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Issue #3166: Make long -> float (and int -> float) conversions 

correctly rounded, using round-half-to-even.  This ensures that the
value of float(n) doesn't depend on whether we're using 15-bit digits
or 30-bit digits for Python longs.
This commit is contained in:
Mark Dickinson 2009-04-20 21:13:33 +00:00
parent cccfc825e4
commit 6736cf8d20
5 changed files with 318 additions and 29 deletions
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34
Lib/test/test_int.py
View File

@ -275,6 +275,40 @@ class IntTestCases(unittest.TestCase):
self.assertEqual((a+1).bit_length(), i+1)
self.assertEqual((-a-1).bit_length(), i+1)
@unittest.skipUnless(float.__getformat__("double").startswith("IEEE"),
"test requires IEEE 754 doubles")
def test_float_conversion(self):
# values exactly representable as floats
exact_values = [-2, -1, 0, 1, 2, 2**52, 2**53-1, 2**53, 2**53+2,
2**53+4, 2**54-4, 2**54-2, 2**63, -2**63, 2**64,
-2**64, 10**20, 10**21, 10**22]
for value in exact_values:
self.assertEqual(int(float(int(value))), value)
# test round-half-to-even
self.assertEqual(int(float(2**53+1)), 2**53)
self.assertEqual(int(float(2**53+2)), 2**53+2)
self.assertEqual(int(float(2**53+3)), 2**53+4)
self.assertEqual(int(float(2**53+5)), 2**53+4)
self.assertEqual(int(float(2**53+6)), 2**53+6)
self.assertEqual(int(float(2**53+7)), 2**53+8)
self.assertEqual(int(float(-2**53-1)), -2**53)
self.assertEqual(int(float(-2**53-2)), -2**53-2)
self.assertEqual(int(float(-2**53-3)), -2**53-4)
self.assertEqual(int(float(-2**53-5)), -2**53-4)
self.assertEqual(int(float(-2**53-6)), -2**53-6)
self.assertEqual(int(float(-2**53-7)), -2**53-8)
self.assertEqual(int(float(2**54-2)), 2**54-2)
self.assertEqual(int(float(2**54-1)), 2**54)
self.assertEqual(int(float(2**54+2)), 2**54)
self.assertEqual(int(float(2**54+3)), 2**54+4)
self.assertEqual(int(float(2**54+5)), 2**54+4)
self.assertEqual(int(float(2**54+6)), 2**54+8)
self.assertEqual(int(float(2**54+10)), 2**54+8)
self.assertEqual(int(float(2**54+11)), 2**54+12)
def test_intconversion(self):
# Test __int__()
class ClassicMissingMethods:

59
Lib/test/test_long.py
View File

@ -645,6 +645,65 @@ class LongTest(unittest.TestCase):
else:
self.assertRaises(TypeError, pow,longx, longy, long(z))
@unittest.skipUnless(float.__getformat__("double").startswith("IEEE"),
"test requires IEEE 754 doubles")
def test_float_conversion(self):
import sys
DBL_MAX = sys.float_info.max
DBL_MAX_EXP = sys.float_info.max_exp
DBL_MANT_DIG = sys.float_info.mant_dig
exact_values = [0L, 1L, 2L,
long(2**53-3),
long(2**53-2),
long(2**53-1),
long(2**53),
long(2**53+2),
long(2**54-4),
long(2**54-2),
long(2**54),
long(2**54+4)]
for x in exact_values:
self.assertEqual(long(float(x)), x)
self.assertEqual(long(float(-x)), -x)
# test round-half-even
for x, y in [(1, 0), (2, 2), (3, 4), (4, 4), (5, 4), (6, 6), (7, 8)]:
for p in xrange(15):
self.assertEqual(long(float(2L**p*(2**53+x))), 2L**p*(2**53+y))
for x, y in [(0, 0), (1, 0), (2, 0), (3, 4), (4, 4), (5, 4), (6, 8),
(7, 8), (8, 8), (9, 8), (10, 8), (11, 12), (12, 12),
(13, 12), (14, 16), (15, 16)]:
for p in xrange(15):
self.assertEqual(long(float(2L**p*(2**54+x))), 2L**p*(2**54+y))
# behaviour near extremes of floating-point range
long_dbl_max = long(DBL_MAX)
top_power = 2**DBL_MAX_EXP
halfway = (long_dbl_max + top_power)/2
self.assertEqual(float(long_dbl_max), DBL_MAX)
self.assertEqual(float(long_dbl_max+1), DBL_MAX)
self.assertEqual(float(halfway-1), DBL_MAX)
self.assertRaises(OverflowError, float, halfway)
self.assertEqual(float(1-halfway), -DBL_MAX)
self.assertRaises(OverflowError, float, -halfway)
self.assertRaises(OverflowError, float, top_power-1)
self.assertRaises(OverflowError, float, top_power)
self.assertRaises(OverflowError, float, top_power+1)
self.assertRaises(OverflowError, float, 2*top_power-1)
self.assertRaises(OverflowError, float, 2*top_power)
self.assertRaises(OverflowError, float, top_power*top_power)
for p in xrange(100):
x = long(2**p * (2**53 + 1) + 1)
y = long(2**p * (2**53+ 2))
self.assertEqual(long(float(x)), y)
x = long(2**p * (2**53 + 1))
y = long(2**p * 2**53)
self.assertEqual(long(float(x)), y)
def test_float_overflow(self):
import math

3
Misc/NEWS
View File

@ -12,6 +12,9 @@ What's New in Python 2.7 alpha 1
Core and Builtins
-----------------
- Issue #3166: Make long -> float (and int -> float) conversions
correctly rounded.
- Issue #5787: object.__getattribute__(some_type, "__bases__") segfaulted on
some builtin types.

80
Objects/intobject.c
View File

@ -3,6 +3,7 @@
#include "Python.h"
#include <ctype.h>
#include <float.h>
static PyObject *int_int(PyIntObject *v);
@ -928,12 +929,78 @@ int_long(PyIntObject *v)
return PyLong_FromLong((v -> ob_ival));
}
static const unsigned 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
};
static int
bits_in_ulong(unsigned long d)
{
int d_bits = 0;
while (d >= 32) {
d_bits += 6;
d >>= 6;
}
d_bits += (int)BitLengthTable[d];
return d_bits;
}
#if 8*SIZEOF_LONG-1 <= DBL_MANT_DIG
/* Every Python int can be exactly represented as a float. */
static PyObject *
int_float(PyIntObject *v)
{
return PyFloat_FromDouble((double)(v -> ob_ival));
}
#else
/* Here not all Python ints are exactly representable as floats, so we may
have to round. We do this manually, since the C standards don't specify
whether converting an integer to a float rounds up or down */
static PyObject *
int_float(PyIntObject *v)
{
unsigned long abs_ival, lsb;
int round_up;
if (v->ob_ival < 0)
abs_ival = 0U-(unsigned long)v->ob_ival;
else
abs_ival = (unsigned long)v->ob_ival;
if (abs_ival < (1L << DBL_MANT_DIG))
/* small integer; no need to round */
return PyFloat_FromDouble((double)v->ob_ival);
/* Round abs_ival to MANT_DIG significant bits, using the
round-half-to-even rule. abs_ival & lsb picks out the 'rounding'
bit: the first bit after the most significant MANT_DIG bits of
abs_ival. We round up if this bit is set, provided that either:
(1) abs_ival isn't exactly halfway between two floats, in which
case at least one of the bits following the rounding bit must be
set; i.e., abs_ival & lsb-1 != 0, or:
(2) the resulting rounded value has least significant bit 0; or
in other words the bit above the rounding bit is set (this is the
'to-even' bit of round-half-to-even); i.e., abs_ival & 2*lsb != 0
The condition "(1) or (2)" equates to abs_ival & 3*lsb-1 != 0. */
lsb = 1L << (bits_in_ulong(abs_ival)-DBL_MANT_DIG-1);
round_up = (abs_ival & lsb) && (abs_ival & (3*lsb-1));
abs_ival &= -2*lsb;
if (round_up)
abs_ival += 2*lsb;
return PyFloat_FromDouble(v->ob_ival < 0 ?
-(double)abs_ival :
(double)abs_ival);
}
#endif
static PyObject *
int_oct(PyIntObject *v)
{
@ -1139,16 +1206,10 @@ int__format__(PyObject *self, PyObject *args)
return NULL;
}
static const unsigned 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
};
static PyObject *
int_bit_length(PyIntObject *v)
{
unsigned long n;
long r = 0;
if (v->ob_ival < 0)
/* avoid undefined behaviour when v->ob_ival == -LONG_MAX-1 */
@ -1156,12 +1217,7 @@ int_bit_length(PyIntObject *v)
else
n = (unsigned long)v->ob_ival;
while (n >= 32) {
r += 6;
n >>= 6;
}
r += (long)(BitLengthTable[n]);
return PyInt_FromLong(r);
return PyInt_FromLong(bits_in_ulong(n));
}
PyDoc_STRVAR(int_bit_length_doc,

171
Objects/longobject.c
View File

@ -8,6 +8,7 @@
#include "longintrepr.h"
#include "structseq.h"
#include <float.h>
#include <ctype.h>
#include <stddef.h>
@ -38,6 +39,9 @@
if (PyErr_CheckSignals()) PyTryBlock \
}
/* forward declaration */
static int bits_in_digit(digit d);
/* Normalize (remove leading zeros from) a long int object.
Doesn't attempt to free the storage--in most cases, due to the nature
of the algorithms used, this could save at most be one word anyway. */
@ -729,33 +733,166 @@ _PyLong_AsScaledDouble(PyObject *vv, int *exponent)
#undef NBITS_WANTED
}
/* Get a C double from a long int object. */
/* Get a C double from a long int object. Rounds to the nearest double,
using the round-half-to-even rule in the case of a tie. */
double
PyLong_AsDouble(PyObject *vv)
{
int e = -1;
PyLongObject *v = (PyLongObject *)vv;
Py_ssize_t rnd_digit, rnd_bit, m, n;
digit lsb, *d;
int round_up = 0;
double x;
if (vv == NULL || !PyLong_Check(vv)) {
PyErr_BadInternalCall();
return -1;
}
x = _PyLong_AsScaledDouble(vv, &e);
if (x == -1.0 && PyErr_Occurred())
return -1.0;
/* 'e' initialized to -1 to silence gcc-4.0.x, but it should be
set correctly after a successful _PyLong_AsScaledDouble() call */
assert(e >= 0);
if (e > INT_MAX / PyLong_SHIFT)
goto overflow;
errno = 0;
x = ldexp(x, e * PyLong_SHIFT);
if (Py_OVERFLOWED(x))
goto overflow;
return x;
}
overflow:
/* Notes on the method: for simplicity, assume v is positive and >=
2**DBL_MANT_DIG. (For negative v we just ignore the sign until the
end; for small v no rounding is necessary.) Write n for the number
of bits in v, so that 2**(n-1) <= v < 2**n, and n > DBL_MANT_DIG.
Some terminology: the *rounding bit* of v is the 1st bit of v that
will be rounded away (bit n - DBL_MANT_DIG - 1); the *parity bit*
is the bit immediately above. The round-half-to-even rule says
that we round up if the rounding bit is set, unless v is exactly
halfway between two floats and the parity bit is zero.
Write d[0] ... d[m] for the digits of v, least to most significant.
Let rnd_bit be the index of the rounding bit, and rnd_digit the
index of the PyLong digit containing the rounding bit. Then the
bits of the digit d[rnd_digit] look something like:
rounding bit
|
v
msb -> sssssrttttttttt <- lsb
^
|
parity bit
where 's' represents a 'significant bit' that will be included in
the mantissa of the result, 'r' is the rounding bit, and 't'
represents a 'trailing bit' following the rounding bit. Note that
if the rounding bit is at the top of d[rnd_digit] then the parity
bit will be the lsb of d[rnd_digit+1]. If we set
lsb = 1 << (rnd_bit % PyLong_SHIFT)
then d[rnd_digit] & (PyLong_BASE - 2*lsb) selects just the
significant bits of d[rnd_digit], d[rnd_digit] & (lsb-1) gets the
trailing bits, and d[rnd_digit] & lsb gives the rounding bit.
We initialize the double x to the integer given by digits
d[rnd_digit:m-1], but with the rounding bit and trailing bits of
d[rnd_digit] masked out. So the value of x comes from the top
DBL_MANT_DIG bits of v, multiplied by 2*lsb. Note that in the loop
that produces x, all floating-point operations are exact (assuming
that FLT_RADIX==2). Now if we're rounding down, the value we want
to return is simply
x * 2**(PyLong_SHIFT * rnd_digit).
and if we're rounding up, it's
(x + 2*lsb) * 2**(PyLong_SHIFT * rnd_digit).
Under the round-half-to-even rule, we round up if, and only
if, the rounding bit is set *and* at least one of the
following three conditions is satisfied:
(1) the parity bit is set, or
(2) at least one of the trailing bits of d[rnd_digit] is set, or
(3) at least one of the digits d[i], 0 <= i < rnd_digit
is nonzero.
Finally, we have to worry about overflow. If v >= 2**DBL_MAX_EXP,
or equivalently n > DBL_MAX_EXP, then overflow occurs. If v <
2**DBL_MAX_EXP then we're usually safe, but there's a corner case
to consider: if v is very close to 2**DBL_MAX_EXP then it's
possible that v is rounded up to exactly 2**DBL_MAX_EXP, and then
again overflow occurs.
*/
if (Py_SIZE(v) == 0)
return 0.0;
m = ABS(Py_SIZE(v)) - 1;
d = v->ob_digit;
assert(d[m]); /* v should be normalized */
/* fast path for case where 0 < abs(v) < 2**DBL_MANT_DIG */
if (m < DBL_MANT_DIG / PyLong_SHIFT ||
(m == DBL_MANT_DIG / PyLong_SHIFT &&
d[m] < (digit)1 << DBL_MANT_DIG%PyLong_SHIFT)) {
x = d[m];
while (--m >= 0)
x = x*PyLong_BASE + d[m];
return Py_SIZE(v) < 0 ? -x : x;
}
/* if m is huge then overflow immediately; otherwise, compute the
number of bits n in v. The condition below implies n (= #bits) >=
m * PyLong_SHIFT + 1 > DBL_MAX_EXP, hence v >= 2**DBL_MAX_EXP. */
if (m > (DBL_MAX_EXP-1)/PyLong_SHIFT)
goto overflow;
n = m * PyLong_SHIFT + bits_in_digit(d[m]);
if (n > DBL_MAX_EXP)
goto overflow;
/* find location of rounding bit */
assert(n > DBL_MANT_DIG); /* dealt with |v| < 2**DBL_MANT_DIG above */
rnd_bit = n - DBL_MANT_DIG - 1;
rnd_digit = rnd_bit/PyLong_SHIFT;
lsb = (digit)1 << (rnd_bit%PyLong_SHIFT);
/* Get top DBL_MANT_DIG bits of v. Assumes PyLong_SHIFT <
DBL_MANT_DIG, so we'll need bits from at least 2 digits of v. */
x = d[m];
assert(m > rnd_digit);
while (--m > rnd_digit)
x = x*PyLong_BASE + d[m];
x = x*PyLong_BASE + (d[m] & (PyLong_BASE-2*lsb));
/* decide whether to round up, using round-half-to-even */
assert(m == rnd_digit);
if (d[m] & lsb) { /* if (rounding bit is set) */
digit parity_bit;
if (lsb == PyLong_BASE/2)
parity_bit = d[m+1] & 1;
else
parity_bit = d[m] & 2*lsb;
if (parity_bit)
round_up = 1;
else if (d[m] & (lsb-1))
round_up = 1;
else {
while (--m >= 0) {
if (d[m]) {
round_up = 1;
break;
}
}
}
}
/* and round up if necessary */
if (round_up) {
x += 2*lsb;
if (n == DBL_MAX_EXP &&
x == ldexp((double)(2*lsb), DBL_MANT_DIG)) {
/* overflow corner case */
goto overflow;
}
}
/* shift, adjust for sign, and return */
x = ldexp(x, rnd_digit*PyLong_SHIFT);
return Py_SIZE(v) < 0 ? -x : x;
overflow:
PyErr_SetString(PyExc_OverflowError,
"long int too large to convert to float");
return -1.0;