from examples.performance import unidiff_output, print_kernelPerformance optimization overview
The purpose of this tutorial is two-fold
- Illustrate the performance optimizations applied to the code generated by an
Operator. - Describe the options Devito provides to users to steer the optimization process.
As we shall see, most optimizations are automatically applied as they’re known to systematically improve performance.
An Operator has several preset optimization levels; the fundamental ones are noop and advanced. With noop, no performance optimizations are introduced. With advanced, several flop-reducing and data locality optimization passes are applied. Examples of flop-reducing optimization passes are common sub-expressions elimination and factorization, while examples of data locality optimization passes are loop fusion and cache blocking. Optimization levels in Devito are conceptually akin to the -O2, -O3, ... flags in classic C/C++/Fortran compilers.
An optimization pass may provide knobs, or options, for fine-grained tuning. As explained in the next sections, some of these options are given at compile-time, others at run-time.
** Remark **
Parallelism – both shared-memory (e.g., OpenMP) and distributed-memory (MPI) – is by default disabled and is not controlled via the optimization level. In this tutorial we will also show how to enable OpenMP parallelism (you’ll see it’s trivial!). Another mini-guide about parallelism in Devito and related aspects is available here.
****
Outline
API
The optimization level may be changed in various ways:
- globally, through the
DEVITO_OPTenvironment variable. For example, to disable all optimizations on allOperator’s, one could run with
DEVITO_OPT=noop python ...- programmatically, adding the following lines to a program
from devito import configuration
configuration['opt'] = 'noop'- locally, as an
Operatorargument
Operator(..., opt='noop')Local takes precedence over programmatic, and programmatic takes precedence over global.
The optimization options, instead, may only be changed locally. The syntax to specify an option is
Operator(..., opt=('advanced', {<optimization options>})A concrete example (you can ignore the meaning for now) is
Operator(..., opt=('advanced', {'blocklevels': 2})That is, options are to be specified together with the optimization level (advanced).
Default values
By default, all Operators are run with the optimization level set to advanced. So this
Operator(Eq(...))is equivalent to
Operator(Eq(...), opt='advanced')and obviously also to
Operator(Eq(...), opt=('advanced', {}))In virtually all scenarios, regardless of application and underlying architecture, this ensures very good performance – but not necessarily the very best.
Misc
The following functions will be used throughout the notebook for printing generated code.
The following cell is only needed for Continuous Integration. But actually it’s also an example of how “programmatic takes precedence over global” (see API section).
from devito import configuration
configuration['language'] = 'C'
configuration['platform'] = 'bdw' # Optimize for an Intel Broadwell
configuration['opt-options']['par-collapse-ncores'] = 1 # Maximize use loop collapsingRunning example
Throughout the notebook we will generate Operator’s for the following time-marching Eq.
from devito import Eq, Grid, Operator, Function, TimeFunction, sin
grid = Grid(shape=(80, 80, 80))
f = Function(name='f', grid=grid)
u = TimeFunction(name='u', grid=grid, space_order=4)
eq = Eq(u.forward, f**2*sin(f)*u.dy.dy)Despite its simplicity, this Eq is all we need to showcase the key components of the Devito optimization engine.
OpenMP parallelism
There are several ways to enable OpenMP parallelism. The one we use here consists of supplying an option to an Operator. The next cell illustrates the difference between two Operator’s generated with the noop optimization level, but with OpenMP enabled on the latter one.
op0 = Operator(eq, opt=('noop'))
op0_omp = Operator(eq, opt=('noop', {'openmp': True}))
# print(op0)
# print(unidiff_output(str(op0), str(op0_omp))) # Uncomment to print out the diff only
print_kernel(op0_omp)for (int time = time_m, t0 = (time)%(2), t1 = (time + 1)%(2); time <= time_M; time += 1, t0 = (time)%(2), t1 = (time + 1)%(2))
{
START(section0)
#pragma omp parallel num_threads(nthreads)
{
#pragma omp for collapse(3) schedule(dynamic,1)
for (int x = x_m; x <= x_M; x += 1)
{
for (int y = y_m; y <= y_M; y += 1)
{
for (int z = z_m; z <= z_M; z += 1)
{
u[t1][x + 4][y + 4][z + 4] = ((-6.66666667e-1F/h_y)*(8.33333333e-2F*u[t0][x + 4][y + 1][z + 4]/h_y - 6.66666667e-1F*u[t0][x + 4][y + 2][z + 4]/h_y + 6.66666667e-1F*u[t0][x + 4][y + 4][z + 4]/h_y - 8.33333333e-2F*u[t0][x + 4][y + 5][z + 4]/h_y) + (-8.33333333e-2F/h_y)*(8.33333333e-2F*u[t0][x + 4][y + 4][z + 4]/h_y - 6.66666667e-1F*u[t0][x + 4][y + 5][z + 4]/h_y + 6.66666667e-1F*u[t0][x + 4][y + 7][z + 4]/h_y - 8.33333333e-2F*u[t0][x + 4][y + 8][z + 4]/h_y) + (8.33333333e-2F/h_y)*(8.33333333e-2F*u[t0][x + 4][y][z + 4]/h_y - 6.66666667e-1F*u[t0][x + 4][y + 1][z + 4]/h_y + 6.66666667e-1F*u[t0][x + 4][y + 3][z + 4]/h_y - 8.33333333e-2F*u[t0][x + 4][y + 4][z + 4]/h_y) + (6.66666667e-1F/h_y)*(8.33333333e-2F*u[t0][x + 4][y + 3][z + 4]/h_y - 6.66666667e-1F*u[t0][x + 4][y + 4][z + 4]/h_y + 6.66666667e-1F*u[t0][x + 4][y + 6][z + 4]/h_y - 8.33333333e-2F*u[t0][x + 4][y + 7][z + 4]/h_y))*sinf(f[x + 1][y + 1][z + 1])*pow(f[x + 1][y + 1][z + 1], 2);
}
}
}
}
STOP(section0,timers)
}The OpenMP-ized op0_omp Operator includes:
- the header file
"omp.h" - a
#pragma omp parallel num_threads(nthreads)directive - a
#pragma omp for collapse(...) schedule(dynamic,1)directive
More complex Operator’s will have more directives, more types of directives, different iteration scheduling strategies based on heuristics and empirical tuning (e.g., static instead of dynamic), etc.
We note how the OpenMP pass introduces a new symbol, nthreads. This allows users to explicitly control the number of threads with which an Operator is run.
op0_omp.apply(time_M=0) # Picks up `nthreads` from the standard environment variable OMP_NUM_THREADS
op0_omp.apply(time_M=0, nthreads=2) # Runs with 2 threads per parallel loopA few optimization options are available for this pass (but not on all platforms, see here), though in our experience the default values do a fine job:
par-collapse-ncores: use a collapse clause only if the number of available physical cores is greater than this value (default=4).par-collapse-work: use a collapse clause only if the trip count of the collapsable loops is statically known to exceed this value (default=100).par-chunk-nonaffine: a coefficient to adjust the chunk size in non-affine parallel loops. The larger the coefficient, the smaller the chunk size (default=3).par-dynamic-work: use dynamic scheduling if the operation count per iteration exceeds this value. Otherwise, use static scheduling (default=10).par-nested: use nested parallelism if the number of hyperthreads per core is greater than this value (default=2).
So, for instance, we could switch to static scheduling by constructing the following Operator
op0_b0_omp = Operator(eq, opt=('noop', {'openmp': True, 'par-dynamic-work': 100}))
print_kernel(op0_b0_omp)for (int time = time_m, t0 = (time)%(2), t1 = (time + 1)%(2); time <= time_M; time += 1, t0 = (time)%(2), t1 = (time + 1)%(2))
{
START(section0)
#pragma omp parallel num_threads(nthreads)
{
#pragma omp for collapse(3) schedule(static,1)
for (int x = x_m; x <= x_M; x += 1)
{
for (int y = y_m; y <= y_M; y += 1)
{
for (int z = z_m; z <= z_M; z += 1)
{
u[t1][x + 4][y + 4][z + 4] = ((-6.66666667e-1F/h_y)*(8.33333333e-2F*u[t0][x + 4][y + 1][z + 4]/h_y - 6.66666667e-1F*u[t0][x + 4][y + 2][z + 4]/h_y + 6.66666667e-1F*u[t0][x + 4][y + 4][z + 4]/h_y - 8.33333333e-2F*u[t0][x + 4][y + 5][z + 4]/h_y) + (-8.33333333e-2F/h_y)*(8.33333333e-2F*u[t0][x + 4][y + 4][z + 4]/h_y - 6.66666667e-1F*u[t0][x + 4][y + 5][z + 4]/h_y + 6.66666667e-1F*u[t0][x + 4][y + 7][z + 4]/h_y - 8.33333333e-2F*u[t0][x + 4][y + 8][z + 4]/h_y) + (8.33333333e-2F/h_y)*(8.33333333e-2F*u[t0][x + 4][y][z + 4]/h_y - 6.66666667e-1F*u[t0][x + 4][y + 1][z + 4]/h_y + 6.66666667e-1F*u[t0][x + 4][y + 3][z + 4]/h_y - 8.33333333e-2F*u[t0][x + 4][y + 4][z + 4]/h_y) + (6.66666667e-1F/h_y)*(8.33333333e-2F*u[t0][x + 4][y + 3][z + 4]/h_y - 6.66666667e-1F*u[t0][x + 4][y + 4][z + 4]/h_y + 6.66666667e-1F*u[t0][x + 4][y + 6][z + 4]/h_y - 8.33333333e-2F*u[t0][x + 4][y + 7][z + 4]/h_y))*sinf(f[x + 1][y + 1][z + 1])*pow(f[x + 1][y + 1][z + 1], 2);
}
}
}
}
STOP(section0,timers)
}The advanced mode
The default optimization level in Devito is advanced. This mode performs several compilation passes to optimize the Operator for computation (number of flops), working set size, and data locality. In the next paragraphs we’ll dissect the advanced mode to analyze, one by one, some of its key passes.
Loop blocking
The next cell creates a new Operator that adds loop blocking to what we had in op0_omp.
op1_omp = Operator(eq, opt=('blocking', {'openmp': True}))
# print(op1_omp) # Uncomment to see the *whole* generated code
print_kernel(op1_omp)for (int time = time_m, t0 = (time)%(2), t1 = (time + 1)%(2); time <= time_M; time += 1, t0 = (time)%(2), t1 = (time + 1)%(2))
{
START(section0)
#pragma omp parallel num_threads(nthreads)
{
#pragma omp for collapse(2) schedule(dynamic,1)
for (int x0_blk0 = x_m; x0_blk0 <= x_M; x0_blk0 += x0_blk0_size)
{
for (int y0_blk0 = y_m; y0_blk0 <= y_M; y0_blk0 += y0_blk0_size)
{
for (int x = x0_blk0; x <= MIN(x_M, x0_blk0 + x0_blk0_size - 1); x += 1)
{
for (int y = y0_blk0; y <= MIN(y_M, y0_blk0 + y0_blk0_size - 1); y += 1)
{
for (int z = z_m; z <= z_M; z += 1)
{
u[t1][x + 4][y + 4][z + 4] = ((-6.66666667e-1F/h_y)*(8.33333333e-2F*u[t0][x + 4][y + 1][z + 4]/h_y - 6.66666667e-1F*u[t0][x + 4][y + 2][z + 4]/h_y + 6.66666667e-1F*u[t0][x + 4][y + 4][z + 4]/h_y - 8.33333333e-2F*u[t0][x + 4][y + 5][z + 4]/h_y) + (-8.33333333e-2F/h_y)*(8.33333333e-2F*u[t0][x + 4][y + 4][z + 4]/h_y - 6.66666667e-1F*u[t0][x + 4][y + 5][z + 4]/h_y + 6.66666667e-1F*u[t0][x + 4][y + 7][z + 4]/h_y - 8.33333333e-2F*u[t0][x + 4][y + 8][z + 4]/h_y) + (8.33333333e-2F/h_y)*(8.33333333e-2F*u[t0][x + 4][y][z + 4]/h_y - 6.66666667e-1F*u[t0][x + 4][y + 1][z + 4]/h_y + 6.66666667e-1F*u[t0][x + 4][y + 3][z + 4]/h_y - 8.33333333e-2F*u[t0][x + 4][y + 4][z + 4]/h_y) + (6.66666667e-1F/h_y)*(8.33333333e-2F*u[t0][x + 4][y + 3][z + 4]/h_y - 6.66666667e-1F*u[t0][x + 4][y + 4][z + 4]/h_y + 6.66666667e-1F*u[t0][x + 4][y + 6][z + 4]/h_y - 8.33333333e-2F*u[t0][x + 4][y + 7][z + 4]/h_y))*sinf(f[x + 1][y + 1][z + 1])*pow(f[x + 1][y + 1][z + 1], 2);
}
}
}
}
}
}
STOP(section0,timers)
}** Remark **
'blocking' is not an optimization level – it rather identifies a specific compilation pass. In other words, the advanced mode defines an ordered sequence of passes, and blocking is one such pass.
****
The blocking pass creates additional loops over blocks. In this simple Operator there’s just one loop nest, so only a pair of additional loops are created. In more complex Operator’s, several loop nests may individually be blocked, whereas others may be left unblocked – this is decided by the Devito compiler according to certain heuristics. The size of a block is represented by the symbols x0_blk0_size and y0_blk0_size, which are runtime parameters akin to nthreads.
By default, Devito applies 2D blocking and sets the default block shape to 8x8. There are two ways to set a different block shape:
- passing an explicit value. For instance, below we run with a 24x8 block shape
op1_omp.apply(..., x0_blk0_size=24)- letting the autotuner pick up a better block shape for us. There are several autotuning modes. A short summary is available here
op1_omp.apply(..., autotune='aggressive')Loop blocking also provides two optimization options:
blockinner={False, True}– to enable 3D (or any nD, n>2) blockingblocklevels={int}– to enable hierarchical blocking, to exploit multiple levels of the cache hierarchy
In the example below, we construct an Operator with six-dimensional loop blocking: the first three loops represent outer blocks, whereas the second three loops represent inner blocks within an outer block.
op1_omp_6D = Operator(eq, opt=('blocking', {'blockinner': True, 'blocklevels': 2, 'openmp': True}))
# print(op1_omp_6D) # Uncomment to see the *whole* generated code
print_kernel(op1_omp_6D)for (int time = time_m, t0 = (time)%(2), t1 = (time + 1)%(2); time <= time_M; time += 1, t0 = (time)%(2), t1 = (time + 1)%(2))
{
START(section0)
#pragma omp parallel num_threads(nthreads)
{
#pragma omp for collapse(3) schedule(dynamic,1)
for (int x0_blk0 = x_m; x0_blk0 <= x_M; x0_blk0 += x0_blk0_size)
{
for (int y0_blk0 = y_m; y0_blk0 <= y_M; y0_blk0 += y0_blk0_size)
{
for (int z0_blk0 = z_m; z0_blk0 <= z_M; z0_blk0 += z0_blk0_size)
{
for (int x0_blk1 = x0_blk0; x0_blk1 <= MIN(x_M, x0_blk0 + x0_blk0_size - 1); x0_blk1 += x0_blk1_size)
{
for (int y0_blk1 = y0_blk0; y0_blk1 <= MIN(y_M, y0_blk0 + y0_blk0_size - 1); y0_blk1 += y0_blk1_size)
{
for (int z0_blk1 = z0_blk0; z0_blk1 <= MIN(z_M, z0_blk0 + z0_blk0_size - 1); z0_blk1 += z0_blk1_size)
{
for (int x = x0_blk1; x <= MIN(x_M, x0_blk1 + x0_blk1_size - 1); x += 1)
{
for (int y = y0_blk1; y <= MIN(y_M, y0_blk1 + y0_blk1_size - 1); y += 1)
{
for (int z = z0_blk1; z <= MIN(z_M, z0_blk1 + z0_blk1_size - 1); z += 1)
{
u[t1][x + 4][y + 4][z + 4] = ((-6.66666667e-1F/h_y)*(8.33333333e-2F*u[t0][x + 4][y + 1][z + 4]/h_y - 6.66666667e-1F*u[t0][x + 4][y + 2][z + 4]/h_y + 6.66666667e-1F*u[t0][x + 4][y + 4][z + 4]/h_y - 8.33333333e-2F*u[t0][x + 4][y + 5][z + 4]/h_y) + (-8.33333333e-2F/h_y)*(8.33333333e-2F*u[t0][x + 4][y + 4][z + 4]/h_y - 6.66666667e-1F*u[t0][x + 4][y + 5][z + 4]/h_y + 6.66666667e-1F*u[t0][x + 4][y + 7][z + 4]/h_y - 8.33333333e-2F*u[t0][x + 4][y + 8][z + 4]/h_y) + (8.33333333e-2F/h_y)*(8.33333333e-2F*u[t0][x + 4][y][z + 4]/h_y - 6.66666667e-1F*u[t0][x + 4][y + 1][z + 4]/h_y + 6.66666667e-1F*u[t0][x + 4][y + 3][z + 4]/h_y - 8.33333333e-2F*u[t0][x + 4][y + 4][z + 4]/h_y) + (6.66666667e-1F/h_y)*(8.33333333e-2F*u[t0][x + 4][y + 3][z + 4]/h_y - 6.66666667e-1F*u[t0][x + 4][y + 4][z + 4]/h_y + 6.66666667e-1F*u[t0][x + 4][y + 6][z + 4]/h_y - 8.33333333e-2F*u[t0][x + 4][y + 7][z + 4]/h_y))*sinf(f[x + 1][y + 1][z + 1])*pow(f[x + 1][y + 1][z + 1], 2);
}
}
}
}
}
}
}
}
}
}
STOP(section0,timers)
}SIMD vectorization
Devito enforces SIMD vectorization through OpenMP pragmas.
op2_omp = Operator(eq, opt=('blocking', 'simd', {'openmp': True}))
# print(op2_omp) # Uncomment to see the generated code
# print(unidiff_output(str(op1_omp), str(op2_omp))) # Uncomment to print out the diff onlyCode motion
The advanced mode has a code motion pass. In explicit PDE solvers, this is most commonly used to lift expensive time-invariant sub-expressions out of the inner loops. The pass is however quite general in that it is not restricted to the concept of time-invariance – any sub-expression invariant with respect to a subset of Dimensions is a code motion candidate. In our running example, sinf(f) gets hoisted out of the inner loops since it is determined to be an expensive invariant sub-expression. In other words, the compiler trades the redundant computation of sinf(f) for additional storage (the r0[...] array).
op3_omp = Operator(eq, opt=('lift', {'openmp': True}))
print_kernel(op3_omp)float r1 = 1.0F/h_y;
START(section0)
#pragma omp parallel num_threads(nthreads)
{
#pragma omp for collapse(3) schedule(static,1)
for (int x = x_m; x <= x_M; x += 1)
{
for (int y = y_m; y <= y_M; y += 1)
{
for (int z = z_m; z <= z_M; z += 1)
{
r0[x][y][z] = sinf(f[x + 1][y + 1][z + 1]);
}
}
}
}
STOP(section0,timers)
for (int time = time_m, t0 = (time)%(2), t1 = (time + 1)%(2); time <= time_M; time += 1, t0 = (time)%(2), t1 = (time + 1)%(2))
{
START(section1)
#pragma omp parallel num_threads(nthreads)
{
#pragma omp for collapse(3) schedule(dynamic,1)
for (int x = x_m; x <= x_M; x += 1)
{
for (int y = y_m; y <= y_M; y += 1)
{
for (int z = z_m; z <= z_M; z += 1)
{
u[t1][x + 4][y + 4][z + 4] = ((-6.66666667e-1F*r1)*(8.33333333e-2F*r1*u[t0][x + 4][y + 1][z + 4] - 6.66666667e-1F*r1*u[t0][x + 4][y + 2][z + 4] + 6.66666667e-1F*r1*u[t0][x + 4][y + 4][z + 4] - 8.33333333e-2F*r1*u[t0][x + 4][y + 5][z + 4]) + (-8.33333333e-2F*r1)*(8.33333333e-2F*r1*u[t0][x + 4][y + 4][z + 4] - 6.66666667e-1F*r1*u[t0][x + 4][y + 5][z + 4] + 6.66666667e-1F*r1*u[t0][x + 4][y + 7][z + 4] - 8.33333333e-2F*r1*u[t0][x + 4][y + 8][z + 4]) + (8.33333333e-2F*r1)*(8.33333333e-2F*r1*u[t0][x + 4][y][z + 4] - 6.66666667e-1F*r1*u[t0][x + 4][y + 1][z + 4] + 6.66666667e-1F*r1*u[t0][x + 4][y + 3][z + 4] - 8.33333333e-2F*r1*u[t0][x + 4][y + 4][z + 4]) + (6.66666667e-1F*r1)*(8.33333333e-2F*r1*u[t0][x + 4][y + 3][z + 4] - 6.66666667e-1F*r1*u[t0][x + 4][y + 4][z + 4] + 6.66666667e-1F*r1*u[t0][x + 4][y + 6][z + 4] - 8.33333333e-2F*r1*u[t0][x + 4][y + 7][z + 4]))*pow(f[x + 1][y + 1][z + 1], 2)*r0[x][y][z];
}
}
}
}
STOP(section1,timers)
}Basic flop-reducing transformations
Among the simpler flop-reducing transformations applied by the advanced mode we find:
- “classic” common sub-expressions elimination (CSE),
- factorization,
- optimization of powers
The cell below shows how the computation of u changes by incrementally applying these three passes. First of all, we observe how the reciprocal of the symbolic spacing h_y gets assigned to a temporary, r0, as it appears in several sub-expressions. This is the effect of CSE.
op4_omp = Operator(eq, opt=('cse', {'openmp': True}))
print(unidiff_output(str(op0_omp), str(op4_omp)))---
+++
@@ -42,7 +42,8 @@
{
for (int z = z_m; z <= z_M; z += 1)
{
- u[t1][x + 4][y + 4][z + 4] = ((-6.66666667e-1F/h_y)*(8.33333333e-2F*u[t0][x + 4][y + 1][z + 4]/h_y - 6.66666667e-1F*u[t0][x + 4][y + 2][z + 4]/h_y + 6.66666667e-1F*u[t0][x + 4][y + 4][z + 4]/h_y - 8.33333333e-2F*u[t0][x + 4][y + 5][z + 4]/h_y) + (-8.33333333e-2F/h_y)*(8.33333333e-2F*u[t0][x + 4][y + 4][z + 4]/h_y - 6.66666667e-1F*u[t0][x + 4][y + 5][z + 4]/h_y + 6.66666667e-1F*u[t0][x + 4][y + 7][z + 4]/h_y - 8.33333333e-2F*u[t0][x + 4][y + 8][z + 4]/h_y) + (8.33333333e-2F/h_y)*(8.33333333e-2F*u[t0][x + 4][y][z + 4]/h_y - 6.66666667e-1F*u[t0][x + 4][y + 1][z + 4]/h_y + 6.66666667e-1F*u[t0][x + 4][y + 3][z + 4]/h_y - 8.33333333e-2F*u[t0][x + 4][y + 4][z + 4]/h_y) + (6.66666667e-1F/h_y)*(8.33333333e-2F*u[t0][x + 4][y + 3][z + 4]/h_y - 6.66666667e-1F*u[t0][x + 4][y + 4][z + 4]/h_y + 6.66666667e-1F*u[t0][x + 4][y + 6][z + 4]/h_y - 8.33333333e-2F*u[t0][x + 4][y + 7][z + 4]/h_y))*sinf(f[x + 1][y + 1][z + 1])*pow(f[x + 1][y + 1][z + 1], 2);
+ float r0 = 1.0F/h_y;
+ u[t1][x + 4][y + 4][z + 4] = ((-6.66666667e-1F*r0)*(8.33333333e-2F*r0*u[t0][x + 4][y + 1][z + 4] - 6.66666667e-1F*r0*u[t0][x + 4][y + 2][z + 4] + 6.66666667e-1F*r0*u[t0][x + 4][y + 4][z + 4] - 8.33333333e-2F*r0*u[t0][x + 4][y + 5][z + 4]) + (-8.33333333e-2F*r0)*(8.33333333e-2F*r0*u[t0][x + 4][y + 4][z + 4] - 6.66666667e-1F*r0*u[t0][x + 4][y + 5][z + 4] + 6.66666667e-1F*r0*u[t0][x + 4][y + 7][z + 4] - 8.33333333e-2F*r0*u[t0][x + 4][y + 8][z + 4]) + (8.33333333e-2F*r0)*(8.33333333e-2F*r0*u[t0][x + 4][y][z + 4] - 6.66666667e-1F*r0*u[t0][x + 4][y + 1][z + 4] + 6.66666667e-1F*r0*u[t0][x + 4][y + 3][z + 4] - 8.33333333e-2F*r0*u[t0][x + 4][y + 4][z + 4]) + (6.66666667e-1F*r0)*(8.33333333e-2F*r0*u[t0][x + 4][y + 3][z + 4] - 6.66666667e-1F*r0*u[t0][x + 4][y + 4][z + 4] + 6.66666667e-1F*r0*u[t0][x + 4][y + 6][z + 4] - 8.33333333e-2F*r0*u[t0][x + 4][y + 7][z + 4]))*sinf(f[x + 1][y + 1][z + 1])*pow(f[x + 1][y + 1][z + 1], 2);
}
}
}
The factorization pass makes sure r0 is collected to reduce the number of multiplications.
op5_omp = Operator(eq, opt=('cse', 'factorize', {'openmp': True}))
print(unidiff_output(str(op4_omp), str(op5_omp)))---
+++
@@ -43,7 +43,7 @@
for (int z = z_m; z <= z_M; z += 1)
{
float r0 = 1.0F/h_y;
- u[t1][x + 4][y + 4][z + 4] = ((-6.66666667e-1F*r0)*(8.33333333e-2F*r0*u[t0][x + 4][y + 1][z + 4] - 6.66666667e-1F*r0*u[t0][x + 4][y + 2][z + 4] + 6.66666667e-1F*r0*u[t0][x + 4][y + 4][z + 4] - 8.33333333e-2F*r0*u[t0][x + 4][y + 5][z + 4]) + (-8.33333333e-2F*r0)*(8.33333333e-2F*r0*u[t0][x + 4][y + 4][z + 4] - 6.66666667e-1F*r0*u[t0][x + 4][y + 5][z + 4] + 6.66666667e-1F*r0*u[t0][x + 4][y + 7][z + 4] - 8.33333333e-2F*r0*u[t0][x + 4][y + 8][z + 4]) + (8.33333333e-2F*r0)*(8.33333333e-2F*r0*u[t0][x + 4][y][z + 4] - 6.66666667e-1F*r0*u[t0][x + 4][y + 1][z + 4] + 6.66666667e-1F*r0*u[t0][x + 4][y + 3][z + 4] - 8.33333333e-2F*r0*u[t0][x + 4][y + 4][z + 4]) + (6.66666667e-1F*r0)*(8.33333333e-2F*r0*u[t0][x + 4][y + 3][z + 4] - 6.66666667e-1F*r0*u[t0][x + 4][y + 4][z + 4] + 6.66666667e-1F*r0*u[t0][x + 4][y + 6][z + 4] - 8.33333333e-2F*r0*u[t0][x + 4][y + 7][z + 4]))*sinf(f[x + 1][y + 1][z + 1])*pow(f[x + 1][y + 1][z + 1], 2);
+ u[t1][x + 4][y + 4][z + 4] = pow(r0, 2)*(6.94444444e-3F*(u[t0][x + 4][y][z + 4] + u[t0][x + 4][y + 8][z + 4]) + 4.44444444e-1F*(u[t0][x + 4][y + 2][z + 4] + u[t0][x + 4][y + 6][z + 4]) + 1.11111111e-1F*(-u[t0][x + 4][y + 1][z + 4] + u[t0][x + 4][y + 3][z + 4] + u[t0][x + 4][y + 5][z + 4] - u[t0][x + 4][y + 7][z + 4]) - 9.02777778e-1F*u[t0][x + 4][y + 4][z + 4])*sinf(f[x + 1][y + 1][z + 1])*pow(f[x + 1][y + 1][z + 1], 2);
}
}
}
Finally, opt-pows turns costly pow calls into multiplications.
op6_omp = Operator(eq, opt=('cse', 'factorize', 'opt-pows', {'openmp': True}))
print(unidiff_output(str(op5_omp), str(op6_omp)))---
+++
@@ -43,7 +43,7 @@
for (int z = z_m; z <= z_M; z += 1)
{
float r0 = 1.0F/h_y;
- u[t1][x + 4][y + 4][z + 4] = pow(r0, 2)*(6.94444444e-3F*(u[t0][x + 4][y][z + 4] + u[t0][x + 4][y + 8][z + 4]) + 4.44444444e-1F*(u[t0][x + 4][y + 2][z + 4] + u[t0][x + 4][y + 6][z + 4]) + 1.11111111e-1F*(-u[t0][x + 4][y + 1][z + 4] + u[t0][x + 4][y + 3][z + 4] + u[t0][x + 4][y + 5][z + 4] - u[t0][x + 4][y + 7][z + 4]) - 9.02777778e-1F*u[t0][x + 4][y + 4][z + 4])*sinf(f[x + 1][y + 1][z + 1])*pow(f[x + 1][y + 1][z + 1], 2);
+ u[t1][x + 4][y + 4][z + 4] = (r0*r0)*(f[x + 1][y + 1][z + 1]*f[x + 1][y + 1][z + 1])*(6.94444444e-3F*(u[t0][x + 4][y][z + 4] + u[t0][x + 4][y + 8][z + 4]) + 4.44444444e-1F*(u[t0][x + 4][y + 2][z + 4] + u[t0][x + 4][y + 6][z + 4]) + 1.11111111e-1F*(-u[t0][x + 4][y + 1][z + 4] + u[t0][x + 4][y + 3][z + 4] + u[t0][x + 4][y + 5][z + 4] - u[t0][x + 4][y + 7][z + 4]) - 9.02777778e-1F*u[t0][x + 4][y + 4][z + 4])*sinf(f[x + 1][y + 1][z + 1]);
}
}
}
Cross-iteration redundancy elimination (CIRE)
This is perhaps the most advanced among the optimization passes in Devito. CIRE [1] searches for redundancies across consecutive loop iterations. These are often induced by a mixture of nested, high-order, partial derivatives. The underlying idea is very simple. Consider:
r0 = a[i-1] + a[i] + a[i+1]at i=1, we have
r0 = a[0] + a[1] + a[2]at i=2, we have
r0 = a[1] + a[2] + a[3]So the sub-expression a[1] + a[2] is computed twice, by two consecutive iterations. What makes CIRE complicated is the generalization to arbitrary expressions, the presence of multiple dimensions, the scheduling strategy due to the trade-off between redundant compute and working set, and the co-existance with other optimizations (e.g., blocking, vectorization). All these aspects won’t be treated here. What instead we will show is the effect of CIRE in our running example and the optimization options at our disposal to drive the detection and scheduling of the captured redundancies.
In our running example, some cross-iteration redundancies are induced by the nested first-order derivatives along y. As we see below, these redundancies are captured and assigned to the two-dimensional temporary r0.
Note: the name cire-sops means “Apply CIRE to sum-of-product expressions”. A sum-of-product is what a finite difference derivative produces.
op7_omp = Operator(eq, opt=('cire-sops', 'simd', {'openmp': True}))
print_kernel(op7_omp)
# print(unidiff_output(str(op7_omp), str(op0_omp))) # Uncomment to print out the diff onlyfor (int time = time_m, t0 = (time)%(2), t1 = (time + 1)%(2); time <= time_M; time += 1, t0 = (time)%(2), t1 = (time + 1)%(2))
{
START(section0)
#pragma omp parallel num_threads(nthreads)
{
const int tid = omp_get_thread_num();
float (*restrict r0)[z_size] __attribute__ ((aligned (64))) = (float (*)[z_size]) pr0[tid];
#pragma omp for schedule(dynamic,1)
for (int x = x_m; x <= x_M; x += 1)
{
for (int y = y_m - 2; y <= y_M + 2; y += 1)
{
#pragma omp simd aligned(u:32)
for (int z = z_m; z <= z_M; z += 1)
{
r0[y + 2][z] = 8.33333333e-2F*u[t0][x + 4][y + 2][z + 4]/h_y - 6.66666667e-1F*u[t0][x + 4][y + 3][z + 4]/h_y + 6.66666667e-1F*u[t0][x + 4][y + 5][z + 4]/h_y - 8.33333333e-2F*u[t0][x + 4][y + 6][z + 4]/h_y;
}
}
for (int y = y_m; y <= y_M; y += 1)
{
#pragma omp simd aligned(f,u:32)
for (int z = z_m; z <= z_M; z += 1)
{
u[t1][x + 4][y + 4][z + 4] = ((-6.66666667e-1F/h_y)*r0[y + 1][z] + (-8.33333333e-2F/h_y)*r0[y + 4][z] + (8.33333333e-2F/h_y)*r0[y][z] + (6.66666667e-1F/h_y)*r0[y + 3][z])*sinf(f[x + 1][y + 1][z + 1])*pow(f[x + 1][y + 1][z + 1], 2);
}
}
}
}
STOP(section0,timers)
}We note that since there are no redundancies along x, the compiler is smart to figure out that r0 and u can safely be computed within the same x loop. This is nice – not only is the reuse distance decreased, but also a grid-sized temporary avoided.
The min-storage option
Let’s now consider a variant of our running example
eq = Eq(u.forward, f**2*sin(f)*(u.dy.dy + u.dx.dx))
op7_b0_omp = Operator(eq, opt=('cire-sops', {'openmp': True}))
print_kernel(op7_b0_omp)for (int time = time_m, t0 = (time)%(2), t1 = (time + 1)%(2); time <= time_M; time += 1, t0 = (time)%(2), t1 = (time + 1)%(2))
{
START(section0)
#pragma omp parallel num_threads(nthreads)
{
#pragma omp for collapse(3) schedule(dynamic,1)
for (int x = x_m - 2; x <= x_M + 2; x += 1)
{
for (int y = y_m - 2; y <= y_M + 2; y += 1)
{
for (int z = z_m; z <= z_M; z += 1)
{
r0[x + 2][y + 2][z] = 8.33333333e-2F*u[t0][x + 2][y + 4][z + 4]/h_x - 6.66666667e-1F*u[t0][x + 3][y + 4][z + 4]/h_x + 6.66666667e-1F*u[t0][x + 5][y + 4][z + 4]/h_x - 8.33333333e-2F*u[t0][x + 6][y + 4][z + 4]/h_x;
r1[x + 2][y + 2][z] = 8.33333333e-2F*u[t0][x + 4][y + 2][z + 4]/h_y - 6.66666667e-1F*u[t0][x + 4][y + 3][z + 4]/h_y + 6.66666667e-1F*u[t0][x + 4][y + 5][z + 4]/h_y - 8.33333333e-2F*u[t0][x + 4][y + 6][z + 4]/h_y;
}
}
}
}
#pragma omp parallel num_threads(nthreads)
{
#pragma omp for collapse(3) schedule(dynamic,1)
for (int x = x_m; x <= x_M; x += 1)
{
for (int y = y_m; y <= y_M; y += 1)
{
for (int z = z_m; z <= z_M; z += 1)
{
u[t1][x + 4][y + 4][z + 4] = ((-6.66666667e-1F/h_x)*r0[x + 1][y + 2][z] + (-8.33333333e-2F/h_x)*r0[x + 4][y + 2][z] + (8.33333333e-2F/h_x)*r0[x][y + 2][z] + (6.66666667e-1F/h_x)*r0[x + 3][y + 2][z] + (-6.66666667e-1F/h_y)*r1[x + 2][y + 1][z] + (-8.33333333e-2F/h_y)*r1[x + 2][y + 4][z] + (8.33333333e-2F/h_y)*r1[x + 2][y][z] + (6.66666667e-1F/h_y)*r1[x + 2][y + 3][z])*sinf(f[x + 1][y + 1][z + 1])*pow(f[x + 1][y + 1][z + 1], 2);
}
}
}
}
STOP(section0,timers)
}As expected, there are now two temporaries, one stemming from u.dy.dy and the other from u.dx.dx. A key difference with respect to op7_omp here is that both are grid-size temporaries. This might seem odd at first – why should the u.dy.dy temporary, that is r1, now be a three-dimensional temporary when we know already, from what seen in the previous cell, it could be a two-dimensional temporary? This is merely a compiler heuristic: by adding an extra dimension to r1, both temporaries can be scheduled within the same loop nest, thus augmenting data reuse and potentially enabling further cross-expression optimizations. We can disable this heuristic through the min-storage option.
op7_b1_omp = Operator(eq, opt=('cire-sops', 'simd', {'openmp': True, 'min-storage': True}))
print_kernel(op7_b1_omp)for (int time = time_m, t0 = (time)%(2), t1 = (time + 1)%(2); time <= time_M; time += 1, t0 = (time)%(2), t1 = (time + 1)%(2))
{
START(section0)
#pragma omp parallel num_threads(nthreads)
{
#pragma omp for collapse(2) schedule(dynamic,1)
for (int x = x_m - 2; x <= x_M + 2; x += 1)
{
for (int y = y_m; y <= y_M; y += 1)
{
#pragma omp simd aligned(u:32)
for (int z = z_m; z <= z_M; z += 1)
{
r0[x + 2][y][z] = 8.33333333e-2F*u[t0][x + 2][y + 4][z + 4]/h_x - 6.66666667e-1F*u[t0][x + 3][y + 4][z + 4]/h_x + 6.66666667e-1F*u[t0][x + 5][y + 4][z + 4]/h_x - 8.33333333e-2F*u[t0][x + 6][y + 4][z + 4]/h_x;
}
}
}
}
#pragma omp parallel num_threads(nthreads)
{
const int tid = omp_get_thread_num();
float (*restrict r1)[z_size] __attribute__ ((aligned (64))) = (float (*)[z_size]) pr1[tid];
#pragma omp for schedule(dynamic,1)
for (int x = x_m; x <= x_M; x += 1)
{
for (int y = y_m - 2; y <= y_M + 2; y += 1)
{
#pragma omp simd aligned(u:32)
for (int z = z_m; z <= z_M; z += 1)
{
r1[y + 2][z] = 8.33333333e-2F*u[t0][x + 4][y + 2][z + 4]/h_y - 6.66666667e-1F*u[t0][x + 4][y + 3][z + 4]/h_y + 6.66666667e-1F*u[t0][x + 4][y + 5][z + 4]/h_y - 8.33333333e-2F*u[t0][x + 4][y + 6][z + 4]/h_y;
}
}
for (int y = y_m; y <= y_M; y += 1)
{
#pragma omp simd aligned(f,u:32)
for (int z = z_m; z <= z_M; z += 1)
{
u[t1][x + 4][y + 4][z + 4] = ((-6.66666667e-1F/h_x)*r0[x + 1][y][z] + (-8.33333333e-2F/h_x)*r0[x + 4][y][z] + (8.33333333e-2F/h_x)*r0[x][y][z] + (6.66666667e-1F/h_x)*r0[x + 3][y][z] + (-6.66666667e-1F/h_y)*r1[y + 1][z] + (-8.33333333e-2F/h_y)*r1[y + 4][z] + (8.33333333e-2F/h_y)*r1[y][z] + (6.66666667e-1F/h_y)*r1[y + 3][z])*sinf(f[x + 1][y + 1][z + 1])*pow(f[x + 1][y + 1][z + 1], 2);
}
}
}
}
STOP(section0,timers)
}The cire-mingain option
So far so good – we’ve seen that Devito can capture and schedule cross-iteration redundancies. But what if, for example, we actually do not want certain redundancies to be captured? There are a few reasons we may like that way, for example we’re allocating too much extra memory for the tensor temporaries, and we rather prefer to avoid that. For this, the user can tell Devito what the minimum gain, or simply mingain, which stems from optimizing a CIRE candidate must be in order to trigger the transformation. Such mingain is an integer number reflecting the number and type of arithmetic operations in a CIRE candidate. The Devito compiler computes the gain of a CIRE candidate, compares it to the mingain, and acts as follows:
- if
gain < mingain, then the CIRE candidate is discarded; - if
mingain <= gain <= mingain*3, then the CIRE candidate is optimized if and only if it doesn’t lead to an increase in working set size; - if
gain > mingain*3, then the CIRE candidate is optimized.
The default mingain is set to 10, a value that was determined empirically. The coefficient 3 in the relationals above was also determined empirically. The user can change the mingain’s default value via the cire-mingain optimization option.
To compute the gain of a CIRE candidate, Devito applies the following rules:
- Any integer arithmetic operation (e.g. for array indexing) has a cost of 0.
- A basic arithmetic operation such as
+and*has a cost of 1. - A
/has a cost of 5. - A power with non-integer exponent has a cost of 50.
- A power with non-negative integer exponent
nhas a cost ofn-1(i.e., the number of*it will be converted into). - A trascendental function (
sin,cos, etc.) has a cost of 100.
Further, if a CIRE candidate is invariant with respect to one or more dimensions, then the gain is proportionately scaled up.
Let’s now take a look at cire-mingain in action.
eq = Eq(u.forward, f**2*sin(f)*u.dy.dy) # Back to original running example
op8_omp = Operator(eq, opt=('cire-sops', {'openmp': True, 'cire-mingain': 31}))
print_kernel(op8_omp)for (int time = time_m, t0 = (time)%(2), t1 = (time + 1)%(2); time <= time_M; time += 1, t0 = (time)%(2), t1 = (time + 1)%(2))
{
START(section0)
#pragma omp parallel num_threads(nthreads)
{
#pragma omp for collapse(3) schedule(dynamic,1)
for (int x = x_m; x <= x_M; x += 1)
{
for (int y = y_m; y <= y_M; y += 1)
{
for (int z = z_m; z <= z_M; z += 1)
{
u[t1][x + 4][y + 4][z + 4] = ((-6.66666667e-1F/h_y)*(8.33333333e-2F*u[t0][x + 4][y + 1][z + 4]/h_y - 6.66666667e-1F*u[t0][x + 4][y + 2][z + 4]/h_y + 6.66666667e-1F*u[t0][x + 4][y + 4][z + 4]/h_y - 8.33333333e-2F*u[t0][x + 4][y + 5][z + 4]/h_y) + (-8.33333333e-2F/h_y)*(8.33333333e-2F*u[t0][x + 4][y + 4][z + 4]/h_y - 6.66666667e-1F*u[t0][x + 4][y + 5][z + 4]/h_y + 6.66666667e-1F*u[t0][x + 4][y + 7][z + 4]/h_y - 8.33333333e-2F*u[t0][x + 4][y + 8][z + 4]/h_y) + (8.33333333e-2F/h_y)*(8.33333333e-2F*u[t0][x + 4][y][z + 4]/h_y - 6.66666667e-1F*u[t0][x + 4][y + 1][z + 4]/h_y + 6.66666667e-1F*u[t0][x + 4][y + 3][z + 4]/h_y - 8.33333333e-2F*u[t0][x + 4][y + 4][z + 4]/h_y) + (6.66666667e-1F/h_y)*(8.33333333e-2F*u[t0][x + 4][y + 3][z + 4]/h_y - 6.66666667e-1F*u[t0][x + 4][y + 4][z + 4]/h_y + 6.66666667e-1F*u[t0][x + 4][y + 6][z + 4]/h_y - 8.33333333e-2F*u[t0][x + 4][y + 7][z + 4]/h_y))*sinf(f[x + 1][y + 1][z + 1])*pow(f[x + 1][y + 1][z + 1], 2);
}
}
}
}
STOP(section0,timers)
}We observe how setting cire-mingain=31 makes the tensor temporary produced by op7_omp disappear. Any cire-mingain greater than 31 will have the same effect. Let’s see why. The CIRE candidate in question is u.dy, which expands to:
8.33333333e-2F*u[t0][x + 4][y + 2][z + 4]/h_y - 6.66666667e-1F*u[t0][x + 4][y + 3][z + 4]/h_y + 6.66666667e-1F*u[t0][x + 4][y + 5][z + 4]/h_y - 8.33333333e-2F*u[t0][x + 4][y + 6][z + 4]/h_y- Three
+(or-) => 3*1 = 3 - Four
/byh_y=> 4*5 = 20 - Four two-way
*with operands(1/h_y, u)=> 8
So we have a cost of 31. Since the CIRE candidate stems from .dy, and since u’s space order is 4, the CIRE candidates appears 4 times in total, with shifted indices along y. It follows that the gain due to optimizing it away through a tensor temporary would be 31*(4 - 1) = 93. However, mingain*3 = 31*3 = 93, so mingain < gain <= mingain*3, and since there would be an increase in working set size (the new tensor temporary to be allocated), then the CIRE candidate is discarded.
As shown below, any cire-mingain smaller than 31 will lead to capturing the tensor temporary.
op8_b1_omp = Operator(eq, opt=('cire-sops', 'simd', {'openmp': True, 'cire-mingain': 30}))
print_kernel(op8_b1_omp)for (int time = time_m, t0 = (time)%(2), t1 = (time + 1)%(2); time <= time_M; time += 1, t0 = (time)%(2), t1 = (time + 1)%(2))
{
START(section0)
#pragma omp parallel num_threads(nthreads)
{
const int tid = omp_get_thread_num();
float (*restrict r0)[z_size] __attribute__ ((aligned (64))) = (float (*)[z_size]) pr0[tid];
#pragma omp for schedule(dynamic,1)
for (int x = x_m; x <= x_M; x += 1)
{
for (int y = y_m - 2; y <= y_M + 2; y += 1)
{
#pragma omp simd aligned(u:32)
for (int z = z_m; z <= z_M; z += 1)
{
r0[y + 2][z] = 8.33333333e-2F*u[t0][x + 4][y + 2][z + 4]/h_y - 6.66666667e-1F*u[t0][x + 4][y + 3][z + 4]/h_y + 6.66666667e-1F*u[t0][x + 4][y + 5][z + 4]/h_y - 8.33333333e-2F*u[t0][x + 4][y + 6][z + 4]/h_y;
}
}
for (int y = y_m; y <= y_M; y += 1)
{
#pragma omp simd aligned(f,u:32)
for (int z = z_m; z <= z_M; z += 1)
{
u[t1][x + 4][y + 4][z + 4] = ((-6.66666667e-1F/h_y)*r0[y + 1][z] + (-8.33333333e-2F/h_y)*r0[y + 4][z] + (8.33333333e-2F/h_y)*r0[y][z] + (6.66666667e-1F/h_y)*r0[y + 3][z])*sinf(f[x + 1][y + 1][z + 1])*pow(f[x + 1][y + 1][z + 1], 2);
}
}
}
}
STOP(section0,timers)
}Dimension-invariant sub-expressions
Cross-iteration redundancies may also be searched across dimension-invariant sub-expressions, typically time-invariants.
eq = Eq(u.forward, f**2*sin(f)*u.dy.dy) # Back to original running example
op11_omp = Operator(eq, opt=('lift', {'openmp': True}))
# print_kernel(op11_omp) # Uncomment to see the generated codeThe lift pass triggers CIRE for dimension-invariant sub-expressions. As seen before, this leads to producing one tensor temporary. By setting cire-mingain to a larger value, we can avoid a grid-size temporary to be allocated, in exchange for a trascendental function (sinf(...)) to be computed at each iteration.
op12_omp = Operator(eq, opt=('lift', {'openmp': True, 'cire-mingain': 34}))
print_kernel(op12_omp)for (int time = time_m, t0 = (time)%(2), t1 = (time + 1)%(2); time <= time_M; time += 1, t0 = (time)%(2), t1 = (time + 1)%(2))
{
START(section0)
#pragma omp parallel num_threads(nthreads)
{
#pragma omp for collapse(3) schedule(dynamic,1)
for (int x = x_m; x <= x_M; x += 1)
{
for (int y = y_m; y <= y_M; y += 1)
{
for (int z = z_m; z <= z_M; z += 1)
{
u[t1][x + 4][y + 4][z + 4] = ((-6.66666667e-1F/h_y)*(8.33333333e-2F*u[t0][x + 4][y + 1][z + 4]/h_y - 6.66666667e-1F*u[t0][x + 4][y + 2][z + 4]/h_y + 6.66666667e-1F*u[t0][x + 4][y + 4][z + 4]/h_y - 8.33333333e-2F*u[t0][x + 4][y + 5][z + 4]/h_y) + (-8.33333333e-2F/h_y)*(8.33333333e-2F*u[t0][x + 4][y + 4][z + 4]/h_y - 6.66666667e-1F*u[t0][x + 4][y + 5][z + 4]/h_y + 6.66666667e-1F*u[t0][x + 4][y + 7][z + 4]/h_y - 8.33333333e-2F*u[t0][x + 4][y + 8][z + 4]/h_y) + (8.33333333e-2F/h_y)*(8.33333333e-2F*u[t0][x + 4][y][z + 4]/h_y - 6.66666667e-1F*u[t0][x + 4][y + 1][z + 4]/h_y + 6.66666667e-1F*u[t0][x + 4][y + 3][z + 4]/h_y - 8.33333333e-2F*u[t0][x + 4][y + 4][z + 4]/h_y) + (6.66666667e-1F/h_y)*(8.33333333e-2F*u[t0][x + 4][y + 3][z + 4]/h_y - 6.66666667e-1F*u[t0][x + 4][y + 4][z + 4]/h_y + 6.66666667e-1F*u[t0][x + 4][y + 6][z + 4]/h_y - 8.33333333e-2F*u[t0][x + 4][y + 7][z + 4]/h_y))*sinf(f[x + 1][y + 1][z + 1])*pow(f[x + 1][y + 1][z + 1], 2);
}
}
}
}
STOP(section0,timers)
}The cire-maxpar option
Sometimes it’s possible to trade storage for parallelism (i.e., for more parallel dimensions). For this, Devito provides the cire-maxpar option which is by default set to:
- False on CPU backends
- True on GPU backends
Let’s see what happens when we switch it on
op13_omp = Operator(eq, opt=('cire-sops', {'openmp': True, 'cire-maxpar': True}))
print_kernel(op13_omp)for (int time = time_m, t0 = (time)%(2), t1 = (time + 1)%(2); time <= time_M; time += 1, t0 = (time)%(2), t1 = (time + 1)%(2))
{
START(section0)
#pragma omp parallel num_threads(nthreads)
{
#pragma omp for collapse(3) schedule(dynamic,1)
for (int x = x_m; x <= x_M; x += 1)
{
for (int y = y_m - 2; y <= y_M + 2; y += 1)
{
for (int z = z_m; z <= z_M; z += 1)
{
r0[x][y + 2][z] = 8.33333333e-2F*u[t0][x + 4][y + 2][z + 4]/h_y - 6.66666667e-1F*u[t0][x + 4][y + 3][z + 4]/h_y + 6.66666667e-1F*u[t0][x + 4][y + 5][z + 4]/h_y - 8.33333333e-2F*u[t0][x + 4][y + 6][z + 4]/h_y;
}
}
}
}
#pragma omp parallel num_threads(nthreads)
{
#pragma omp for collapse(3) schedule(dynamic,1)
for (int x = x_m; x <= x_M; x += 1)
{
for (int y = y_m; y <= y_M; y += 1)
{
for (int z = z_m; z <= z_M; z += 1)
{
u[t1][x + 4][y + 4][z + 4] = ((-6.66666667e-1F/h_y)*r0[x][y + 1][z] + (-8.33333333e-2F/h_y)*r0[x][y + 4][z] + (8.33333333e-2F/h_y)*r0[x][y][z] + (6.66666667e-1F/h_y)*r0[x][y + 3][z])*sinf(f[x + 1][y + 1][z + 1])*pow(f[x + 1][y + 1][z + 1], 2);
}
}
}
}
STOP(section0,timers)
}The generated code uses a three-dimensional temporary that gets written and subsequently read in two separate x-y-z loop nests. Now, both loops can safely be openmp-collapsed, which is vital when running on GPUs.
Impact of CIRE in the advanced mode
The advanced mode triggers all of the passes we’ve seen so far… and in fact, many more! Some of them, however, aren’t visible in our running example (e.g., all of the MPI-related optimizations). These will be treated in a future notebook.
Obviously, all of the optimization options (e.g. cire-mingain, blocklevels) are applicable and composable in any arbitrary way.
op14_omp = Operator(eq, openmp=True)
print(op14_omp)
# op14_b0_omp = Operator(eq, opt=('advanced', {'min-storage': True}))#define _POSIX_C_SOURCE 200809L
#define START(S) struct timeval start_ ## S , end_ ## S ; gettimeofday(&start_ ## S , NULL);
#define STOP(S,T) gettimeofday(&end_ ## S, NULL); T->S += (double)(end_ ## S .tv_sec-start_ ## S.tv_sec)+(double)(end_ ## S .tv_usec-start_ ## S .tv_usec)/1000000;
#define MIN(a,b) (((a) < (b)) ? (a) : (b))
#include "stdlib.h"
#include "math.h"
#include "sys/time.h"
#include "xmmintrin.h"
#include "pmmintrin.h"
#include "omp.h"
struct dataobj
{
void *restrict data;
unsigned long * size;
unsigned long * npsize;
unsigned long * dsize;
int * hsize;
int * hofs;
int * oofs;
void * dmap;
} ;
struct profiler
{
double section0;
double section1;
} ;
int Kernel(struct dataobj *restrict f_vec, struct dataobj *restrict u_vec, const float h_y, const int time_M, const int time_m, const int x0_blk0_size, const int x_M, const int x_m, const int y0_blk0_size, const int y_M, const int y_m, const int z_M, const int z_m, const int nthreads, const int z_size, const int x_size, const int y_size, struct profiler * timers)
{
float **restrict pr2_vec __attribute__ ((aligned (64)));
posix_memalign((void**)(&pr2_vec),64,nthreads*sizeof(float*));
float *restrict r0_vec __attribute__ ((aligned (64)));
posix_memalign((void**)(&r0_vec),64,x_size*y_size*z_size*sizeof(float));
#pragma omp parallel num_threads(nthreads)
{
const int tid = omp_get_thread_num();
posix_memalign((void**)(&(pr2_vec[tid])),64,z_size*(y0_blk0_size + 4)*sizeof(float));
}
float (*restrict f)[f_vec->size[1]][f_vec->size[2]] __attribute__ ((aligned (64))) = (float (*)[f_vec->size[1]][f_vec->size[2]]) f_vec->data;
float **pr2 = (float**) pr2_vec;
float (*restrict r0)[y_size][z_size] __attribute__ ((aligned (64))) = (float (*)[y_size][z_size]) r0_vec;
float (*restrict u)[u_vec->size[1]][u_vec->size[2]][u_vec->size[3]] __attribute__ ((aligned (64))) = (float (*)[u_vec->size[1]][u_vec->size[2]][u_vec->size[3]]) u_vec->data;
/* Flush denormal numbers to zero in hardware */
_MM_SET_DENORMALS_ZERO_MODE(_MM_DENORMALS_ZERO_ON);
_MM_SET_FLUSH_ZERO_MODE(_MM_FLUSH_ZERO_ON);
float r1 = 1.0F/h_y;
START(section0)
#pragma omp parallel num_threads(nthreads)
{
#pragma omp for collapse(2) schedule(static,1)
for (int x = x_m; x <= x_M; x += 1)
{
for (int y = y_m; y <= y_M; y += 1)
{
#pragma omp simd aligned(f:32)
for (int z = z_m; z <= z_M; z += 1)
{
r0[x][y][z] = sinf(f[x + 1][y + 1][z + 1]);
}
}
}
}
STOP(section0,timers)
for (int time = time_m, t0 = (time)%(2), t1 = (time + 1)%(2); time <= time_M; time += 1, t0 = (time)%(2), t1 = (time + 1)%(2))
{
START(section1)
#pragma omp parallel num_threads(nthreads)
{
const int tid = omp_get_thread_num();
float (*restrict r2)[z_size] __attribute__ ((aligned (64))) = (float (*)[z_size]) pr2[tid];
#pragma omp for collapse(2) schedule(dynamic,1)
for (int x0_blk0 = x_m; x0_blk0 <= x_M; x0_blk0 += x0_blk0_size)
{
for (int y0_blk0 = y_m; y0_blk0 <= y_M; y0_blk0 += y0_blk0_size)
{
for (int x = x0_blk0; x <= MIN(x_M, x0_blk0 + x0_blk0_size - 1); x += 1)
{
for (int y = y0_blk0 - 2, ys = 0; y <= MIN(y_M + 2, y0_blk0 + y0_blk0_size + 1); y += 1, ys += 1)
{
#pragma omp simd aligned(u:32)
for (int z = z_m; z <= z_M; z += 1)
{
r2[ys][z] = r1*(8.33333333e-2F*(u[t0][x + 4][y + 2][z + 4] - u[t0][x + 4][y + 6][z + 4]) + 6.66666667e-1F*(-u[t0][x + 4][y + 3][z + 4] + u[t0][x + 4][y + 5][z + 4]));
}
}
for (int y = y0_blk0, ys = 0; y <= MIN(y_M, y0_blk0 + y0_blk0_size - 1); y += 1, ys += 1)
{
#pragma omp simd aligned(f,u:32)
for (int z = z_m; z <= z_M; z += 1)
{
u[t1][x + 4][y + 4][z + 4] = r1*(f[x + 1][y + 1][z + 1]*f[x + 1][y + 1][z + 1])*(8.33333333e-2F*(r2[ys][z] - r2[ys + 4][z]) + 6.66666667e-1F*(-r2[ys + 1][z] + r2[ys + 3][z]))*r0[x][y][z];
}
}
}
}
}
}
STOP(section1,timers)
}
#pragma omp parallel num_threads(nthreads)
{
const int tid = omp_get_thread_num();
free(pr2_vec[tid]);
}
free(pr2_vec);
free(r0_vec);
return 0;
}
A crucial observation here is that CIRE is applied on top of loop blocking – the r2 temporary is computed within the same block as u, which in turn requires an additional iteration at the block edge along y to be performed (the first y loop starts at y0_blk0 - 2, while the second one at y0_blk0). Further, the size of r2 is now a function of the block shape. Altogether, this implements what in the literature is often referred to as “overlapped tiling” (or “overlapped blocking”): data reuse across consecutive loop nests is obtained by cross-loop blocking, which in turn requires a certain degree of redundant computation at the block edge. Clearly, there’s a tension between the block shape and the amount of redundant computation. For example, a small block shape guarantees a small(er) working set, and thus better data reuse, but also requires more redundant computation.
The cire-rotate option
So far we’ve seen two ways to compute the tensor temporaries:
- The temporary dimensions span the whole grid;
- The temporary dimensions span a block created by the loop blocking pass.
There are a few other ways, and in particular there’s a third way supported in Devito, enabled through the cire-rotate option:
- The temporary outermost-dimension is a function of the stencil radius; all other temporary dimensions are a function of the loop blocking shape.
Let’s jump straight into an example
op15_omp = Operator(eq, opt=('advanced', {'openmp': True, 'cire-rotate': True}))
print_kernel(op15_omp)float r1 = 1.0F/h_y;
START(section0)
#pragma omp parallel num_threads(nthreads)
{
#pragma omp for collapse(2) schedule(static,1)
for (int x = x_m; x <= x_M; x += 1)
{
for (int y = y_m; y <= y_M; y += 1)
{
#pragma omp simd aligned(f:32)
for (int z = z_m; z <= z_M; z += 1)
{
r0[x][y][z] = sinf(f[x + 1][y + 1][z + 1]);
}
}
}
}
STOP(section0,timers)
for (int time = time_m, t0 = (time)%(2), t1 = (time + 1)%(2); time <= time_M; time += 1, t0 = (time)%(2), t1 = (time + 1)%(2))
{
START(section1)
#pragma omp parallel num_threads(nthreads)
{
const int tid = omp_get_thread_num();
float (*restrict r2)[z_size] __attribute__ ((aligned (64))) = (float (*)[z_size]) pr2[tid];
#pragma omp for collapse(2) schedule(dynamic,1)
for (int x0_blk0 = x_m; x0_blk0 <= x_M; x0_blk0 += x0_blk0_size)
{
for (int y0_blk0 = y_m; y0_blk0 <= y_M; y0_blk0 += y0_blk0_size)
{
for (int x = x0_blk0; x <= MIN(x_M, x0_blk0 + x0_blk0_size - 1); x += 1)
{
for (int y = y0_blk0, ys = 0, yr0 = (ys)%(5), yr1 = (ys + 3)%(5), yr2 = (ys + 4)%(5), yr3 = (ys + 1)%(5), yii = -2; y <= MIN(y_M, y0_blk0 + y0_blk0_size - 1); y += 1, ys += 1, yr0 = (ys)%(5), yr1 = (ys + 3)%(5), yr2 = (ys + 4)%(5), yr3 = (ys + 1)%(5), yii = 2)
{
for (int yc = yii, yi = (yc + ys + 2)%(5); yc <= 2; yc += 1, yi = (yc + ys + 2)%(5))
{
#pragma omp simd aligned(u:32)
for (int z = z_m; z <= z_M; z += 1)
{
r2[yi][z] = r1*(8.33333333e-2F*(u[t0][x + 4][y + yc + 2][z + 4] - u[t0][x + 4][y + yc + 6][z + 4]) + 6.66666667e-1F*(-u[t0][x + 4][y + yc + 3][z + 4] + u[t0][x + 4][y + yc + 5][z + 4]));
}
}
#pragma omp simd aligned(f,u:32)
for (int z = z_m; z <= z_M; z += 1)
{
u[t1][x + 4][y + 4][z + 4] = r1*(f[x + 1][y + 1][z + 1]*f[x + 1][y + 1][z + 1])*(8.33333333e-2F*(r2[yr0][z] - r2[yr2][z]) + 6.66666667e-1F*(r2[yr1][z] - r2[yr3][z]))*r0[x][y][z];
}
}
}
}
}
}
STOP(section1,timers)
}There are two key things to notice here:
- The
r2temporary is a pointer to a two-dimensional array of size[2][z_size]. It’s obtained via casting ofpr2[tid], which in turn is defined as
print(op15_omp.body.allocs[3])posix_memalign((void**)(&r0_vec),64,x_size*y_size*z_size*sizeof(float));- Within the
yloop there are several iteration variables, some of which (yr0,yr1, …) employ modulo increment to cyclically produce the indices 0 and 1.
In essence, with cire-rotate, instead of computing an entire slice of y values, at each y iteration we only keep track of the values that are strictly necessary to evaluate u at y – only two values in this case. This results in a working set reduction, at the price of turning one parallel loop (y) into a sequential one.
The advanced-fsg mode
The alternative advanced-fsg optimization level applies the same passes as advanced, but in a different order. The key difference is that -fsg does not generate overlapped blocking code across CIRE-generated loop nests.
eq = Eq(u.forward, f**2*sin(f)*u.dy.dy) # Back to original running example
op17_omp = Operator(eq, opt=('advanced-fsg', {'openmp': True}))
print(op17_omp)#define _POSIX_C_SOURCE 200809L
#define START(S) struct timeval start_ ## S , end_ ## S ; gettimeofday(&start_ ## S , NULL);
#define STOP(S,T) gettimeofday(&end_ ## S, NULL); T->S += (double)(end_ ## S .tv_sec-start_ ## S.tv_sec)+(double)(end_ ## S .tv_usec-start_ ## S .tv_usec)/1000000;
#include "stdlib.h"
#include "math.h"
#include "sys/time.h"
#include "xmmintrin.h"
#include "pmmintrin.h"
#include "omp.h"
struct dataobj
{
void *restrict data;
unsigned long * size;
unsigned long * npsize;
unsigned long * dsize;
int * hsize;
int * hofs;
int * oofs;
void * dmap;
} ;
struct profiler
{
double section0;
double section1;
} ;
int Kernel(struct dataobj *restrict f_vec, struct dataobj *restrict u_vec, const float h_y, const int time_M, const int time_m, const int x_M, const int x_m, const int y_M, const int y_m, const int z_M, const int z_m, const int nthreads, const int z_size, const int x_size, const int y_size, struct profiler * timers)
{
float **restrict pr2_vec __attribute__ ((aligned (64)));
posix_memalign((void**)(&pr2_vec),64,nthreads*sizeof(float*));
float *restrict r0_vec __attribute__ ((aligned (64)));
posix_memalign((void**)(&r0_vec),64,x_size*y_size*z_size*sizeof(float));
#pragma omp parallel num_threads(nthreads)
{
const int tid = omp_get_thread_num();
posix_memalign((void**)(&(pr2_vec[tid])),64,z_size*(y_size + 4)*sizeof(float));
}
float (*restrict f)[f_vec->size[1]][f_vec->size[2]] __attribute__ ((aligned (64))) = (float (*)[f_vec->size[1]][f_vec->size[2]]) f_vec->data;
float **pr2 = (float**) pr2_vec;
float (*restrict r0)[y_size][z_size] __attribute__ ((aligned (64))) = (float (*)[y_size][z_size]) r0_vec;
float (*restrict u)[u_vec->size[1]][u_vec->size[2]][u_vec->size[3]] __attribute__ ((aligned (64))) = (float (*)[u_vec->size[1]][u_vec->size[2]][u_vec->size[3]]) u_vec->data;
/* Flush denormal numbers to zero in hardware */
_MM_SET_DENORMALS_ZERO_MODE(_MM_DENORMALS_ZERO_ON);
_MM_SET_FLUSH_ZERO_MODE(_MM_FLUSH_ZERO_ON);
float r1 = 1.0F/h_y;
START(section0)
#pragma omp parallel num_threads(nthreads)
{
#pragma omp for collapse(2) schedule(static,1)
for (int x = x_m; x <= x_M; x += 1)
{
for (int y = y_m; y <= y_M; y += 1)
{
#pragma omp simd aligned(f:32)
for (int z = z_m; z <= z_M; z += 1)
{
r0[x][y][z] = sinf(f[x + 1][y + 1][z + 1]);
}
}
}
}
STOP(section0,timers)
for (int time = time_m, t0 = (time)%(2), t1 = (time + 1)%(2); time <= time_M; time += 1, t0 = (time)%(2), t1 = (time + 1)%(2))
{
START(section1)
#pragma omp parallel num_threads(nthreads)
{
const int tid = omp_get_thread_num();
float (*restrict r2)[z_size] __attribute__ ((aligned (64))) = (float (*)[z_size]) pr2[tid];
#pragma omp for schedule(dynamic,1)
for (int x = x_m; x <= x_M; x += 1)
{
for (int y = y_m - 2; y <= y_M + 2; y += 1)
{
#pragma omp simd aligned(u:32)
for (int z = z_m; z <= z_M; z += 1)
{
r2[y + 2][z] = r1*(8.33333333e-2F*(u[t0][x + 4][y + 2][z + 4] - u[t0][x + 4][y + 6][z + 4]) + 6.66666667e-1F*(-u[t0][x + 4][y + 3][z + 4] + u[t0][x + 4][y + 5][z + 4]));
}
}
for (int y = y_m; y <= y_M; y += 1)
{
#pragma omp simd aligned(f,u:32)
for (int z = z_m; z <= z_M; z += 1)
{
u[t1][x + 4][y + 4][z + 4] = r1*(f[x + 1][y + 1][z + 1]*f[x + 1][y + 1][z + 1])*(8.33333333e-2F*(r2[y][z] - r2[y + 4][z]) + 6.66666667e-1F*(-r2[y + 1][z] + r2[y + 3][z]))*r0[x][y][z];
}
}
}
}
STOP(section1,timers)
}
#pragma omp parallel num_threads(nthreads)
{
const int tid = omp_get_thread_num();
free(pr2_vec[tid]);
}
free(pr2_vec);
free(r0_vec);
return 0;
}
The x loop here is still shared by the two loop nests, but the y one isn’t. Analogously, if we consider the alternative eq already used in op7_b0_omp, we get two completely separate, and therefore individually blocked, loop nests.
eq = Eq(u.forward, f**2*sin(f)*(u.dy.dy + u.dx.dx))
op17_b0 = Operator(eq, opt=('advanced-fsg', {'openmp': True}))
print(op17_b0)#define _POSIX_C_SOURCE 200809L
#define START(S) struct timeval start_ ## S , end_ ## S ; gettimeofday(&start_ ## S , NULL);
#define STOP(S,T) gettimeofday(&end_ ## S, NULL); T->S += (double)(end_ ## S .tv_sec-start_ ## S.tv_sec)+(double)(end_ ## S .tv_usec-start_ ## S .tv_usec)/1000000;
#define MIN(a,b) (((a) < (b)) ? (a) : (b))
#include "stdlib.h"
#include "math.h"
#include "sys/time.h"
#include "xmmintrin.h"
#include "pmmintrin.h"
#include "omp.h"
struct dataobj
{
void *restrict data;
unsigned long * size;
unsigned long * npsize;
unsigned long * dsize;
int * hsize;
int * hofs;
int * oofs;
void * dmap;
} ;
struct profiler
{
double section0;
double section1;
} ;
int Kernel(struct dataobj *restrict f_vec, struct dataobj *restrict u_vec, const float h_x, const float h_y, const int time_M, const int time_m, const int x0_blk0_size, const int x1_blk0_size, const int x_M, const int x_m, const int y0_blk0_size, const int y1_blk0_size, const int y_M, const int y_m, const int z_M, const int z_m, const int nthreads, const int x_size, const int y_size, const int z_size, struct profiler * timers)
{
float *restrict r0_vec __attribute__ ((aligned (64)));
posix_memalign((void**)(&r0_vec),64,x_size*y_size*z_size*sizeof(float));
float *restrict r3_vec __attribute__ ((aligned (64)));
posix_memalign((void**)(&r3_vec),64,z_size*(x_size + 4)*(y_size + 4)*sizeof(float));
float *restrict r4_vec __attribute__ ((aligned (64)));
posix_memalign((void**)(&r4_vec),64,z_size*(x_size + 4)*(y_size + 4)*sizeof(float));
float (*restrict f)[f_vec->size[1]][f_vec->size[2]] __attribute__ ((aligned (64))) = (float (*)[f_vec->size[1]][f_vec->size[2]]) f_vec->data;
float (*restrict r0)[y_size][z_size] __attribute__ ((aligned (64))) = (float (*)[y_size][z_size]) r0_vec;
float (*restrict r3)[y_size + 4][z_size] __attribute__ ((aligned (64))) = (float (*)[y_size + 4][z_size]) r3_vec;
float (*restrict r4)[y_size + 4][z_size] __attribute__ ((aligned (64))) = (float (*)[y_size + 4][z_size]) r4_vec;
float (*restrict u)[u_vec->size[1]][u_vec->size[2]][u_vec->size[3]] __attribute__ ((aligned (64))) = (float (*)[u_vec->size[1]][u_vec->size[2]][u_vec->size[3]]) u_vec->data;
/* Flush denormal numbers to zero in hardware */
_MM_SET_DENORMALS_ZERO_MODE(_MM_DENORMALS_ZERO_ON);
_MM_SET_FLUSH_ZERO_MODE(_MM_FLUSH_ZERO_ON);
float r1 = 1.0F/h_x;
float r2 = 1.0F/h_y;
START(section0)
#pragma omp parallel num_threads(nthreads)
{
#pragma omp for collapse(2) schedule(static,1)
for (int x = x_m; x <= x_M; x += 1)
{
for (int y = y_m; y <= y_M; y += 1)
{
#pragma omp simd aligned(f:32)
for (int z = z_m; z <= z_M; z += 1)
{
r0[x][y][z] = sinf(f[x + 1][y + 1][z + 1]);
}
}
}
}
STOP(section0,timers)
for (int time = time_m, t0 = (time)%(2), t1 = (time + 1)%(2); time <= time_M; time += 1, t0 = (time)%(2), t1 = (time + 1)%(2))
{
START(section1)
#pragma omp parallel num_threads(nthreads)
{
#pragma omp for collapse(2) schedule(dynamic,1)
for (int x0_blk0 = x_m - 2; x0_blk0 <= x_M + 2; x0_blk0 += x0_blk0_size)
{
for (int y0_blk0 = y_m - 2; y0_blk0 <= y_M + 2; y0_blk0 += y0_blk0_size)
{
for (int x = x0_blk0; x <= MIN(x_M + 2, x0_blk0 + x0_blk0_size - 1); x += 1)
{
for (int y = y0_blk0; y <= MIN(y_M + 2, y0_blk0 + y0_blk0_size - 1); y += 1)
{
#pragma omp simd aligned(u:32)
for (int z = z_m; z <= z_M; z += 1)
{
r3[x + 2][y + 2][z] = r1*(8.33333333e-2F*(u[t0][x + 2][y + 4][z + 4] - u[t0][x + 6][y + 4][z + 4]) + 6.66666667e-1F*(-u[t0][x + 3][y + 4][z + 4] + u[t0][x + 5][y + 4][z + 4]));
r4[x + 2][y + 2][z] = r2*(8.33333333e-2F*(u[t0][x + 4][y + 2][z + 4] - u[t0][x + 4][y + 6][z + 4]) + 6.66666667e-1F*(-u[t0][x + 4][y + 3][z + 4] + u[t0][x + 4][y + 5][z + 4]));
}
}
}
}
}
}
#pragma omp parallel num_threads(nthreads)
{
#pragma omp for collapse(2) schedule(dynamic,1)
for (int x1_blk0 = x_m; x1_blk0 <= x_M; x1_blk0 += x1_blk0_size)
{
for (int y1_blk0 = y_m; y1_blk0 <= y_M; y1_blk0 += y1_blk0_size)
{
for (int x = x1_blk0; x <= MIN(x_M, x1_blk0 + x1_blk0_size - 1); x += 1)
{
for (int y = y1_blk0; y <= MIN(y_M, y1_blk0 + y1_blk0_size - 1); y += 1)
{
#pragma omp simd aligned(f,u:32)
for (int z = z_m; z <= z_M; z += 1)
{
u[t1][x + 4][y + 4][z + 4] = (f[x + 1][y + 1][z + 1]*f[x + 1][y + 1][z + 1])*(r1*(8.33333333e-2F*(r3[x][y + 2][z] - r3[x + 4][y + 2][z]) + 6.66666667e-1F*(-r3[x + 1][y + 2][z] + r3[x + 3][y + 2][z])) + r2*(8.33333333e-2F*(r4[x + 2][y][z] - r4[x + 2][y + 4][z]) + 6.66666667e-1F*(-r4[x + 2][y + 1][z] + r4[x + 2][y + 3][z])))*r0[x][y][z];
}
}
}
}
}
}
STOP(section1,timers)
}
free(r0_vec);
free(r3_vec);
free(r4_vec);
return 0;
}
References
[1] Fabio Luporini, Mathias Louboutin, Michael Lange, Navjot Kukreja, Philipp Witte, Jan Hückelheim, Charles Yount, Paul H. J. Kelly, Felix J. Herrmann, and Gerard J. Gorman. 2020. Architecture and Performance of Devito, a System for Automated Stencil Computation. ACM Trans. Math. Softw. 46, 1, Article 6 (April 2020), 28 pages. DOI:https://doi.org/10.1145/3374916