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From: Oscar Benjamin <oscar.j.benjamin@gmail.com>
Date: Fri, 6 Dec 2013 20:57:49 +0000
Subject: Re: Does Python optimize low-power functions?
To: John Ladasky <john_ladasky@sbcglobal.net>
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On 6 December 2013 18:16, John Ladasky <john_ladasky@sbcglobal.net> wrote:
> The following two functions return the same result:
>
>     x**2
>     x*x
>
> But they may be computed in different ways.  The first choice can accommo=
date non-integer powers and so it would logically proceed by taking a logar=
ithm, multiplying by the power (in this case, 2), and then taking the anti-=
logarithm.  But for a trivial value for the power like 2, this is clearly a=
 wasteful choice.  Just multiply x by itself, and skip the expensive log an=
d anti-log steps.
>
> My question is, what do Python interpreters do with power operators where=
 the power is a small constant, like 2?  Do they know to take the shortcut?

As mentioned this will depend on the interpreter and on the type of x.
Python's integer arithmetic is exact and unbounded so switching to
floating point and using approximate logarithms is a no go if x is an
int object.

For CPython specifically, you can see here:
http://hg.python.org/cpython/file/07ef52e751f3/Objects/floatobject.c#l741
that for floats x**2 will be equivalent to x**2.0 and will be handled
by the pow function from the underlying C math library. If you read
the comments around that line you'll see that different inconsistent
math libraries can do things very differently leading to all kinds of
different problems.

For CPython if x is an int (long) then as mentioned before it is
handled by the HAC algorithm:
http://hg.python.org/cpython/file/07ef52e751f3/Objects/longobject.c#l3934

For CPython if x is a complex then it is handled roughly as you say:
for x**n if n is between -100 and 100 then multiplication is performed
using the "bit-mask exponentiation" algorithm. Otherwise it is
computed by converting to polar exponential form and using logs (see
also the two functions above this one):
http://hg.python.org/cpython/file/07ef52e751f3/Objects/complexobject.c#l151


Oscar