```// Halide tutorial lesson 8: Scheduling multi-stage pipelines

// On linux, you can compile and run it like so:
// g++ lesson_08*.cpp -g -std=c++17 -I <path/to/Halide.h> -L <path/to/libHalide.so> -lHalide -lpthread -ldl -o lesson_08
// LD_LIBRARY_PATH=<path/to/libHalide.so> ./lesson_08

// On os x:
// g++ lesson_08*.cpp -g -std=c++17 -I <path/to/Halide.h> -L <path/to/libHalide.so> -lHalide -o lesson_08
// DYLD_LIBRARY_PATH=<path/to/libHalide.dylib> ./lesson_08

// If you have the entire Halide source tree, you can also build it by
// running:
//    make tutorial_lesson_08_scheduling_2
// in a shell with the current directory at the top of the halide
// source tree.

#include "Halide.h"
#include <stdio.h>

using namespace Halide;

int main(int argc, char **argv) {
// First we'll declare some Vars to use below.
Var x("x"), y("y");

// Let's examine various scheduling options for a simple two stage
{
Func producer("producer_default"), consumer("consumer_default");

// The first stage will be some simple pointwise math similar
// to our familiar gradient function. The value at position x,
// y is the sin of product of x and y.
producer(x, y) = sin(x * y);

// Now we'll add a second stage which averages together multiple
// points in the first stage.
consumer(x, y) = (producer(x, y) +
producer(x, y + 1) +
producer(x + 1, y) +
producer(x + 1, y + 1)) / 4;

// We'll turn on tracing for both functions.
consumer.trace_stores();
producer.trace_stores();

// And evaluate it over a 4x4 box.
printf("\nEvaluating producer-consumer pipeline with default schedule\n");
consumer.realize({4, 4});
// Click to show output ...

// There were no messages about computing values of the
// producer. This is because the default schedule fully
// inlines 'producer' into 'consumer'. It is as if we had
// written the following code instead:

// consumer(x, y) = (sin(x * y) +
//                   sin(x * (y + 1)) +
//                   sin((x + 1) * y) +
//                   sin((x + 1) * (y + 1))/4);

// All calls to 'producer' have been replaced with the body of
// 'producer', with the arguments substituted in for the
// variables.

// The equivalent C code is:
float result;
for (int y = 0; y < 4; y++) {
for (int x = 0; x < 4; x++) {
result[y][x] = (sin(x * y) +
sin(x * (y + 1)) +
sin((x + 1) * y) +
sin((x + 1) * (y + 1))) / 4;
}
}
printf("\n");

// If we look at the loop nest, the producer doesn't appear
// at all. It has been inlined into the consumer.
printf("Pseudo-code for the schedule:\n");
consumer.print_loop_nest();
printf("\n");
// Click to show output ...
}

// Next we'll examine the next simplest option - computing all
// values required in the producer before computing any of the
// consumer. We call this schedule "root".
{
Func producer("producer_root"), consumer("consumer_root");
producer(x, y) = sin(x * y);
consumer(x, y) = (producer(x, y) +
producer(x, y + 1) +
producer(x + 1, y) +
producer(x + 1, y + 1)) / 4;

// Tell Halide to evaluate all of producer before any of consumer.
producer.compute_root();

// Turn on tracing.
consumer.trace_stores();
producer.trace_stores();

// Compile and run.
printf("\nEvaluating producer.compute_root()\n");
consumer.realize({4, 4});
// Click to show output ...

// Reading the output we can see that:
// A) There were stores to producer.
// B) They all happened before any stores to consumer.

// See below for a visualization.
// The producer is on the left and the consumer is on the
// right. Stores are marked in orange and loads are marked in
// blue. // Equivalent C:

float result;

// Allocate some temporary storage for the producer.
float producer_storage;

// Compute the producer.
for (int y = 0; y < 5; y++) {
for (int x = 0; x < 5; x++) {
producer_storage[y][x] = sin(x * y);
}
}

// Compute the consumer. Skip the prints this time.
for (int y = 0; y < 4; y++) {
for (int x = 0; x < 4; x++) {
result[y][x] = (producer_storage[y][x] +
producer_storage[y + 1][x] +
producer_storage[y][x + 1] +
producer_storage[y + 1][x + 1]) / 4;
}
}

// Note that consumer was evaluated over a 4x4 box, so Halide
// automatically inferred that producer was needed over a 5x5
// box. This is the same 'bounds inference' logic we saw in
// the previous lesson, where it was used to detect and avoid
// out-of-bounds reads from an input image.

// If we print the loop nest, we'll see something very
// similar to the C above.
printf("Pseudo-code for the schedule:\n");
consumer.print_loop_nest();
printf("\n");
// Click to show output ...
}

// Let's compare the two approaches above from a performance
// perspective.

// Full inlining (the default schedule):
// - Temporary memory allocated: 0
// - Stores: 16
// - Calls to sin: 64

// producer.compute_root():
// - Temporary memory allocated: 25 floats
// - Stores: 41
// - Calls to sin: 25

// There's a trade-off here. Full inlining used minimal temporary
// memory and memory bandwidth, but did a whole bunch of redundant
// expensive math (calling sin). It evaluated most points in
// 'producer' four times. The second schedule,
// producer.compute_root(), did the mimimum number of calls to
// sin, but used more temporary memory and more memory bandwidth.

// In any given situation the correct choice can be difficult to
// make. If you're memory-bandwidth limited, or don't have much
// memory (e.g. because you're running on an old cell-phone), then
// it can make sense to do redundant math. On the other hand, sin
// is expensive, so if you're compute-limited then fewer calls to
// multi-core parallelism tilts the scales in favor of doing
// redundant work, because firing up multiple cpu cores increases
// the amount of math you can do per second, but doesn't increase
// your system memory bandwidth or capacity.

// We can make choices in between full inlining and
// compute_root. Next we'll alternate between computing the
// producer and consumer on a per-scanline basis:
{
Func producer("producer_y"), consumer("consumer_y");
producer(x, y) = sin(x * y);
consumer(x, y) = (producer(x, y) +
producer(x, y + 1) +
producer(x + 1, y) +
producer(x + 1, y + 1)) / 4;

// Tell Halide to evaluate producer as needed per y coordinate
// of the consumer:
producer.compute_at(consumer, y);

// This places the code that computes the producer just
// *inside* the consumer's for loop over y, as in the
// equivalent C below.

// Turn on tracing.
producer.trace_stores();
consumer.trace_stores();

// Compile and run.
printf("\nEvaluating producer.compute_at(consumer, y)\n");
consumer.realize({4, 4});
// Click to show output ...

// See below for a visualization. // Reading the log or looking at the figure you should see
// that producer and consumer alternate on a per-scanline
// basis. Let's look at the equivalent C:

float result;

// There's an outer loop over scanlines of consumer:
for (int y = 0; y < 4; y++) {

// Allocate space and compute enough of the producer to
// satisfy this single scanline of the consumer. This
// means a 5x2 box of the producer.
float producer_storage;
for (int py = y; py < y + 2; py++) {
for (int px = 0; px < 5; px++) {
producer_storage[py - y][px] = sin(px * py);
}
}

// Compute a scanline of the consumer.
for (int x = 0; x < 4; x++) {
result[y][x] = (producer_storage[x] +
producer_storage[x] +
producer_storage[x + 1] +
producer_storage[x + 1]) / 4;
}
}

// Again, if we print the loop nest, we'll see something very
// similar to the C above.
printf("Pseudo-code for the schedule:\n");
consumer.print_loop_nest();
printf("\n");
// Click to show output ...

// The performance characteristics of this strategy are in
// between inlining and compute root. We still allocate some
// temporary memory, but less than compute_root, and with
// better locality (we load from it soon after writing to it,
// so for larger images, values should still be in cache). We
// still do some redundant work, but less than full inlining:

// producer.compute_at(consumer, y):
// - Temporary memory allocated: 10 floats
// - Stores: 56
// - Calls to sin: 40
}

// We could also say producer.compute_at(consumer, x), but this
// would be very similar to full inlining (the default
// schedule). Instead let's distinguish between the loop level at
// which we allocate storage for producer, and the loop level at
// which we actually compute it. This unlocks a few optimizations.
{
Func producer("producer_root_y"), consumer("consumer_root_y");
producer(x, y) = sin(x * y);
consumer(x, y) = (producer(x, y) +
producer(x, y + 1) +
producer(x + 1, y) +
producer(x + 1, y + 1)) / 4;

// Tell Halide to make a buffer to store all of producer at
// the outermost level:
producer.store_root();
// ... but compute it as needed per y coordinate of the
// consumer.
producer.compute_at(consumer, y);

producer.trace_stores();
consumer.trace_stores();

printf("\nEvaluating producer.store_root().compute_at(consumer, y)\n");
consumer.realize({4, 4});
// Click to show output ...

// See below for a
// visualization. // Reading the log or looking at the figure you should see
// that producer and consumer again alternate on a
// per-scanline basis. It computes a 5x2 box of the producer
// to satisfy the first scanline of the consumer, but after
// that it only computes a 5x1 box of the output for each new
// scanline of the consumer!
//
// Halide has detected that for all scanlines except for the
// first, it can reuse the values already sitting in the
// buffer we've allocated for producer. Let's look at the
// equivalent C:

float result;

// producer.store_root() implies that storage goes here:
float producer_storage;

// There's an outer loop over scanlines of consumer:
for (int y = 0; y < 4; y++) {

// Compute enough of the producer to satisfy this scanline
// of the consumer.
for (int py = y; py < y + 2; py++) {

// Skip over rows of producer that we've already
// computed in a previous iteration.
if (y > 0 && py == y) continue;

for (int px = 0; px < 5; px++) {
producer_storage[py][px] = sin(px * py);
}
}

// Compute a scanline of the consumer.
for (int x = 0; x < 4; x++) {
result[y][x] = (producer_storage[y][x] +
producer_storage[y + 1][x] +
producer_storage[y][x + 1] +
producer_storage[y + 1][x + 1]) / 4;
}
}

printf("Pseudo-code for the schedule:\n");
consumer.print_loop_nest();
printf("\n");
// Click to show output ...

// The performance characteristics of this strategy are pretty
// good! The numbers are similar to compute_root, except locality
// is better. We're doing the minimum number of sin calls,
// and we load values soon after they are stored, so we're
// probably making good use of the cache:

// producer.store_root().compute_at(consumer, y):
// - Temporary memory allocated: 10 floats
// - Stores: 41
// - Calls to sin: 25

// Note that my claimed amount of memory allocated doesn't
// match the reference C code. Halide is performing one more
// optimization under the hood. It folds the storage for the
// producer down into a circular buffer of two
// scanlines. Equivalent C would actually look like this:

{
// Actually store 2 scanlines instead of 5
float producer_storage;
for (int y = 0; y < 4; y++) {
for (int py = y; py < y + 2; py++) {
if (y > 0 && py == y) continue;
for (int px = 0; px < 5; px++) {
// Stores to producer_storage have their y coordinate bit-masked.
producer_storage[py & 1][px] = sin(px * py);
}
}

// Compute a scanline of the consumer.
for (int x = 0; x < 4; x++) {
result[y][x] = (producer_storage[y & 1][x] +
producer_storage[(y + 1) & 1][x] +
producer_storage[y & 1][x + 1] +
producer_storage[(y + 1) & 1][x + 1]) / 4;
}
}
}
}

// We can do even better, by leaving the storage in the outermost
// loop, but moving the computation into the innermost loop:
{
Func producer("producer_root_x"), consumer("consumer_root_x");
producer(x, y) = sin(x * y);
consumer(x, y) = (producer(x, y) +
producer(x, y + 1) +
producer(x + 1, y) +
producer(x + 1, y + 1)) / 4;

// Store outermost, compute innermost.
producer.store_root().compute_at(consumer, x);

producer.trace_stores();
consumer.trace_stores();

printf("\nEvaluating producer.store_root().compute_at(consumer, x)\n");
consumer.realize({4, 4});
// Click to show output ...

// See below for a
// visualization. // You should see that producer and consumer now alternate on
// a per-pixel basis. Here's the equivalent C:

float result;

// producer.store_root() implies that storage goes here, but
// we can fold it down into a circular buffer of two
// scanlines:
float producer_storage;

// For every pixel of the consumer:
for (int y = 0; y < 4; y++) {
for (int x = 0; x < 4; x++) {

// Compute enough of the producer to satisfy this
// pixel of the consumer, but skip values that we've
if (y == 0 && x == 0) {
producer_storage[y & 1][x] = sin(x * y);
}
if (y == 0) {
producer_storage[y & 1][x + 1] = sin((x + 1) * y);
}
if (x == 0) {
producer_storage[(y + 1) & 1][x] = sin(x * (y + 1));
}
producer_storage[(y + 1) & 1][x + 1] = sin((x + 1) * (y + 1));

result[y][x] = (producer_storage[y & 1][x] +
producer_storage[(y + 1) & 1][x] +
producer_storage[y & 1][x + 1] +
producer_storage[(y + 1) & 1][x + 1]) / 4;
}
}

printf("Pseudo-code for the schedule:\n");
consumer.print_loop_nest();
printf("\n");
// Click to show output ...

// The performance characteristics of this strategy are the
// best so far. One of the four values of the producer we need
// is probably still sitting in a register, so I won't count
// producer.store_root().compute_at(consumer, x):
// - Temporary memory allocated: 10 floats
// - Stores: 41
// - Calls to sin: 25
}

// So what's the catch? Why not always do
// producer.store_root().compute_at(consumer, x) for this type of
// code?
//
// The answer is parallelism. In both of the previous two
// strategies we've assumed that values computed in previous
// iterations are lying around for us to reuse. This assumes that
// previous values of x or y happened earlier in time and have
// finished. This is not true if you parallelize or vectorize
// either loop. Darn. If you parallelize, Halide won't inject the
// optimizations that skip work already done if there's a parallel
// loop in between the store_at level and the compute_at level,
// and won't fold the storage down into a circular buffer either,
// which makes our store_root pointless.

// We're running out of options. We can make new ones by
// splitting. We can store_at or compute_at at the natural
// variables of the consumer (x and y), or we can split x or y
// into new inner and outer sub-variables and then schedule with
// respect to those. We'll use this to express fusion in tiles:
{
Func producer("producer_tile"), consumer("consumer_tile");
producer(x, y) = sin(x * y);
consumer(x, y) = (producer(x, y) +
producer(x, y + 1) +
producer(x + 1, y) +
producer(x + 1, y + 1)) / 4;

// We'll compute 8x8 of the consumer, in 4x4 tiles.
Var x_outer, y_outer, x_inner, y_inner;
consumer.tile(x, y, x_outer, y_outer, x_inner, y_inner, 4, 4);

// Compute the producer per tile of the consumer
producer.compute_at(consumer, x_outer);

// Notice that I wrote my schedule starting from the end of
// the pipeline (the consumer). This is because the schedule
// for the producer refers to x_outer, which we introduced
// when we tiled the consumer. You can write it in the other
// order, but it tends to be harder to read.

// Turn on tracing.
producer.trace_stores();
consumer.trace_stores();

printf("\nEvaluating:\n"
"consumer.tile(x, y, x_outer, y_outer, x_inner, y_inner, 4, 4);\n"
"producer.compute_at(consumer, x_outer);\n");
consumer.realize({8, 8});
// Click to show output ...

// See below for a visualization. // The producer and consumer now alternate on a per-tile
// basis. Here's the equivalent C:

float result;

// For every tile of the consumer:
for (int y_outer = 0; y_outer < 2; y_outer++) {
for (int x_outer = 0; x_outer < 2; x_outer++) {
// Compute the x and y coords of the start of this tile.
int x_base = x_outer * 4;
int y_base = y_outer * 4;

// Compute enough of producer to satisfy this tile. A
// 4x4 tile of the consumer requires a 5x5 tile of the
// producer.
float producer_storage;
for (int py = y_base; py < y_base + 5; py++) {
for (int px = x_base; px < x_base + 5; px++) {
producer_storage[py - y_base][px - x_base] = sin(px * py);
}
}

// Compute this tile of the consumer
for (int y_inner = 0; y_inner < 4; y_inner++) {
for (int x_inner = 0; x_inner < 4; x_inner++) {
int x = x_base + x_inner;
int y = y_base + y_inner;
result[y][x] =
(producer_storage[y - y_base][x - x_base] +
producer_storage[y - y_base + 1][x - x_base] +
producer_storage[y - y_base][x - x_base + 1] +
producer_storage[y - y_base + 1][x - x_base + 1]) / 4;
}
}
}
}

printf("Pseudo-code for the schedule:\n");
consumer.print_loop_nest();
printf("\n");
// Click to show output ...

// Tiling can make sense for problems like this one with
// stencils that reach outwards in x and y. Each tile can be
// computed independently in parallel, and the redundant work
// done by each tile isn't so bad once the tiles get large
// enough.
}

// Let's try a mixed strategy that combines what we have done with
// splitting, parallelizing, and vectorizing. This is one that
// often works well in practice for large images. If you
// understand this schedule, then you understand 95% of scheduling
// in Halide.
{
Func producer("producer_mixed"), consumer("consumer_mixed");
producer(x, y) = sin(x * y);
consumer(x, y) = (producer(x, y) +
producer(x, y + 1) +
producer(x + 1, y) +
producer(x + 1, y + 1)) / 4;

// Split the y coordinate of the consumer into strips of 16 scanlines:
Var yo, yi;
consumer.split(y, yo, yi, 16);
// Compute the strips using a thread pool and a task queue.
consumer.parallel(yo);
// Vectorize across x by a factor of four.
consumer.vectorize(x, 4);

// Now store the producer per-strip. This will be 17 scanlines
// of the producer (16+1), but hopefully it will fold down
// into a circular buffer of two scanlines:
producer.store_at(consumer, yo);
// Within each strip, compute the producer per scanline of the
// consumer, skipping work done on previous scanlines.
producer.compute_at(consumer, yi);
// Also vectorize the producer (because sin is vectorizable on x86 using SSE).
producer.vectorize(x, 4);

// Let's leave tracing off this time, because we're going to
// evaluate over a larger image.
// consumer.trace_stores();
// producer.trace_stores();

Buffer<float> halide_result = consumer.realize({160, 160});

// See below for a visualization.

Your browser does not support the video tag :(

// Here's the equivalent (serial) C:

float c_result;

// For every strip of 16 scanlines (this loop is parallel in
// the Halide version)
for (int yo = 0; yo < 160 / 16; yo++) {
int y_base = yo * 16;

// Allocate a two-scanline circular buffer for the producer
float producer_storage;

// For every scanline in the strip of 16:
for (int yi = 0; yi < 16; yi++) {
int y = y_base + yi;

for (int py = y; py < y + 2; py++) {
if (yi > 0 && py == y) continue;

// Compute this scanline of the producer in 4-wide vectors
for (int x_vec = 0; x_vec < 160 / 4 + 1; x_vec++) {
int x_base = x_vec * 4;
// 4 doesn't divide 161, so push the last vector left
// (see lesson 05).
if (x_base > 161 - 4) x_base = 161 - 4;
// If you're on x86, Halide generates SSE code for this part:
int x[] = {x_base, x_base + 1, x_base + 2, x_base + 3};
float vec = {sinf(x * py), sinf(x * py),
sinf(x * py), sinf(x * py)};
producer_storage[py & 1][x] = vec;
producer_storage[py & 1][x] = vec;
producer_storage[py & 1][x] = vec;
producer_storage[py & 1][x] = vec;
}
}

// Now compute consumer for this scanline:
for (int x_vec = 0; x_vec < 160 / 4; x_vec++) {
int x_base = x_vec * 4;
// Again, Halide's equivalent here uses SSE.
int x[] = {x_base, x_base + 1, x_base + 2, x_base + 3};
float vec[] = {
(producer_storage[y & 1][x] +
producer_storage[(y + 1) & 1][x] +
producer_storage[y & 1][x + 1] +
producer_storage[(y + 1) & 1][x + 1]) /
4,
(producer_storage[y & 1][x] +
producer_storage[(y + 1) & 1][x] +
producer_storage[y & 1][x + 1] +
producer_storage[(y + 1) & 1][x + 1]) /
4,
(producer_storage[y & 1][x] +
producer_storage[(y + 1) & 1][x] +
producer_storage[y & 1][x + 1] +
producer_storage[(y + 1) & 1][x + 1]) /
4,
(producer_storage[y & 1][x] +
producer_storage[(y + 1) & 1][x] +
producer_storage[y & 1][x + 1] +
producer_storage[(y + 1) & 1][x + 1]) /
4};

c_result[y][x] = vec;
c_result[y][x] = vec;
c_result[y][x] = vec;
c_result[y][x] = vec;
}
}
}
printf("Pseudo-code for the schedule:\n");
consumer.print_loop_nest();
printf("\n");
// Click to show output ...

// Look on my code, ye mighty, and despair!

// Let's check the C result against the Halide result. Doing
// this I found several bugs in my C implementation, which
// should tell you something.
for (int y = 0; y < 160; y++) {
for (int x = 0; x < 160; x++) {
float error = halide_result(x, y) - c_result[y][x];
// It's floating-point math, so we'll allow some slop:
if (error < -0.001f || error > 0.001f) {
printf("halide_result(%d, %d) = %f instead of %f\n",
x, y, halide_result(x, y), c_result[y][x]);
return -1;
}
}
}
}

// This stuff is hard. We ended up in a three-way trade-off
// between memory bandwidth, redundant work, and
// parallelism. Halide can't make the correct choice for you
// automatically (sorry). Instead it tries to make it easier for
// you to explore various options, without messing up your
// program. In fact, Halide promises that scheduling calls like
// compute_root won't change the meaning of your algorithm -- you
// should get the same bits back no matter how you schedule
// things.

// So be empirical! Experiment with various schedules and keep a
// log of performance. Form hypotheses and then try to prove
// yourself wrong. Don't assume that you just need to vectorize
// your code by a factor of four and run it on eight cores and
// you'll get 32x faster. This almost never works. Modern systems
// are complex enough that you can't predict performance reliably

// We suggest you start by scheduling all of your non-trivial
// stages compute_root, and then work from the end of the pipeline
// upwards, inlining, parallelizing, and vectorizing each stage in
// turn until you reach the top.

// Halide is not just about vectorizing and parallelizing your
// code. That's not enough to get you very far. Halide is about