# Schedule Primitives in TVM¶

Author: Ziheng Jiang

TVM is a domain specific language for efficient kernel construction.

In this tutorial, we will show you how to schedule the computation by various primitives provided by TVM.

```from __future__ import absolute_import, print_function

import tvm
from tvm import te
import numpy as np
```

There often exist several methods to compute the same result, however, different methods will result in different locality and performance. So TVM asks user to provide how to execute the computation called Schedule.

A Schedule is a set of transformation of computation that transforms the loop of computations in the program.

```# declare some variables for use later
n = te.var("n")
m = te.var("m")
```

A schedule can be created from a list of ops, by default the schedule computes tensor in a serial manner in a row-major order.

```# declare a matrix element-wise multiply
A = te.placeholder((m, n), name="A")
B = te.placeholder((m, n), name="B")
C = te.compute((m, n), lambda i, j: A[i, j] * B[i, j], name="C")

s = te.create_schedule([C.op])
# lower will transform the computation from definition to the real
# callable function. With argument `simple_mode=True`, it will
# return you a readable C like statement, we use it here to print the
# schedule result.
print(tvm.lower(s, [A, B, C], simple_mode=True))
```
```# from tvm.script import ir as I
# from tvm.script import tir as T

@I.ir_module
class Module:
@T.prim_func
def main(A: T.handle, B: T.handle, C: T.handle):
T.func_attr({"from_legacy_te_schedule": T.bool(True), "tir.noalias": T.bool(True)})
m, n = T.int32(), T.int32()
A_1 = T.match_buffer(A, (m, n), strides=("stride", "stride"), buffer_type="auto")
B_1 = T.match_buffer(B, (m, n), strides=("stride", "stride"), buffer_type="auto")
C_1 = T.match_buffer(C, (m, n), strides=("stride", "stride"), buffer_type="auto")
for i, j in T.grid(m, n):
C_2 = T.Buffer((C_1.strides[0] * m,), data=C_1.data, buffer_type="auto")
A_2 = T.Buffer((A_1.strides[0] * m,), data=A_1.data, buffer_type="auto")
B_2 = T.Buffer((B_1.strides[0] * m,), data=B_1.data, buffer_type="auto")
C_2[i * C_1.strides[0] + j * C_1.strides[1]] = A_2[i * A_1.strides[0] + j * A_1.strides[1]] * B_2[i * B_1.strides[0] + j * B_1.strides[1]]
```

One schedule is composed by multiple stages, and one Stage represents schedule for one operation. We provide various methods to schedule every stage.

## split¶

`split` can split a specified axis into two axes by `factor`.

```A = te.placeholder((m,), name="A")
B = te.compute((m,), lambda i: A[i] * 2, name="B")

s = te.create_schedule(B.op)
xo, xi = s[B].split(B.op.axis[0], factor=32)
print(tvm.lower(s, [A, B], simple_mode=True))
```
```# from tvm.script import ir as I
# from tvm.script import tir as T

@I.ir_module
class Module:
@T.prim_func
def main(A: T.handle, B: T.handle):
T.func_attr({"from_legacy_te_schedule": T.bool(True), "tir.noalias": T.bool(True)})
m = T.int32()
A_1 = T.match_buffer(A, (m,), strides=("stride",), buffer_type="auto")
B_1 = T.match_buffer(B, (m,), strides=("stride",), buffer_type="auto")
for i_outer, i_inner in T.grid((m + 31) // 32, 32):
if T.likely(i_outer * 32 + i_inner < m):
B_2 = T.Buffer((B_1.strides[0] * m,), data=B_1.data, buffer_type="auto")
A_2 = T.Buffer((A_1.strides[0] * m,), data=A_1.data, buffer_type="auto")
cse_var_1: T.int32 = i_outer * 32 + i_inner
B_2[cse_var_1 * B_1.strides[0]] = A_2[cse_var_1 * A_1.strides[0]] * T.float32(2)
```

You can also split a axis by `nparts`, which splits the axis contrary with `factor`.

```A = te.placeholder((m,), name="A")
B = te.compute((m,), lambda i: A[i], name="B")

s = te.create_schedule(B.op)
bx, tx = s[B].split(B.op.axis[0], nparts=32)
print(tvm.lower(s, [A, B], simple_mode=True))
```
```# from tvm.script import ir as I
# from tvm.script import tir as T

@I.ir_module
class Module:
@T.prim_func
def main(A: T.handle, B: T.handle):
T.func_attr({"from_legacy_te_schedule": T.bool(True), "tir.noalias": T.bool(True)})
m = T.int32()
A_1 = T.match_buffer(A, (m,), strides=("stride",), buffer_type="auto")
B_1 = T.match_buffer(B, (m,), strides=("stride",), buffer_type="auto")
for i_outer, i_inner in T.grid(32, (m + 31) // 32):
if T.likely(i_inner + i_outer * ((m + 31) // 32) < m):
B_2 = T.Buffer((B_1.strides[0] * m,), data=B_1.data, buffer_type="auto")
A_2 = T.Buffer((A_1.strides[0] * m,), data=A_1.data, buffer_type="auto")
B_2[(i_inner + i_outer * ((m + 31) // 32)) * B_1.strides[0]] = A_2[(i_inner + i_outer * ((m + 31) // 32)) * A_1.strides[0]]
```

## tile¶

`tile` help you execute the computation tile by tile over two axes.

```A = te.placeholder((m, n), name="A")
B = te.compute((m, n), lambda i, j: A[i, j], name="B")

s = te.create_schedule(B.op)
xo, yo, xi, yi = s[B].tile(B.op.axis[0], B.op.axis[1], x_factor=10, y_factor=5)
print(tvm.lower(s, [A, B], simple_mode=True))
```
```# from tvm.script import ir as I
# from tvm.script import tir as T

@I.ir_module
class Module:
@T.prim_func
def main(A: T.handle, B: T.handle):
T.func_attr({"from_legacy_te_schedule": T.bool(True), "tir.noalias": T.bool(True)})
m, n = T.int32(), T.int32()
A_1 = T.match_buffer(A, (m, n), strides=("stride", "stride"), buffer_type="auto")
B_1 = T.match_buffer(B, (m, n), strides=("stride", "stride"), buffer_type="auto")
for i_outer, j_outer, i_inner in T.grid((m + 9) // 10, (n + 4) // 5, 10):
if T.likely(i_outer * 10 + i_inner < m):
for j_inner in range(5):
if T.likely(j_outer * 5 + j_inner < n):
cse_var_2: T.int32 = j_outer * 5 + j_inner
cse_var_1: T.int32 = i_outer * 10 + i_inner
B_2 = T.Buffer((B_1.strides[0] * m,), data=B_1.data, buffer_type="auto")
A_2 = T.Buffer((A_1.strides[0] * m,), data=A_1.data, buffer_type="auto")
B_2[cse_var_1 * B_1.strides[0] + cse_var_2 * B_1.strides[1]] = A_2[cse_var_1 * A_1.strides[0] + cse_var_2 * A_1.strides[1]]
```

## fuse¶

`fuse` can fuse two consecutive axes of one computation.

```A = te.placeholder((m, n), name="A")
B = te.compute((m, n), lambda i, j: A[i, j], name="B")

s = te.create_schedule(B.op)
# tile to four axes first: (i.outer, j.outer, i.inner, j.inner)
xo, yo, xi, yi = s[B].tile(B.op.axis[0], B.op.axis[1], x_factor=10, y_factor=5)
# then fuse (i.inner, j.inner) into one axis: (i.inner.j.inner.fused)
fused = s[B].fuse(xi, yi)
print(tvm.lower(s, [A, B], simple_mode=True))
```
```# from tvm.script import ir as I
# from tvm.script import tir as T

@I.ir_module
class Module:
@T.prim_func
def main(A: T.handle, B: T.handle):
T.func_attr({"from_legacy_te_schedule": T.bool(True), "tir.noalias": T.bool(True)})
m, n = T.int32(), T.int32()
A_1 = T.match_buffer(A, (m, n), strides=("stride", "stride"), buffer_type="auto")
B_1 = T.match_buffer(B, (m, n), strides=("stride", "stride"), buffer_type="auto")
for i_outer, j_outer, i_inner_j_inner_fused in T.grid((m + 9) // 10, (n + 4) // 5, 50):
if T.likely(i_outer * 10 + i_inner_j_inner_fused // 5 < m):
if T.likely(j_outer * 5 + i_inner_j_inner_fused % 5 < n):
cse_var_2: T.int32 = j_outer * 5 + i_inner_j_inner_fused % 5
cse_var_1: T.int32 = i_outer * 10 + i_inner_j_inner_fused // 5
B_2 = T.Buffer((B_1.strides[0] * m,), data=B_1.data, buffer_type="auto")
A_2 = T.Buffer((A_1.strides[0] * m,), data=A_1.data, buffer_type="auto")
B_2[cse_var_1 * B_1.strides[0] + cse_var_2 * B_1.strides[1]] = A_2[cse_var_1 * A_1.strides[0] + cse_var_2 * A_1.strides[1]]
```

## reorder¶

`reorder` can reorder the axes in the specified order.

```A = te.placeholder((m, n), name="A")
B = te.compute((m, n), lambda i, j: A[i, j], name="B")

s = te.create_schedule(B.op)
# tile to four axes first: (i.outer, j.outer, i.inner, j.inner)
xo, yo, xi, yi = s[B].tile(B.op.axis[0], B.op.axis[1], x_factor=10, y_factor=5)
# then reorder the axes: (i.inner, j.outer, i.outer, j.inner)
s[B].reorder(xi, yo, xo, yi)
print(tvm.lower(s, [A, B], simple_mode=True))
```
```# from tvm.script import ir as I
# from tvm.script import tir as T

@I.ir_module
class Module:
@T.prim_func
def main(A: T.handle, B: T.handle):
T.func_attr({"from_legacy_te_schedule": T.bool(True), "tir.noalias": T.bool(True)})
m, n = T.int32(), T.int32()
A_1 = T.match_buffer(A, (m, n), strides=("stride", "stride"), buffer_type="auto")
B_1 = T.match_buffer(B, (m, n), strides=("stride", "stride"), buffer_type="auto")
for i_inner, j_outer, i_outer in T.grid(10, (n + 4) // 5, (m + 9) // 10):
if T.likely(i_outer * 10 + i_inner < m):
for j_inner in range(5):
if T.likely(j_outer * 5 + j_inner < n):
cse_var_2: T.int32 = j_outer * 5 + j_inner
cse_var_1: T.int32 = i_outer * 10 + i_inner
B_2 = T.Buffer((B_1.strides[0] * m,), data=B_1.data, buffer_type="auto")
A_2 = T.Buffer((A_1.strides[0] * m,), data=A_1.data, buffer_type="auto")
B_2[cse_var_1 * B_1.strides[0] + cse_var_2 * B_1.strides[1]] = A_2[cse_var_1 * A_1.strides[0] + cse_var_2 * A_1.strides[1]]
```

## bind¶

`bind` can bind a specified axis with a thread axis, often used in gpu programming.

```A = te.placeholder((n,), name="A")
B = te.compute(A.shape, lambda i: A[i] * 2, name="B")

s = te.create_schedule(B.op)
bx, tx = s[B].split(B.op.axis[0], factor=64)
print(tvm.lower(s, [A, B], simple_mode=True))
```
```# from tvm.script import ir as I
# from tvm.script import tir as T

@I.ir_module
class Module:
@T.prim_func
def main(A: T.handle, B: T.handle):
T.func_attr({"from_legacy_te_schedule": T.bool(True), "tir.noalias": T.bool(True)})
n = T.int32()
A_1 = T.match_buffer(A, (n,), strides=("stride",), buffer_type="auto")
B_1 = T.match_buffer(B, (n,), strides=("stride",), buffer_type="auto")
blockIdx_x = T.launch_thread("blockIdx.x", (n + 63) // 64)
if T.likely(blockIdx_x * 64 + threadIdx_x < n):
B_2 = T.Buffer((B_1.strides[0] * n,), data=B_1.data, buffer_type="auto")
A_2 = T.Buffer((A_1.strides[0] * n,), data=A_1.data, buffer_type="auto")
B_2[(blockIdx_x * 64 + threadIdx_x) * B_1.strides[0]] = A_2[(blockIdx_x * 64 + threadIdx_x) * A_1.strides[0]] * T.float32(2)
```

## compute_at¶

For a schedule that consists of multiple operators, TVM will compute tensors at the root separately by default.

```A = te.placeholder((m,), name="A")
B = te.compute((m,), lambda i: A[i] + 1, name="B")
C = te.compute((m,), lambda i: B[i] * 2, name="C")

s = te.create_schedule(C.op)
print(tvm.lower(s, [A, B, C], simple_mode=True))
```
```# from tvm.script import ir as I
# from tvm.script import tir as T

@I.ir_module
class Module:
@T.prim_func
def main(A: T.handle, B: T.handle, C: T.handle):
T.func_attr({"from_legacy_te_schedule": T.bool(True), "tir.noalias": T.bool(True)})
m = T.int32()
A_1 = T.match_buffer(A, (m,), strides=("stride",), buffer_type="auto")
B_1 = T.match_buffer(B, (m,), strides=("stride",), buffer_type="auto")
C_1 = T.match_buffer(C, (m,), strides=("stride",), buffer_type="auto")
B_2 = T.Buffer((B_1.strides[0] * m,), data=B_1.data, buffer_type="auto")
for i in range(m):
A_2 = T.Buffer((A_1.strides[0] * m,), data=A_1.data, buffer_type="auto")
B_2[i * B_1.strides[0]] = A_2[i * A_1.strides[0]] + T.float32(1)
for i in range(m):
C_2 = T.Buffer((C_1.strides[0] * m,), data=C_1.data, buffer_type="auto")
C_2[i * C_1.strides[0]] = B_2[i * B_1.strides[0]] * T.float32(2)
```

`compute_at` can move computation of B into the first axis of computation of C.

```A = te.placeholder((m,), name="A")
B = te.compute((m,), lambda i: A[i] + 1, name="B")
C = te.compute((m,), lambda i: B[i] * 2, name="C")

s = te.create_schedule(C.op)
s[B].compute_at(s[C], C.op.axis[0])
print(tvm.lower(s, [A, B, C], simple_mode=True))
```
```# from tvm.script import ir as I
# from tvm.script import tir as T

@I.ir_module
class Module:
@T.prim_func
def main(A: T.handle, B: T.handle, C: T.handle):
T.func_attr({"from_legacy_te_schedule": T.bool(True), "tir.noalias": T.bool(True)})
m = T.int32()
A_1 = T.match_buffer(A, (m,), strides=("stride",), buffer_type="auto")
B_1 = T.match_buffer(B, (m,), strides=("stride",), buffer_type="auto")
C_1 = T.match_buffer(C, (m,), strides=("stride",), buffer_type="auto")
for i in range(m):
B_2 = T.Buffer((B_1.strides[0] * m,), data=B_1.data, buffer_type="auto")
A_2 = T.Buffer((A_1.strides[0] * m,), data=A_1.data, buffer_type="auto")
B_2[i * B_1.strides[0]] = A_2[i * A_1.strides[0]] + T.float32(1)
C_2 = T.Buffer((C_1.strides[0] * m,), data=C_1.data, buffer_type="auto")
C_2[i * C_1.strides[0]] = B_2[i * B_1.strides[0]] * T.float32(2)
```

## compute_inline¶

`compute_inline` can mark one stage as inline, then the body of computation will be expanded and inserted at the address where the tensor is required.

```A = te.placeholder((m,), name="A")
B = te.compute((m,), lambda i: A[i] + 1, name="B")
C = te.compute((m,), lambda i: B[i] * 2, name="C")

s = te.create_schedule(C.op)
s[B].compute_inline()
print(tvm.lower(s, [A, B, C], simple_mode=True))
```
```# from tvm.script import ir as I
# from tvm.script import tir as T

@I.ir_module
class Module:
@T.prim_func
def main(A: T.handle, B: T.handle, C: T.handle):
T.func_attr({"from_legacy_te_schedule": T.bool(True), "tir.noalias": T.bool(True)})
m = T.int32()
A_1 = T.match_buffer(A, (m,), strides=("stride",), buffer_type="auto")
B_1 = T.match_buffer(B, (m,), strides=("stride",), buffer_type="auto")
C_1 = T.match_buffer(C, (m,), strides=("stride",), buffer_type="auto")
for i in range(m):
C_2 = T.Buffer((C_1.strides[0] * m,), data=C_1.data, buffer_type="auto")
A_2 = T.Buffer((A_1.strides[0] * m,), data=A_1.data, buffer_type="auto")
C_2[i * C_1.strides[0]] = (A_2[i * A_1.strides[0]] + T.float32(1)) * T.float32(2)
```

## compute_root¶

`compute_root` can move computation of one stage to the root.

```A = te.placeholder((m,), name="A")
B = te.compute((m,), lambda i: A[i] + 1, name="B")
C = te.compute((m,), lambda i: B[i] * 2, name="C")

s = te.create_schedule(C.op)
s[B].compute_at(s[C], C.op.axis[0])
s[B].compute_root()
print(tvm.lower(s, [A, B, C], simple_mode=True))
```
```# from tvm.script import ir as I
# from tvm.script import tir as T

@I.ir_module
class Module:
@T.prim_func
def main(A: T.handle, B: T.handle, C: T.handle):
T.func_attr({"from_legacy_te_schedule": T.bool(True), "tir.noalias": T.bool(True)})
m = T.int32()
A_1 = T.match_buffer(A, (m,), strides=("stride",), buffer_type="auto")
B_1 = T.match_buffer(B, (m,), strides=("stride",), buffer_type="auto")
C_1 = T.match_buffer(C, (m,), strides=("stride",), buffer_type="auto")
B_2 = T.Buffer((B_1.strides[0] * m,), data=B_1.data, buffer_type="auto")
for i in range(m):
A_2 = T.Buffer((A_1.strides[0] * m,), data=A_1.data, buffer_type="auto")
B_2[i * B_1.strides[0]] = A_2[i * A_1.strides[0]] + T.float32(1)
for i in range(m):
C_2 = T.Buffer((C_1.strides[0] * m,), data=C_1.data, buffer_type="auto")
C_2[i * C_1.strides[0]] = B_2[i * B_1.strides[0]] * T.float32(2)
```

## Summary¶

This tutorial provides an introduction to schedule primitives in tvm, which permits users schedule the computation easily and flexibly.

In order to get a good performance kernel implementation, the general workflow often is:

• Describe your computation via series of operations.

• Try to schedule the computation with primitives.

• Compile and run to see the performance difference.