2016-04-13-doug-lind-example.sagews

area = dblquad(lambda x, y: log(abs( 4 - 2*cos(2*pi*x) - 2*cos(2*pi*y))), 0, 1, lambda x:0, lambda x:1)

area

(1.1662436161232819, 7.180621168198628e-09)

Another speed trick is fast_float.

This is something Robert Bradshaw wrote, originally to make 3d plotting faster. It takes a symbolic epxression and "compiles" it (very quickly) into an internal form that can be evaluated quickly.

See http://doc.sagemath.org/html/en/reference/misc/sage/ext/fast_eval.html

# how long does yours take?  about 1s -- pretty bad?
from scipy.integrate import quad, dblquad
%time area = dblquad(lambda x, y: log(abs( 4 - 2*cos(2*pi*x) - 2*cos(2*pi*y))), 0, 1, lambda x:0, lambda x:1)

CPU time: 0.77 s, Wall time: 0.81 s

# try that trick I mentioned... and get a MASSIVE speedup.
from scipy.integrate import quad, dblquad
f(x,y) = log(abs( 4 - 2*cos(2*pi*x) - 2*cos(2*pi*y)))
g = fast_float(f)
%time area = dblquad(g, 0r, 1r, lambda x:0r, lambda x:1r)

CPU time: 0.00 s, Wall time: 0.00 s

# How much better?
%timeit area0 = dblquad(g, 0r, 1r, lambda x:0r, lambda x:1r)
%timeit area1 = dblquad(lambda x, y: log(abs( 4 - 2*cos(2*pi*x) - 2*cos(2*pi*y))), 0, 1, lambda x:0, lambda x:1)

625 loops, best of 3: 1.34 ms per loop 5 loops, best of 3: 780 ms per loop

780/1.34

582.089552238806

# Speedup by a factor of more than 500x!


︠dff64924-b26d-44fa-b3f3-91ddd385d4cei︠
%md
**Exercise: ** Make up your own symbolic functions by copying how we wrote `f(x,y) = log(abs( 4 - 2*cos(2*pi*x) - 2*cos(2*pi*y)))` above,
and compare the speed of evaluation using fast_float to the speed of a Python function `def f(x,y): ...` that does the same thing.

**Exercise: ** Make up your own symbolic functions by copying how we wrote f(x,y) = log(abs( 4 - 2*cos(2*pi*x) - 2*cos(2*pi*y))) above, and compare the speed of evaluation using fast_float to the speed of a Python function def f(x,y): ... that does the same thing.

f(x,y) = x+y  # change this
f0 = fast_float(f)
def g(x,y):
    return x+y    # change this

# rerun this after changing the above...
x = float(5); y = float(7)
%timeit f(x,y)
%timeit f0(x,y)
%timeit g(x,y)

# How much faster?!