Issue 3439: create a numbits() method for int and long types (original) (raw)
Created on 2008-07-24 17:20 by fredrikj, last changed 2022-04-11 14:56 by admin. This issue is now closed.
Messages (95)
Author: Fredrik Johansson (fredrikj)
Date: 2008-07-24 17:20
Python 3.0b2 (r30b2:65106, Jul 18 2008, 18:44:17) [MSC v.1500 32 bit (Intel)] on win32 Type "help", "copyright", "credits" or "license" for more information.
import math math.frexp(10100) (0.5714936956411375, 333) math.frexp(101000) Traceback (most recent call last): File "", line 1, in OverflowError: Python int too large to convert to C double
(Same behavior in Python 2.5.2, and presumably 2.6 although I haven't checked the latter.)
I think it should be easy to make frexp work for large integers by calling PyLong_AsScaledDouble and adding the exponents. It would be logical to fix this since math.log(n) already works for large integers.
My reason for requesting this change is that math.frexp is the fastest way I know of to accurately count the number of bits in a Python integer (it is more robust than math.log(n,2) and makes it easy to verify that the computed size is exact) and this is something I need to do a lot.
Actually, it would be even more useful to have a standard function to directly obtain the bit size of an integer. If I'm not mistaken, PyLong_NumBits does precisely this, and would just have to be wrapped. Aside from my own needs (which don't reflect those of the Python community), there is at least one place in the standard library where this would be useful: decimal.py contains an inefficient implementation (_nbits) that could removed.
Author: Martin v. Löwis (loewis) * 
Date: 2008-07-24 19:41
Would you like to work on a patch?
Author: Raymond Hettinger (rhettinger) *
Date: 2008-07-24 19:47
I prefer your idea to expose PyLong_Numbits(). IMO, frexp() is very much a floating point concept and should probably remain that way.
Author: Raymond Hettinger (rhettinger) *
Date: 2008-07-24 21:39
Another reason to leave frexp() untouched is that it is tightly coupled to ldexp() as its inverse, for a lossless roundtrip:
assert ldexp(*frexp(pi)) == pi
This relationship is bound to get mucked-up or confused if frexp starts accepting large integers that are no exactly representable as floats (i.e. 2**100+1).
Author: Fredrik Johansson (fredrikj)
Date: 2008-07-24 23:12
Raymond, yes, I think that a separate numbits function would better, although exposing this functionality would not prevent also changing the behavior of frexp. As I said, math.log already knows about long integers, so handling long integers similarly in frexp would not be all that unnatural. (On the other hand, it is true that math.sqrt, math.pow, math.cos, etc could all theoretically be "fixed" to work with larger-than-double input, and that slippery slope is probably better avoided.)
Good point about roundtripping, but the problem with that argument is that frexp already accepts integers that cannot be represented exactly, e.g.:
ldexp(*frexp(10100)) == 10100 False
Anyway, if there is support for exposing _PyLong_Numbits, should it be a method or a function? (And if a function, placed where? Should it accept floating-point input?)
I'm attaching a patch (for the trunk) that adds a numbits method to the int and long types. My C-fu is limited, and I've never hacked on Python before, so the code is probably broken or otherwise bad in several ways (but in that case you can tell me about it and I will learn something :-). I did not bother to optimize the implementation for int, and the tests may be redundant/incomplete/placed wrongly.
A slight problem is that _PyLong_NumBits is restricted to a size_t, so it raises an OverflowError on 32-bit platforms for some easily physically representable numbers:
(1<<3*10**9).numbits() Traceback (most recent call last): File "", line 1, in OverflowError: long int too large to convert to int
This would need to be fixed somehow.
If numbits becomes a method, should it also be added to the Integral ABC? GMPY's mpz type, by the way, defines a method numdigits(self, base). This generalization would possibly be overkill, but it's worth considering.
If it's too late to add this method/function for 2.6 and 3.0, please update the issue version field as appropriate.
Author: Raymond Hettinger (rhettinger) *
Date: 2008-07-25 00:11
numbers.Integral is already way too fat of an API. Am -1 on expanding it further. Recommend sticking with the simplest, least invasive, least pervasive version of your request, a numbits() method for ints.
FWIW, in Py2.6 you can already write:
def numbits(x): return len(bin(abs(x))) - 2
Author: Mark Dickinson (mark.dickinson) *
Date: 2008-07-27 06:29
I'd also be interested in having _PyLong_NumBits exposed to Python in some way or another. It's something I've needed many times before, and it's used in the decimal module, for example.
My favorite workaround uses hex instead of bin:
4*len('%x'%x) - correction_dictionary[first_hexdigit_of_x]
but this is still O(log x) rather than O(1).
Author: Mark Dickinson (mark.dickinson) *
Date: 2008-08-14 09:22
With the patch, the following code causes a non-keyboard-interruptible interpreter hang.
from sys import maxint (-maxint-1).numbits()
[... interpreter hang ...]
The culprit is, of course, the statement
if (n < 0) n = -n;
in int_numbits: LONG_MIN is negated to itself (this may even be undefined behaviour according to the C standards).
The patch also needs documentation, and that documentation should clearly spell out what happens for zero and for negative numbers. It's not at all clear that everyone will expect (0).numbits() to be 0, though I agree that this is probably the most useful definition in practice.
One could make a case for (0).numbits() raising ValueError: for some algorithms, what one wants is an integer k such that 2**(k-1) <= abs(n) < 2**k; when n == 0 no such integer exists.
Other than those two things, I think the patch looks fine.
Author: Mark Dickinson (mark.dickinson) *
Date: 2008-08-14 09:47
One possible fix would be to compute the absolute value of n as an unsigned long. I think the following is portable and avoids any undefined behaviour coming from signed arithmetic overflow.
unsigned long absn; if (n < 0) absn = 1 + (unsigned long)(-1-n); else absn = (unsigned long)n;
Might this work?
Perhaps it would also be worth changing the tests in test_int from e.g.
self.assertEqual((-a).numbits(), i+1)
to
self.assertEqual(int(-a).numbits(), i+1)
This would have caught the -LONG_MAX error.
Author: Fredrik Johansson (fredrikj)
Date: 2008-08-18 20:41
Wow, newbie error. Thanks for spotting!
In every case I can think of, I've wanted (0).numbits() to be 0. The explanation in the docstring can probably be improved. What other documentation is needed (where)?
Author: Mark Dickinson (mark.dickinson) *
Date: 2008-08-18 21:28
In every case I can think of, I've wanted (0).numbits() to be 0.
Me too, in most cases, though I've encountered the occasional case where raising ValueError would be more appropriate and would catch some bugs that might otherwise pass silently.
So I agree that (0).numbits() should be 0, but I think there's enough
potential ambiguity that there should be a sentence in the documentation
making this explicit. Two of the most obvious (wrong) formulas for
numbits are: (1) numbits(n) = ceiling(log_2(n)), and (2) numbits(n) =
len(bin(n))-2, but neither of these formulas gives the right result for
0, or for negative numbers for that matter.
The explanation in the docstring can probably be improved. What other documentation is needed (where)?
The docstring looked okay to me. There should be more comprehensive ReST documentation in the Doc/ directory somewhere, probably in Doc/library/stdtypes.rst
Author: Terry J. Reedy (terry.reedy) *
Date: 2008-10-06 17:41
To add support to the proposal: there is currently yet another thread on c.l.p on how to calculate numbits efficiently. The OP needs it for prototyping cryptographic algorithms and found Python-level code slower than he wanted.
Author: Fredrik Johansson (fredrikj)
Date: 2008-10-14 10:38
Some elaboration (that perhaps could be adapted into the documentation or at least source comments).
There are two primary uses for numbits, both of which justify (0).numbits() == 0.
The first is that for positive k, n = k.numbits() gives the minimum width of a register that can hold k, where a register can hold the 2n integers 0, 1, ..., 2n-1 (inclusive). This definition continues to make sense for k = 0, n = 0 (the empty register holds the 2**0 = 1 values 0).
In Python terms, one could say that self.numbits() "returns the smallest n such that abs(self) is in range(2**n)". Perhaps this would make a clearer docstring?
Second, k.numbits() (plus/minus 1, or perhaps multiplied by a constant factor) measures the number of steps required to solve a problem of size k using various divide-and-conquer algorithms. The problem of size k = 0 is trivial and therefore requires (0).numbits() == 0 steps.
In particular, if L is a sorted list, then len(L).numbits() exactly gives the maximum number of comparisons required to find an insertion point in L using binary search.
Finally, the convention (-k).numbits() == k.numbits() is useful in contexts where the number k itself is the input to a mathematical function. For example, in a function for multiplying two integers, one might want to choose a different algorithm depending on the sizes of the inputs, and this choice is likely to be independent of signs (if not, one probably needs to check signs anyway.)
Author: STINNER Victor (vstinner) *
Date: 2008-10-14 11:07
I changed the title since I agree that numbits() with long integer is not related to floats.
Author: Mark Dickinson (mark.dickinson) *
Date: 2008-10-14 15:07
Accidentally removed the following message from Victor Stinner;
apologies. (Time to turn off tap-to-click on my trackpad, methinks.)
See also issue #3724 which proposes to support long integers for math.log2().
One other note: in Fredrik's patch there's commented out code for a numbits property (rather than a method). Is there any good reason to make this a property? I don't have a good feeling for when something should be a method and when it should be a property, but in this case I'd be inclined to leave numbits as a method.
Are there general guidelines for making things properties?
Author: STINNER Victor (vstinner) *
Date: 2008-10-14 15:13
Accidentally removed the following message from Victor Stinner
No problem.
Is there any good reason to make this a property?
Since numbits() cost is O(n) with n: number of digits. I prefer a method than a property because, IMHO, reading a property should be O(1) (read an attribute is different than compute a value).
Author: Fredrik Johansson (fredrikj)
Date: 2008-10-14 15:38
One other note: in Fredrik's patch there's commented out code for a numbits property (rather than a method). Is there any good reason to make this a property?
Aesthetically, I think numbits as a function would make more sense. (Maybe if the hypothetical imath module comes along...)
Since numbits() cost is O(n) with n: number of digits. I prefer a method than a property because, IMHO, reading a property should be O(1) (read an attribute is different than compute a value).
Unless I missed something, numbits() is O(1). Only the topmost word in a number needs to be examined.
reading a property should be O(1) (read an attribute is different than compute a value).
O(1) is necessary but not sufficient. My sense is that an attribute should access an existing "part" of an object while an operation that involves creating a "new" object should be a method. Compare complex.real/.imag and complex.conjugate().
Author: STINNER Victor (vstinner) *
Date: 2008-10-14 16:20
Unless I missed something, numbits() is O(1).
Ooops, you're right. I looked quickly at the patch and I read "while(n)" but n is a digit, not the number of digits! So it's very quick to compute number of bits.
Author: Terry J. Reedy (terry.reedy) *
Date: 2008-10-14 17:09
I consider .numbits to be an internal property of ints and would prefer it accessed that way. To me, this sort of thing is what property() is for.
Guido has said that the nuisance of tacking on otherwise unnecessary empty parens is a warning to the user that getting the answer might take a long time.
Another tack is to notice that numbits is the length of the bit sequence representation of an int (excepting 0) and give ints a .len method ;-). I would not expect that suggestion to fly very far, though.
Author: Fredrik Johansson (fredrikj)
Date: 2008-10-14 17:19
Another tack is to notice that numbits is the length of the bit sequence representation of an int (excepting 0) and give ints a .len method ;-). I would not expect that suggestion to fly very far, though.
FWIW, I'm one of the people who'd additionally find indexing and slicing of the bits of integers very useful. It's not going to happen, though!
Author: STINNER Victor (vstinner) *
Date: 2008-10-14 17:31
A property /looks/ like an attribute and an user might try to change its value: "x=1; x.numbits = 2" (gives 3 or 4 ? :-))
Author: Raymond Hettinger (rhettinger) *
Date: 2008-10-14 18:03
Properties can be made read-only. Also, there is a good precedent: c=4+5j; print c.real
Author: Mark Dickinson (mark.dickinson) *
Date: 2008-10-14 19:29
One more minor deficiency in the patch: it gives incorrect results for very large integers. For example, on a 32-bit build of the trunk:
x = 1 << 231-1 x <<= 231-1 x.numbits() # expect 4294967295 4294967295L x <<= 2 x.numbits() # expect 4294967297 4294967295L
It would be nicer if the OverflowError from _PyLong_NumBits were propagated, so that the second case raises OverflowError instead of giving an incorrect result.
Alternatively, in case of OverflowError one could recompute numbits
correctly, without overflow, by using Python longs instead of a C size_t;
but this would mean adding little-used, and probably little-tested, extra
code for what must be a very rare special case. Probably not worth it.
Author: STINNER Victor (vstinner) *
Date: 2008-11-04 18:15
It would be nicer if the OverflowError from _PyLong_NumBits were propagated, so that the second case raises OverflowError instead of giving an incorrect result
Why not, but I prefer your second proposition: return a long integer. Attached patch implements this solution.
x=1<<(231-1) n=x.numbits(); n, n.numbits() (2147483648L, 32L) x<<=(231-1) n=x.numbits(); n, n.numbits() (4294967295L, 32L) x<<=1 n=x.numbits(); n, n.numbits() (4294967296L, 33L) # yeah!
With my patch, there are two functions:
- _PyLong_NumBits(long)->size_t: may overflow
- long_numbits(long)->long: don't raise overflow error, but may raise other errors like memory error
Author: Mark Dickinson (mark.dickinson) *
Date: 2008-11-11 17:15
Hi, Victor! Thanks for the updated patch.
Your patch still hangs on:
from sys import maxint (-maxint-1).numbits()
on my 32-bit machine.
Author: STINNER Victor (vstinner) *
Date: 2008-11-11 18:15
(-maxint-1).numbits() hangs
Ooops, nice catch! It was an integer overflow, already present in fredrikj's original patch. The new patch fixes this bug but also included a documentation patch ;-)
Author: Mark Dickinson (mark.dickinson) *
Date: 2008-11-11 23:00
The latest patch from Victor looks good. A few comments:
(1) the number of bits should be computed first directly using C arithmetic, and only recomputed using PyLong arithmetic if the C computations overflow. For one thing, overflow is going to be very rare in practice; for another, in the sort of applications that use .numbits(), speed of numbits() is often critical.
(2) Just as a matter of style, I think "if (x == NULL)" is preferable to "if (!x)". At any rate, the former style is much more common in Python source.
(3) the reference counting all looks good.
(4) I quite like the idea of having numbits be a property rather than a method---might still be worth considering?
Author: Fredrik Johansson (fredrikj)
Date: 2008-11-12 00:02
In stdtypes.rst, x.numbits should be listed in the table under "Bit-string Operations on Integer Types" and not in the table of operations supported by all numeric types.
(1) the number of bits should be computed first directly using C arithmetic, and only recomputed using PyLong arithmetic if the C computations overflow.
+1
(4) I quite like the idea of having numbits be a property rather than a method---might still be worth considering?
I'm not against.
Author: STINNER Victor (vstinner) *
Date: 2008-11-12 00:33
[Mark Dickinson]
(1) the number of bits should be computed first directly using ...
_PyLong_NumBits() : done. But the result type is always long.
(2) Just as a matter of style, I think "if (x == NULL)" is preferable
done
(4) I quite like the idea of having numbits be a property rather than a method---might still be worth considering?
I agree, so in the new patch, numbits is now a property.
[Fredrik Johansson]
In stdtypes.rst, x.numbits should be listed in the table under "Bit-string Operations on Integer Types"
done
--
10
x=1023L; x.numbits 10L x=2**(2**10); n=x.numbits; n, n.numbits (1025L, 11L)
Author: STINNER Victor (vstinner) *
Date: 2008-12-07 12:35
I plan to change my patch to come back to a method (x.numbits()) because other integer implementations like Decimal may be slower.
Author: STINNER Victor (vstinner) *
Date: 2008-12-09 12:28
New version of my patch using a method instead of a property.
Author: Mark Dickinson (mark.dickinson) *
Date: 2008-12-11 21:12
Victor,
Thanks for the updated patch.
I think you need to do a PyErr_Clear after the 'return PyLong_FromSize_t' line.
To be safe, you should probably check the exception type first, and either do a PyErr_Clear and continue if it's an OverflowError, or just return NULL if it's some other exception. It's true that _PyLong_NumBits can't raise anything other than OverflowError at the moment, but you never know when that might change. :)
I'm also getting a 'malformed table' warning when building the documentation.
Author: Mark Dickinson (mark.dickinson) *
Date: 2008-12-11 21:22
Alternatively, you could just ignore _PyLong_NumBits entirely and put the whole calculation into long_numbits. You're already halfway there with the calculation of msd_bits anyway, and I suspect the code will look cleaner (and run faster) that way.
Author: STINNER Victor (vstinner) *
Date: 2008-12-11 23:05
Oops, you're right Mark: PyErr_Clear() is needed! New patch:
- don't catch error different than PyExc_OverflowError in long_numbits()
- clear error in long_numbits()
- fix doc indentation
About the indentation: i'm using >>svndiff='svn diff --diff-cmd="/usr/bin/diff" -x "-ub"'<< instead of svn diff because my editor removes trailing spaces.
I prefer to keep the "fast path" (use _PyLong_NumBits) as you proposed in another message: Message75767.
Author: Mark Dickinson (mark.dickinson) *
Date: 2008-12-12 16:50
Victor,
This looks good. If it's okay with you, I'll work on the documentation a little (we need an entry in whatsnew, and a description of the semantics of numbits for 0 and negative integers.) Then I think this will be ready.
I prefer to keep the "fast path" (use _PyLong_NumBits) as you proposed in another message: Message75767.
Sorry---I wasn't clear. I wasn't suggesting getting rid of the fast path---I was suggesting inlining the _PyLong_NumBits code in long_numbits (leaving _PyLong_NumBits itself intact, of course). It would eliminate all the messing around with exceptions. But I think the current code is fine.
Author: STINNER Victor (vstinner) *
Date: 2008-12-12 16:56
I'll work on the documentation
Ok, cool.
a description of the semantics of numbits for 0 and negative integers
About the negative integer: the documentation should be "number of bits of the absolute value of x" and "by convention, 0.numbits() is 0". But you're right, the documentation is not clear.
Author: Mark Dickinson (mark.dickinson) *
Date: 2008-12-12 17:26
Here's Victor's patch, but with extra documentation. No code has been changed.
Two more questions about the code, Victor; specifically about long_numbits:
why do you use PyLong_FromSize_t rather than PyInt_FromSize_t?
isn't the if (ndigits == 0) check redundant?
Author: Mark Dickinson (mark.dickinson) *
Date: 2008-12-12 17:26
Oops. Here's the patch.
Author: STINNER Victor (vstinner) *
Date: 2008-12-12 18:03
- why do you use PyLong_FromSize_t rather than PyInt_FromSize_t?
I choosed to use consistent result type. But now I would prefer int :-) I see that PyInt_FromSize_t() returns a long if the value doesn't fit in an int. So it's correct.
- isn't the if (ndigits == 0) check redundant?
I'm paranoid: if _PyLong_NumBits() fails but the number is zero, the function may crash. But this case is impossible: _PyLong_NumBits() can only fails with an OverflowError.
Can you fix these two issues (use PyInt_FromSize_t and remove the if)?
Author: Mark Dickinson (mark.dickinson) *
Date: 2008-12-12 18:10
I choosed to use consistent result type. But now I would prefer int :-) I see that PyInt_FromSize_t() returns a long if the value doesn't fit in an int. So it's correct.
Cool. I think int is better, simply because the result is almost always going to fit into an int in practice, and because calculations with ints are faster.
I'm paranoid
Yeah, me too. I removed the check, but also changed the assert to check that Py_SIZE is nonzero.
Can you fix these two issues (use PyInt_FromSize_t and remove the if)?
Done.
Author: Mark Dickinson (mark.dickinson) *
Date: 2008-12-12 19:50
As far as I'm concerned, this patch is ready to be applied.
Raymond, care to give a second opinion?
Author: Raymond Hettinger (rhettinger) *
Date: 2008-12-12 22:47
The code looks fine.
For speed, you could insert the following just before the while-loop:
while (n >= 256) {
n >>= 8;
result += 8;
}Am not sure the "numbits" is the right-name. Is there precedent in other languages or texts on bit-twiddling techniques? For numbits(0b0100), I could forsee people predicting a result of 4, 3, or 1 depending on their world-view of what numbits might mean. Personally, I tend to think in term of high-bit location or somesuch.
Author: Raymond Hettinger (rhettinger) *
Date: 2008-12-13 04:23
FWIW, here is a link to faster algorithms:
http://www-graphics.stanford.edu/~seander/bithacks.html#IntegerLog
Author: Mark Dickinson (mark.dickinson) *
Date: 2008-12-13 09:26
About the name:
Java's Bignum has a 'significantBits' method, which apparently returns the position of the MSB (i.e., numbits - 1).
GMP has mpz_sizeinbase; mpz_sizeinbase(n, 2) is almost the same as numbits, except that mpz_sizeinbase(0, 2) is 1, not 0.
Mathematica has BitLength. I quite like this.
Googling for 'ilog2' returns a good few relevant hits; again, this is really numbits - 1, not numbits.
How about n.bitlength?
Author: Raymond Hettinger (rhettinger) *
Date: 2008-12-13 09:57
Perhaps n.bit_length() with an underscore. The name sounds good to my ear and sensibilities.
Author: Mark Dickinson (mark.dickinson) *
Date: 2008-12-13 10:35
Perhaps n.bit_length() with an underscore.
I thought I recalled Guido having a preference for float.fromhex over float.from_hex when that got implemented. But either spelling seems good to me.
Author: Mark Dickinson (mark.dickinson) *
Date: 2008-12-13 10:43
I thought I recalled Guido having a preference for float.fromhex over
Can't find anything to back this up. It looks like I'm just making this up.
Author: Raymond Hettinger (rhettinger) *
Date: 2008-12-13 10:53
It was probably true. The issue is how well the words self-separate visually and whether an underscore would create a detrimental mental pause. So fromkeys() and fromhex() read fine and would feel awkward with an underscore. But bitlength causes me a mental double-take when my mind searches for the word separation. It that case, PEP 8 suggests that an underscore be added for clarity. This is probably why Mathematica chose BitLength in titlecase instead of Bitlength. Logic aside, bit_length() just looks and feels better to me.
Did you look at the O(lg n) algorithm yet? Any thoughts?
Author: Mark Dickinson (mark.dickinson) *
Date: 2008-12-13 11:19
Did you look at the O(lg n) algorithm yet? Any thoughts?
So there are two places this could be applied: the int_numbits method and _PyLong_NumBits. I'd be quite surprised if this offered any noticeable speedup in either case. Might be worth a try, though---I'll see if I can get some timings.
For int_numbits, it would have to be implemented in a way that works for 32-bit and 64-bit C longs. And if it also just happens to work for other sizes, that would be good, too. I'd probably try something like:
while (v > 0xFFFFFFFF) { bitcount += 32; v >>= 32; }
before doing a 32-bit bitcount on what's left. On a 32-bit machine, the whole while loop should just be optimized away to nothing.
Similarly, in _PyLong_NumBits it would be good if it could be implemented in a way that doesn't have to change if/when PyLong_SHIFT is changed.
I prefer the second variant given (the non-branching version).
Author: Mark Dickinson (mark.dickinson) *
Date: 2008-12-13 13:10
Three files: (1) bit_length7.patch just does the numbits->bit_length renaming; otherwise it's the same as numbits-6b.patch. (2) bit_length7_opt.patch uses the fast bitcount method that Raymond pointed out. (3) bit_length_pybench.patch adds a test for bit_length to pybench.
On my system (OS X 10.5.5/Core 2 Duo), on a 32-bit non-debug build of the trunk, pybench is showing me a 4-5% speedup for the optimized version.
(I also tried a version that uses gcc's __builtin_clzl function; this was around 10% faster than the basic version, but of course it's not portable.)
Author: Antoine Pitrou (pitrou) *
Date: 2008-12-13 13:57
Well, I don't think bit_length() warrants a specific test which will slow down the whole pybench suite. pybench tracks interpreter speed for common and generally useful operations, while I think bit_length() is only a niche method.
Author: Mark Dickinson (mark.dickinson) *
Date: 2008-12-13 14:54
Well, I don't think bit_length() warrants a specific test which will slow down the whole pybench suite.
Me neither. It's a temporary patch to pybench, to establish whether it's worth applying optimizations to bit_length. I didn't intend that it should be applied to the trunk.
Author: Mark Dickinson (mark.dickinson) *
Date: 2008-12-13 15:02
Slightly improved optimized version.
Author: Mark Dickinson (mark.dickinson) *
Date: 2008-12-13 15:06
Added another test to pybench; it makes sense to consider cases where the input n is uniformly distributed in [0, 2**31) as well as cases where the output n.bit_length() is uniformly distributed in [0, 32).
Author: Mark Dickinson (mark.dickinson) *
Date: 2008-12-13 15:34
I think I've found the happy medium:
bit_length7_opt2.patch achieves nearly the same performance gains as bit_length7_opt.patch (around 7% for uniformly-distributed inputs, 3.5% for uniformly-distributed outputs), but is much more straightforward and readable. It's pretty much what Raymond suggested in the first place, plus a table lookup.
Author: Raymond Hettinger (rhettinger) *
Date: 2008-12-13 16:04
If there's not a hurry, would like to review this a bit more when I get back early next week.
Author: Fredrik Johansson (fredrikj)
Date: 2008-12-13 16:43
FYI, Brent & Zimmermann call this function nbits(n) in "Modern Computer Arithmetic": http://www.loria.fr/~zimmerma/mca/pub226.html
I don't really care much about the name though.
In the documentation, should "same value as x.bit_length" not be
"same value as x.bit_length()"?
Author: Mark Dickinson (mark.dickinson) *
Date: 2008-12-13 17:50
In the documentation, should "same value as
x.bit_length" not be "same value asx.bit_length()"?
Yes, of course. Thanks!
Author: Mark Dickinson (mark.dickinson) *
Date: 2008-12-14 10:24
Update both patches:
(1) change PyLong_FromLong(ndigits-1) to PyLong_FromSsize_t(ndigits-1) in both patches (it's possible to have a 32-bit long and 64-bit Py_ssize_t), and
(2) in the optimized patch, add the table lookup optimization for long_bit_length.
Author: Raymond Hettinger (rhettinger) *
Date: 2008-12-16 12:41
In the whatsnew docs, add two lines showing the binary representation of 37. That will provide clarification as to why the answer is six:
>>> n = 37>>> bin(37)'0b100101' >>> n.bit_length() 6
Also, the main entry in the docs should be built-out just a bit. I like that the current entry provides a precise mathematical spec that is easily validated, but it should also mention the more intuitive notion of the "number of bits in a number's binary representation excluding leading zeros (for example the decimal number 37, which is 0b100101 in binary, has six binary digits)."
Possibly, the BitLengthTables can be dropped in favor of the old shift-one, add-one code (to be inserted right after the shift-eight, add-eight code). The two tables consume a lot of memory and the relevant entry is not likely to be in cache when needed (cache misses are very slow). It is simpler and possibly faster to skip the table lookup unless the table is made much smaller. Plus, it makes the code a bit more maintainable and self-evidently correct (not requiring as extensive test coverage to validate the whole table).
My own crack at optimization looks like this:
static const char BitLengthTable[32] =
{0, 1, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4,
5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5};
while (n >= 32) { r += 5; n >>= 5; }
r += (long)(BitLengthTable[n]);If you don't adopt mine, at least consider making a table of chars instead of longs (this will give a 4:1 or 8:1 table compression, reducing the memory footprint and increasing the likelihood of a cache hit).
I would feel better about the test code if it also directly validated the definitions in docs and thoroughly exercised the tables:
for x in range(-65000, 65000): k = x.numbits if x > 0: assert 2 ** (k-1) <= x < 2**k assert k == 1 + math.floor(math.log(x) / math.log(2)) elif x == 0: assert k == 0 else: assert k == (-x).numbits()
Other than that, I'm basically happy with the patch.
Author: Raymond Hettinger (rhettinger) *
Date: 2008-12-16 12:47
Oops, forgot one part of the doc definition in the proposed additional tests:
for x in range(-65000, 65000): k = x.numbits() assert k == len(bin(x).lstrip('-0b')) if x > 0: assert 2 ** (k-1) <= x < 2**k assert k == 1 + math.floor(math.log(x) / math.log(2)) elif x == 0: assert k == 0 else: assert k == (-x).numbits()
Author: STINNER Victor (vstinner) *
Date: 2008-12-16 13:56
x.numbits() is: math.ceil(math.log(abs(x)) / math.log(2)) if x != 0 0 otherwise
and not 1 + math.floor(math.log(x) / math.log(2))
(16).numbits() is 4, not 5.
Author: Antoine Pitrou (pitrou) *
Date: 2008-12-16 13:57
Le mardi 16 décembre 2008 à 13:56 +0000, STINNER Victor a écrit :
(16).numbits() is 4, not 5.
Well, I do hope (16).numbits() returns 5...
Author: STINNER Victor (vstinner) *
Date: 2008-12-16 14:00
Ooops, you're right! (15).numbits() -> 4 and (16).numbits() -> 5. I'm tired.
Author: Raymond Hettinger (rhettinger) *
Date: 2008-12-16 14:04
One other thought on the docs. I would like to continue the style of supplying pure python equivalents to help precisely explain what a function does (see the itertools docs for an example, also a few builtins are explained that way and namedtuples too):
- Equivalent to::
def numbits(x):'Number of binary bits necessary to represent the integer n'return len(bin(x).lstrip('-0b'))
Author: Fredrik Johansson (fredrikj)
Date: 2008-12-16 14:11
When did the name change back to numbits? Anyway, I think this reference implementation is better:
def numbits(x): x = abs(x) n = 0 while x: n += 1 x //= 2 return n
(//= 2 could be changed to >>= 1)
Author: STINNER Victor (vstinner) *
Date: 2008-12-16 14:53
Please don't promote this ugly code "len(bin(x).lstrip('-0b'))". It's not the best way to compute the number of bits... I prefer fredrikj's proposition (with // 2, few people understand >> 1).
Author: Skip Montanaro (skip.montanaro) * 
Date: 2008-12-16 18:33
Regarding the last few posts:
Raymond's implementation, while ugly, provides a completely orthogonal way to test compute numbits, useful in unit tests if nothing else.
Using x >> 1 in a reference implementation is perfectly reasonable. If the person using the reference implementation to produce a real C-based implementation doesn't understand the equivalence of x // 2 and x >> 1, heaven help us.
Skip
Author: Mark Dickinson (mark.dickinson) *
Date: 2008-12-16 19:41
Thanks for all the comments. Here's an updated patch, with quite a few changes:
code:
- drop lookup tables in favour of "while(x) {x >>= 1; count += 1;}"
- add example to docstring, and use PyDoc_STRVAR macro for docstrings
docs:
- add "bin(37)" to whatsnew example
- move main documentation out of the bit operations table into its own subsection, so that it can be indexed properly.
- expand main documentation with examples, informal definition, equivalent Python code
- I dropped the 1 + math.floor(...) definition from the docs, judging that 1 informal, 1 formal and 1 Python definition should be enough.
tests:
- as proposed by Raymond, directly check equivalence with formal definition, equivalent Python code, and two other possible definitions.
(FWIW I prefer 'x>>1' over 'x//2' in the Python code, because it's more immediately related to the intended sense: the operation should be thought of as a bit shift rather than a division.)
Author: Raymond Hettinger (rhettinger) *
Date: 2008-12-16 19:44
Of course, the name should have been bit_length() instead of numbits().
For the code equivalent, I'm aiming for something less process oriented and more focused on what it does. bit_length IS the number of bits in a binary representation without the sign or leading zeroes -- that definition IS a correct mental picture and does not require special cases for zero or for negatives.
The purpose of the code equivalent is not efficiency or beauty; it to help communicate was a function does. If you want it to be more beautiful, it can be broken into multiple lines.
I don't think you gain ANY explanatory power with code that says: bit_length is the number of right-shifts (or floor divisions by two) of the absolute value of a number until that number becomes zero. Tell that description to a high school student and prepare for a blank stare.
FWIW, I think having a mental picture that was too process oriented was behind last night's mistake of thinking the (16).bit_length() was 5 instead of 4. I theorize that error wouldn't have occurred if the mental picture was of len('10000') instead of power-of-two mental model where people (including some of the commenter here) tend to get the edge cases wrong.
It would be better to have no code equivalent at all than to present the repeated //2 method as a definition. That is a process, but not a useful mental picture to help explain what bit_length is all about. Think about it, bit_length() is about the length in bits of a binary representation -- any code provided needs to translate directly to that definition.
def bit_length(x): '''Length of a binary representation without the sign or leading zeroes:
>>> (-37).numbits()
6
'''
s = bin(x) # binary representation: bin(-37) --> '-0b100101'
s = s.lstrip('-0b') # remove leading zeros and sign
return len(s)Author: Mark Dickinson (mark.dickinson) *
Date: 2008-12-16 19:51
Okay; I don't have strong feelings about the form the Python code takes;
I'll let you guys argue it out and then fix things accordingly
Author: Raymond Hettinger (rhettinger) *
Date: 2008-12-16 20:16
IMO, the choices are something like my version or none at all. The repeated floor division by two of abs(x) has ZERO explanatory power and may even detract from a beginner's ability to understand what the method does. Show that code to most finance people and they will avoid the method entirely.
Anyone who disagrees needs to show both code fragments to some junior programmers and see which best leads to understanding the method and being able to correctly predict the edge cases bordering powers of two, the zero case, and how negatives are handled.
No fair trying this experiment on assembly language programmers ;-)
Author: Antoine Pitrou (pitrou) *
Date: 2008-12-16 20:20
Show that code to most finance people and they will avoid the method entirely.
Why would finance people be interested in bit_length()?
I think this discussion begins to feel like bikeshedding. Documentation can always be improved afterwards.
Author: Raymond Hettinger (rhettinger) *
Date: 2008-12-16 20:30
Antoine, it's not bike-shedding at all. Communicative docs are important to users other than assembly language programmers. BTW, I am a finance person (a CPA).
Yes, you're correct, I can fix-up the docs after the patch is posted.
Author: Mark Dickinson (mark.dickinson) *
Date: 2008-12-16 20:34
Updated patch.
Author: Mark Dickinson (mark.dickinson) *
Date: 2008-12-16 20:39
Bah. Fix test_int so that it actually works.
Author: Mark Dickinson (mark.dickinson) *
Date: 2008-12-16 20:45
...and use proper unittest methods instead of asserts...
Author: Raymond Hettinger (rhettinger) *
Date: 2008-12-16 21:00
Looks good. Marking as accepted.
Before applying, consider adding back the part of the docs with the '1 + floor(...' definition. I think it provides a useful alternative way to look at what the method does. Also, it gives a useful mathematical expression that can be used in reasoning about invariants. More importantly, we should provide it because it is so easy to make a mistake when rolling your own version of the formula (i.e. using ceil instead of floor or having an off by one error).
Author: Mark Dickinson (mark.dickinson) *
Date: 2008-12-16 21:38
Before applying, consider adding back the part of the docs with the '1 + floor(...' definition.
My only (minor) objection to this definition is that a straight Python translation of it doesn't work, thanks to roundoff error and the limited precision of floating-point:
from math import floor, log n = 2101 n.bit_length() 102 1 + floor(log(n)/log(2)) 101.0 n = 280-1 n.bit_length() 80 1 + floor(log(n)/log(2)) 81.0
But as you say, it provides another perspective; I'm fine with putting it back in.
Author: Raymond Hettinger (rhettinger) *
Date: 2008-12-16 21:45
Also, consider writing it in the two argument form:
1 + floor(log(n, 2))
and using the word "approximately" or somesuch.
Author: Mark Dickinson (mark.dickinson) *
Date: 2008-12-17 16:20
Committed to trunk in r67822, py3k in r67824.
Author: STINNER Victor (vstinner) *
Date: 2008-12-17 18:44
Committed
Really? YEAH! Great job everybody ;-) I read the code and it looks valid. Micro optimisation (for smaller memory footprint): would it be possible to share the 32 int table (BitLengthTable) between int and long (in Python 2.x)?
Author: Raymond Hettinger (rhettinger) *
Date: 2008-12-17 21:23
32 bytes isn't worth sharing between modules.
Author: Raymond Hettinger (rhettinger) *
Date: 2008-12-19 09:13
Posted some doc cleanups in r67850 and r67851.
Author: Mark Dickinson (mark.dickinson) *
Date: 2008-12-19 16:57
About the footnote:
floor(log(n, 2)) is poor code. This is not supposed to be a dramatic statement, just a statement of fact. Its correctness is dependent on minute details of floating point. It is poor code in exactly the same way that "while x < 1.0: x += 0.1" is poor code---behaviour in boundary cases is almost entirely unpredictable.
If 1 + floor(log(n, 2)) happens to give the correct result in the common corner case where x is a power of 2, then that's due to little more than sheer luck. Correct rounding by itself is nowhere near enough to guarantee correct results.
In the case of IEEE 754 doubles, a large part of the luck is that the closest double to log(2) just happens to be smaller than log(2) itself, so that the implicit division by log(2) in log(x, 2) tends to give a larger result than the true one; if things were the other way around, the formula above would likely fail for many (even small) n.
So I don't like seeing this poor code in the Python reference manual, for two reasons: (1) it might get propagated to real world code, and (2) its presence in the docs reflects poorly on the numerical competence of the Python developers.
IMO, either: (1) the warning needs to be stronger, or (2) the formulation should be given purely mathematically, without any explicit code, or (3) the formula should be left out of the docs altogether.
Mark
Author: Raymond Hettinger (rhettinger) *
Date: 2008-12-19 18:16
Other possible wording:
... so that k is approximately 1 + int(log(abs(x), 2)).
Author: Mark Dickinson (mark.dickinson) *
Date: 2008-12-19 22:19
... so that
kis approximately1 + int(log(abs(x), 2)).
I guess that could work.
One other thing: the docs for the trunk seem to suggest that we should be using trunc here, rather than int. I'm looking at:
http://docs.python.org/dev/library/stdtypes.html#numeric-types-int- float-long-complex
and particularly the note (2), that says of int(x) and long(x):
"Deprecated since version 2.6: Instead, convert floats to long explicitly with trunc()."
Author: Zooko O'Whielacronx (zooko)
Date: 2010-06-21 22:04
There is a small mistake in the docs:
def bit_length(x): 'Number of bits necessary to represent self in binary.' s = bin(x) # binary representation: bin(-37) --> '-0b100101' s = s.lstrip('-0b') # remove leading zeros and minus sign return len(s) # len('100101') --> 6
is probably supposed to be:
def bit_length(x): 'Number of bits necessary to represent x in binary.' s = bin(x) # binary representation: bin(-37) --> '-0b100101' s = s.lstrip('-0b') # remove leading zeros and minus sign return len(s) # len('100101') --> 6
Author: Senthil Kumaran (orsenthil) *
Date: 2010-06-22 02:45
There is a small mistake in the docs:
Yes there was. Fixed in 82146.
Author: Terry J. Reedy (terry.reedy) *
Date: 2010-06-22 17:13
Minor addendum to Mark's last message: in the near release version of 2.7 (rc2), note 2 in 5.4. "Numeric Types — int, float, long, complex" now starts "Conversion from floats using int() or long() truncates toward zero like the related function, math.trunc()" and no longer marks the usage of long as deprecated.
Author: Mark Dickinson (mark.dickinson) *
Date: 2010-06-22 18:00
So there are two issues here:
- deprecation of int(my_float) and long(my_float)
- removal of long in 3.x
I'm not sure which Terry is referring to here.
On the first, I don't think use of int() with float arguments actually is deprecated in any meaningful way. At one point there was a push (related to PEP 3141) to deprecate truncating uses of int and introduce a new builtin trunk, but it never really took hold (and trunc ended up being relegated to the math module0; I certainly don't expect to see such deprecation happen within the lifetime of Python 3.x, so I don't think it would be appropriate to mention it in the 2.x docs.
On the second, it's possible that there should be a mention somewhere in the 2.x docs that long() no longer exists in 3.x, and that for almost all uses int() works just as well. A separate issue should probably be opened for this.
Author: Mark Dickinson (mark.dickinson) *
Date: 2010-06-22 18:00
Aargh! 'a new builtin trunk' -> 'a new builtin trunc'
Author: STINNER Victor (vstinner) *
Date: 2010-06-22 21:09
Please open a new issue for the documentation problems, it's no more related to "numbits()" method and this issue is closed.
Author: Terry J. Reedy (terry.reedy) *
Date: 2010-06-22 21:34
Whoops, sorry to create confusion when I was trying to put this issue to rest completely. Let me try better.
In his last message, Raymond said "Other possible wording:
... so that k is approximately 1 + int(log(abs(x), 2))."
That is what the current 2.7rc2 doc says.
In response, Mark said "I guess that could work." but quoted footnote 2, which implied that 'int' should be changed to 'trunc' in the example above.
This implied to me that there was a lingering .numbits doc issue. But footnote 2 is now changed, so I not longer think there is a doc issue, so I reported that, so no one else would think so.
I hope this is the end of this.
Author: Mark Dickinson (mark.dickinson) *
Date: 2010-06-23 07:53
Ah; sorry for misunderstanding. Thanks for the explanation, Terry!