel) of output matrix C.
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* @param[in] ldc Offset between elements in adjacent columns in C.
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* @param[in] alpha Scalar factor.
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* @param[in] offset Number of columns of A and rows of B to skip (for triangular matrices).
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* @param[in] off Running offset for handling triangular matrices.
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*/
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static inline void GEBP_block(BLASLONG m, BLASLONG n,
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BLASLONG first_row,
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const FLOAT * restrict A, BLASLONG k,
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const FLOAT * restrict B,
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FLOAT *restrict C, BLASLONG ldc,
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FLOAT alpha,
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BLASLONG offset, BLASLONG off)
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{
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if (trmm && left)
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off = offset + first_row;
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A += first_row * k;
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C += first_row;
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if (trmm) {
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if (backwards) {
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A += off * m;
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B += off * n;
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k -= off;
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} else {
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if (left) {
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k = off + m;
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} else {
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k = off + n;
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}
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}
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}
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#define BLOCK(bm, bn) \
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if (m == bm && n == bn) { \
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GEBP_block_##bm##_##bn(A, k, B, C, ldc, alpha); \
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return; \
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}
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#if UNROLL_M == 16
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BLOCK(16, 4); BLOCK(16, 2); BLOCK(16, 1);
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#endif
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#if UNROLL_N == 8
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BLOCK(8, 8); BLOCK(4, 8);
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#endif
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BLOCK(8, 4); BLOCK(8, 2); BLOCK(8, 1);
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BLOCK(4, 4); BLOCK(4, 2); BLOCK(4, 1);
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#ifdef DOUBLE
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BLOCK(2, 4);
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BLOCK(2, 2);
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#endif
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#undef BLOCK
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/* simple implementation for smaller block sizes: */
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FLOAT Caux[m][n] __attribute__ ((aligned (16)));
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/*
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* Peel off first iteration (i.e., column of A) for initializing Caux
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*/
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for (BLASLONG i = 0; i < m; i++)
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for (BLASLONG j = 0; j < n; j++)
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Caux[i][j] = A[i] * B[j];
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for (BLASLONG kk = 1; kk < k; kk++)
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for (BLASLONG i = 0; i < m; i++)
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for (BLASLONG j = 0; j < n; j++)
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Caux[i][j] += A[i + kk * m] * B[j + kk * n];
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for (BLASLONG i = 0; i < m; i++)
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for (BLASLONG j = 0; j < n; j++)
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if (trmm) {
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C[i + j * ldc] = alpha * Caux[i][j];
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} else {
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C[i + j * ldc] += alpha * Caux[i][j];
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}
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}
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/**
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* Handle a column block (panel) of C and B while calculating C += alpha(A * B).
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*
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* @param[in] num_cols Number of columns in the block (in C and B).
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* @param[in] first_col First column of the current block (in C and B).
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* @param[in] A Pointer to input matrix A.
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* @param[in] bk Number of columns in A and rows in B.
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* @param[in] B Pointer to input matrix B (note: all of it).
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* @param[in] bm Number of rows in C and A.
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* @param[inout] C Pointer to output matrix C (note: all of it).
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* @param[in] ldc Offset between elements in adjacent columns in C.
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* @param[in] alpha Scalar factor.
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* @param[in] offset Number of columns of A and rows of B to skip (for triangular matrices).
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*/
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static inline void GEBP_column_block(BLASLONG num_cols, BLASLONG first_col,
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const FLOAT *restrict A, BLASLONG bk,
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const FLOAT *restrict B, BLASLONG bm,
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FLOAT *restrict C, BLASLONG ldc,
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FLOAT alpha,
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BLASLONG const offset) {
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FLOAT *restrict C_i = C + first_col * ldc;
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/*
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* B is in column-order with n_r packed row elements, which does
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* not matter -- we always move in full such blocks of
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* column*pack
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*/
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const FLOAT *restrict B_i = B + first_col * bk;
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BLASLONG off = 0;
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if (trmm) {
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if (left) {
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off = offset;
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} else {
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off = -offset + first_col;
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}
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}
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/*
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* Calculate C_aux := A * B_j
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* then unpack C_i += alpha * C_aux.
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*
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* For that purpose, further partition C_aux and A into blocks
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* of m_r (unroll_m) rows, or powers-of-2 if smaller.
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*/
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BLASLONG row = 0;
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for (BLASLONG block_size = unroll_m; block_size > 0; block_size /= 2)
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for (; bm - row >= block_size; row += block_size)
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GEBP_block(block_size, num_cols, row, A, bk, B_i, C_i,
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ldc, alpha, offset, off);
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}
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/**
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* Inner kernel for matrix-matrix multiplication. C += alpha (A * B)
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* where C is an m-by-n matrix, A is m-by-k and B is k-by-n. Note that A, B, and
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* C are pointers to submatrices of the actual matrices.
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*
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* For triangular matrix multiplication, calculate B := alpha (A * B) where A
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* or B can be triangular (in case of A, the macro LEFT will be defined).
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*
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* @param[in] bm Number of rows in C and A.
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* @param[in] bn Number of columns in C and B.
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* @param[in] bk Number of columns in A and rows in B.
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* @param[in] alpha Scalar factor.
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* @param[in] ba Pointer to input matrix A.
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* @param[in] bb Pointer to input matrix B.
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* @param[inout] C Pointer to output matrix C.
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* @param[in] ldc Offset between elements in adjacent columns in C.
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* @param[in] offset Number of columns of A and rows of B to skip (for triangular matrices).
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* @returns 0 on success.
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*/
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int CNAME(BLASLONG bm, BLASLONG bn, BLASLONG bk, FLOAT alpha,
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FLOAT *restrict ba, FLOAT *restrict bb,
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FLOAT *restrict C, BLASLONG ldc
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#ifdef TRMMKERNEL
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, BLASLONG offset
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#endif
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)
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{
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if ( (bm == 0) || (bn == 0) || (bk == 0) || (alpha == ZERO))
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return 0;
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/*
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* interface code allocates buffers for ba and bb at page
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* granularity (i.e., using mmap(MAP_ANONYMOUS), so enable the compiler
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* to make use of the fact in vector load operations.
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*/
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ba = __builtin_assume_aligned(ba, 16);
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bb = __builtin_assume_aligned(bb, 16);
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/*
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* Use offset and off even when compiled as SGEMMKERNEL to simplify
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* function signatures and function calls.
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*/
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#ifndef TRMMKERNEL
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BLASLONG const offset = 0;
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#endif
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/*
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* Partition B and C into blocks of n_r (unroll_n) columns, called B_i
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* and C_i. For each partition, calculate C_i += alpha * (A * B_j).
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*
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* For remaining columns that do not fill up a block of n_r, iteratively
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* use smaller block sizes of powers of 2.
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*/
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BLASLONG col = 0;
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for (BLASLONG block_size = unroll_n; block_size > 0; block_size /= 2)
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for (; bn - col >= block_size; col += block_size)
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GEBP_column_block(block_size, col, ba, bk, bb, bm, C, ldc, alpha, offset);
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return 0;
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}
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