From c630039edd7526642ffdaeee3bfa81352b5c98ae Mon Sep 17 00:00:00 2001 From: Mark Dickinson Date: Mon, 20 Apr 2009 21:38:00 +0000 Subject: Merged revisions 71772 via svnmerge from svn+ssh://pythondev@svn.python.org/python/trunk ........ r71772 | mark.dickinson | 2009-04-20 22:13:33 +0100 (Mon, 20 Apr 2009) | 5 lines 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. ........ --- Lib/test/test_long.py | 59 ++++++++++++++++++ Misc/NEWS | 3 + Objects/longobject.c | 167 +++++++++++++++++++++++++++++++++++++++++++++----- 3 files changed, 214 insertions(+), 15 deletions(-) diff --git a/Lib/test/test_long.py b/Lib/test/test_long.py index 92285b2..53d3e6b 100644 --- a/Lib/test/test_long.py +++ b/Lib/test/test_long.py @@ -620,6 +620,65 @@ class LongTest(unittest.TestCase): else: self.assertRaises(TypeError, pow,longx, longy, int(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 = [0, 1, 2, + 2**53-3, + 2**53-2, + 2**53-1, + 2**53, + 2**53+2, + 2**54-4, + 2**54-2, + 2**54, + 2**54+4] + for x in exact_values: + self.assertEqual(float(x), x) + self.assertEqual(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 range(15): + self.assertEqual(int(float(2**p*(2**53+x))), 2**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 range(15): + self.assertEqual(int(float(2**p*(2**54+x))), 2**p*(2**54+y)) + + # behaviour near extremes of floating-point range + int_dbl_max = int(DBL_MAX) + top_power = 2**DBL_MAX_EXP + halfway = (int_dbl_max + top_power)//2 + self.assertEqual(float(int_dbl_max), DBL_MAX) + self.assertEqual(float(int_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 range(100): + x = 2**p * (2**53 + 1) + 1 + y = 2**p * (2**53 + 2) + self.assertEqual(int(float(x)), y) + + x = 2**p * (2**53 + 1) + y = 2**p * 2**53 + self.assertEqual(int(float(x)), y) + def test_float_overflow(self): import math diff --git a/Misc/NEWS b/Misc/NEWS index 523166c..ca4a864 100644 --- a/Misc/NEWS +++ b/Misc/NEWS @@ -12,6 +12,9 @@ What's New in Python 3.1 beta 1? Core and Builtins ----------------- +- Issue #3166: Make long -> float (and int -> float) conversions + correctly rounded. + - Issue #1869 (and many duplicates): make round(x, n) correctly rounded for a float x, by using the decimal <-> binary conversions from Python/dtoa.c. As a consequence, (e.g.) round(x, 2) now diff --git a/Objects/longobject.c b/Objects/longobject.c index 598c873..81af4ea 100644 --- a/Objects/longobject.c +++ b/Objects/longobject.c @@ -6,6 +6,7 @@ #include "longintrepr.h" #include "structseq.h" +#include #include #include @@ -100,6 +101,9 @@ maybe_small_long(PyLongObject *v) 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. */ @@ -962,33 +966,166 @@ _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) + } + + /* 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; - errno = 0; - x = ldexp(x, e * PyLong_SHIFT); - if (Py_OVERFLOWED(x)) + n = m * PyLong_SHIFT + bits_in_digit(d[m]); + if (n > DBL_MAX_EXP) goto overflow; - return x; -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, "Python int too large to convert to C double"); return -1.0; -- cgit v0.12