More Loop Optimizations
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1 Spring 2010 More Loop Optimizations
2 5 Outline Strength Reduction Loop Test Replacement Loop Invariant Code Motion SIMDization with SSE Saman Amarasinghe MIT Fall 1998
3 Strength Reduction Replace expensive operations in an expression using cheaper ones Not a data-flow problem Algebraic simplification Example: a*4 a << 2 Saman Amarasinghe MIT Fall 1998
4 Strength Reduction In loops reduce expensive operations in expressions in to cheaper ones by using the previously calculated value Saman Amarasinghe MIT Fall 1998
5 t = 202 for j = 1 to 100 t = t - 2 A(j) = t Strength Reduction Saman Amarasinghe MIT Fall 1998
6 t = 202 for j = 1 to 100 t = t - 2 Strength Reduction *(abase + 4*j) = t Saman Amarasinghe MIT Fall 1998
7 t = 202 for j = 1 to 100 t = t - 2 Strength Reduction *(abase + 4*j) = t Basic Induction variable: J = 1, 2, 3, 4,.. Induction variable 200-2*j t = 202, 200, 198, 196,.. Induction variable abase+4*j: abase+4*j = abase+4, abase+8, abase+12, abase+14,. Saman Amarasinghe MIT Fall 1998
8 t = 202 for j = 1 to 100 t = t - 2 Strength Reduction *(abase + 4*j) = t Basic Induction variable: J = 1, 2, 3, 4, Induction variable 200-2*j t = 202, 200, 198, 196,.. Induction variable abase+ 4*j: abase+4*j = abase+4, abase+8, abase+12, abase+14,. Saman Amarasinghe MIT Fall 1998
9 t = 202 for j = 1 to 100 t = t - 2 Strength Reduction *(abase + 4*j) = t Basic Induction variable: J = 1, 2, 3, 4, Induction variable 200-2*j t = 202, 200, 198, 196, Induction variable abase+4*j: abase+4*j = abase+4, abase+8, abase+12, abase+14,. Saman Amarasinghe MIT Fall 1998
10 t = 202 for j = 1 to 100 t = t - 2 Strength Reduction *(abase + 4*j) = t Basic Induction variable: J = 1, 2, 3, 4, Induction variable 200-2*j t = 202, 200, 198, 196, Induction variable abase+4*j: abase+4*j = abase+4, abase+8, abase+12, abase+14, Saman Amarasinghe MIT Fall 1998
11 Strength Reduction Algorithm For a dependent induction variable k = a*j + b for j = 1 to 100 *(abase + 4*j) = j Saman Amarasinghe MIT Fall 1998
12 Strength Reduction Algorithm For a dependent induction variable k = a*j + b Add a pre-header k = a*jinit + b t = abase + 4*1 for j = 1 to 100 *(abase + 4*j) = j Saman Amarasinghe MIT Fall 1998
13 Strength Reduction Algorithm For a dependent induction variable k = a*j + b Add a pre-header k = a*jinit + b Next to j = j + c add k =k + a*c t = abase + 4*1 for j = 1 to 100 *(abase + 4*j) = j t = t + 4 Saman Amarasinghe MIT Fall 1998
14 Strength Reduction Algorithm For a dependent induction variable k = a*j + b Add a pre-header k = a*jinit + b Next to j = j + c add k =k + a*c Use k instead of k t = abase + 4*1 for j = 1 to 100 *(t) = j t = t + 4 Saman Amarasinghe MIT Fall 1998
15 Example double A[256], B[256][256] j = 1 while(j>100) A[j] = B[j][j] j = j + 2 Saman Amarasinghe MIT Fall 2006
16 Example double A[256], B[256][256] j = 1 while(j>100) *(&A + 4*j) = *(&B + 4*(256*j + j)) j = j + 2 Saman Amarasinghe MIT Fall 2006
17 Example double A[256], B[256][256] j = 1 while(j>100) *(&A + 4*j) = *(&B + 4*(256*j + j)) j = j + 2 Base Induction Variable: j Saman Amarasinghe MIT Fall 2006
18 Example double A[256], B[256][256] j = 1 while(j>100) *(&A + 4*j) = *(&B + 4*(256*j + j)) j = j + 2 Base Induction Variable: j Dependent Induction Variable: a = &A + 4*j Saman Amarasinghe MIT Fall 2006
19 Example double A[256], B[256][256] j = 1 a = &A + 4 while(j>100) *(&A + 4*j) = *(&B + 4*(256*j + j)) j = j + 2 Base Induction Variable: j Dependent Induction Variable: a = &A + 4*j Saman Amarasinghe MIT Fall 2006
20 Example double A[256], B[256][256] j = 1 a = &A + 4 while(j>100) *(&A + 4*j) = *(&B + 4*(256*j + j)) j = j + 2 a = a + 8 Base Induction Variable: j Dependent Induction Variable: a = &A + 4*j Saman Amarasinghe MIT Fall 2006
21 Example double A[256], B[256][256] j = 1 a = &A + 4 while(j>100) *a = *(&B + 4*(256*j + j)) j = j + 2 a = a + 8 Base Induction Variable: j Dependent Induction Variable: a = &A + 4*j Saman Amarasinghe MIT Fall 2006
22 Example double A[256], B[256][256] j = 1 a = &A + 4 while(j>100) *a = *(&B + 4*(256*j + j)) j = j + 2 a = a + 8 Base Induction Variable: j Dependent Induction Variable: b = &B + 4*257*j Saman Amarasinghe MIT Fall 2006
23 Example double A[256], B[256][256] j = 1 a = &A + 4 b = &B while(j>100) *a = *(&B + 4*(256*j + j)) j = j + 2 a = a + 8 Base Induction Variable: j Dependent Induction Variable: b = &B + 4*257*j Saman Amarasinghe MIT Fall 2006
24 Example double A[256], B[256][256] j = 1 a = &A + 4 b = &B while(j>100) *a = *(&B + 4*(256*j + j)) j = j + 2 a = a + 8 b = b Base Induction Variable: j Dependent Induction Variable: b = &B + 4*257*j Saman Amarasinghe MIT Fall 2006
25 Example double A[256], B[256][256] j = 1 a = &A + 4 b = &B while(j>100) *a = *b j = j + 2 a = a + 8 b = b Base Induction Variable: j Dependent Induction Variable: b = &B + 4*257*j Saman Amarasinghe MIT Fall 2006
26 Example double A[256], B[256][256] j = 1 a = &A + 4 b = &B while(j>100) *a = *b j = j + 2 a = a + 8 b = b Saman Amarasinghe MIT Fall 2006
27 5 Outline Strength Reduction Loop Test Replacement Loop Invariant Code Motion SIMDization with SSE Saman Amarasinghe MIT Fall 1998
28 Loop Test Replacement Eliminate basic induction variable used only for calculating other induction variables Saman Amarasinghe MIT Fall 1998
29 Loop Test Replacement Eliminate basic induction variable used only for calculating other induction variables double A[256], B[256][256] j = 1 while(j>100) A[j] = B[j][j] Saman Amarasinghe MIT Fall 1998
30 Loop Test Replacement Eliminate basic induction variable used only for calculating other induction variables double A[256], B[256][256] j = 1 a = &A + 4 b = &B while(j>100) *a = *b j = j + 2 a = a + 8 b = b Saman Amarasinghe MIT Fall 1998
31 Loop Test Replacement Eliminate basic induction variable used only for calculating other induction variables double A[256], B[256][256] j = 1 a = &A + 4 b = &B while(j>100) *a = *b j = j + 2 a = a + 8 b = b Saman Amarasinghe MIT Fall 1998 J is only used for the loop bound
32 Loop Test Replacement Eliminate basic induction variable used only for calculating other induction variables double A[256], B[256][256] j = 1 a = &A + 4 b = &B J is only used for the loop bound while(j>100) Use a dependent IV (a or b) *a = *b j = j + 2 a = a + 8 b = b Saman Amarasinghe MIT Fall 1998
33 Loop Test Replacement Eliminate basic induction variable used only for calculating other induction variables double A[256], B[256][256] j = 1 a = &A + 4 b = &B J is only used for the loop bound while(j>100) Use a dependent IV (a or b) *a = *b Lets choose a j = j + 2 a = a + 8 b = b Saman Amarasinghe MIT Fall 1998
34 Loop Test Replacement Eliminate basic induction variable used only for calculating other induction variables double A[256], B[256][256] j = 1 a = &A + 4 b = &B J is only used for the loop bound while(j>100) Use a dependent IV (a or b) *a = *b Lets choose a j = j + 2 j > 100 a > &A a = a + 8 b = b Saman Amarasinghe MIT Fall 1998
35 Loop Test Replacement Eliminate basic induction variable used only for calculating other induction variables double A[256], B[256][256] j = 1 a = &A + 4 b = &B while(a>&a+800) J is only used for the loop bound while(a>&a+800) Use a dependent IV (a or b) *a = *b Lets choose a j = j + 2 j > 100 a > &A a = a + 8 Replace the loop condition b = b Saman Amarasinghe MIT Fall 1998
36 Loop Test Replacement Eliminate basic induction variable used only for calculating other induction variables double A[256], B[256][256] a = &A + 4 b = &B while(a>&a+800) J is only used for the loop bound while(a>&a+800) Use a dependent IV (a or b) *a = *b Lets choose a j > 100 a > &A a = a + 8 Replace the loop condition b = b Get rid of j Saman Amarasinghe MIT Fall 1998
37 Loop Test Replacement Eliminate basic induction variable used only for calculating other induction variables double A[256], B[256][256] a = &A + 4 b = &B while(a>&a+800) *a = *b a = a + 8 b = b Saman Amarasinghe MIT Fall 1998
38 Loop Test Replacement Algorithm If basic induction variable J is only used for calculating other induction variables Select an induction variable k in the family of J (K = a*j + b) Replace a comparison such as by if (J > X) goto L1 if(k > a*x + b) goto L1 if a is positive if(k < a*x + b) goto L1 if a is negative If J is live at any exit from loop, recompute J = (K - b)/a Saman Amarasinghe MIT Fall 1998
39 5 Outline Strength Reduction Loop Test Replacement Loop Invariant Code Motion SIMDization with SSE Saman Amarasinghe MIT Fall 1998
40 Loop Invariant Code Motion If a computation produces the same value in every loop iteration, move it out of the loop Same idea as with induction variables Variables not updated in the loop are loop invariant Expressions of loop invariant variables are loop invariant Variables assigned only loop invariant expressions are loop invariant Saman Amarasinghe MIT Fall 1998
41 Loop Invariant Code Motion If a computation produces the same value in every loop iteration, move it out of the loop for i = 1 to N x = x + 1 for j = 1 to N a(i,j) = 100*N + 10*i + j + x Saman Amarasinghe MIT Fall 1998
42 Loop Invariant Code Motion If a computation produces the same value in every loop iteration, move it out of the loop for i = 1 to N x = x + 1 for j = 1 to N a(i,j) = 100*N + 10*i + j + x Saman Amarasinghe MIT Fall 1998
43 Loop Invariant Code Motion If a computation produces the same value in every loop iteration, move it out of the loop t1 = 100*N for i = 1 to N x = x + 1 for j = 1 to N a(i,j) = 100*N + 10*i + j + x Saman Amarasinghe MIT Fall 1998
44 Loop Invariant Code Motion If a computation produces the same value in every loop iteration, move it out of the loop t1 = 100*N for i = 1 to N x = x + 1 for j = 1 to N a(i,j) = t1 + 10*i + j + x Saman Amarasinghe MIT Fall 1998
45 Loop Invariant Code Motion If a computation produces the same value in every loop iteration, move it out of the loop t1 = 100*N for i = 1 to N x = x + 1 for j = 1 to N a(i,j) = t1 + 10*i + j + x Saman Amarasinghe MIT Fall 1998
46 Loop Invariant Code Motion If a computation produces the same value in every loop iteration, move it out of the loop t1 = 100*N for i = 1 to N x = x + 1 for j = 1 to N a(i,j) = t1 + 10*i + j + x Saman Amarasinghe MIT Fall 1998
47 Loop Invariant Code Motion If a computation produces the same value in every loop iteration, move it out of the loop t1 = 100*N for i = 1 to N x = x + 1 t2 = t1 + 10*i + x for j = 1 to N a(i,j) = t1 + 10*i + j + x Saman Amarasinghe MIT Fall 1998
48 Loop Invariant Code Motion If a computation produces the same value in every loop iteration, move it out of the loop t1 = 100*N for i = 1 to N x = x + 1 t2 = t1 + 10*i + x for j = 1 to N a(i,j) = t2 + j Saman Amarasinghe MIT Fall 1998
49 5 Outline Strength Reduction Loop Test Replacement Loop Invariant Code Motion SIMDization with SSE Saman Amarasinghe MIT Fall 1998
50 SIMD Through SSE extensions Single Instruction Multiple Data Compute multiple identical operations in a single instruction Exploit fine grained parallelism Saman Amarasinghe MIT Fall 1998
51 SSE Registers bit registers: %xmm0 to %xmm16 Multiple interpretations for each register Each arithmetic operation comes in multiple versions 128 bit Double Quadword 64 bit Quadword 64 bit Quadword 32 bit Doubleword 32 bit Doubleword 32 bit Doubleword 32 bit Doubleword 16 bit word 16 bit word 16 bit word 16 bit word 16 bit word 16 bit word 16 bit word 16 bit word Saman Amarasinghe MIT Fall 1998
52 Data Transfer Moving Data From Memory or xmm registers MOVDQA OP1, OP2 Move aligned Double Quadword Can read or write to memory in 128 bit chunks If OP1 or OP2 are registers, they must be xmm registers Memory locations in OP1 or OP2 must be multiples of 16 MOVDQU OP1, OP2 Move unaligned Double Quadword Same as MOVDQA but memory addresses don t have to be multiples of 16 Saman Amarasinghe MIT Fall 1998
53 Data Transfer Moving Data From 64-bit registers MOVQ OP1, OP2 Move Double Quadword Can move from 64 bit register to xmm register or viceversa Writes to/reads from the lower 64 bits of xmm register Can also be used to read a 64-bit chunk to/from memory MOVQ %r11, %xmm1 %r11 %xmm Saman Amarasinghe MIT Fall 1998
54 Data Reordering Unpack and Interleave PUNPCKLDQ Low Doublewords Saman Amarasinghe MIT Fall 1998
55 Data Reordering Unpack and Interleave PUNPCKLQDQ Low Quadwords Saman Amarasinghe MIT Fall 1998
56 Arithmetic Arithmetic operations come in many flavors based on the datatype of the register specified in the instruction sufix Example: Addition PADDQ Add 64-bit Quadwords PADDD Add 32-bit Doublewords PADDW Add 16-bit words Example: Subtraction PSUBQ Subtract 64-bit Quadwords PSUBD Subtract 32-bit Doublewords PSUBW Subtract 16-bit words Saman Amarasinghe MIT Fall 1998
57 Putting It All Together Source Code for i = 1 to N A[i] = A[i] * b After Unrolling loop: mov (%rdi,%rax), %r10 mov (%rdi,%rbx), %rcx imul %r11, %r10 imul %r11, %rcx mov %r10, (%rdi,%rax) sub $8, %rax mov %rcx, (%rdi,%rbx) sub $8, %rbx jz loop mov (%rdi,%rax), %r10 Reading from consecutive addresses Mult by a loop invariant Writing to consecutive addresses Saman Amarasinghe MIT Fall 1998
58 Putting it all together Original Version loop: mov (%rdi,%rax), %r10 mov (%rdi,%rbx), %rcx imul %r11, %r10 imul %r11, %rcx mov %r10, (%rdi,%rax) sub $8, %rax mov %rcx, (%rdi,%rbx) sub $8, %rbx jz loop SSE Version movq %r11, %xmm2 punpckldq %xmm2, %xmm2 loop: movdqa (%rdi,%rax), %xmm0 pmuludq %xmm2, %xmm0 movdqa %xmm0, (%rdi, %rax) sub $ 8, % rax jz loop Saman Amarasinghe MIT Fall 1998
59 SSE Version Putting it all together movq %r11, %xmm2 punpckldq %xmm2, %xmm2 loop: movdqa (%rdi,%rax), %xmm0 pmuludq %xmm2, %xmm0 movdqa %xmm0, (%rdi, %rax) sub $8, % a jz loop Populate xmm2 with copies of %r11 sub $8, %rax Only one index is needed Saman Amarasinghe MIT Fall 1998
60 Conditions for SIMDization Consecutive iterations reading and writing from consecutive locations Consecutive iterations are independent of each other The easiest thing is to pattern match at the basic block level after unrolling loops Saman Amarasinghe MIT Fall 1998
61 MIT OpenCourseWare Computer Language Engineering Spring 2010 For information about citing these materials or our Terms of Use, visit:
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