ah*bh and al*bl. This is much easier than explaining why that's true
for (ah+al)*(bh+bl), and follows directly from the simple part of the
(ah+al)*(bh+bl) explanation.
space is no longer needed, so removed the code. It was only possible when
a degenerate (ah->ob_size == 0) split happened, but after that fix went
in I added k_lopsided_mul(), which saves the body of k_mul() from seeing
a degenerate split. So this removes code, and adds a honking long comment
block explaining why spilling out of bounds isn't possible anymore. Note:
ff we end up spilling out of bounds anyway <wink>, an assert in v_iadd()
is certain to trigger.
k_mul() when inputs have vastly different sizes, and a little more
efficient when they're close to a factor of 2 out of whack.
I consider this done now, although I'll set up some more correctness
tests to run overnight.
cases, overflow the allocated result object by 1 bit. In such cases,
it would have been brought back into range if we subtracted al*bl and
ah*bh from it first, but I don't want to do that because it hurts cache
behavior. Instead we just ignore the excess bit when it appears -- in
effect, this is forcing unsigned mod BASE**(asize + bsize) arithmetic
in a case where that doesn't happen all by itself.
algorithm. MSVC 6 wasn't impressed <wink>.
Something odd: the x_mul algorithm appears to get substantially worse
than quadratic time as the inputs grow larger:
bits in each input x_mul time k_mul time
------------------ ---------- ----------
15360 0.01 0.00
30720 0.04 0.01
61440 0.16 0.04
122880 0.64 0.14
245760 2.56 0.40
491520 10.76 1.23
983040 71.28 3.69
1966080 459.31 11.07
That is, x_mul is perfectly quadratic-time until a little burp at
2.56->10.76, and after that goes to hell in a hurry. Under Karatsuba,
doubling the input size "should take" 3 times longer instead of 4, and
that remains the case throughout this range. I conclude that my "be nice
to the cache" reworkings of k_mul() are paying.
correct now, so added some final comments, did some cleanup, and enabled
it for all long-int multiplies. The KARAT envar no longer matters,
although I left some #if 0'ed code in there for my own use (temporary).
k_mul() is still much slower than x_mul() if the inputs have very
differenent sizes, and that still needs to be addressed.
(it's possible, but should be harmless -- this requires more thought,
and allocating enough space in advance to prevent it requires exactly
as much thought, to know exactly how much that is -- the end result
certainly fits in the allocated space -- hmm, but that's really all
the thought it needs! borrows/carries out of the high digits really
are harmless).
k_mul(): This didn't allocate enough result space when one input had
more than twice as many bits as the other. This was partly hidden by
that x_mul() didn't normalize its result.
The Karatsuba recurrence is pretty much hosed if the inputs aren't
roughly the same size. If one has at least twice as many bits as the
other, we get a degenerate case where the "high half" of the smaller
input is 0. Added a special case for that, for speed, but despite that
it helped, this can still be much slower than the "grade school" method.
It seems to take a really wild imbalance to trigger that; e.g., a
2**22-bit input times a 1000-bit input on my box runs about twice as slow
under k_mul than under x_mul. This still needs to be addressed.
I'm also not sure that allocating a->ob_size + b->ob_size digits is
enough, given that this is computing k = (ah+al)*(bh+bl) instead of
k = (ah-al)*(bl-bh); i.e., it's certainly enough for the final result,
but it's vaguely possible that adding in the "artificially" large k may
overflow that temporarily. If so, an assert will trigger in the debug
build, but we'll probably compute the right result anyway(!).
addition and subtraction. Reworked the tail end of k_mul() to use them.
This saves oodles of one-shot longobject allocations (this is a triply-
recursive routine, so saving one allocation in the body saves 3**n
allocations at depth n; we actually save 2 allocations in the body).
SF 560379: Karatsuba multiplication.
Lots of things were changed from that. This needs a lot more testing,
for correctness and speed, the latter especially when bit lengths are
unbalanced. For now, the Karatsuba code gets invoked if and only if
envar KARAT exists.
The staticforward define was needed to support certain broken C
compilers (notably SCO ODT 3.0, perhaps early AIX as well) botched the
static keyword when it was used with a forward declaration of a static
initialized structure. Standard C allows the forward declaration with
static, and we've decided to stop catering to broken C compilers. (In
fact, we expect that the compilers are all fixed eight years later.)
I'm leaving staticforward and statichere defined in object.h as
static. This is only for backwards compatibility with C extensions
that might still use it.
XXX I haven't updated the documentation.
many types were subclassable but had a xxx_dealloc function that
called PyObject_DEL(self) directly instead of deferring to
self->ob_type->tp_free(self). It is permissible to set tp_free in the
type object directly to _PyObject_Del, for non-GC types, or to
_PyObject_GC_Del, for GC types. Still, PyObject_DEL was a tad faster,
so I'm fearing that our pystone rating is going down again. I'm not
sure if doing something like
void xxx_dealloc(PyObject *self)
{
if (PyXxxCheckExact(self))
PyObject_DEL(self);
else
self->ob_type->tp_free(self);
}
is any faster than always calling the else branch, so I haven't
attempted that -- however those types whose own dealloc is fancier
(int, float, unicode) do use this pattern.
Generalize PyLong_AsLongLong to accept int arguments too. The real point
is so that PyArg_ParseTuple's 'L' code does too. That code was
undocumented (AFAICT), so documented it.
Both int and long multiplication are changed to be more careful in
their assumptions about when one of the arguments is a sequence: the
assumption that at least one of the arguments must be an int (or long,
respectively) is still held, but the assumption that these don't smell
like sequences is no longer true: a subtype of int or long may well
have a sequence-repeat thingie!
Given an immutable type M, and an instance I of a subclass of M, the
constructor call M(I) was just returning I as-is; but it should return a
new instance of M. This fixes it for M in {int, long}. Strings, floats
and tuples remain to be done.
Added new macros PyInt_CheckExact and PyLong_CheckExact, to more easily
distinguish between "is" and "is a" (i.e., only an int passes
PyInt_CheckExact, while any sublass of int passes PyInt_Check).
Added private API function _PyLong_Copy.
the fiddling is simply due to that no caller of PyLong_AsDouble ever
checked for failure (so that's fixing old bugs). PyLong_AsDouble is much
faster for big inputs now too, but that's more of a happy consequence
than a design goal.
but will be the foundation for Good Things:
+ Speed PyLong_AsDouble.
+ Give PyLong_AsDouble the ability to detect overflow.
+ Make true division of long/long nearly as accurate as possible (no
spurious infinities or NaNs).
+ Return non-insane results from math.log and math.log10 when passing a
long that can't be approximated by a double better than HUGE_VAL.
PEP 238. Changes:
- add a new flag variable Py_DivisionWarningFlag, declared in
pydebug.h, defined in object.c, set in main.c, and used in
{int,long,float,complex}object.c. When this flag is set, the
classic division operator issues a DeprecationWarning message.
- add a new API PyRun_SimpleStringFlags() to match
PyRun_SimpleString(). The main() function calls this so that
commands run with -c can also benefit from -Dnew.
- While I was at it, I changed the usage message in main() somewhat:
alphabetized the options, split it in *four* parts to fit in under
512 bytes (not that I still believe this is necessary -- doc strings
elsewhere are much longer), and perhaps most visibly, don't display
the full list of options on each command line error. Instead, the
full list is only displayed when -h is used, and otherwise a brief
reminder of -h is displayed. When -h is used, write to stdout so
that you can do `python -h | more'.
Notes:
- I don't want to use the -W option to control whether the classic
division warning is issued or not, because the machinery to decide
whether to display the warning or not is very expensive (it involves
calling into the warnings.py module). You can use -Werror to turn
the warnings into exceptions though.
- The -Dnew option doesn't select future division for all of the
program -- only for the __main__ module. I don't know if I'll ever
change this -- it would require changes to the .pyc file magic
number to do it right, and a more global notion of compiler flags.
- You can usefully combine -Dwarn and -Dnew: this gives the __main__
module new division, and warns about classic division everywhere
else.
- Do not compile unicodeobject, unicodectype, and unicodedata if Unicode is disabled
- check for Py_USING_UNICODE in all places that use Unicode functions
- disables unicode literals, and the builtin functions
- add the types.StringTypes list
- remove Unicode literals from most tests.
This introduces:
- A new operator // that means floor division (the kind of division
where 1/2 is 0).
- The "future division" statement ("from __future__ import division)
which changes the meaning of the / operator to implement "true
division" (where 1/2 is 0.5).
- New overloadable operators __truediv__ and __floordiv__.
- New slots in the PyNumberMethods struct for true and floor division,
new abstract APIs for them, new opcodes, and so on.
I emphasize that without the future division statement, the semantics
of / will remain unchanged until Python 3.0.
Not yet implemented are warnings (default off) when / is used with int
or long arguments.
This has been on display since 7/31 as SF patch #443474.
Flames to /dev/null.
