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Title: pow(a,b,c) should accept b<0
Type: enhancement Stage:
Components: Library (Lib) Versions:
Status: closed Resolution: rejected
Dependencies: Superseder:
Assigned To: gvanrossum Nosy List: gvanrossum, nobody, phr, thomasahle, tim.peters
Priority: normal Keywords:

Created on 2001-08-31 01:17 by anonymous, last changed 2022-04-10 16:04 by admin. This issue is now closed.

Messages (17)
msg6266 - (view) Author: Nobody/Anonymous (nobody) Date: 2001-08-31 01:17
You should be able to raise integers to negative powers
mod n.  If b<0, pow(a,b,c)==pow(pow(a,-1,c),-b,c)
where pow(a,-1,c) is a's multiplicative inverse mod c,
computed with the extended Euclid algorithm.  This
would be in Python's spirit of math completeness and
would save people from the trouble of having to figure
out the algorithm over and over.  

I can come up with a patch for this if it's agreed on
as desirable.
msg6267 - (view) Author: Guido van Rossum (gvanrossum) * (Python committer) Date: 2001-08-31 01:46
Logged In: YES 


In 2.2a2, currently, pow(a, b, c) for ints a, b, c where b <
0 is defined as pow(float(a), float(b), float(c)), IOW
(1.0/(a**b))%c. This doesn't make a lot of sense for the
three-argument version though, because the result tends to
be between 0.0 and 1.0...  But it is consistent with the
(future) rule that operations on integers and floats should
give results with the same value only a different type.

Assigning to Tim, whose mathematical sensibilities are more
refined than mine...
msg6268 - (view) Author: Guido van Rossum (gvanrossum) * (Python committer) Date: 2001-08-31 01:49
Logged In: YES 

Of course I meant (1.0/(a**-b))%c. Sorry!
msg6269 - (view) Author: Tim Peters (tim.peters) * (Python committer) Date: 2001-08-31 05:23
Logged In: YES 

The desire is understandable but this isn't the right way 
to do it, so I'm rejecting this.  While 2.2a2 changed the 
specifics, the general rule remains that

pow(a, b, c) == a**b % c

except that the LHS may be very much faster for large 
integer arguments.

"The right way" to do modular arithmetic is to define a 
class for it, and do the full job (+ - * /, not just 
modular pow).  egcd is easy to code in Python, and because 
it's an obscure speciality need (it gets asked about maybe 
twice per year on doesn't really belong in the core 
even if it weren't.  I'm not even sure how 3-argument pow 
got in, but am grateful it did and don't want to press our 
luck <wink>.
msg6270 - (view) Author: Nobody/Anonymous (nobody) Date: 2001-09-01 06:34
Logged In: NO 

Making a 3-arg integer pow return a tiny floating point
number seems weird to me.  I don't see any situation
where I'd want that.  If I call pow with b<0 without
expecting it to use the egcd to get the integer answer mod
c, I've almost certainly made a mistake.  So my first
preference is still egcd, but second preference is to stay
with the old behavior of throwing an exception rather than
have my integer calculation suddenly turn into a floating
point calculation without my realizing it.  

I'm also enthused about changing / to turn 2/3 into a
float, but at least I understand the argument that 2/3=0
confuses newbies.  But newbies won't be using 3-arg pow(),
so we needn't worry about confusing them.  IMHO anyone using
3-arg pow on integers will expect an integer result.

By the way (off topic), 3-arg pow with ~150 decimal digits
is about 5x slower in Python than in carefully optimized
asm on an x86, which is pretty good.  But on a StrongARM
it appears to be about 30x slower :-(.  This can't really
be fixed without adding a lot more code.  Sigh.
msg6271 - (view) Author: paul rubin (phr) Date: 2001-09-01 06:38
Logged In: YES 

Argh.  I meant to say I'm NOT enthused about changing /.
This item is jinxed :-).
msg6272 - (view) Author: Guido van Rossum (gvanrossum) * (Python committer) Date: 2001-09-01 18:19
Logged In: YES 

Hm. There's something to say for making 3-arg pow() only
work for ints (and longs), and then the egcd would make
sense. But that would mean removing the 3-arg pow() for
floats. Why would anyone use 3-arg pow() with floats? I
don't know, but that doesn't mean it doesn't happen. *If*
there are no community objections against making 3-arg pow()
raise a TypeError on float or complex arguments, I'm OK with
the egcd interpretation. But this is PEP material -- that's
the only way to find out. phr, would you mind writing a PEP?
It can be short and sweet.
msg6273 - (view) Author: Tim Peters (tim.peters) * (Python committer) Date: 2001-09-01 20:04
Logged In: YES 

Changed Resolution to None since this was opened again.

I still don't like this.  It's a wart no matter how you cut 
it:  implement the egcd meaning, and it's still a wart, 
because the "multiplicative inverse" meaning doesn't always 
make sense.  For example, pow(4, -1, 6) -- 4 has no 
multiplicative inverse mod 6.  The best it can return is 2, 
i.e. the best pow(i, -1, k) can return is an integer x s.t. 
i*x is congruent to gcd(i, k) mod k.  But Python provides 
no way to get the gcd, so there's no way (short of 
computing gcd separately) to know what the result of pow
(i, -1, k) really means (and it doesn't mean "inverse" 
unless the gcd is 1; OTOH, raise an exception if the gcd 
isn't one, and then you've artificially ruled out 
legitimate uses of egcd apparently not related to Paul's 
particular interest today).

The natural way to spell egcd as a library routine would 
return the gcd too; e.g.,

def egcd(aorig, borig):
.    """Return (g, i) s.t. g=gcd(aorig, borig) and i*aorig 
% borig = g."""
.    a, b = aorig, borig
.    a1, a2 = 1, 0
.    while b:
.        q, r = divmod(a, b)
.        a1, a2 = a2, a1-q*a2
.        a, b = b, r
.    if __debug__:
.        b1, r = divmod(a - a1*aorig, borig)
.        assert r == 0
.        assert a1*aorig + b1*borig == a
.    return a, a1
msg6274 - (view) Author: Guido van Rossum (gvanrossum) * (Python committer) Date: 2001-09-01 20:12
Logged In: YES 

The resolution remains Rejected -- apparently selecting
"None" signals a "no change" to SF. :-(

I don't like it either -- my suggestion to write a PEP was a
passive-aggressive way to reject the proposal. :-)

Still, it's unclear whether 3-arg pow() makes any sense at
all for floats. Maybe that *should* be rejected. And then we
could reject 3-arg() pow with negative second arg as well.
msg6275 - (view) Author: paul rubin (phr) Date: 2001-09-01 21:05
Logged In: YES 

If b<0 uses egcd, then pow(4,-1,6) should definitely throw a
value error, like dividing by 0.  Pow isn't advertised as
computing gcd's.  It just happens that egcd is a way of
computing inverses mod n.

I'm fine with 3-arg pow throwing an error on non-integer
args.  I like that better than unexpected conversions.

How about continuing to throw an error on b<0, but adding
an egcd function to the math library?

What got me started on this was wanting a modular inverse,
not remembering how egcd worked and having to figure it
out again, and realizing I've been thru that same exercise
many times over the years :-).
msg6276 - (view) Author: Tim Peters (tim.peters) * (Python committer) Date: 2001-09-01 21:20
Logged In: YES 

Well, speaking as an old fp number-cruncher, mod makes 
little sense for floats on its own, and even less so 
combining it with pow (math.fmod makes sense for floats, 
but that's a different operation than the builtin float 
%).  As a practical matter, x%1.0 is sometimes used to get 
at the fractional bits of a float x >= 0, but I haven't 
seen that combined with pow (in practice -- "in theory" pow
(theta, n, 1.0) has interesting properties for irrational 
theta, but fp arithmetic isn't precise enough to exploit 

OTOH, I can't doubt that some existing code uses integers 
that just happen to be in fp format, and then e.g. pow(3., 
4., 7.) makes as much sense as pow(3, 4, 7).  If you want a 
model where it's a number's value that matters-- not 
especially its type --that's worth preserving.  But 
then "something should be done about" stuff like this:

>>> pow(3., 500., 7.)
>>> pow(3, 500, 7)

So, as currently implemented, floats in 3-arg pow are 
surprising even when they denote whole integers.

3-arg pow makes clear sense for ints and longs when the 
power is non-negative, and compelling sense for "go fast" 
reasons, so I'd like to see it restricted to that.  We 
already complain about 3-arg pow with a complex argument.  
I haven't found any actual examples of 3-arg float pow on 
the web ... but who knows?  Let's break it in 2.a3 and see 
whether anyone screams?  I can ask on too.
msg6277 - (view) Author: Guido van Rossum (gvanrossum) * (Python committer) Date: 2001-09-01 21:27
Logged In: YES 

OK.  Sounds like a good plan: break 3-arg pow() for all
float args in 2.2a3, and see in anybody screams. I predict
dead silence.

msg6278 - (view) Author: paul rubin (phr) Date: 2001-09-01 21:35
Logged In: YES 

Sounds good to me re breaking float pow.  It's doing
really weird in 2.2.1 anyway.  pow(3.,500.,7.) returns
2, pow(3,5000,7) returns 2, pow(3.,5000.,7.) returns 4.0,
but 3.**5000. returns inf.  pow(3.,50000.,7.) returns NaN.

The roundoff errors though don't bother me especially
more than any other float roundoff errors, e.g.
 3.**99+1-3.**99 = 0.

msg6279 - (view) Author: Guido van Rossum (gvanrossum) * (Python committer) Date: 2001-09-05 18:14
Logged In: YES 

Can we close this now that it's been rejected and we've made
pow(a,b,c) illegal for float args?
msg6280 - (view) Author: Tim Peters (tim.peters) * (Python committer) Date: 2001-09-05 19:20
Logged In: YES 

That's up to you -- I closed it before, and you opened it 
again.  Since I'm not clear on why it was reopened, 
assigning back to you.  Disallowing 3-arg pow for floats 
didn't preclude the possibility of giving 3-arg integer pow 
with negative 2nd arg a "modular inverse" meaning (which is 
the substance of the feature request, not float behavior), 
and I've got nothing new to say about that.
msg6281 - (view) Author: Guido van Rossum (gvanrossum) * (Python committer) Date: 2001-09-05 19:40
Logged In: YES 

I reopened it because there was a different action item then
(forbid triple-float-arg pow()). Since that has been taken
care of now, I'm closing it again.
msg151764 - (view) Author: Thomas Dybdahl Ahle (thomasahle) * Date: 2012-01-22 09:24
For anyone who finds this through google,
if you are finding the inverse mod a prime, you can use fermats little theorem: pow(a, -1, mod) = pow(a, a-2, mod).
(You also need that mod doesn't divide a).
Date User Action Args
2022-04-10 16:04:23adminsetgithub: 35082
2012-01-22 09:24:14thomasahlesetnosy: + thomasahle
messages: + msg151764
2001-08-31 01:17:23anonymouscreate