Not only is Mojo great for writing high-performance code, but it also allows us to leverage the huge Python ecosystem of libraries and tools. With seamless Python interoperability, Mojo can use Python for what it’s good at, especially GUIs, without sacrificing performance in critical code. Let’s take the classic Mandelbrot set algorithm and implement it in Mojo.
This tutorial shows two aspects of Mojo. First, it shows that Mojo can be used to develop fast programs for irregular applications. It also shows how we can leverage Python for visualizing the results.
from benchmark import Benchmark
from complex import ComplexSIMD, ComplexFloat64
from math import iota
from python import Python
from runtime.llcl import num_cores, Runtime
from algorithm import parallelize, vectorize
from tensor import Tensor
from utils.index import Index
alias float_type = DType.float64
alias simd_width = 2 * simdwidthof[float_type]()First set some parameters, you can try changing these to see different results:
alias width = 960
alias height = 960
alias MAX_ITERS = 200
alias min_x = -2.0
alias max_x = 0.6
alias min_y = -1.5
alias max_y = 1.5The core Mandelbrot algorithm involves computing an iterative complex function for each pixel until it “escapes” the complex circle of radius 2, counting the number of iterations to escape:
\[z_{i+1} = z_i^2 + c\]
# Compute the number of steps to escape.
def mandelbrot_kernel(c: ComplexFloat64) -> Int:
z = c
for i in range(MAX_ITERS):
z = z * z + c
if z.squared_norm() > 4:
return i
return MAX_ITERS
def compute_mandelbrot() -> Tensor[float_type]:
# create a matrix. Each element of the matrix corresponds to a pixel
t = Tensor[float_type](height, width)
dx = (max_x - min_x) / width
dy = (max_y - min_y) / height
y = min_y
for row in range(height):
x = min_x
for col in range(width):
t[Index(row, col)] = mandelbrot_kernel(ComplexFloat64(x, y))
x += dx
y += dy
return tPlotting the number of iterations to escape with some color gives us the canonical Mandelbrot set plot. To render it we can directly leverage Python’s matplotlib right from Mojo!
def show_plot(tensor: Tensor[float_type]):
alias scale = 10
alias dpi = 64
np = Python.import_module("numpy")
plt = Python.import_module("matplotlib.pyplot")
colors = Python.import_module("matplotlib.colors")
numpy_array = np.zeros((height, width), np.float64)
for row in range(height):
for col in range(width):
numpy_array.itemset((col, row), tensor[col, row])
fig = plt.figure(1, [scale, scale * height // width], dpi)
ax = fig.add_axes([0.0, 0.0, 1.0, 1.0], False, 1)
light = colors.LightSource(315, 10, 0, 1, 1, 0)
image = light.shade(numpy_array, plt.cm.hot, colors.PowerNorm(0.3), "hsv", 0, 0, 1.5)
plt.imshow(image)
plt.axis("off")
plt.show()
show_plot(compute_mandelbrot())
We showed a naive implementation of the Mandelbrot algorithm, but there are two things we can do to speed it up. We can early-stop the loop iteration when a pixel is known to have escaped, and we can leverage Mojo’s access to hardware by vectorizing the loop, computing multiple pixels simultaneously. To do that we will use the vectorize higher order generator.
We start by defining our main iteration loop in a vectorized fashion
fn mandelbrot_kernel_SIMD[
simd_width: Int
](c: ComplexSIMD[float_type, simd_width]) -> SIMD[float_type, simd_width]:
"""A vectorized implementation of the inner mandelbrot computation."""
let cx = c.re
let cy = c.im
var x = SIMD[float_type, simd_width](0)
var y = SIMD[float_type, simd_width](0)
var y2 = SIMD[float_type, simd_width](0)
var iters = SIMD[float_type, simd_width](0)
var t: SIMD[DType.bool, simd_width] = True
for i in range(MAX_ITERS):
if not t.reduce_or():
break
y2 = y*y
y = x.fma(y + y, cy)
t = x.fma(x, y2) <= 4
x = x.fma(x, cx - y2)
iters = t.select(iters + 1, iters)
return itersThe above function is parameterized on the simd_width and processes simd_width pixels. It only escapes once all pixels within the vector lane are done. We can use the same iteration loop as above, but this time we vectorize within each row instead. We use the vectorize generator to make this a simple function call.
fn vectorized():
let t = Tensor[float_type](height, width)
@parameter
fn worker(row: Int):
let scale_x = (max_x - min_x) / width
let scale_y = (max_y - min_y) / height
@parameter
fn compute_vector[simd_width: Int](col: Int):
"""Each time we oeprate on a `simd_width` vector of pixels."""
let cx = min_x + (col + iota[float_type, simd_width]()) * scale_x
let cy = min_y + row * scale_y
let c = ComplexSIMD[float_type, simd_width](cx, cy)
t.data().simd_store[simd_width](row * width + col, mandelbrot_kernel_SIMD[simd_width](c))
# Vectorize the call to compute_vector where call gets a chunk of pixels.
vectorize[simd_width, compute_vector](width)
@parameter
fn bench[simd_width: Int]():
for row in range(height):
worker(row)
let vectorized = Benchmark().run[bench[simd_width]]() / 1e6
print("Vectorized", ":", vectorized, "ms")
try:
_ = show_plot(t)
except e:
print("failed to show plot:", e.value)
vectorized()Vectorized : 12.177345000000001 ms
While the vectorized implementation above is efficient, we can get better performance by parallelizing on the cols. This again is simple in Mojo using the parallelize higher order function. Only the function that performs the invocation needs to change.
fn parallelized():
let t = Tensor[float_type](height, width)
@parameter
fn worker(row: Int):
let scale_x = (max_x - min_x) / width
let scale_y = (max_y - min_y) / height
@parameter
fn compute_vector[simd_width: Int](col: Int):
"""Each time we oeprate on a `simd_width` vector of pixels."""
let cx = min_x + (col + iota[float_type, simd_width]()) * scale_x
let cy = min_y + row * scale_y
let c = ComplexSIMD[float_type, simd_width](cx, cy)
t.data().simd_store[simd_width](row * width + col, mandelbrot_kernel_SIMD[simd_width](c))
# Vectorize the call to compute_vector where call gets a chunk of pixels.
vectorize[simd_width, compute_vector](width)
with Runtime() as rt:
@parameter
fn bench_parallel[simd_width: Int]():
parallelize[worker](rt, height, height)
let parallelized = Benchmark().run[bench_parallel[simd_width]]() / 1e6
print("Parallelized:", parallelized, "ms")
try:
_ = show_plot(t)
except e:
print("failed to show plot:", e.value)
parallelized()Parallelized: 1.4245639999999999 ms
In this section we increase the size to 4096x4096 and run 1000 iterations for a larger test to stress the CPU
fn compare():
let t = Tensor[float_type](height, width)
@parameter
fn worker(row: Int):
let scale_x = (max_x - min_x) / width
let scale_y = (max_y - min_y) / height
@parameter
fn compute_vector[simd_width: Int](col: Int):
"""Each time we oeprate on a `simd_width` vector of pixels."""
let cx = min_x + (col + iota[float_type, simd_width]()) * scale_x
let cy = min_y + row * scale_y
let c = ComplexSIMD[float_type, simd_width](cx, cy)
t.data().simd_store[simd_width](row * width + col, mandelbrot_kernel_SIMD[simd_width](c))
# Vectorize the call to compute_vector where call gets a chunk of pixels.
vectorize[simd_width, compute_vector](width)
@parameter
fn bench[simd_width: Int]():
for row in range(height):
worker(row)
let vectorized = Benchmark().run[bench[simd_width]]() / 1e6
print("Number of threads:", num_cores())
print("Vectorized:", vectorized, "ms")
# Parallelized
with Runtime() as rt:
@parameter
fn bench_parallel[simd_width: Int]():
parallelize[worker](rt, height, height)
let parallelized = Benchmark().run[bench_parallel[simd_width]]() / 1e6
print("Parallelized:", parallelized, "ms")
print("Parallel speedup:", vectorized / parallelized)
_ = t # Make sure tensor isn't destroyed before benchmark is finishedcompare()Number of threads: 16
Vectorized: 12.171849 ms
Parallelized: 1.3043979999999999 ms
Parallel speedup: 9.3313919524562294