Outline:
We want to integrate the function $f(x) = x^4 - 3x$.
In [1]:
def f(x):
y = x**4 - 3*x
return y
def integrate_f(a, b, n):
dx = (b - a) / n
dx2 = dx / 2
s = f(a) * dx2
for i in range(1, n):
s += f(a + i * dx) * dx
s += f(b) * dx2
return s
Now, let's time this:
In [2]:
from scipy.integrate import quad
In [3]:
%timeit quad(f, -100,100)
The slowest run took 4.72 times longer than the fastest. This could mean that an intermediate result is being cached
100000 loops, best of 3: 15.6 µs per loop
In [4]:
%timeit integrate_f(-100, 100, int(1e5))
10 loops, best of 3: 52.4 ms per loop
Not too bad, but this can add up. Let's see if Cython can do better:
In [5]:
%load_ext cython
In [4]:
%%cython
def f2(x):
y = x**4 - 3*x
return y
def integrate_f2(a, b, n):
dx = (b - a) / n
dx2 = dx / 2
s = f2(a) * dx2
for i in range(1, n):
s += f2(a + i * dx) * dx
s += f2(b) * dx2
return s
In [16]:
%timeit integrate_f2(-100, 100, int(1e5))
10 loops, best of 3: 37.6 ms per loop
That's a little bit faster, which is nice since all we did was to call Cython on the exact same code. But can we do better?
In [17]:
%%cython
def f3(double x):
y = x**4 - 3*x
return y
def integrate_f3(double a, double b, int n):
dx = (b - a) / n
dx2 = dx / 2
s = f3(a) * dx2
for i in range(1, n):
s += f3(a + i * dx) * dx
s += f3(b) * dx2
return s
In [18]:
%timeit integrate_f3(-100, 100, int(1e5))
10 loops, best of 3: 23.8 ms per loop
The final bit of "easy" Cython optimization is "declaring" the variables inside the function:
In [19]:
%%cython
def f4(double x):
y = x**4 - 3*x
return y
def integrate_f4(double a, double b, int n):
cdef:
double dx = (b - a) / n
double dx2 = dx / 2
double s = f4(a) * dx2
int i = 0
for i in range(1, n):
s += f4(a + i * dx) * dx
s += f4(b) * dx2
return s
In [21]:
%timeit integrate_f4(-100, 100, int(1e5))
100 loops, best of 3: 14.8 ms per loop
4X speedup with so little effort is pretty nice. What else can we do?
Cython has a nice "-a" flag (for annotation) that can provide clues about why your code is slow.
In [22]:
%%cython -a
def f4(double x):
y = x**4 - 3*x
return y
def integrate_f4(double a, double b, int n):
cdef:
double dx = (b - a) / n
double dx2 = dx / 2
double s = f4(a) * dx2
int i = 0
for i in range(1, n):
s += f4(a + i * dx) * dx
s += f4(b) * dx2
return s
Out[22]:
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That's a lot of yellow still! How do we reduce this?
In [53]:
%%cython -a
#cython: cdivision=True
#import cython
cdef double f5(double x):
y = x**4 - 3*x
return y
def integrate_f6(double a, double b, int n):
cdef:
double dx = (b - a) / n
double dx2 = dx / 2
double s = f5(a) * dx2
int i = 0
for i in range(1, n):
s += f5(a + i * dx) * dx
s += f5(b) * dx2
return s
Out[53]:
Cython: _cython_magic_866acf3a78d320cf35b8a8e8296ecbb8.pyx
Generated by Cython 0.22.1
Yellow lines hint at Python interaction.
Click on a line that starts with a "+" to see the C code that Cython generated for it.
01: #cython: cdivision=True
02: #import cython
03:
+04: cdef double f5(double x):
static double __pyx_f_46_cython_magic_866acf3a78d320cf35b8a8e8296ecbb8_f5(double __pyx_v_x) {
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+06: return y
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goto __pyx_L0;
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/* Python wrapper */
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09: cdef:
+10: double dx = (b - a) / n
__pyx_v_dx = ((__pyx_v_b - __pyx_v_a) / __pyx_v_n);
+11: double dx2 = dx / 2
__pyx_v_dx2 = (__pyx_v_dx / 2.0);
+12: double s = f5(a) * dx2
__pyx_v_s = (__pyx_f_46_cython_magic_866acf3a78d320cf35b8a8e8296ecbb8_f5(__pyx_v_a) * __pyx_v_dx2);
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for (__pyx_t_2 = 1; __pyx_t_2 < __pyx_t_1; __pyx_t_2+=1) {
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goto __pyx_L0;
In [54]:
%timeit integrate_f6(-100, 100, int(1e5))
100 loops, best of 3: 7.18 ms per loop
This is a very small subset of Python. Most scientific application deal not with single values, but with arrays of data.
In [55]:
import numpy as np
def mean3filter(arr):
arr_out = np.empty_like(arr)
for i in range(1, arr.shape[0] - 1):
arr_out[i] = np.sum(arr[i-1 : i+1]) / 3
arr_out[0] = (arr[0] + arr[1]) / 2
arr_out[-1] = (arr[-1] + arr[-2]) / 2
return arr_out
In [56]:
%timeit mean3filter(np.random.rand(1e5))
1 loops, best of 3: 628 ms per loop
In [57]:
%%cython
import cython
import numpy as np
@cython.boundscheck(False)
def mean3filter2(double[::1] arr):
cdef double[::1] arr_out = np.empty_like(arr)
cdef int i
for i in range(1, arr.shape[0]-1):
arr_out[i] = np.sum(arr[i-1 : i+1]) / 3
arr_out[0] = (arr[0] + arr[1]) / 2
arr_out[-1] = (arr[-1] + arr[-2]) / 2
return np.asarray(arr_out)
In [77]:
%timeit np.convolve(np.random.rand(1e5), np.array([1.,1.,1.]), 'same')
100 loops, best of 3: 3.11 ms per loop
In [58]:
%timeit mean3filter2(np.random.rand(1e5))
1 loops, best of 3: 1.17 s per loop
Rubbish! How do we fix this?
In [75]:
%%cython -a
import cython
import numpy as np
@cython.boundscheck(False)
def mean3filter2a(double[::1] arr):
# ::1 means that the array is contiguous
cdef double[::1] arr_out = np.empty_like(arr)
cdef int i
for i in range(1, arr.shape[0]-1):
#for j in range(3):
arr_out[i] = arr[i-1] + arr[i] + arr[i+1]
arr_out[i] *= 0.333333333333333333333333
#arr_out[i] = np.sum(arr[i-1 : i+1]) / 3
arr_out[0] = (arr[0] + arr[1]) / 2
arr_out[-1] = (arr[-1] + arr[-2]) / 2
return np.asarray(arr_out)
Out[75]:
Cython: _cython_magic_91ce2d70c0de6735a44d76fd658e8707.pyx
Generated by Cython 0.22.1
Yellow lines hint at Python interaction.
Click on a line that starts with a "+" to see the C code that Cython generated for it.
+01: import cython
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04: @cython.boundscheck(False)
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In [76]:
%timeit mean3filter2a(np.random.rand(1e5))
1000 loops, best of 3: 1.56 ms per loop
Warning:: Dragons afoot.
In [82]:
%%cython -a
import cython
from cython.parallel import prange
import numpy as np
@cython.boundscheck(False)
def mean3filter3a(double[::1] arr, double[::1] out):
cdef int i, j, k = arr.shape[0]-1
for i in range(1, k-1):
for j in range(i-1, i+1):
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out[i] /= 3
out[0] = (arr[0] + arr[1]) / 2
out[-1] = (arr[-1] + arr[-2]) / 2
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Out[82]:
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Yellow lines hint at Python interaction.
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In [85]:
%%cython -a
import cython
from cython.parallel import prange
import numpy as np
@cython.boundscheck(False)
def mean3filter3(double[::1] arr, double[::1] out):
cdef int i, j, k = arr.shape[0]-1
with nogil:
for i in prange(1, k-1, schedule='static',
chunksize=(k-2) // 2, num_threads=4):
for j in range(i-1, i+1):
out[i] += arr[j]
out[i] /= 3
out[0] = (arr[0] + arr[1]) / 2
out[-1] = (arr[-1] + arr[-2]) / 2
return np.asarray(out)
Error compiling Cython file:
------------------------------------------------------------
...
import numpy as np
@cython.boundscheck(False)
def mean3filter3(double[::1] arr, double[::1] out):
cdef int i, j, k = arr.shape[0]-1
for i in prange(1, k-1, schedule='static',
^
------------------------------------------------------------
/home/student/.cache/ipython/cython/_cython_magic_01f16d263991d5b51fc55754d032c5d4.pyx:8:19: prange() can only be used without the GIL
In [86]:
%%cython -a
import cython
from cython.parallel import prange
import numpy as np
@cython.boundscheck(False)
def mean3filter3b(double[::1] arr, double[::1] out):
cdef int i, j, k = arr.shape[0]-1
for i in range(1, k-1):
for j in range(i-1, i+1):
out[i] += arr[j]
with nogil:
for i in prange(1, k-1, schedule='static',
chunksize=(k-2) // 2, num_threads=4):
out[i] /= 3
out[0] = (arr[0] + arr[1]) / 2
out[-1] = (arr[-1] + arr[-2]) / 2
return np.asarray(out)
Out[86]:
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In [91]:
del rin, rout
In [89]:
rin = np.random.rand(1e8)
rout = np.empty_like(rin)
In [90]:
%timeit mean3filter3b(rin, rout)
The slowest run took 14.04 times longer than the fastest. This could mean that an intermediate result is being cached
1 loops, best of 3: 683 ms per loop
In [84]:
%timeit mean3filter3(rin, rout)
10 loops, best of 3: 45.6 ms per loop
In [ ]:
In [92]:
%%cython -a
# distutils: language=c++
import cython
from libcpp.vector cimport vector
@cython.boundscheck(False)
def build_list_with_vector(double[::1] in_arr):
cdef vector[double] out
cdef int i
for i in range(in_arr.shape[0]):
out.push_back(in_arr[i])
return out
Out[92]:
Cython: _cython_magic_9e62336531da4dc6d78b22908dd4ed1c.pyx
Generated by Cython 0.22.1
Yellow lines hint at Python interaction.
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02: import cython
03: from libcpp.vector cimport vector
04:
05:
06: @cython.boundscheck(False)
+07: def build_list_with_vector(double[::1] in_arr):
/* Python wrapper */
static PyObject *__pyx_pw_46_cython_magic_9e62336531da4dc6d78b22908dd4ed1c_1build_list_with_vector(PyObject *__pyx_self, PyObject *__pyx_arg_in_arr); /*proto*/
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08: cdef vector[double] out
09: cdef int i
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In [93]:
build_list_with_vector(np.random.rand(10))
Out[93]:
[0.26085286315839795,
0.9763933277734224,
0.9676751436173779,
0.6887305413123229,
0.009623500824120224,
0.2690635362697693,
0.46649754307129254,
0.9523794831061152,
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In [94]:
%%cython -a
#distutils: language=c++
from cython.operator cimport dereference as deref, preincrement as inc
from libcpp.vector cimport vector
from libcpp.map cimport map as cppmap
cdef class Graph:
cdef cppmap[int, vector[int]] _adj
cpdef int has_node(self, int node):
return self._adj.find(node) != self._adj.end()
cdef void add_node(self, int new_node):
cdef vector[int] out
if not self.has_node(new_node):
self._adj[new_node] = out
def add_edge(self, int u, int v):
self.add_node(u)
self.add_node(v)
self._adj[u].push_back(v)
self._adj[v].push_back(u)
def __getitem__(self, int u):
return self._adj[u]
cdef vector[int] _degrees(self):
cdef vector[int] deg
cdef int first = 0
cdef vector[int] edges
cdef cppmap[int, vector[int]].iterator it = self._adj.begin()
while it != self._adj.end():
deg.push_back(deref(it).second.size())
it = inc(it)
return deg
def degrees(self):
return self._degrees()
Out[94]:
Cython: _cython_magic_207f54e3e5ad780456f48229a083fca4.pyx
Generated by Cython 0.22.1
Yellow lines hint at Python interaction.
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+01: #distutils: language=c++
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03:
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05: from libcpp.map cimport map as cppmap
06:
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struct __pyx_vtabstruct_46_cython_magic_207f54e3e5ad780456f48229a083fca4_Graph *__pyx_vtab;
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39:
In [95]:
g0 = Graph()
In [96]:
g0.add_edge(1, 5)
g0.add_edge(1, 6)
In [97]:
g0[1]
Out[97]:
[5, 6]
In [98]:
g0.has_node(1)
Out[98]:
1
In [99]:
g0.degrees()
Out[99]:
[2, 1, 1]
In [100]:
import networkx as nx
g = nx.barabasi_albert_graph(100000, 6)
with open('graph.txt', 'w') as fout:
for u, v in g.edges_iter():
fout.write('%i,%i\n' % (u, v))
In [101]:
%timeit list(g.degree())
10 loops, best of 3: 51.7 ms per loop
In [102]:
myg = Graph()
def line2edges(line):
u, v = map(int, line.rstrip().split(','))
return u, v
edges = map(line2edges, open('graph.txt'))
for u, v in edges:
myg.add_edge(u, v)
In [103]:
%timeit mydeg = myg.degrees()
100 loops, best of 3: 9.83 ms per loop
Use setup.py to build your Cython files.
from distutils.core import setup
from distutils.extension import Extension
from Cython.Distutils import build_ext
import numpy as np
setup(
cmdclass = {'build_ext': build_ext},
ext_modules = [
Extension("prange_demo", ["prange_demo.pyx"],
include_dirs=[np.get_include()],
extra_compile_args=['-fopenmp'],
extra_link_args=['-fopenmp', '-lgomp']),
]
)
In [2]:
import numpy as np
from mean3 import mean3filter
mean3filter(np.random.rand(10))
Out[2]:
array([ 0.6960285 , 0.56840097, 0.38207445, 0.28536657, 0.37981228,
0.52849249, 0.60342301, 0.59280659, 0.59454272, 0.67086538])
In [ ]:
Content source: dboonz/aspp2015
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