15213 Recitation Section C• Loop Unrolling• BlockingShimin ChenOct. 21, 2002Outline215213 Recitation C Shimin ChenLab 4 Reminders• Due: Thursday, Oct 24, 11:59pm• Submission is online *NOT* automatic– Class web page > Labs > L4– http://www.cs.cmu.edu/afs/cs/academic/class/15213-f02/www/L4.html315213 Recitation C Shimin ChenLoop Unrolling• Combine multiple iterations into single loop body• Amortizes loop overhead across multiple iterations• Finish extras at endvoid combine5(vec_ptr v, int *dest){int length = vec_length(v);int limit = length-2;int *data = get_vec_start(v);int sum = 0;int i;/* Combine 3 elements at a time */for (i = 0; i < limit; i+=3) {sum += data[i] + data[i+1]+ data[i+2];}/* Finish any remaining elements */for (; i < length; i++) {sum += data[i];}*dest = sum;}415213 Recitation C Shimin ChenPractice Problem• Problem 5.12 and 5.13515213 Recitation C Shimin ChenSolution 5.12void inner5(vec_ptr u, vec_ptr v, data_t *dest){ int i;int length = vec_length(u);int limit = length-3;data_t *udata = get_vec_start(u);data_t *vdata = get_vec_start(v);data_t sum = (data_t) 0;/* Do four elements at a time */for (i = 0; i < limit; i+=4) {sum += udata[i]*vdata[i] + udata[i+1]*vdata[i+1] + udata[i+2]*vdata[i+2] + udata[i+3]*vdata[i+3];}/* Finish off any remaining elements */for (; i < length; i++) sum += udata[i] * vdata[i]; *dest = sum;}615213 Recitation C Shimin ChenSolution 5.12A. We must perform two loads per element to read values for udata and vdata. There is only one unit to perform these loads, and it requires one cycle.B. The performance for floating point is still limited by the 3 cycle latency of the floating-point adder.715213 Recitation C Shimin ChenSolution 5.13void inner6(vec_ptr u, vec_ptr v, data_t *dest) { int i;int length = vec_length(u);int limit = length-3;data_t *udata = get_vec_start(u);data_t *vdata = get_vec_start(v);data_t sum0 = (data_t) 0;data_t sum1 = (data_t) 0;/* Do four elements at a time */for (i = 0; i < limit; i+=4) {sum0 += udata[i] * vdata[i];sum1 += udata[i+1] * vdata[i+1];sum0 += udata[i+2] * vdata[i+2];sum1 += udata[i+3] * vdata[i+3];}/* Finish off any remaining elements */for (; i < length; i++) sum0 = sum0 + udata[i] * vdata[i];*dest = sum0 + sum1;}815213 Recitation C Shimin ChenSolution 5.13• For each element, we must perform two loads with a unit that can only load one value per clock cycle. • We must also perform one floating-point multiplication with a unit that can only perform one multiplication every two clock cycles. • Both of these factors limit the CPE to 2.915213 Recitation C Shimin ChenSummary of Matrix Multiplicationfor (i=0; i<n; i++) {for (j=0; j<n; j++) {sum = 0.0;for (k=0; k<n; k++) sum += a[i][k] * b[k][j];c[i][j] = sum;}} ijk (& jik):• 2 loads, 0 stores• misses/iter = 1.25for (k=0; k<n; k++) {for (i=0; i<n; i++) {r = a[i][k];for (j=0; j<n; j++)c[i][j] += r * b[k][j]; }}for (j=0; j<n; j++) {for (k=0; k<n; k++) {r = b[k][j];for (i=0; i<n; i++)c[i][j] += a[i][k] * r;}}kij (& ikj):• 2 loads, 1 store• misses/iter = 0.5jki (& kji):• 2 loads, 1 store• misses/iter = 2.01015213 Recitation C Shimin ChenImproving Temporal Locality by Blocking• Example: Blocked matrix multiplication– “block” (in this context) does not mean “cache block”.– Instead, it mean a sub-block within the matrix.– Example: N = 8; sub-block size = 4C11= A11B11+ A12B21 C12= A11B12+ A12B22C21= A21B11+ A22B21 C22= A21B12+ A22B22A11A12A21A22B11B12B21B22X= C11C12C21C22Key idea: Sub-blocks (i.e., Axy) can be treated just like scalars.1115213 Recitation C Shimin ChenBlocked Matrix Multiply (bijk)for (jj=0; jj<n; jj+=bsize) {for (i=0; i<n; i++)for (j=jj; j < min(jj+bsize,n); j++)c[i][j] = 0.0;for (kk=0; kk<n; kk+=bsize) { for (i=0; i<n; i++) {for (j=jj; j < min(jj+bsize,n); j++) { sum = 0.0for (k=kk; k < min(kk+bsize,n); k++) {sum += a[i][k] * b[k][j];}c[i][j] += sum;}}}}– Provides temporal locality as block is reused multiple times– Constant cache performance1215213 Recitation C Shimin ChenBlocked Matrix Multiply Analysis– Innermost loop pair multiplies a 1 X bsize sliver of A by a bsize X bsize block of B and accumulates into 1 X bsize sliver of C– Loop over i steps through n row slivers of A & C, using same BA B Cblock reused ntimes in successionrow sliver accessedbsize timesUpdate successiveelements of sliveri ikkkk jjjjfor (i=0; i<n; i++) {for (j=jj; j < min(jj+bsize,n); j++) { sum = 0.0for (k=kk; k < min(kk+bsize,n); k++) {sum += a[i][k] * b[k][j];}c[i][j] += sum;}1315213 Recitation C Shimin ChenPentium Blocked Matrix Multiply Performance• Blocking (bijk and bikj) improves performance by a factor of two over unblocked versions (ijk and jik)– relatively insensitive to array size.0102030405060255075100125150175200225250275300325350375400Array size (n)Cycles/iterationkjijkikijikjjikijkbijk (bsize = 25)bikj (bsize = 25)1415213 Recitation C Shimin ChenSummaryAll systems favor “cache friendly code”• Can get most of the advantage with generic optimizations:– Keep working set reasonably small (temporal locality)– Use small strides (spatial locality)• Getting absolute optimum performance is very platform specific– Cache sizes, Line sizes, Associativities,
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