Index: Include/longobject.h =================================================================== --- Include/longobject.h (revision 81495) +++ Include/longobject.h (working copy) @@ -101,6 +101,14 @@ */ PyAPI_FUNC(size_t) _PyLong_NumBits(PyObject *v); +/* _PyLong_Divmod_Near. Given integers a and b, compute the nearest + integer q to the exact quotient a / b, rounding to the nearest even integer + in the case of a tie. Return (q, r), where r = a - q*b. The remainder r + will satisfy abs(r) <= abs(b)/2, with equality possible only if q is + even. +*/ +PyAPI_FUNC(PyObject *) _PyLong_Divmod_Near(PyObject *, PyObject *); + /* _PyLong_FromByteArray: View the n unsigned bytes as a binary integer in base 256, and return a Python long with the same numeric value. If n is 0, the integer is 0. Else: Index: Objects/longobject.c =================================================================== --- Objects/longobject.c (revision 81495) +++ Objects/longobject.c (working copy) @@ -4201,140 +4201,169 @@ PyUnicode_GET_SIZE(format_spec)); } -static PyObject * -long_round(PyObject *self, PyObject *args) +/* Return a pair (q, r) such that a = b * q + r, and + abs(r) <= abs(b)/2, with equality possible only if q is even. + In other words, q == a / b, rounded to the nearest integer using + round-half-to-even. */ + +PyObject * +_PyLong_Divmod_Near(PyObject *a, PyObject *b) { - PyObject *o_ndigits=NULL, *temp; - PyLongObject *pow=NULL, *q=NULL, *r=NULL, *ndigits=NULL, *one; - int errcode; - digit q_mod_4; + PyLongObject *quo = NULL, *rem = NULL; + PyObject *one = NULL, *twice_rem, *result, *temp; + int cmp, quo_is_odd, quo_is_neg; - /* Notes on the algorithm: to round to the nearest 10**n (n positive), - the straightforward method is: + /* Equivalent Python code: - (1) divide by 10**n - (2) round to nearest integer (round to even in case of tie) - (3) multiply result by 10**n. + def divmod_near(a, b): + q, r = divmod(a, b) + # round up if either r / b > 0.5, or r / b == 0.5 and q is odd. + # The expression r / b > 0.5 is equivalent to 2 * r > b if b is + # positive, 2 * r < b if b negative. + greater_than_half = 2*r > b if b > 0 else 2*r < b + exactly_half = 2*r == b + if greater_than_half or exactly_half and q % 2 == 1: + q += 1 + r -= b + return q, r - But the rounding step involves examining the fractional part of the - quotient to see whether it's greater than 0.5 or not. Since we - want to do the whole calculation in integer arithmetic, it's - simpler to do: + */ + if (!PyLong_Check(a) || !PyLong_Check(b)) { + PyErr_SetString(PyExc_TypeError, + "non-integer arguments in division"); + return NULL; + } - (1) divide by (10**n)/2 - (2) round to nearest multiple of 2 (multiple of 4 in case of tie) - (3) multiply result by (10**n)/2. + /* Do a and b have different signs? If so, quotient is negative. */ + quo_is_neg = (Py_SIZE(a) < 0) != (Py_SIZE(b) < 0); - Then all we need to know about the fractional part of the quotient - arising in step (2) is whether it's zero or not. + one = PyLong_FromLong(1L); + if (one == NULL) + return NULL; - Doing both a multiplication and division is wasteful, and is easily - avoided if we just figure out how much to adjust the original input - by to do the rounding. + if (long_divrem((PyLongObject*)a, (PyLongObject*)b, &quo, &rem) < 0) + goto error; - Here's the whole algorithm expressed in Python. + /* compare twice the remainder with the divisor, to see + if we need to adjust the quotient and remainder */ + twice_rem = long_lshift((PyObject *)rem, one); + if (twice_rem == NULL) + goto error; + if (quo_is_neg) { + temp = long_neg((PyLongObject*)twice_rem); + Py_DECREF(twice_rem); + twice_rem = temp; + if (twice_rem == NULL) + goto error; + } + cmp = long_compare((PyLongObject *)twice_rem, (PyLongObject *)b); + Py_DECREF(twice_rem); - def round(self, ndigits = None): - """round(int, int) -> int""" - if ndigits is None or ndigits >= 0: - return self - pow = 10**-ndigits >> 1 - q, r = divmod(self, pow) - self -= r - if (q & 1 != 0): - if (q & 2 == r == 0): - self -= pow - else: - self += pow - return self + quo_is_odd = Py_SIZE(quo) != 0 && ((quo->ob_digit[0] & 1) != 0); + if ((Py_SIZE(b) < 0 ? cmp < 0 : cmp > 0) || (cmp == 0 && quo_is_odd)) { + /* fix up quotient */ + if (quo_is_neg) + temp = long_sub(quo, (PyLongObject *)one); + else + temp = long_add(quo, (PyLongObject *)one); + Py_DECREF(quo); + quo = (PyLongObject *)temp; + if (quo == NULL) + goto error; + /* and remainder */ + if (quo_is_neg) + temp = long_add(rem, (PyLongObject *)b); + else + temp = long_sub(rem, (PyLongObject *)b); + Py_DECREF(rem); + rem = (PyLongObject *)temp; + if (rem == NULL) + goto error; + } + Py_DECREF(one); - */ + result = PyTuple_New(2); + if (result == NULL) + goto error; + + /* PyTuple_SET_ITEM steals references */ + PyTuple_SET_ITEM(result, 0, (PyObject *)quo); + PyTuple_SET_ITEM(result, 1, (PyObject *)rem); + return result; + + error: + Py_XDECREF(one); + Py_XDECREF(quo); + Py_XDECREF(rem); + return NULL; +} + +static PyObject * +long_round(PyObject *self, PyObject *args) +{ + PyObject *o_ndigits=NULL, *temp, *result, *ndigits; + + /* To round an integer m to the nearest 10**n (n positive), we make use of + * the divmod_near operation, defined by: + * + * divmod_near(a, b) = (q, r) + * + * where q is the nearest integer to the quotient a / b (the + * nearest even integer in the case of a tie) and r == a - q * b. + * Hence q * b = a - r is the nearest multiple of b to a, + * preferring even multiples in the case of a tie. + * + * So the nearest multiple of 10**n to m is: + * + * m - divmod_near(m, 10**n)[1]. + */ if (!PyArg_ParseTuple(args, "|O", &o_ndigits)) return NULL; if (o_ndigits == NULL) return long_long(self); - ndigits = (PyLongObject *)PyNumber_Index(o_ndigits); + ndigits = PyNumber_Index(o_ndigits); if (ndigits == NULL) return NULL; + /* if ndigits >= 0 then no rounding is necessary; return self unchanged */ if (Py_SIZE(ndigits) >= 0) { Py_DECREF(ndigits); return long_long(self); } - Py_INCREF(self); /* to keep refcounting simple */ - /* we now own references to self, ndigits */ - - /* pow = 10 ** -ndigits >> 1 */ - pow = (PyLongObject *)PyLong_FromLong(10L); - if (pow == NULL) - goto error; - temp = long_neg(ndigits); + /* result = self - divmod_near(self, 10 ** -ndigits)[1] */ + temp = long_neg((PyLongObject*)ndigits); Py_DECREF(ndigits); - ndigits = (PyLongObject *)temp; + ndigits = temp; if (ndigits == NULL) - goto error; - temp = long_pow((PyObject *)pow, (PyObject *)ndigits, Py_None); - Py_DECREF(pow); - pow = (PyLongObject *)temp; - if (pow == NULL) - goto error; - assert(PyLong_Check(pow)); /* check long_pow returned a long */ - one = (PyLongObject *)PyLong_FromLong(1L); - if (one == NULL) - goto error; - temp = long_rshift(pow, one); - Py_DECREF(one); - Py_DECREF(pow); - pow = (PyLongObject *)temp; - if (pow == NULL) - goto error; + return NULL; - /* q, r = divmod(self, pow) */ - errcode = l_divmod((PyLongObject *)self, pow, &q, &r); - if (errcode == -1) - goto error; + result = PyLong_FromLong(10L); + if (result == NULL) { + Py_DECREF(ndigits); + return NULL; + } - /* self -= r */ - temp = long_sub((PyLongObject *)self, r); - Py_DECREF(self); - self = temp; - if (self == NULL) - goto error; + temp = long_pow(result, ndigits, Py_None); + Py_DECREF(ndigits); + Py_DECREF(result); + result = temp; + if (result == NULL) + return NULL; - /* get value of quotient modulo 4 */ - if (Py_SIZE(q) == 0) - q_mod_4 = 0; - else if (Py_SIZE(q) > 0) - q_mod_4 = q->ob_digit[0] & 3; - else - q_mod_4 = (PyLong_BASE-q->ob_digit[0]) & 3; + temp = _PyLong_Divmod_Near(self, result); + Py_DECREF(result); + result = temp; + if (result == NULL) + return NULL; - if ((q_mod_4 & 1) == 1) { - /* q is odd; round self up or down by adding or subtracting pow */ - if (q_mod_4 == 1 && Py_SIZE(r) == 0) - temp = (PyObject *)long_sub((PyLongObject *)self, pow); - else - temp = (PyObject *)long_add((PyLongObject *)self, pow); - Py_DECREF(self); - self = temp; - if (self == NULL) - goto error; - } - Py_DECREF(q); - Py_DECREF(r); - Py_DECREF(pow); - Py_DECREF(ndigits); - return self; + temp = long_sub((PyLongObject *)self, + (PyLongObject *)PyTuple_GET_ITEM(result, 1)); + Py_DECREF(result); + result = temp; - error: - Py_XDECREF(q); - Py_XDECREF(r); - Py_XDECREF(pow); - Py_XDECREF(self); - Py_XDECREF(ndigits); - return NULL; + return result; } static PyObject *