public static int Main(string[] args)
{
// First we'll declare some Vars to use below.
var x = new HSVar("x");
var y = new HSVar("y");
// Let's examine various scheduling options for a simple two stage
// pipeline. We'll start with the default schedule:
{
var producer = new HSFunc("producer_default");
var consumer = new HSFunc("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] = HSMath.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.TraceStores();
producer.TraceStores();
// And evaluate it over a 4x4 box.
Console.WriteLine("\nEvaluating producer-consumer pipeline with default schedule");
consumer.Realize <float>(4, 4);
// 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:
var result = new float[4, 4];
for (int yy = 0; yy < 4; yy++)
{
for (int xx = 0; xx < 4; xx++)
{
result[yy, xx] = (float)((Math.Sin(xx * yy) +
Math.Sin(xx * (yy + 1)) +
Math.Sin((xx + 1) * yy) +
Math.Sin((xx + 1) * (yy + 1))) / 4);
}
}
Console.WriteLine();
// If we look at the loop nest, the producer doesn't appear
// at all. It has been inlined into the consumer.
Console.WriteLine("Pseudo-code for the schedule:");
consumer.PrintLoopNest();
Console.WriteLine();
}
// 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".
{
// Start with the same function definitions:
var producer = new HSFunc("producer_root");
var consumer = new HSFunc("consumer_root");
producer[x, y] = HSMath.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.ComputeRoot();
// Turn on tracing.
consumer.TraceStores();
producer.TraceStores();
// Compile and run.
Console.WriteLine("\nEvaluating producer.compute_root()");
consumer.Realize <float>(4, 4);
// Reading the output we can see that:
// A) There were stores to producer.
// B) They all happened before any stores to consumer.
// See figures/lesson_08_compute_root.gif 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:
var result = new float[4, 4];
// Allocate some temporary storage for the producer.
var producer_storage = new float[5, 5];
// Compute the producer.
for (int yy = 0; yy < 5; yy++)
{
for (int xx = 0; xx < 5; xx++)
{
producer_storage[yy, xx] = (float)Math.Sin(xx * yy);
}
}
// Compute the consumer. Skip the prints this time.
for (int yy = 0; yy < 4; yy++)
{
for (int xx = 0; xx < 4; xx++)
{
result[yy, xx] = (producer_storage[yy, xx] +
producer_storage[yy + 1, xx] +
producer_storage[yy, xx + 1] +
producer_storage[yy + 1, xx + 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.
Console.WriteLine("Pseudo-code for the schedule:");
consumer.PrintLoopNest();
Console.WriteLine();
}
// Let's compare the two approaches above from a performance
// perspective.
// Full inlining (the default schedule):
// - Temporary memory allocated: 0
// - Loads: 0
// - Stores: 16
// - Calls to sin: 64
// producer.compute_root():
// - Temporary memory allocated: 25 floats
// - Loads: 64
// - 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
// sin will make your program faster. Adding vectorization or
// 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:
{
// Start with the same function definitions:
var producer = new HSFunc("producer_y");
var consumer = new HSFunc("consumer_y");
producer[x, y] = HSMath.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.ComputeAt(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.TraceStores();
consumer.TraceStores();
// Compile and run.
Console.WriteLine("\nEvaluating producer.ComputeAt(consumer, y)");
consumer.Realize <float>(4, 4);
// See figures/lesson_08_compute_y.gif 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:
var result = new float[4, 4];
// There's an outer loop over scanlines of consumer:
for (int yy = 0; yy < 4; yy++)
{
// Allocate space and compute enough of the producer to
// satisfy this single scanline of the consumer. This
// means a 5x2 box of the producer.
var producer_storage = new float[2, 5];
for (int py = yy; py < yy + 2; py++)
{
for (int px = 0; px < 5; px++)
{
producer_storage[py - yy, px] = (float)Math.Sin(px * py);
}
}
// Compute a scanline of the consumer.
for (int xx = 0; xx < 4; xx++)
{
result[yy, xx] = (producer_storage[0, xx] +
producer_storage[1, xx] +
producer_storage[0, xx + 1] +
producer_storage[1, xx + 1]) / 4;
}
}
// Again, if we print the loop nest, we'll see something very
// similar to the C above.
Console.WriteLine("Pseudo-code for the schedule:");
consumer.PrintLoopNest();
Console.WriteLine();
// The performance characteristics of this strategy are in
// between inlining and compute root. We still allocate some
// temporary memory, but less that 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.ComputeAt(consumer, y):
// - Temporary memory allocated: 10 floats
// - Loads: 64
// - Stores: 56
// - Calls to sin: 40
}
// We could also say producer.ComputeAt(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.
{
var producer = new HSFunc("producer_root_y");
var consumer = new HSFunc("consumer_root_y");
producer[x, y] = HSMath.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.StoreRoot();
// ... but compute it as needed per y coordinate of the
// consumer.
producer.ComputeAt(consumer, y);
producer.TraceStores();
consumer.TraceStores();
Console.WriteLine("\nEvaluating producer.store_root().ComputeAt(consumer, y)");
consumer.Realize <float>(4, 4);
// See figures/lesson_08_store_root_compute_y.gif 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:
var result = new float[4, 4];
{
// producer.store_root() implies that storage goes here:
var producer_storage = new float[5, 5];
// There's an outer loop over scanlines of consumer:
for (int yy = 0; yy < 4; yy++)
{
// Compute enough of the producer to satisfy this scanline
// of the consumer.
for (int py = yy; py < yy + 2; py++)
{
// Skip over rows of producer that we've already
// computed in a previous iteration.
if (yy > 0 && py == yy)
{
continue;
}
for (int px = 0; px < 5; px++)
{
producer_storage[py, px] = (float)Math.Sin(px * py);
}
}
// Compute a scanline of the consumer.
for (int xx = 0; xx < 4; xx++)
{
result[yy, xx] = (producer_storage[yy, xx] +
producer_storage[yy + 1, xx] +
producer_storage[yy, xx + 1] +
producer_storage[yy + 1, xx + 1]) / 4;
}
}
}
Console.WriteLine("Pseudo-code for the schedule:");
consumer.PrintLoopNest();
Console.WriteLine();
// The performance characteristics of this strategy are pretty
// good! The numbers are similar 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().ComputeAt(consumer, y):
// - Temporary memory allocated: 10 floats
// - Loads: 64
// - Stores: 39
// - 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
var producer_storage = new float[2, 5];
for (int yy = 0; yy < 4; yy++)
{
for (int py = yy; py < yy + 2; py++)
{
if (yy > 0 && py == yy)
{
continue;
}
for (int px = 0; px < 5; px++)
{
// Stores to producer_storage have their y coordinate bit-masked.
producer_storage[py & 1, px] = (float)Math.Sin(px * py);
}
}
// Compute a scanline of the consumer.
for (int xx = 0; xx < 4; xx++)
{
// Loads from producer_storage have their y coordinate bit-masked.
result[yy, xx] = (producer_storage[yy & 1, xx] +
producer_storage[(yy + 1) & 1, xx] +
producer_storage[yy & 1, xx + 1] +
producer_storage[(yy + 1) & 1, xx + 1]) / 4;
}
}
}
}
// We can do even better, by leaving the storage outermost, but
// moving the computation into the innermost loop:
{
var producer = new HSFunc("producer_root_x");
var consumer = new HSFunc("consumer_root_x");
producer[x, y] = HSMath.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.StoreRoot().ComputeAt(consumer, x);
producer.TraceStores();
consumer.TraceStores();
Console.WriteLine("\nEvaluating producer.store_root().ComputeAt(consumer, x)");
consumer.Realize <float>(4, 4);
// See figures/lesson_08_store_root_compute_x.gif for a
// visualization.
// You should see that producer and consumer now alternate on
// a per-pixel basis. Here's the equivalent C:
var result = new float[4, 4];
// producer.store_root() implies that storage goes here, but
// we can fold it down into a circular buffer of two
// scanlines:
var producer_storage = new float[2, 5];
// For every pixel of the consumer:
for (int yy = 0; yy < 4; yy++)
{
for (int xx = 0; xx < 4; xx++)
{
// Compute enough of the producer to satisfy this
// pixel of the consumer, but skip values that we've
// already computed:
if (yy == 0 && xx == 0)
{
producer_storage[yy & 1, xx] = (float)Math.Sin(xx * yy);
}
if (yy == 0)
{
producer_storage[yy & 1, xx + 1] = (float)Math.Sin((xx + 1) * yy);
}
if (xx == 0)
{
producer_storage[(yy + 1) & 1, xx] = (float)Math.Sin(xx * (yy + 1));
}
producer_storage[(yy + 1) & 1, xx + 1] = (float)Math.Sin((xx + 1) * (yy + 1));
result[yy, xx] = (producer_storage[yy & 1, xx] +
producer_storage[(yy + 1) & 1, xx] +
producer_storage[yy & 1, xx + 1] +
producer_storage[(yy + 1) & 1, xx + 1]) / 4;
}
}
Console.WriteLine("Pseudo-code for the schedule:");
consumer.PrintLoopNest();
Console.WriteLine();
// 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
// it as a load:
// producer.store_root().ComputeAt(consumer, x):
// - Temporary memory allocated: 10 floats
// - Loads: 48
// - Stores: 56
// - Calls to sin: 40
}
// So what's the catch? Why not always do
// producer.store_root().ComputeAt(consumer, x) for this type of
// code?
//
// The answer is parallelism. In both of the previous two
// strategies we've assumed that values computed on 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 ComputeAt 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 ComputeAt 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:
{
var producer = new HSFunc("producer_tile");
var consumer = new HSFunc("consumer_tile");
producer[x, y] = HSMath.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 = new HSVar("x_outer");
var y_outer = new HSVar("y_outer");
var x_inner = new HSVar("x_inner");
var y_inner = new HSVar("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.ComputeAt(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.TraceStores();
consumer.TraceStores();
Console.WriteLine("\nEvaluating:");
Console.WriteLine("consumer.tile(x, y, x_outer, y_outer, x_inner, y_inner, 4, 4);");
Console.WriteLine("producer.ComputeAt(consumer, x_outer);");
consumer.Realize <float>(8, 8);
// See figures/lesson_08_tile.gif for a visualization.
// The producer and consumer now alternate on a per-tile
// basis. Here's the equivalent C:
var result = new float[8, 8];
// For every tile of the consumer:
for (int yy_outer = 0; yy_outer < 2; yy_outer++)
{
for (int xx_outer = 0; xx_outer < 2; xx_outer++)
{
// Compute the x and y coords of the start of this tile.
int x_base = xx_outer * 4;
int y_base = yy_outer * 4;
// Compute enough of producer to satisfy this tile. A
// 4x4 tile of the consumer requires a 5x5 tile of the
// producer.
var producer_storage = new float[5, 5];
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] = (float)Math.Sin(px * py);
}
}
// Compute this tile of the consumer
for (int yy_inner = 0; yy_inner < 4; yy_inner++)
{
for (int xx_inner = 0; xx_inner < 4; xx_inner++)
{
int xx = x_base + xx_inner;
int yy = y_base + yy_inner;
result[yy, xx] =
(producer_storage[yy - y_base, xx - x_base] +
producer_storage[yy - y_base + 1, xx - x_base] +
producer_storage[yy - y_base, xx - x_base + 1] +
producer_storage[yy - y_base + 1, xx - x_base + 1]) / 4;
}
}
}
}
Console.WriteLine("Pseudo-code for the schedule:");
consumer.PrintLoopNest();
Console.WriteLine();
// 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.
{
var producer = new HSFunc("producer_mixed");
var consumer = new HSFunc("consumer_mixed");
producer[x, y] = HSMath.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 = new HSVar("yo");
var yi = new HSVar("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.StoreAt(consumer, yo);
// Within each strip, compute the producer per scanline of the
// consumer, skipping work done on previous scanlines.
producer.ComputeAt(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.TraceStores();
// producer.TraceStores();
var halide_result = consumer.Realize <float>(160, 160);
// See figures/lesson_08_mixed.mp4 for a visualization.
// Here's the equivalent (serial) C:
var c_result = new float[160, 160];
// For every strip of 16 scanlines (this loop is parallel in
// the Halide version)
for (int yyo = 0; yyo < 160 / 16 + 1; yyo++)
{
// 16 doesn't divide 160, so push the last slice upwards
// to fit within [0, 159] (see lesson 05).
int y_base = yyo * 16;
if (y_base > 160 - 16)
{
y_base = 160 - 16;
}
// Allocate a two-scanline circular buffer for the producer
var producer_storage = new float[2, 161];
// For every scanline in the strip of 16:
for (int yyi = 0; yyi < 16; yyi++)
{
int yy = y_base + yyi;
for (int py = yy; py < yy + 2; py++)
{
// Skip scanlines already computed *within this task*
if (yyi > 0 && py == yy)
{
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[] xx = { x_base, x_base + 1, x_base + 2, x_base + 3 };
float[] vec = { (float)Math.Sin(xx[0] * py), (float)Math.Sin(xx[1] * py),
(float)Math.Sin(xx[2] * py), (float)Math.Sin(xx[3] * py) };
producer_storage[py & 1, xx[0]] = vec[0];
producer_storage[py & 1, xx[1]] = vec[1];
producer_storage[py & 1, xx[2]] = vec[2];
producer_storage[py & 1, xx[3]] = vec[3];
}
}
// 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[] xx = { x_base, x_base + 1, x_base + 2, x_base + 3 };
float[] vec =
{
(producer_storage[yy & 1, xx[0]] +
producer_storage[(yy + 1) & 1, xx[0]] +
producer_storage[yy & 1, xx[0] + 1] +
producer_storage[(yy + 1) & 1, xx[0] + 1]) / 4,
(producer_storage[yy & 1, xx[1]] +
producer_storage[(yy + 1) & 1, xx[1]] +
producer_storage[yy & 1, xx[1] + 1] +
producer_storage[(yy + 1) & 1, xx[1] + 1]) / 4,
(producer_storage[yy & 1, xx[2]] +
producer_storage[(yy + 1) & 1, xx[2]] +
producer_storage[yy & 1, xx[2] + 1] +
producer_storage[(yy + 1) & 1, xx[2] + 1]) / 4,
(producer_storage[yy & 1, xx[3]] +
producer_storage[(yy + 1) & 1, xx[3]] +
producer_storage[yy & 1, xx[3] + 1] +
producer_storage[(yy + 1) & 1, xx[3] + 1]) / 4
};
c_result[yy, xx[0]] = vec[0];
c_result[yy, xx[1]] = vec[1];
c_result[yy, xx[2]] = vec[2];
c_result[yy, xx[3]] = vec[3];
}
}
}
Console.WriteLine("Pseudo-code for the schedule:");
consumer.PrintLoopNest();
Console.WriteLine();
// 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 yy = 0; yy < 160; yy++)
{
for (int xx = 0; xx < 160; xx++)
{
float error = halide_result[xx, yy] - c_result[yy, xx];
// It's floating-point math, so we'll allow some slop:
if (error < -0.001f || error > 0.001f)
{
Console.WriteLine("halide_result(%d, %d) = %f instead of %f",
xx, yy, halide_result[xx, yy], c_result[yy, xx]);
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
// without running your code.
// 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
// giving you tools that help you quickly explore different
// trade-offs between locality, redundant work, and parallelism,
// without messing up the actual result you're trying to compute.
Console.WriteLine("Success!");
return(0);
}
public static int Main(string[] args)
{
// We're going to define and schedule our gradient function in
// several different ways, and see what order pixels are computed
// in.
var x = new HSVar("x");
var y = new HSVar("y");
// First we observe the default ordering.
{
var gradient = new HSFunc("gradient");
gradient[x, y] = x + y;
gradient.TraceStores();
// By default we walk along the rows and then down the
// columns. This means x varies quickly, and y varies
// slowly. x is the column and y is the row, so this is a
// row-major traversal.
Console.WriteLine("Evaluating gradient row-major");
var output = gradient.Realize <int>(4, 4);
// See figures/lesson_05_row_major.gif for a visualization of
// what this did.
// The equivalent C is:
Console.WriteLine("Equivalent C:");
for (int yy = 0; yy < 4; yy++)
{
for (int xx = 0; xx < 4; xx++)
{
Console.WriteLine($"Evaluating at x = {xx}, y = {yy}: {xx + yy}");
}
}
Console.WriteLine("\n");
// Tracing is one useful way to understand what a schedule is
// doing. You can also ask Halide to print out pseudocode
// showing what loops Halide is generating:
Console.WriteLine("Pseudo-code for the schedule:");
gradient.PrintLoopNest();
Console.WriteLine();
// Because we're using the default ordering, it should print:
// compute gradient:
// for y:
// for x:
// gradient(...) = ...
}
// Reorder variables.
{
var gradient = new HSFunc("gradient_col_major");
gradient[x, y] = x + y;
gradient.TraceStores();
// If we reorder x and y, we can walk down the columns
// instead. The reorder call takes the arguments of the func,
// and sets a new nesting order for the for loops that are
// generated. The arguments are specified from the innermost
// loop out, so the following call puts y in the inner loop:
gradient.Reorder(y, x);
// This means y (the row) will vary quickly, and x (the
// column) will vary slowly, so this is a column-major
// traversal.
Console.WriteLine("Evaluating gradient column-major");
var output = gradient.Realize <int>(4, 4);
// See figures/lesson_05_col_major.gif for a visualization of
// what this did.
Console.WriteLine("Equivalent C:");
for (int xx = 0; xx < 4; xx++)
{
for (int yy = 0; yy < 4; yy++)
{
Console.WriteLine($"Evaluating at x = {xx}, y = {yy}: {xx + yy}");
}
}
Console.WriteLine();
// If we print pseudo-code for this schedule, we'll see that
// the loop over y is now inside the loop over x.
Console.WriteLine("Pseudo-code for the schedule:");
gradient.PrintLoopNest();
Console.WriteLine();
}
// Split a variable into two.
{
var gradient = new HSFunc("gradient_split");
gradient[x, y] = x + y;
gradient.TraceStores();
// The most powerful primitive scheduling operation you can do
// to a var is to split it into inner and outer sub-variables:
var x_outer = new HSVar("x_outer");
var x_inner = new HSVar("x_inner");
gradient.Split(x, x_outer, x_inner, 2);
// This breaks the loop over x into two nested loops: an outer
// one over x_outer, and an inner one over x_inner. The last
// argument to split was the "split factor". The inner loop
// runs from zero to the split factor. The outer loop runs
// from zero to the extent required of x (4 in this case)
// divided by the split factor. Within the loops, the old
// variable is defined to be outer * factor + inner. If the
// old loop started at a value other than zero, then that is
// also added within the loops.
Console.WriteLine("Evaluating gradient with x split into x_outer and x_inner ");
var output = gradient.Realize <int>(4, 4);
Console.WriteLine("Equivalent C:");
for (int yy = 0; yy < 4; yy++)
{
for (int xOuter = 0; xOuter < 2; xOuter++)
{
for (int xInner = 0; xInner < 2; xInner++)
{
int xx = xOuter * 2 + xInner;
Console.WriteLine($"Evaluating at x = {xx}, y = {yy}: {xx + yy}");
}
}
}
Console.WriteLine();
Console.WriteLine("Pseudo-code for the schedule:");
gradient.PrintLoopNest();
Console.WriteLine();
// Note that the order of evaluation of pixels didn't actually
// change! Splitting by itself does nothing, but it does open
// up all of the scheduling possibilities that we will explore
// below.
}
// Fuse two variables into one.
{
var gradient = new HSFunc("gradient_fused");
gradient[x, y] = x + y;
// The opposite of splitting is 'fusing'. Fusing two variables
// merges the two loops into a single for loop over the
// product of the extents. Fusing is less important than
// splitting, but it also sees use (as we'll see later in this
// lesson). Like splitting, fusing by itself doesn't change
// the order of evaluation.
var fused = new HSVar("fused");
gradient.Fuse(x, y, fused);
Console.WriteLine("Evaluating gradient with x and y fused");
var output = gradient.Realize <int>(4, 4);
Console.WriteLine("Equivalent C:");
for (int f = 0; f < 4 * 4; f++)
{
int yy = f / 4;
int xx = f % 4;
Console.WriteLine($"Evaluating at x = {xx}, y = {yy}: {xx + yy}");
}
Console.WriteLine();
Console.WriteLine("Pseudo-code for the schedule:");
gradient.PrintLoopNest();
Console.WriteLine();
}
// Evaluating in tiles.
{
var gradient = new HSFunc("gradient_tiled");
gradient[x, y] = x + y;
gradient.TraceStores();
// Now that we can both split and reorder, we can do tiled
// evaluation. Let's split both x and y by a factor of four,
// and then reorder the vars to express a tiled traversal.
//
// A tiled traversal splits the domain into small rectangular
// tiles, and outermost iterates over the tiles, and within
// that iterates over the points within each tile. It can be
// good for performance if neighboring pixels use overlapping
// input data, for example in a blur. We can express a tiled
// traversal like so:
var x_outer = new HSVar("x_outer");
var x_inner = new HSVar("x_inner");
var y_outer = new HSVar("y_outer");
var y_inner = new HSVar("y_inner");
gradient.Split(x, x_outer, x_inner, 4);
gradient.Split(y, y_outer, y_inner, 4);
gradient.Reorder(x_inner, y_inner, x_outer, y_outer);
// This pattern is common enough that there's a shorthand for it:
// gradient.tile(x, y, x_outer, y_outer, x_inner, y_inner, 4, 4);
Console.WriteLine("Evaluating gradient in 4x4 tiles");
var output = gradient.Realize <int>(8, 8);
// See figures/lesson_05_tiled.gif for a visualization of this
// schedule.
Console.WriteLine("Equivalent C:");
for (int yOuter = 0; yOuter < 2; yOuter++)
{
for (int xOuter = 0; xOuter < 2; xOuter++)
{
for (int yInner = 0; yInner < 4; yInner++)
{
for (int xInner = 0; xInner < 4; xInner++)
{
int xx = xOuter * 4 + xInner;
int yy = yOuter * 4 + yInner;
Console.WriteLine($"Evaluating at x = {xx}, y = {yy}: {xx + yy}");
}
}
}
}
Console.WriteLine();
Console.WriteLine("Pseudo-code for the schedule:");
gradient.PrintLoopNest();
Console.WriteLine();
}
// Evaluating in vectors.
{
var gradient = new HSFunc("gradient_in_vectors");
gradient[x, y] = x + y;
gradient.TraceStores();
// The nice thing about splitting is that it guarantees the
// inner variable runs from zero to the split factor. Most of
// the time the split-factor will be a compile-time constant,
// so we can replace the loop over the inner variable with a
// single vectorized computation. This time we'll split by a
// factor of four, because on X86 we can use SSE to compute in
// 4-wide vectors.
var x_outer = new HSVar("x_outer");
var x_inner = new HSVar("x_inner");
gradient.Split(x, x_outer, x_inner, 4);
gradient.Vectorize(x_inner);
// Splitting and then vectorizing the inner variable is common
// enough that there's a short-hand for it. We could have also
// said:
//
// gradient.vectorize(x, 4);
//
// which is equivalent to:
//
// gradient.split(x, x, x_inner, 4);
// gradient.vectorize(x_inner);
//
// Note that in this case we reused the name 'x' as the new
// outer variable. Later scheduling calls that refer to x
// will refer to this new outer variable named x.
// This time we'll evaluate over an 8x4 box, so that we have
// more than one vector of work per scanline.
Console.WriteLine("Evaluating gradient with x_inner vectorized ");
var output = gradient.Realize <int>(8, 4);
// See figures/lesson_05_vectors.gif for a visualization.
Console.WriteLine("Equivalent C:");
for (int yy = 0; yy < 4; yy++)
{
for (int xOuter = 0; xOuter < 2; xOuter++)
{
// The loop over x_inner has gone away, and has been
// replaced by a vectorized version of the
// expression. On x86 processors, Halide generates SSE
// for all of this.
int[] x_vec = { xOuter * 4 + 0,
xOuter * 4 + 1,
xOuter * 4 + 2,
xOuter * 4 + 3 };
int[] val = { x_vec[0] + yy,
x_vec[1] + yy,
x_vec[2] + yy,
x_vec[3] + yy };
Console.WriteLine($"Evaluating at " +
$"<{x_vec[0]}, {x_vec[1]}, {x_vec[2]}, {x_vec[3]}>, " +
$"<{yy}, {yy}, {yy}, {yy}>: " +
$"<{val[0]}, {val[1]}, {val[2]}, {val[3]}>");
}
}
Console.WriteLine();
Console.WriteLine("Pseudo-code for the schedule:");
gradient.PrintLoopNest();
Console.WriteLine();
}
// Unrolling a loop.
{
var gradient = new HSFunc("gradient_unroll");
gradient[x, y] = x + y;
gradient.TraceStores();
// If multiple pixels share overlapping data, it can make
// sense to unroll a computation so that shared values are
// only computed or loaded once. We do this similarly to how
// we expressed vectorizing. We split a dimension and then
// fully unroll the loop of the inner variable. Unrolling
// doesn't change the order in which things are evaluated.
var x_outer = new HSVar("x_outer");
var x_inner = new HSVar("x_inner");
gradient.Split(x, x_outer, x_inner, 2);
gradient.Unroll(x_inner);
// The shorthand for this is:
// gradient.unroll(x, 2);
Console.WriteLine("Evaluating gradient unrolled by a factor of two");
var result = gradient.Realize <int>(4, 4);
Console.WriteLine("Equivalent C:");
for (int yy = 0; yy < 4; yy++)
{
for (int xOuter = 0; xOuter < 2; xOuter++)
{
// Instead of a for loop over x_inner, we get two
// copies of the innermost statement.
{
int xInner = 0;
int xx = xOuter * 2 + xInner;
Console.WriteLine($"Evaluating at x = {xx}, y = {yy}: {xx + yy}");
}
{
int xInner = 1;
int xx = xOuter * 2 + xInner;
Console.WriteLine($"Evaluating at x = {xx}, y = {yy}: {xx + yy}");
}
}
}
Console.WriteLine();
Console.WriteLine("Pseudo-code for the schedule:");
gradient.PrintLoopNest();
Console.WriteLine();
}
// Splitting by factors that don't divide the extent.
{
var gradient = new HSFunc("gradient_split_7x2");
gradient[x, y] = x + y;
gradient.TraceStores();
// Splitting guarantees that the inner loop runs from zero to
// the split factor, which is important for the uses we saw
// above. So what happens when the total extent we wish to
// evaluate x over isn't a multiple of the split factor? We'll
// split by a factor three, and we'll evaluate gradient over a
// 7x2 box instead of the 4x4 box we've been using.
var x_outer = new HSVar("x_outer");
var x_inner = new HSVar("x_inner");
gradient.Split(x, x_outer, x_inner, 3);
Console.WriteLine("Evaluating gradient over a 7x2 box with x split by three ");
var output = gradient.Realize <int>(7, 2);
// See figures/lesson_05_split_7_by_3.gif for a visualization
// of what happened. Note that some points get evaluated more
// than once!
Console.WriteLine("Equivalent C:");
for (int yy = 0; yy < 2; yy++)
{
for (int xOuter = 0; xOuter < 3; xOuter++) // Now runs from 0 to 2
{
for (int xInner = 0; xInner < 3; xInner++)
{
int xx = xOuter * 3;
// Before we add x_inner, make sure we don't
// evaluate points outside of the 7x2 box. We'll
// clamp x to be at most 4 (7 minus the split
// factor).
if (xx > 4)
{
xx = 4;
}
xx += xInner;
Console.WriteLine($"Evaluating at x = {xx}, y = {yy}: {xx + yy}");
}
}
}
Console.WriteLine();
Console.WriteLine("Pseudo-code for the schedule:");
gradient.PrintLoopNest();
Console.WriteLine();
// If you read the output, you'll see that some coordinates
// were evaluated more than once. That's generally OK, because
// pure Halide functions have no side-effects, so it's safe to
// evaluate the same point multiple times. If you're calling
// out to C functions like we are, it's your responsibility to
// make sure you can handle the same point being evaluated
// multiple times.
// The general rule is: If we require x from x_min to x_min + x_extent, and
// we split by a factor 'factor', then:
//
// x_outer runs from 0 to (x_extent + factor - 1)/factor
// x_inner runs from 0 to factor
// x = min(x_outer * factor, x_extent - factor) + x_inner + x_min
//
// In our example, x_min was 0, x_extent was 7, and factor was 3.
// However, if you write a Halide function with an update
// definition (see lesson 9), then it is not safe to evaluate
// the same point multiple times, so we won't apply this
// trick. Instead the range of values computed will be rounded
// up to the next multiple of the split factor.
}
// Fusing, tiling, and parallelizing.
{
// We saw in the previous lesson that we can parallelize
// across a variable. Here we combine it with fusing and
// tiling to express a useful pattern - processing tiles in
// parallel.
// This is where fusing shines. Fusing helps when you want to
// parallelize across multiple dimensions without introducing
// nested parallelism. Nested parallelism (parallel for loops
// within parallel for loops) is supported by Halide, but
// often gives poor performance compared to fusing the
// parallel variables into a single parallel for loop.
var gradient = new HSFunc("gradient_fused_tiles");
gradient[x, y] = x + y;
gradient.TraceStores();
// First we'll tile, then we'll fuse the tile indices and
// parallelize across the combination.
var x_outer = new HSVar("x_outer");
var y_outer = new HSVar("y_outer");
var x_inner = new HSVar("x_inner");
var y_inner = new HSVar("y_inner");
var tile_index = new HSVar("tile_index");
gradient.Tile(x, y, x_outer, y_outer, x_inner, y_inner, 4, 4);
gradient.Fuse(x_outer, y_outer, tile_index);
gradient.Parallel(tile_index);
// The scheduling calls all return a reference to the Func, so
// you can also chain them together into a single statement to
// make things slightly clearer:
//
// gradient
// .tile(x, y, x_outer, y_outer, x_inner, y_inner, 2, 2)
// .fuse(x_outer, y_outer, tile_index)
// .parallel(tile_index);
Console.WriteLine("Evaluating gradient tiles in parallel");
var output = gradient.Realize <int>(8, 8);
// The tiles should occur in arbitrary order, but within each
// tile the pixels will be traversed in row-major order. See
// figures/lesson_05_parallel_tiles.gif for a visualization.
Console.WriteLine("Equivalent (serial) C:\n");
// This outermost loop should be a parallel for loop, but that's hard in C.
for (int ti = 0; ti < 4; ti++)
{
int yOuter = ti / 2;
int xOuter = ti % 2;
for (int j_inner = 0; j_inner < 4; j_inner++)
{
for (int i_inner = 0; i_inner < 4; i_inner++)
{
int j = yOuter * 4 + j_inner;
int i = xOuter * 4 + i_inner;
Console.WriteLine($"Evaluating at x = {i}, y = {j}: {i + j}");
}
}
}
Console.WriteLine();
Console.WriteLine("Pseudo-code for the schedule:");
gradient.PrintLoopNest();
Console.WriteLine();
}
// Putting it all together.
{
// Are you ready? We're going to use all of the features above now.
var gradient_fast = new HSFunc("gradient_fast");
gradient_fast[x, y] = x + y;
// We'll process 64x64 tiles in parallel.
var x_outer = new HSVar("x_outer");
var y_outer = new HSVar("y_outer");
var x_inner = new HSVar("x_inner");
var y_inner = new HSVar("y_inner");
var tile_index = new HSVar("tile_index");
gradient_fast
.Tile(x, y, x_outer, y_outer, x_inner, y_inner, 64, 64)
.Fuse(x_outer, y_outer, tile_index)
.Parallel(tile_index);
// We'll compute two scanlines at once while we walk across
// each tile. We'll also vectorize in x. The easiest way to
// express this is to recursively tile again within each tile
// into 4x2 subtiles, then vectorize the subtiles across x and
// unroll them across y:
var x_inner_outer = new HSVar("x_inner_outer");
var y_inner_outer = new HSVar("y_inner_outer");
var x_vectors = new HSVar("x_vectors");
var y_pairs = new HSVar("y_pairs");
gradient_fast
.Tile(x_inner, y_inner, x_inner_outer, y_inner_outer, x_vectors, y_pairs, 4, 2)
.Vectorize(x_vectors)
.Unroll(y_pairs);
// Note that we didn't do any explicit splitting or
// reordering. Those are the most important primitive
// operations, but mostly they are buried underneath tiling,
// vectorizing, or unrolling calls.
// Now let's evaluate this over a range which is not a
// multiple of the tile size.
// If you like you can turn on tracing, but it's going to
// produce a lot of printfs. Instead we'll compute the answer
// both in C and Halide and see if the answers match.
var result = gradient_fast.Realize <int>(350, 250);
// See figures/lesson_05_fast.mp4 for a visualization.
Console.WriteLine("Checking Halide result against equivalent C...");
for (int tileIndex = 0; tileIndex < 6 * 4; tileIndex++)
{
int yOuter = tileIndex / 4;
int xOuter = tileIndex % 4;
for (int yInnerOuter = 0; yInnerOuter < 64 / 2; yInnerOuter++)
{
for (int xInnerOuter = 0; xInnerOuter < 64 / 4; xInnerOuter++)
{
// We're vectorized across x
int xx = Math.Min(xOuter * 64, 350 - 64) + xInnerOuter * 4;
int[] xVec = { xx + 0,
xx + 1,
xx + 2,
xx + 3 };
// And we unrolled across y
int yBase = Math.Min(yOuter * 64, 250 - 64) + yInnerOuter * 2;
{
// y_pairs = 0
int yy = yBase + 0;
int[] yVec = { yy, yy, yy, yy };
int[] val = { xVec[0] + yVec[0],
xVec[1] + yVec[1],
xVec[2] + yVec[2],
xVec[3] + yVec[3] };
// Check the result.
for (int i = 0; i < 4; i++)
{
if (result[xVec[i], yVec[i]] != val[i])
{
Console.WriteLine($"There was an error at {xVec[i]} {yVec[i]}!");
return(-1);
}
}
}
{
// y_pairs = 1
int yy = yBase + 1;
int[] yVec = { yy, yy, yy, yy };
int[] val = { xVec[0] + yVec[0],
xVec[1] + yVec[1],
xVec[2] + yVec[2],
xVec[3] + yVec[3] };
// Check the result.
for (int i = 0; i < 4; i++)
{
if (result[xVec[i], yVec[i]] != val[i])
{
Console.WriteLine($"There was an error at {xVec[i]} {yVec[i]}!");
return(-1);
}
}
}
}
}
}
Console.WriteLine();
Console.WriteLine("Pseudo-code for the schedule:");
gradient_fast.PrintLoopNest();
Console.WriteLine();
// Note that in the Halide version, the algorithm is specified
// once at the top, separately from the optimizations, and there
// aren't that many lines of code total. Compare this to the C
// version. There's more code (and it isn't even parallelized or
// vectorized properly). More annoyingly, the statement of the
// algorithm (the result is x plus y) is buried in multiple places
// within the mess. This C code is hard to write, hard to read,
// hard to debug, and hard to optimize further. This is why Halide
// exists.
}
Console.WriteLine("Success!");
return(0);
}