CS 33. Caches. CS33 Intro to Computer Systems XVIII 1 Copyright 2017 Thomas W. Doeppner. All rights reserved.

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CS 33 Caches CS33 Intro to Computer Systems XVIII 1 Copyright 2017 Thomas W. Doeppner. All rights reserved.

Cache Performance Metrics Miss rate fraction of memory references not found in cache (misses / accesses) = 1 hit rate typical numbers (in percentages): Hit time» 3-10% for L1» can be quite small (e.g., < 1%) for L2, depending on size, etc. time to deliver a line in the cache to the processor» includes time to determine whether the line is in the cache typical numbers: Miss penalty» 1-2 clock cycles for L1» 5-20 clock cycles for L2 additional time required because of a miss» typically 50-200 cycles for main memory (trend: increasing!) CS33 Intro to Computer Systems XVIII 2 Copyright 2017 Thomas W. Doeppner. All rights reserved.

Let s Think About Those Numbers Huge difference between a hit and a miss could be 100x, if just L1 and main memory Would you believe 99% hit rate is twice as good as 97%? consider: cache hit time of 1 cycle miss penalty of 100 cycles average access time: 97% hits:.97 * 1 cycle + 0.03 * 100 cycles 4 cycles 99% hits:.99 * 1 cycle + 0.01 * 100 cycles 2 cycles This is why miss rate is used instead of hit rate CS33 Intro to Computer Systems XVIII 3 Copyright 2017 Thomas W. Doeppner. All rights reserved.

Locality Principle of Locality: programs tend to use data and instructions with addresses near or equal to those they have used recently Temporal locality: recently referenced items are likely to be referenced again in the near future Spatial locality: items with nearby addresses tend to be referenced close together in time CS33 Intro to Computer Systems XVIII 4 Copyright 2017 Thomas W. Doeppner. All rights reserved.

Locality Example sum = 0; for (i = 0; i < n; i++) sum += a[i]; return sum; Data references reference array elements in succession (stride-1 reference pattern) reference variable sum each iteration Instruction references reference instructions in sequence. cycle through loop repeatedly Spatial locality Temporal locality Spatial locality Temporal locality CS33 Intro to Computer Systems XVIII 5 Copyright 2017 Thomas W. Doeppner. All rights reserved.

Qualitative Estimates of Locality Claim: being able to look at code and get a qualitative sense of its locality is a key skill for a professional programmer Question: does this function have good locality with respect to array a? int sum_array_rows(int a[m][n]){ int i, j, sum = 0; for (i = 0; i < M; i++) for (j = 0; j < N; j++) sum += a[i][j]; return sum; CS33 Intro to Computer Systems XVIII 6 Copyright 2017 Thomas W. Doeppner. All rights reserved.

Quiz 1 Does this function have good locality with respect to array a? a) yes b) no int sum_array_cols(int a[m][n]) { int i, j, sum = 0; for (j = 0; j < N; j++) for (i = 0; i < M; i++) sum += a[i][j]; return sum; CS33 Intro to Computer Systems XVIII 7 Copyright 2017 Thomas W. Doeppner. All rights reserved.

Writing Cache-Friendly Code Make the common case go fast focus on the inner loops of the core functions Minimize the misses in the inner loops repeated references to variables are good (temporal locality) stride-1 reference patterns are good (spatial locality) Key idea: our qualitative notion of locality is quantified through our understanding of cache memories CS33 Intro to Computer Systems XVIII 8 Copyright 2017 Thomas W. Doeppner. All rights reserved.

Matrix Multiplication Example Description: multiply N x N matrices O(N 3 ) total operations N reads per source element N values summed per destination» but may be able to hold in register Variable sum /* ijk */ held in register for (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; /* ikj */ for (i=0; i<n; i++) { for (k=0; k<n; k++) { r = a[i][k]; for (j=0; j<n; j++) c[i][j] += r * b[k][j]; CS33 Intro to Computer Systems XVIII 9 Copyright 2017 Thomas W. Doeppner. All rights reserved.

Miss-Rate Analysis for Matrix Multiply Assume: Block size = 32B (big enough for four 64-bit words) matrix dimension (N) is very large» approximate 1/N as 0.0 cache is not big enough to hold multiple rows Analysis method: look at access pattern of inner loop j k j i i = * k C A B CS33 Intro to Computer Systems XVIII 10 Copyright 2017 Thomas W. Doeppner. All rights reserved.

Layout of C Arrays in Memory (review) C arrays allocated in row-major order each row in contiguous memory locations Stepping through columns in one row: for (i = 0; i < N; i++) sum += a[0][i]; accesses successive elements if block size (B) > 4 bytes, exploit spatial locality» compulsory miss rate = 4 bytes / B Stepping through rows in one column: for (i = 0; i < n; i++) sum += a[i][0]; accesses distant elements no spatial locality!» compulsory miss rate = 1 (i.e. 100%) CS33 Intro to Computer Systems XVIII 11 Copyright 2017 Thomas W. Doeppner. All rights reserved.

Matrix Multiplication (ijk) /* ijk */ for (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; Misses per inner loop iteration: A B C 0.25 1.0 0.0 Inner loop: (i,*) (*,j) (i,j) A B C Row-wise Columnwise Fixed CS33 Intro to Computer Systems XVIII 12 Copyright 2017 Thomas W. Doeppner. All rights reserved.

Matrix Multiplication (jik) /* jik */ for (j=0; j<n; j++) { for (i=0; i<n; i++) { sum = 0.0; for (k=0; k<n; k++) sum += a[i][k] * b[k][j]; c[i][j] = sum Misses per inner loop iteration: A B C 0.25 1.0 0.0 Inner loop: (i,*) (*,j) (i,j) A B C Row-wise Columnwise Fixed CS33 Intro to Computer Systems XVIII 13 Copyright 2017 Thomas W. Doeppner. All rights reserved.

Matrix Multiplication (kij) /* kij */ for (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]; Inner loop: (i,k) (k,*) (i,*) A B C Fixed Row-wise Row-wise Misses per inner loop iteration: A B C 0.0 0.25 0.25 CS33 Intro to Computer Systems XVIII 14 Copyright 2017 Thomas W. Doeppner. All rights reserved.

Matrix Multiplication (ikj) /* ikj */ for (i=0; i<n; i++) { for (k=0; k<n; k++) { r = a[i][k]; for (j=0; j<n; j++) c[i][j] += r * b[k][j]; Inner loop: (i,k) (k,*) (i,*) A B C Fixed Row-wise Row-wise Misses per inner loop iteration: A B C 0.0 0.25 0.25 CS33 Intro to Computer Systems XVIII 15 Copyright 2017 Thomas W. Doeppner. All rights reserved.

Matrix Multiplication (jki) /* jki */ 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; Inner loop: (*,k) (k,j) (*,j) A B C Fixed Columnwise Columnwise Misses per inner loop iteration: A B C 1.0 0.0 1.0 CS33 Intro to Computer Systems XVIII 16 Copyright 2017 Thomas W. Doeppner. All rights reserved.

Matrix Multiplication (kji) /* kji */ for (k=0; k<n; k++) { for (j=0; j<n; j++) { r = b[k][j]; for (i=0; i<n; i++) c[i][j] += a[i][k] * r; Misses per inner loop iteration: A B C 1.0 0.0 1.0 Inner loop: (*,k) (k,j) (*,j) A B C Fixed Columnwise Columnwise CS33 Intro to Computer Systems XVIII 17 Copyright 2017 Thomas W. Doeppner. All rights reserved.

Summary of Matrix Multiplication for (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; for (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]; ijk (& jik): 2 loads, 0 stores misses/iter = 1.25 kij (& ikj): 2 loads, 1 store misses/iter = 0.5 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; jki (& kji): 2 loads, 1 store misses/iter = 2.0 CS33 Intro to Computer Systems XVIII 18 Copyright 2017 Thomas W. Doeppner. All rights reserved.

Core i7 Matrix Multiply Performance 60 jki / kji 50 Cycles per inner loop iteration 40 30 20 10 ijk / jik kij / ikj jki kji ijk jik kij ikj 0 50 100 150 200 250 300 350 400 450 500 550 600 650 700 750 Array size (n) CS33 Intro to Computer Systems XVIII 19 Copyright 2017 Thomas W. Doeppner. All rights reserved.

Matrix Multiplication: More Analysis /* Multiply n x n matrices a and b */ void mmm(int n, double a[n][n], double b[n][n], double c[n][n]) { int i, j, k; for (i = 0; i < n; i++) for (j = 0; j < n; j++) for (k = 0; k < n; k++) c[i][j] += a[i][k]*b[k][j]; c = i a * b j CS33 Intro to Computer Systems XVIII 20 Copyright 2017 Thomas W. Doeppner. All rights reserved.

Cache-Miss Analysis Assume: matrix elements are doubles cache block = 8 doubles cache size C << n (much smaller than n) n First iteration: n/8 + n = 9n/8 misses = * afterwards in cache: (schematic) = * 8 wide CS33 Intro to Computer Systems XVIII 21 Copyright 2017 Thomas W. Doeppner. All rights reserved.

Cache-Miss Analysis Assume: matrix elements are doubles cache block = 8 doubles cache size C << n (much smaller than n) Second iteration: again: n/8 + n = 9n/8 misses Total misses: 9n/8 * n 2 = (9/8) * n 3 = * 8 wide n CS33 Intro to Computer Systems XVIII 22 Copyright 2017 Thomas W. Doeppner. All rights reserved.

Blocked Matrix Multiplication /* Multiply n x n matrices a and b */ void mmm(int n, double a[n][n], double b[n][n], double c[n][n]) { int i, j, k; for (i = 0; i < n; i+=b) for (j = 0; j < n; j+=b) for (k = 0; k < n; k+=b) /* B x B mini matrix multiplications */ for (i1 = i; i1 < i+b; i1++) for (j1 = j; j1 < j+b; j1++) for (k1 = k; k1 < k+b; k1++) c[i1][j1] += a[i1][k1]*b[k1][j1]; j1 c = i1 a * b + c Matrix Block size B x B CS33 Intro to Computer Systems XVIII 23 Copyright 2017 Thomas W. Doeppner. All rights reserved.

Cache-Miss Analysis Assume: cache block = 8 doubles cache size C << n (much smaller than n) three matrix blocks fit into cache: 3B 2 < C First (matrix block) iteration: B 2 /8 misses for each block 2n/B * B 2 /8 = nb/4 (omitting matrix c) = * n/b blocks afterwards in cache (schematic) = Matrix Block size B x B * CS33 Intro to Computer Systems XVIII 24 Copyright 2017 Thomas W. Doeppner. All rights reserved.

Cache-Miss Analysis Assume: cache block = 8 doubles cache size C << n (much smaller than n) three matrix blocks fit into cache: 3B 2 < C Second (matrix block) iteration: same as first iteration 2n/B * B 2 /8 = nb/4 = * n/b blocks Total misses: nb/4 * (n/b) 2 = n 3 /(4B) Matrix Block size B x B CS33 Intro to Computer Systems XVIII 25 Copyright 2017 Thomas W. Doeppner. All rights reserved.

Summary No blocking: (9/8) * n 3 Blocking: 1/(4B) * n 3 Suggest largest possible block size B, but limit 3B 2 < C! Reason for dramatic difference: matrix multiplication has inherent temporal locality:» input data: 3n 2, computation 2n 3» every array element used O(n) times! but program has to be written properly CS33 Intro to Computer Systems XVIII 26 Copyright 2017 Thomas W. Doeppner. All rights reserved.

Quiz 2 What is the smallest value of B (in 8-byte doubles) for which the cache-miss analysis works? a) 1 b) 2 c) 4 d) 8 CS33 Intro to Computer Systems XVIII 27 Copyright 2017 Thomas W. Doeppner. All rights reserved.

Blocking vs. ikj j1 c = i1 a * b + c n 3 /(4*8) misses (i,k) (k,*) = (i,*) 2*n 3 /8 misses CS33 Intro to Computer Systems XVIII 28 Copyright 2017 Thomas W. Doeppner. All rights reserved.

Blocking vs. ikj $./matmult_ikj ikj:.608 secs $./matmult_blocked Blocked:.880 secs CS33 Intro to Computer Systems XVIII 29 Copyright 2017 Thomas W. Doeppner. All rights reserved.

Why is ikj Faster? c Prefetching the processor detects sequential (stride-1) accesses to memory and issues loads before they are needed j1 = i1 a * b + c (i,k) = (k,*) (i,*) CS33 Intro to Computer Systems XVIII 30 Copyright 2017 Thomas W. Doeppner. All rights reserved.

Concluding Observations Programmer can optimize for cache performance organize data structures appropriately take care in how data structures are accesses» nested loop structure» blocking is a general technique All systems favor cache-friendly code getting absolute optimum performance is very platform specific» cache sizes, line sizes, associativities, etc. can get most of the advantage with generic code» keep working set reasonably small (temporal locality)» use small strides (spatial locality) CS33 Intro to Computer Systems XVIII 31 Copyright 2017 Thomas W. Doeppner. All rights reserved.

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