Re: [sympy] ufuncify now creates actual ufuncs

Fri, 29 Aug 2014 14:24:07 -0700

Try applying it to the expression tree. Horner's rule is supposed to be optimal for evaluating polynomials. 

On 29 Aug 2014, at 17:04, James Crist wrote:
For handling Pow? horner(x**11) results in x**11. Or were you recommending 
applying horner to an entire expression tree?
On Fri, Aug 29, 2014 at 3:55 PM, Tim Lahey <tim.la...@gmail.com> wrote: 


I recommend that you use the horner function in polys.


On 29 Aug 2014, at 16:48, James Crist wrote:
If I understand correctly, there is no cost in representing pow(x, n) as
x*x*x*x... for any positive integer n, as long as it's done correctly. C 
compilers don't like to change how you write out calculations unless
they're asked too. So x*x*x*x will not generate the same machine code as (x*x)*(x*x). The second case is preferable, as it will result in y = x*x, 
sol = y*y, rather than sol = x*x*x*x. This removes the need for one
computation. Also, the second way apparently results in better precision 
for the end result (not sure why).

I am all for writing all positive integer powers as multiplication,
provided we can get the parenthesis convention correct. So x**4 ->
(x*x)*(x*x), or x**11 -> x*((x*x)*(x*x)*x)*((x*x)*(x*x)*x), etc... If 
others are supportive of this, I'll submit a PR.
On Fri, Aug 29, 2014 at 2:00 PM, Aaron Meurer <asmeu...@gmail.com> wrote: 

I think we should print pow using repeated multiplication. People
might not know about --ffast-math, not realize that we are using pow 
and that it is needed, or not want other optimizations that it
provides.
Is there a reason to put a limit on the power (5 was suggested here, 
10 on the pull request)?

Aaron Meurer

On Fri, Aug 29, 2014 at 9:38 AM, Jason Moore <moorepa...@gmail.com>
wrote:


Sorry, it wasn't merged. He found that the --fast-math flag in the


complier


takes care of this.


Jason
moorepants.info
+01 530-601-9791
On Fri, Aug 29, 2014 at 10:37 AM, Jason Moore <moorepa...@gmail.com> 


wrote:




Here is some work on the pow issue:
https://github.com/sympy/sympy/pull/7519
Looks like it was merged so the ccode printer should print x*x*x... for 
less that 10 x's.


Jason
moorepants.info
+01 530-601-9791
On Fri, Aug 29, 2014 at 7:33 AM, Jason Moore <moorepa...@gmail.com> 


wrote:








Jason
moorepants.info
+01 530-601-9791


On Fri, Aug 29, 2014 at 2:38 AM, James Crist <crist...@umn.edu>
wrote:
I was planning on going to bed, but ended up working on this instead. 


I



have no self control...


Anyway, I've uncovered some things:
1. Addition of the restrict keyword to tell the compiler we're not aliasing offers marginal gains. Gain a couple microseconds here and 


there.
This requires a c99 compiler, but it's 2014, everyone should have one 


by



now.
2. Inlining the function call resulted in smaller gains than 1, but still *slightly* measurable. I suspect that for larger expression 


sizes this



will be negligible to none.
3. Here's the big one: For small powers, pow(c, n) is considerably slower than c*c*c*c... Changing the ccode Pow handler to print all 


pows less



than 5 (arbitrary number) out as multiplication I was able to



match/beat



(slightly) all of jason's benchmarks with the C + numpy ufuncs.
Oh yes! I knew that. In fact, I feel like I read in the current code somewhere. I forget, but that seems like a standard way we should be 
handling pows in C. Nice find!





On Thursday, August 28, 2014 1:38:30 PM UTC-5, Tim Lahey wrote:
On why Fortran is faster, Fortran semantics ensure that function arguments never alias, this allows the optimizer to make assumptions 


about



the function and the arguments. This the main advantage of Fortran



over C.
But, because of this, it can lead to more memory usage. I know that 


the



newer C++ standards have a keyword to mark arguments to indicate



that they
won't be aliased, but that requires that the code generator and the 


compiler



support them.


Cheers,

Tim.
On 2014-08-28, at 2:17 PM, Jason Moore <moore...@gmail.com> wrote: 

Jim and others,


Here are the benchmarks I made yesterday:

http://www.moorepants.info/blog/fast-matrix-eval.html

The working code is here:
https://gist.github.com/moorepants/6ef8ab450252789a1411

Any feedback is welcome.


Jason
moorepants.info
+01 530-601-9791
On Wed, Aug 27, 2014 at 11:44 PM, James Crist <cris...@umn.edu> 
wrote:
I was wondering about that. I wasn't sure if the overhead from 
looping through the inputs multiple times would outweigh


improvements from



fast C loops. Glad that in your case it does.
I've thrown a WIP PR up: https://github.com/sympy/sympy/pull/7929
For some reason, creating the functions in python with numpy calls still seems to be faster (for micro-benchmarks). This probably has 


something



to do with function complexity (the example function above is



simple), but



I'd still think it'd be faster in pure C. I tried inlining the



call, which



was a small improvement, but it was still slower than the pure



numpy-python



version. Something to look into.
On Wed, Aug 27, 2014 at 10:28 PM, Jason Moore <moore...@gmail.com> 
wrote:
Yeh, but if you simply create a ufunc for each expression in a 


matrix



you still get substantial speedups. I wrote a bunch of test cases



that I'll



post to my blog tomorrow.



Jason
moorepants.info
+01 530-601-9791
On Wed, Aug 27, 2014 at 11:26 PM, James Crist <cris...@umn.edu> 
wrote:
Not yet. I wrote it this morning during an extremely boring


meeting,



and haven't had a chance to clean it up. This doesn't solve your



problem



about broadcasting a matrix calculation though...
On Wed, Aug 27, 2014 at 10:23 PM, Jason Moore <moore...@gmail.com> 
wrote:
Awesome. I was working on this today but it looks like you've by 
passed what I had working. Do you have a PR with this?


Jason
moorepants.info
+01 530-601-9791


On Wed, Aug 27, 2014 at 11:11 PM, Matthew Rocklin <


mroc...@gmail.com>



wrote:

Cool
On Wed, Aug 27, 2014 at 8:07 PM, James Crist <cris...@umn.edu> 


wrote:



I still need to do some cleanups and add tests, but I finally have
this working and thought I'd share. I'm really happy with this: 

In [1]: from sympy import *

In [2]: a, b, c = symbols('a, b, c')

In [3]: expr = (sin(a) + sqrt(b)*c**2)/2

In [4]: from sympy.utilities.autowrap import ufuncify

In [5]: func = ufuncify((a, b, c), expr)

In [6]: func(1, 2, 3)
Out[6]: 6.7846965230828769

In [7]: func([1, 2, 3, 4, 5], [6, 7, 8, 9, 10], 3)
Out[7]: array([ 11.44343933,  12.36052961,  12.79848207,
13.12159875,  13.75078733])

In [8]: from numpy import arange

In [9]: a = arange(10).reshape((2, 5))

In [10]: c = arange(10, 20).reshape((2, 5))

In [11]: b = 25

In [12]: func(a, b, c)
Out[12]:
array([[ 250. , 302.92073549, 360.45464871, 422.57056 , 
   489.62159875],
 [ 562.02053786,  639.86029225,  722.8284933 ,  810.49467912,
   902.70605924]])

In [13]: type(func)
Out[13]: numpy.ufunc
This now does everything a numpy `ufunc` does normally, as it *is* 


a



ufunc. Codegen is hooked up to numpy api. Type conversion and



broadcasting



are done automagically.
Caveats: only functions with a single output are accepted (this 


could



be changed to accept multi-output without much effort though).



Also, as with



all unfuncs, input/outputs must all be scalars (no matrix/Indexed



operations



allowed).


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