1. Interprocedural Symbolic Range Propagation for Optimizing Compilers
Hansang Bae and Rudolf Eigenmann
Purdue University

2. Interprocedural Symbolic Range Propagation
Outline
Motivation
Symbolic Range Propagation
Interprocedural Symbolic Range Propagation
Experiments
Conclusions
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Interprocedural Symbolic Range Propagation

3. Interprocedural Symbolic Range Propagation
Motivation
Symbolic analysis is key to static analysis
X=Y+1 better than X=?
Range analysis has been successful
X=[0,10] better than X=?
Relevant questions
How much can we achieve interprocedurally?
Can interprocedural range propagation outperform other alternatives?
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Interprocedural Symbolic Range Propagation

4. Symbolic Range Propagation (Background)
Has been effective for compiler analyses
Abstract interpretation
Provides lower/upper bounds for variables
Sources of information
Variable definition
IF conditionals
Loop variables
Intersect with new source, union at merge
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Interprocedural Symbolic Range Propagation

5. Interprocedural Symbolic Range Propagation
SRP – Example
Example Code
Ranges
 X=[-INF,INF]
 X=[1,1]
 X=[2,2]
 X=[3,3]
 X=[2,3]
X = 1
IF (X.LE.N) THEN
X = 2*X
ELSE
X = X+2
ENDIF …
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Interprocedural Symbolic Range Propagation

6. Interprocedural Symbolic Range Propagation (ISRP)
Propagates ranges across procedure calls
Collects ISR at important program points
Entry to a subroutine
After a call site
SRP as the source of information
Iterative algorithm
Context sensitivity by procedure-cloning
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Interprocedural Symbolic Range Propagation

7. Interprocedural Symbolic Range Propagation
ISRP – Terminology
Symbolic Range
Mapping from a variable to its value
range, V = [LB, UB], where LB and UB
are expressions
Interprocedural Symbolic Range
Symbolic Range valid at relevant
program points - subroutine entries and
call sites (forward/backward)
Jump Function
Set of symbolic ranges expressed in
terms of input variables to a called
subroutine (actual parameters, global
variables)
Return Jump Function
terms of return variables to a calling
subroutine (formal parameters, global
Caller
Jump Function
Backward ISR
Forward ISR
Return Jump Function
Callee
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Interprocedural Symbolic Range Propagation

8. Interprocedural Symbolic Range Propagation
ISRP – Algorithm
Propagate_Interprocedural_Ranges() {
Initialize_Call_Graph()
while (there is any change in ISR) {
foreach Subroutine (bottom-up) {
Get_Backward_Interprocedural_Ranges()
Compute_Jump_Functions()
Compute_Return_Jump_Functions() }
Get_Forward_Interprocedural_Ranges()
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Interprocedural Symbolic Range Propagation

9. Interprocedural Symbolic Range Propagation
ISRP – Algorithm
Propagate_Interprocedural_Ranges()
Initialize_Call_Graph()
while (there is any change in ISR) {
foreach Subroutine (bottom-up) {
Get_Backward_Interprocedural_Ranges()
Compute_Jump_Functions()
Compute_Return_Jump_Functions() }
Get_Forward_Interprocedural_Ranges()
Get_Backward_Interprocedural_Ranges()
Transforms return jump functions to ISRs
Does nothing for leaf nodes of the call graph
Compute_Jump_Functions()
Computes intraprocedural ranges
Discards non-input-variables to the callee
Compute_Return_Jump_Functions()
Discards non-return-variables to the caller
Get_Forward_Interprocedural_Ranges()
Transforms jump functions to ISRs
Clone procedures if necessary
Keeps track of any changes
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Interprocedural Symbolic Range Propagation

10. ISRP – Example (1st iteration)
foreach subroutine (bottom-up)
Get_Backward_ISRs
Compute_Jump_Functions
Compute_Return_Jump_Functions
Get_Forward_ISRs (for call graph)
X=[1]
X=[1],Y=[2]
N=[10,40]
N=[10,40],T=[U+N]
V=[W+M]
PROGRAM MAIN
INTEGER X, Y
X = 1
Y = 2
α CALL A(X, Y)
END
SUBROUTINE A(T, U)
INTEGER N, T, U
DO N = 10, 40
β CALL B(T, U, N)
ENDDO
SUBROUTINE B(V, W, M)
INTEGER M, V, W
V = W + M
J :X=[1],Y=[2]
ISR:T=[1],U=[2]
J :N=[10,40]
ISR:T=[U+N]
ISR:M=[10,40]
RJ :V=[W+M]
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Interprocedural Symbolic Range Propagation

11. ISRP – Example (2nd iteration)
foreach subroutine (bottom-up)
Get_Backward_ISRs
Compute_Jump_Functions
Compute_Return_Jump_Functions
Get_Forward_ISRs (for call graph)
X=[1]
X=[1],Y=[2]
Y=[2]
T=[1],U=[2]
T=[1],U=[2],N=[10,40]
N=[10,40],U=[2]
U=[2]
M=[10,40]
M=[10,40],V=[W+M]
PROGRAM MAIN
INTEGER X, Y
X = 1
Y = 2
α CALL A(X, Y)
END
SUBROUTINE A(T, U)
INTEGER N, T, U
DO N = 10, 40
β CALL B(T, U, N)
ENDDO
SUBROUTINE B(V, W, M)
INTEGER M, V, W
V = W + M
J :X=[1],Y=[2]
ISR:Y=[2]
ISR:T=[1],U=[2]
J :U=[2],N=[10,40]
ISR:N=[10,40],T=[U+N]
RJ :U=[2]
ISR:M=[10,40]
RJ :M=[10,40],V=[W+M]
ISR:T=[1],U=[2]
ISR:M=[10,40],W=[2]
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Interprocedural Symbolic Range Propagation

12. ISRP – Example (3rd iteration)
foreach subroutine (bottom-up)
Get_Backward_ISRs
Compute_Jump_Functions
Compute_Return_Jump_Functions
Get_Forward_ISRs (for call graph)
X=[1]
X=[1],Y=[2]
Y=[2]
T=[1],U=[2]
T=[1],U=[2],N=[10,40]
N=[10,40],U=[2]
U=[2]
M=[10,40],W=[2]
M=[10,40],W=[2],V=[W+M]
J :X=[1],Y=[2]
ISR:Y=[2]
ISR:T=[1],U=[2]
J :U=[2],N=[10,40]
ISR:N=[10,40],U=[2],T=[U+N]
RJ :U=[2]
ISR:M=[10,40],W=[2]
RJ :M=[10,40],W=[2],V=[W+M]
PROGRAM MAIN
INTEGER X, Y
X = 1
Y = 2
α CALL A(X, Y)
END
SUBROUTINE A(T, U)
INTEGER N, T, U
DO N = 10, 40
β CALL B(T, U, N)
ENDDO
SUBROUTINE B(V, W, M)
INTEGER M, V, W
V = W + M
ISR:T=[1],U=[2]
ISR:M=[10,40],W=[2]
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Interprocedural Symbolic Range Propagation

13. Interprocedural Symbolic Range Propagation
Experiments
Efficacy of ISRP for an optimizing compiler (Polaris)
Test elision and dead-code elimination
Data dependence analysis
Other optimizations for parallelization
21 Fortran codes from SPEC CFP and Perfect
Best available optimizations in Polaris as Base
Interprocedural expression propagation
Forward substitution
Intraprocedural symbolic range propagation
Automatic partial inlining
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Interprocedural Symbolic Range Propagation

14. Interprocedural Symbolic Range Propagation
Result – Test Elision
Codes
Base
ISRP
ARC2D
BDNA
DYFESM
FLO52Q
MDG
MIGRATION
OCEAN
QCD2
SPEC77
TRACK
TRFD 4 15 18 2 67 3 5 26 1 6 72 10 8
applu
apsi
fpppp
hydro2d
mgrid
su2cor
swim
tomcatv
turb3d
wupwise
TOTAL 12 9 7 19
176 27
242
ISRP found more or same number of cases for 20 codes
Base made an aggressive decision with hard-wired test elision for fpppp
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Interprocedural Symbolic Range Propagation

15. Result – Data Dependence Analysis
Codes
Base
ISRP
ARC2D
BDNA
DYFESM
FLO52Q
MDG
MIGRATION
OCEAN
QCD2
SPEC77
TRACK
TRFD
801
1124
1388
263
1120
7180
433
17257
2392
2003 93
459
1081
1108
279
908
6219
370
3579
1652
1781 40
applu
apsi
fpppp
hydro2d
mgrid
su2cor
swim
tomcatv
turb3d
wupwise
TOTAL
677
11395
22064
110 83
10915 45
371
2013
81727
8222
20006 66 19
8712
342
1071
56636
ISRP disproved more data dependences for 20 codes
Base benefits from forward substitution for FLO52Q
Data dependence analysis benefits from other improved optimizations
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Interprocedural Symbolic Range Propagation

16. Result – Other Optimizations
IF (((-num)+(-num**2))/2.LE.0.AND.(-num).LE.0) THEN ALLOCATE (xrsiq00(1:morb, 1:num, 1:numthreads)) !$OMP PARALLEL !$OMP+IF(6+((-1)*num+(-1)*num**2)/2.LE.0) !$OMP+DEFAULT(SHARED) !$OMP+PRIVATE(MY_CPU_ID,MRS,MRSIJ1,MI0,MJ0,VAL,MQ,MP,XIJ00,MI,MJ) my_cpu_id = omp_get_thread_num()+1 !$OMP DO DO mrs = 1, (num*(1+num))/2, IF ((num*(1+num))/2.NE.mrs) THEN DO mq = 1, num, DO mi0 = 1, num, CONTINUE xrsiq00(mi0, mq, my_cpu_id) = zero ENDDO ENDDO DO mp = 1, num, DO mq = 1, mp, val = xrspq(mq+(mp**2+(-mp)+(-num)+(-num**2)+mrs*num+mrs*num* **2)/2) IF (zero.NE.val) THEN DO mi0 = 1, num, xrsiq00(mi0, mq, my_cpu_id) = xrsiq00(mi0, mq, my_cpu_id)+v *(mp, mi0)*val xrsiq00(mi0, mp, my_cpu_id) = xrsiq00(mi0, mp, my_cpu_id)+v *(mq, mi0)*val CONTINUE ENDDO ENDIF CONTINUE ENDDO CONTINUE ENDDO mrsij1 = ((-num)+(-num**2)+mrs*num+mrs*num**2)/ DO mi0 = 1, num, DO mj0 = 1, mi0, CONTINUE xij00(mj0) = zero ENDDO DO mq = 1, num, val = xrsiq00(mi0, mq, my_cpu_id) IF (zero.NE.val) THEN DO mj0 = 1, mi0, CONTINUE xij00(mj0) = xij00(mj0)+v(mq, mj0)*val ENDDO ENDIF CONTINUE ENDDO DO mj0 = 1, mi0, CONTINUE xrsij(mj0+(mi0**2+(-mi0))/2+((-num)+(-num**2)+mrs*num+mrs*num ***2)/2) = xij00(mj0) ENDDO CONTINUE ENDDO CONTINUE ELSE DO mq = 1, num, DO mi = 1, num, CONTINUE xrsiq(mi, mq) = zero ENDDO ENDDO DO mp = 1, num, DO mq = 1, mp, val = xrspq(mq+(mp**2+(-mp)+(-num)+(-num**2)+mrs*num+mrs*num* **2)/2) IF (zero.NE.val) THEN DO mi = 1, num, xrsiq(mi, mq) = xrsiq(mi, mq)+v(mp, mi)*val xrsiq(mi, mp) = xrsiq(mi, mp)+v(mq, mi)*val CONTINUE ENDDO ENDIF CONTINUE ENDDO CONTINUE ENDDO mrsij1 = ((-num)+(-num**2)+mrs*num+mrs*num**2)/ DO mi = 1, num, DO mj = 1, mi, CONTINUE xij(mj) = zero ENDDO DO mq = 1, num, val = xrsiq(mi, mq) IF (zero.NE.val) THEN DO mj = 1, mi, CONTINUE xij(mj) = xij(mj)+v(mq, mj)*val ENDDO ENDIF CONTINUE ENDDO DO mj = 1, mi, CONTINUE xrsij(mj+(mi**2+(-mi))/2+((-num)+(-num**2)+mrs*num+mrs*num** *)/2) = xij(mj) ENDDO CONTINUE ENDDO CONTINUE ENDIF ENDDO !$OMP END DO NOWAIT !$OMP END PARALLEL DEALLOCATE (xrsiq00) ELSE DO mrs = 1, (num*(1+num))/2, 1 !$OMP PARALLEL !$OMP+IF(6+(-1)*num.LE.0) !$OMP+DEFAULT(SHARED) !$OMP+PRIVATE(MI) !$OMP DO DO mq = 1, num, DO mi = 1, (num), CONTINUE xrsiq(mi, mq) = zero ENDDO ENDDO !$OMP END DO NOWAIT !$OMP END PARALLEL DO mp = 1, num, DO mq = 1, mp, mrspq = 1+mrspq val = xrspq(mrspq) IF (zero.NE.val) THEN !$OMP PARALLEL !$OMP+IF(6+(-1)*num.LE.0) !$OMP+DEFAULT(SHARED) !$OMP DO DO mi = 1, (num), xrsiq(mi, mq) = xrsiq(mi, mq)+v(mp, mi)*val xrsiq(mi, mp) = xrsiq(mi, mp)+v(mq, mi)*val CONTINUE ENDDO !$OMP END DO NOWAIT !$OMP END PARALLEL ENDIF CONTINUE ENDDO CONTINUE ENDDO mrsij = mrsij DO mi = 1, (num), 1 !$OMP PARALLEL !$OMP+IF(6+(-1)*mi.LE.0) !$OMP+DEFAULT(SHARED) !$OMP DO DO mj = 1, mi, CONTINUE xij(mj) = zero ENDDO !$OMP END DO NOWAIT !$OMP END PARALLEL ALLOCATE (xij1(1:mi, 1:numthreads)) !$OMP PARALLEL !$OMP+IF(6+(-1)*num.LE.0) !$OMP+DEFAULT(SHARED) !$OMP+PRIVATE(MY_CPU_ID,MQ,TPINIT,VAL,MJ) my_cpu_id = omp_get_thread_num() DO tpinit = 1, mi, xij1(tpinit, my_cpu_id) = ENDDO !$OMP DO DO mq = 1, num, val = xrsiq(mi, mq) IF (zero.NE.val) THEN DO mj = 1, mi, CONTINUE xij1(mj, my_cpu_id) = xij1(mj, my_cpu_id)+v(mq, mj)*val ENDDO ENDIF CONTINUE ENDDO !$OMP END DO NOWAIT !$OMP CRITICAL DO tpinit = 1, mi, xij(tpinit) = xij(tpinit)+xij1(tpinit, my_cpu_id) ENDDO !$OMP END CRITICAL !$OMP END PARALLEL DEALLOCATE (xij1) DO mj = 1, mi, mrsij = mrsij CONTINUE xrsij(mrsij) = xij(mj) ENDDO CONTINUE ENDDO mrsij0 = mrsij0+(num*(num+1))/ CONTINUE ENDDO ENDIF
!$OMP PARALLEL !$OMP+DEFAULT(SHARED) !$OMP+PRIVATE(MRSIJ1,MI0,MJ0,VAL,MQ,MP,XIJ00,XRSIQ00) !$OMP DO DO mrs = 1, (num+num**2)/2, IF ((num+num**2)/2.NE.mrs) THEN DO mq = 1, num, DO mi0 = 1, num, CONTINUE xrsiq00(mi0, mq) = zero ENDDO ENDDO DO mp = 1, num, DO mq = 1, mp, val = xrspq(mq+(mp**2+(-mp)+(-num)+(-num**2)+mrs*num+mrs*num** *2)/2) IF (zero.NE.val) THEN DO mi0 = 1, num, xrsiq00(mi0, mq) = xrsiq00(mi0, mq)+v(mp, mi0)*val xrsiq00(mi0, mp) = xrsiq00(mi0, mp)+v(mq, mi0)*val CONTINUE ENDDO ENDIF CONTINUE ENDDO CONTINUE ENDDO mrsij1 = ((-num)+(-num**2)+mrs*num+mrs*num**2)/ DO mi0 = 1, num, DO mj0 = 1, mi0, CONTINUE xij00(mj0) = zero ENDDO DO mq = 1, num, val = xrsiq00(mi0, mq) IF (zero.NE.val) THEN DO mj0 = 1, mi0, CONTINUE xij00(mj0) = xij00(mj0)+v(mq, mj0)*val ENDDO ENDIF CONTINUE ENDDO DO mj0 = 1, mi0, CONTINUE xrsij(mj0+(mi0**2+(-mi0)+(-num)+(-num**2)+mrs*num+mrs*num**2)/ *2) = xij00(mj0) ENDDO CONTINUE ENDDO CONTINUE ELSE DO mq = 1, num, DO mi = 1, num, CONTINUE xrsiq(mi, mq) = zero ENDDO ENDDO DO mp = 1, num, DO mq = 1, mp, val = xrspq(mq+(mp**2+(-mp)+(-num)+(-num**2)+mrs*num+mrs*num** *2)/2) IF (zero.NE.val) THEN DO mi = 1, num, xrsiq(mi, mq) = xrsiq(mi, mq)+v(mp, mi)*val xrsiq(mi, mp) = xrsiq(mi, mp)+v(mq, mi)*val CONTINUE ENDDO ENDIF CONTINUE ENDDO CONTINUE ENDDO mrsij1 = ((-num)+(-num**2)+mrs*num+mrs*num**2)/ DO mi = 1, num, DO mj = 1, mi, CONTINUE xij(mj) = zero ENDDO DO mq = 1, num, val = xrsiq(mi, mq) IF (zero.NE.val) THEN DO mj = 1, mi, CONTINUE xij(mj) = xij(mj)+v(mq, mj)*val ENDDO ENDIF CONTINUE ENDDO DO mj = 1, mi, CONTINUE xrsij(mj+(mi**2+(-mi)+(-num)+(-num**2)+mrs*num+mrs*num**2)/2) *= xij(mj) ENDDO CONTINUE ENDDO CONTINUE ENDIF ENDDO !$OMP END DO NOWAIT !$OMP END PARALLEL
Result – Other Optimizations
Induction Variable Substitution
Reduction Translation
“Yes” to the questions helps the compiler generate better codes
ISRP helped the compiler make better decisions for 5 codes
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Interprocedural Symbolic Range Propagation

17. Interprocedural Symbolic Range Propagation
Conclusions
Interprocedural analysis of symbolic ranges
Based on intraprocedural analysis
Iterative algorithm
ISRP enhances other optimizations
Compilation time increases up to 150%
Exceptions: OCEAN and TRACK
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Interprocedural Symbolic Range Propagation

18. Interprocedural Symbolic Range Propagation
Thank you.
4/8/2019
Interprocedural Symbolic Range Propagation