cpython/Objects/longobject.c

/***********************************************************
Copyright 1991-1995 by Stichting Mathematisch Centrum, Amsterdam,
The Netherlands.

                        All Rights Reserved

Permission to use, copy, modify, and distribute this software and its
documentation for any purpose and without fee is hereby granted,
provided that the above copyright notice appear in all copies and that
both that copyright notice and this permission notice appear in
supporting documentation, and that the names of Stichting Mathematisch
Centrum or CWI or Corporation for National Research Initiatives or
CNRI not be used in advertising or publicity pertaining to
distribution of the software without specific, written prior
permission.

While CWI is the initial source for this software, a modified version
is made available by the Corporation for National Research Initiatives
(CNRI) at the Internet address ftp://ftp.python.org.

STICHTING MATHEMATISCH CENTRUM AND CNRI DISCLAIM ALL WARRANTIES WITH
REGARD TO THIS SOFTWARE, INCLUDING ALL IMPLIED WARRANTIES OF
MERCHANTABILITY AND FITNESS, IN NO EVENT SHALL STICHTING MATHEMATISCH
CENTRUM OR CNRI BE LIABLE FOR ANY SPECIAL, INDIRECT OR CONSEQUENTIAL
DAMAGES OR ANY DAMAGES WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR
PROFITS, WHETHER IN AN ACTION OF CONTRACT, NEGLIGENCE OR OTHER
TORTIOUS ACTION, ARISING OUT OF OR IN CONNECTION WITH THE USE OR
PERFORMANCE OF THIS SOFTWARE.

******************************************************************/

/* Long (arbitrary precision) integer object implementation */

/* XXX The functional organization of this file is terrible */

#include "Python.h"
#include "longintrepr.h"
#include "mymath.h"

#include <assert.h>
#include <ctype.h>

#define ABS(x) ((x) < 0 ? -(x) : (x))

/* Forward */
static PyLongObject *long_normalize Py_PROTO((PyLongObject *));
static PyLongObject *mul1 Py_PROTO((PyLongObject *, wdigit));
static PyLongObject *muladd1 Py_PROTO((PyLongObject *, wdigit, wdigit));
static PyLongObject *divrem1 Py_PROTO((PyLongObject *, wdigit, digit *));
static PyObject *long_format Py_PROTO((PyObject *aa, int base));

static int ticker;	/* XXX Could be shared with ceval? */

#define SIGCHECK(PyTryBlock) \
	if (--ticker < 0) { \
		ticker = 100; \
		if (PyErr_CheckSignals()) { PyTryBlock; } \
	}

/* 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. */

static PyLongObject *
long_normalize(v)
	register PyLongObject *v;
{
	int j = ABS(v->ob_size);
	register int i = j;
	
	while (i > 0 && v->ob_digit[i-1] == 0)
		--i;
	if (i != j)
		v->ob_size = (v->ob_size < 0) ? -(i) : i;
	return v;
}

/* Allocate a new long int object with size digits.
   Return NULL and set exception if we run out of memory. */

PyLongObject *
_PyLong_New(size)
	int size;
{
	return PyObject_NEW_VAR(PyLongObject, &PyLong_Type, size);
}

/* Create a new long int object from a C long int */

PyObject *
PyLong_FromLong(ival)
	long ival;
{
	/* Assume a C long fits in at most 5 'digits' */
	/* Works on both 32- and 64-bit machines */
	PyLongObject *v = _PyLong_New(5);
	if (v != NULL) {
		unsigned long t = ival;
		int i;
		if (ival < 0) {
			t = -ival;
			v->ob_size = -(v->ob_size);
  		}
		for (i = 0; i < 5; i++) {
			v->ob_digit[i] = (digit) (t & MASK);
			t >>= SHIFT;
		}
		v = long_normalize(v);
	}
	return (PyObject *)v;
}

/* Create a new long int object from a C unsigned long int */

PyObject *
PyLong_FromUnsignedLong(ival)
	unsigned long ival;
{
	/* Assume a C long fits in at most 5 'digits' */
	/* Works on both 32- and 64-bit machines */
	PyLongObject *v = _PyLong_New(5);
	if (v != NULL) {
		unsigned long t = ival;
		int i;
		for (i = 0; i < 5; i++) {
			v->ob_digit[i] = (digit) (t & MASK);
			t >>= SHIFT;
		}
		v = long_normalize(v);
	}
	return (PyObject *)v;
}

/* Create a new long int object from a C double */

PyObject *
#ifdef MPW
PyLong_FromDouble(double dval)
#else
PyLong_FromDouble(dval)
	double dval;
#endif /* MPW */
{
	PyLongObject *v;
	double frac;
	int i, ndig, expo, neg;
	neg = 0;
	if (dval < 0.0) {
		neg = 1;
		dval = -dval;
	}
	frac = frexp(dval, &expo); /* dval = frac*2**expo; 0.0 <= frac < 1.0 */
	if (expo <= 0)
		return PyLong_FromLong(0L);
	ndig = (expo-1) / SHIFT + 1; /* Number of 'digits' in result */
	v = _PyLong_New(ndig);
	if (v == NULL)
		return NULL;
	frac = ldexp(frac, (expo-1) % SHIFT + 1);
	for (i = ndig; --i >= 0; ) {
		long bits = (long)frac;
		v->ob_digit[i] = (digit) bits;
		frac = frac - (double)bits;
		frac = ldexp(frac, SHIFT);
	}
	if (neg)
		v->ob_size = -(v->ob_size);
	return (PyObject *)v;
}

/* Get a C long int from a long int object.
   Returns -1 and sets an error condition if overflow occurs. */

long
PyLong_AsLong(vv)
	PyObject *vv;
{
	/* This version by Tim Peters */
	register PyLongObject *v;
	unsigned long x, prev;
	int i, sign;

	if (vv == NULL || !PyLong_Check(vv)) {
		PyErr_BadInternalCall();
		return -1;
	}
	v = (PyLongObject *)vv;
	i = v->ob_size;
	sign = 1;
	x = 0;
	if (i < 0) {
		sign = -1;
		i = -(i);
	}
	while (--i >= 0) {
		prev = x;
		x = (x << SHIFT) + v->ob_digit[i];
		if ((x >> SHIFT) != prev)
			goto overflow;
	}
	/* Haven't lost any bits, but if the sign bit is set we're in
	 * trouble *unless* this is the min negative number.  So,
	 * trouble iff sign bit set && (positive || some bit set other
	 * than the sign bit).
	 */
	if ((long)x < 0 && (sign > 0 || (x << 1) != 0))
		goto overflow;
	return (long)x * sign;

 overflow:
	PyErr_SetString(PyExc_OverflowError,
			"long int too long to convert");
	return -1;
}

/* Get a C long int from a long int object.
   Returns -1 and sets an error condition if overflow occurs. */

unsigned long
PyLong_AsUnsignedLong(vv)
	PyObject *vv;
{
	register PyLongObject *v;
	unsigned long x, prev;
	int i;
	
	if (vv == NULL || !PyLong_Check(vv)) {
		PyErr_BadInternalCall();
		return (unsigned long) -1;
	}
	v = (PyLongObject *)vv;
	i = v->ob_size;
	x = 0;
	if (i < 0) {
		PyErr_SetString(PyExc_OverflowError,
			   "can't convert negative value to unsigned long");
		return (unsigned long) -1;
	}
	while (--i >= 0) {
		prev = x;
		x = (x << SHIFT) + v->ob_digit[i];
		if ((x >> SHIFT) != prev) {
			PyErr_SetString(PyExc_OverflowError,
				"long int too long to convert");
			return (unsigned long) -1;
		}
	}
	return x;
}

/* Get a C double from a long int object. */

double
PyLong_AsDouble(vv)
	PyObject *vv;
{
	register PyLongObject *v;
	double x;
	double multiplier = (double) (1L << SHIFT);
	int i, sign;
	
	if (vv == NULL || !PyLong_Check(vv)) {
		PyErr_BadInternalCall();
		return -1;
	}
	v = (PyLongObject *)vv;
	i = v->ob_size;
	sign = 1;
	x = 0.0;
	if (i < 0) {
		sign = -1;
		i = -(i);
	}
	while (--i >= 0) {
		x = x*multiplier + (double)v->ob_digit[i];
	}
	return x * sign;
}

/* Multiply by a single digit, ignoring the sign. */

static PyLongObject *
mul1(a, n)
	PyLongObject *a;
	wdigit n;
{
	return muladd1(a, n, (digit)0);
}

/* Multiply by a single digit and add a single digit, ignoring the sign. */

static PyLongObject *
muladd1(a, n, extra)
	PyLongObject *a;
	wdigit n;
	wdigit extra;
{
	int size_a = ABS(a->ob_size);
	PyLongObject *z = _PyLong_New(size_a+1);
	twodigits carry = extra;
	int i;
	
	if (z == NULL)
		return NULL;
	for (i = 0; i < size_a; ++i) {
		carry += (twodigits)a->ob_digit[i] * n;
		z->ob_digit[i] = (digit) (carry & MASK);
		carry >>= SHIFT;
	}
	z->ob_digit[i] = (digit) carry;
	return long_normalize(z);
}

/* Divide a long integer by a digit, returning both the quotient
   (as function result) and the remainder (through *prem).
   The sign of a is ignored; n should not be zero. */

static PyLongObject *
divrem1(a, n, prem)
	PyLongObject *a;
	wdigit n;
	digit *prem;
{
	int size = ABS(a->ob_size);
	PyLongObject *z;
	int i;
	twodigits rem = 0;
	
	assert(n > 0 && n <= MASK);
	z = _PyLong_New(size);
	if (z == NULL)
		return NULL;
	for (i = size; --i >= 0; ) {
		rem = (rem << SHIFT) + a->ob_digit[i];
		z->ob_digit[i] = (digit) (rem/n);
		rem %= n;
	}
	*prem = (digit) rem;
	return long_normalize(z);
}

/* Convert a long int object to a string, using a given conversion base.
   Return a string object.
   If base is 8 or 16, add the proper prefix '0' or '0x'.
   External linkage: used in bltinmodule.c by hex() and oct(). */

static PyObject *
long_format(aa, base)
	PyObject *aa;
	int base;
{
	register PyLongObject *a = (PyLongObject *)aa;
	PyStringObject *str;
	int i;
	int size_a = ABS(a->ob_size);
	char *p;
	int bits;
	char sign = '\0';

	if (a == NULL || !PyLong_Check(a)) {
		PyErr_BadInternalCall();
		return NULL;
	}
	assert(base >= 2 && base <= 36);
	
	/* Compute a rough upper bound for the length of the string */
	i = base;
	bits = 0;
	while (i > 1) {
		++bits;
		i >>= 1;
	}
	i = 6 + (size_a*SHIFT + bits-1) / bits;
	str = (PyStringObject *) PyString_FromStringAndSize((char *)0, i);
	if (str == NULL)
		return NULL;
	p = PyString_AS_STRING(str) + i;
	*p = '\0';
	*--p = 'L';
	if (a->ob_size < 0)
		sign = '-';
	
	Py_INCREF(a);
	do {
		digit rem;
		PyLongObject *temp = divrem1(a, (digit)base, &rem);
		if (temp == NULL) {
			Py_DECREF(a);
			Py_DECREF(str);
			return NULL;
		}
		if (rem < 10)
			rem += '0';
		else
			rem += 'A'-10;
		assert(p > PyString_AS_STRING(str));
		*--p = (char) rem;
		Py_DECREF(a);
		a = temp;
		SIGCHECK({
			Py_DECREF(a);
			Py_DECREF(str);
			return NULL;
		})
	} while (ABS(a->ob_size) != 0);
	Py_DECREF(a);
	if (base == 8) {
		if (size_a != 0)
			*--p = '0';
	}
	else if (base == 16) {
		*--p = 'x';
		*--p = '0';
	}
	else if (base != 10) {
		*--p = '#';
		*--p = '0' + base%10;
		if (base > 10)
			*--p = '0' + base/10;
	}
	if (sign)
		*--p = sign;
	if (p != PyString_AS_STRING(str)) {
		char *q = PyString_AS_STRING(str);
		assert(p > q);
		do {
		} while ((*q++ = *p++) != '\0');
		q--;
		_PyString_Resize((PyObject **)&str,
				 (int) (q - PyString_AS_STRING(str)));
	}
	return (PyObject *)str;
}

#if 0
/* Convert a string to a long int object, in a given base.
   Base zero implies a default depending on the number.
   External linkage: used in compile.c and stropmodule.c. */

PyObject *
long_scan(str, base)
	char *str;
	int base;
{
	return PyLong_FromString(str, (char **)NULL, base);
}
#endif

PyObject *
PyLong_FromString(str, pend, base)
	char *str;
	char **pend;
	int base;
{
	int sign = 1;
	char *start;
	PyLongObject *z;
	
	if ((base != 0 && base < 2) || base > 36) {
		PyErr_SetString(PyExc_ValueError,
				"invalid base for long literal");
		return NULL;
	}
	while (*str != '\0' && isspace(Py_CHARMASK(*str)))
		str++;
	if (*str == '+')
		++str;
	else if (*str == '-') {
		++str;
		sign = -1;
	}
	while (*str != '\0' && isspace(Py_CHARMASK(*str)))
		str++;
	if (base == 0) {
		if (str[0] != '0')
			base = 10;
		else if (str[1] == 'x' || str[1] == 'X')
			base = 16;
		else
			base = 8;
	}
	if (base == 16 && str[0] == '0' && (str[1] == 'x' || str[1] == 'X'))
		str += 2;
	z = _PyLong_New(0);
	start = str;
	for ( ; z != NULL; ++str) {
		int k = -1;
		PyLongObject *temp;
		
		if (*str <= '9')
			k = *str - '0';
		else if (*str >= 'a')
			k = *str - 'a' + 10;
		else if (*str >= 'A')
			k = *str - 'A' + 10;
		if (k < 0 || k >= base)
			break;
		temp = muladd1(z, (digit)base, (digit)k);
		Py_DECREF(z);
		z = temp;
	}
	if (str == start) {
		PyErr_SetString(PyExc_ValueError,
				"no digits in long int constant");
		return NULL;
	}
	if (sign < 0 && z != NULL && z->ob_size != 0)
		z->ob_size = -(z->ob_size);
	if (pend)
		*pend = str;
	return (PyObject *) z;
}

static PyLongObject *x_divrem
	Py_PROTO((PyLongObject *, PyLongObject *, PyLongObject **));
static PyObject *long_pos Py_PROTO((PyLongObject *));
static int long_divrem Py_PROTO((PyLongObject *, PyLongObject *,
	PyLongObject **, PyLongObject **));

/* Long division with remainder, top-level routine */

static int
long_divrem(a, b, pdiv, prem)
	PyLongObject *a, *b;
	PyLongObject **pdiv;
	PyLongObject **prem;
{
	int size_a = ABS(a->ob_size), size_b = ABS(b->ob_size);
	PyLongObject *z;
	
	if (size_b == 0) {
		PyErr_SetString(PyExc_ZeroDivisionError, "long division or modulo");
		return -1;
	}
	if (size_a < size_b ||
	    (size_a == size_b &&
	     a->ob_digit[size_a-1] < b->ob_digit[size_b-1])) {
		/* |a| < |b|. */
		*pdiv = _PyLong_New(0);
		Py_INCREF(a);
		*prem = (PyLongObject *) a;
		return 0;
	}
	if (size_b == 1) {
		digit rem = 0;
		z = divrem1(a, b->ob_digit[0], &rem);
		if (z == NULL)
			return -1;
		*prem = (PyLongObject *) PyLong_FromLong((long)rem);
	}
	else {
		z = x_divrem(a, b, prem);
		if (z == NULL)
			return -1;
	}
	/* Set the signs.
	   The quotient z has the sign of a*b;
	   the remainder r has the sign of a,
	   so a = b*z + r. */
	if ((a->ob_size < 0) != (b->ob_size < 0))
		z->ob_size = -(z->ob_size);
	if (a->ob_size < 0 && (*prem)->ob_size != 0)
		(*prem)->ob_size = -((*prem)->ob_size);
	*pdiv = z;
	return 0;
}

/* Unsigned long division with remainder -- the algorithm */

static PyLongObject *
x_divrem(v1, w1, prem)
	PyLongObject *v1, *w1;
	PyLongObject **prem;
{
	int size_v = ABS(v1->ob_size), size_w = ABS(w1->ob_size);
	digit d = (digit) ((twodigits)BASE / (w1->ob_digit[size_w-1] + 1));
	PyLongObject *v = mul1(v1, d);
	PyLongObject *w = mul1(w1, d);
	PyLongObject *a;
	int j, k;
	
	if (v == NULL || w == NULL) {
		Py_XDECREF(v);
		Py_XDECREF(w);
		return NULL;
	}
	
	assert(size_v >= size_w && size_w > 1); /* Assert checks by div() */
	assert(v->ob_refcnt == 1); /* Since v will be used as accumulator! */
	assert(size_w == ABS(w->ob_size)); /* That's how d was calculated */
	
	size_v = ABS(v->ob_size);
	a = _PyLong_New(size_v - size_w + 1);
	
	for (j = size_v, k = a->ob_size-1; a != NULL && k >= 0; --j, --k) {
		digit vj = (j >= size_v) ? 0 : v->ob_digit[j];
		twodigits q;
		stwodigits carry = 0;
		int i;
		
		SIGCHECK({
			Py_DECREF(a);
			a = NULL;
			break;
		})
		if (vj == w->ob_digit[size_w-1])
			q = MASK;
		else
			q = (((twodigits)vj << SHIFT) + v->ob_digit[j-1]) /
				w->ob_digit[size_w-1];
		
		while (w->ob_digit[size_w-2]*q >
				((
					((twodigits)vj << SHIFT)
					+ v->ob_digit[j-1]
					- q*w->ob_digit[size_w-1]
								) << SHIFT)
				+ v->ob_digit[j-2])
			--q;
		
		for (i = 0; i < size_w && i+k < size_v; ++i) {
			twodigits z = w->ob_digit[i] * q;
			digit zz = (digit) (z >> SHIFT);
			carry += v->ob_digit[i+k] - z
				+ ((twodigits)zz << SHIFT);
			v->ob_digit[i+k] = carry & MASK;
			carry = (carry >> SHIFT) - zz;
		}
		
		if (i+k < size_v) {
			carry += v->ob_digit[i+k];
			v->ob_digit[i+k] = 0;
		}
		
		if (carry == 0)
			a->ob_digit[k] = (digit) q;
		else {
			assert(carry == -1);
			a->ob_digit[k] = (digit) q-1;
			carry = 0;
			for (i = 0; i < size_w && i+k < size_v; ++i) {
				carry += v->ob_digit[i+k] + w->ob_digit[i];
				v->ob_digit[i+k] = carry & MASK;
				carry >>= SHIFT;
			}
		}
	} /* for j, k */
	
	if (a == NULL)
		*prem = NULL;
	else {
		a = long_normalize(a);
		*prem = divrem1(v, d, &d);
		/* d receives the (unused) remainder */
		if (*prem == NULL) {
			Py_DECREF(a);
			a = NULL;
		}
	}
	Py_DECREF(v);
	Py_DECREF(w);
	return a;
}

/* Methods */

/* Forward */
static void long_dealloc Py_PROTO((PyObject *));
static PyObject *long_repr Py_PROTO((PyObject *));
static int long_compare Py_PROTO((PyLongObject *, PyLongObject *));
static long long_hash Py_PROTO((PyLongObject *));

static PyObject *long_add Py_PROTO((PyLongObject *, PyLongObject *));
static PyObject *long_sub Py_PROTO((PyLongObject *, PyLongObject *));
static PyObject *long_mul Py_PROTO((PyLongObject *, PyLongObject *));
static PyObject *long_div Py_PROTO((PyLongObject *, PyLongObject *));
static PyObject *long_mod Py_PROTO((PyLongObject *, PyLongObject *));
static PyObject *long_divmod Py_PROTO((PyLongObject *, PyLongObject *));
static PyObject *long_pow
	Py_PROTO((PyLongObject *, PyLongObject *, PyLongObject *));
static PyObject *long_neg Py_PROTO((PyLongObject *));
static PyObject *long_pos Py_PROTO((PyLongObject *));
static PyObject *long_abs Py_PROTO((PyLongObject *));
static int long_nonzero Py_PROTO((PyLongObject *));
static PyObject *long_invert Py_PROTO((PyLongObject *));
static PyObject *long_lshift Py_PROTO((PyLongObject *, PyLongObject *));
static PyObject *long_rshift Py_PROTO((PyLongObject *, PyLongObject *));
static PyObject *long_and Py_PROTO((PyLongObject *, PyLongObject *));
static PyObject *long_xor Py_PROTO((PyLongObject *, PyLongObject *));
static PyObject *long_or Py_PROTO((PyLongObject *, PyLongObject *));

static void
long_dealloc(v)
	PyObject *v;
{
	PyMem_DEL(v);
}

static PyObject *
long_repr(v)
	PyObject *v;
{
	return long_format(v, 10);
}

static int
long_compare(a, b)
	PyLongObject *a, *b;
{
	int sign;
	
	if (a->ob_size != b->ob_size) {
		if (ABS(a->ob_size) == 0 && ABS(b->ob_size) == 0)
			sign = 0;
		else
			sign = a->ob_size - b->ob_size;
	}
	else {
		int i = ABS(a->ob_size);
		while (--i >= 0 && a->ob_digit[i] == b->ob_digit[i])
			;
		if (i < 0)
			sign = 0;
		else {
			sign = (int)a->ob_digit[i] - (int)b->ob_digit[i];
			if (a->ob_size < 0)
				sign = -sign;
		}
	}
	return sign < 0 ? -1 : sign > 0 ? 1 : 0;
}

static long
long_hash(v)
	PyLongObject *v;
{
	long x;
	int i, sign;

	/* This is designed so that Python ints and longs with the
	   same value hash to the same value, otherwise comparisons
	   of mapping keys will turn out weird */
	i = v->ob_size;
	sign = 1;
	x = 0;
	if (i < 0) {
		sign = -1;
		i = -(i);
	}
	while (--i >= 0) {
		/* Force a 32-bit circular shift */
		x = ((x << SHIFT) & ~MASK) | ((x >> (32-SHIFT)) & MASK);
		x += v->ob_digit[i];
	}
	x = x * sign;
	if (x == -1)
		x = -2;
	return x;
}


/* Add the absolute values of two long integers. */

static PyLongObject *x_add Py_PROTO((PyLongObject *, PyLongObject *));
static PyLongObject *
x_add(a, b)
	PyLongObject *a, *b;
{
	int size_a = ABS(a->ob_size), size_b = ABS(b->ob_size);
	PyLongObject *z;
	int i;
	digit carry = 0;
	
	/* Ensure a is the larger of the two: */
	if (size_a < size_b) {
		{ PyLongObject *temp = a; a = b; b = temp; }
		{ int size_temp = size_a;
		  size_a = size_b;
		  size_b = size_temp; }
	}
	z = _PyLong_New(size_a+1);
	if (z == NULL)
		return NULL;
	for (i = 0; i < size_b; ++i) {
		carry += a->ob_digit[i] + b->ob_digit[i];
		z->ob_digit[i] = carry & MASK;
		/* The following assumes unsigned shifts don't
		   propagate the sign bit. */
		carry >>= SHIFT;
	}
	for (; i < size_a; ++i) {
		carry += a->ob_digit[i];
		z->ob_digit[i] = carry & MASK;
		carry >>= SHIFT;
	}
	z->ob_digit[i] = carry;
	return long_normalize(z);
}

/* Subtract the absolute values of two integers. */

static PyLongObject *x_sub Py_PROTO((PyLongObject *, PyLongObject *));
static PyLongObject *
x_sub(a, b)
	PyLongObject *a, *b;
{
	int size_a = ABS(a->ob_size), size_b = ABS(b->ob_size);
	PyLongObject *z;
	int i;
	int sign = 1;
	digit borrow = 0;
	
	/* Ensure a is the larger of the two: */
	if (size_a < size_b) {
		sign = -1;
		{ PyLongObject *temp = a; a = b; b = temp; }
		{ int size_temp = size_a;
		  size_a = size_b;
		  size_b = size_temp; }
	}
	else if (size_a == size_b) {
		/* Find highest digit where a and b differ: */
		i = size_a;
		while (--i >= 0 && a->ob_digit[i] == b->ob_digit[i])
			;
		if (i < 0)
			return _PyLong_New(0);
		if (a->ob_digit[i] < b->ob_digit[i]) {
			sign = -1;
			{ PyLongObject *temp = a; a = b; b = temp; }
		}
		size_a = size_b = i+1;
	}
	z = _PyLong_New(size_a);
	if (z == NULL)
		return NULL;
	for (i = 0; i < size_b; ++i) {
		/* The following assumes unsigned arithmetic
		   works module 2**N for some N>SHIFT. */
		borrow = a->ob_digit[i] - b->ob_digit[i] - borrow;
		z->ob_digit[i] = borrow & MASK;
		borrow >>= SHIFT;
		borrow &= 1; /* Keep only one sign bit */
	}
	for (; i < size_a; ++i) {
		borrow = a->ob_digit[i] - borrow;
		z->ob_digit[i] = borrow & MASK;
		borrow >>= SHIFT;
	}
	assert(borrow == 0);
	if (sign < 0)
		z->ob_size = -(z->ob_size);
	return long_normalize(z);
}

static PyObject *
long_add(a, b)
	PyLongObject *a;
	PyLongObject *b;
{
	PyLongObject *z;
	
	if (a->ob_size < 0) {
		if (b->ob_size < 0) {
			z = x_add(a, b);
			if (z != NULL && z->ob_size != 0)
				z->ob_size = -(z->ob_size);
		}
		else
			z = x_sub(b, a);
	}
	else {
		if (b->ob_size < 0)
			z = x_sub(a, b);
		else
			z = x_add(a, b);
	}
	return (PyObject *)z;
}

static PyObject *
long_sub(a, b)
	PyLongObject *a;
	PyLongObject *b;
{
	PyLongObject *z;
	
	if (a->ob_size < 0) {
		if (b->ob_size < 0)
			z = x_sub(a, b);
		else
			z = x_add(a, b);
		if (z != NULL && z->ob_size != 0)
			z->ob_size = -(z->ob_size);
	}
	else {
		if (b->ob_size < 0)
			z = x_add(a, b);
		else
			z = x_sub(a, b);
	}
	return (PyObject *)z;
}

static PyObject *
long_mul(a, b)
	PyLongObject *a;
	PyLongObject *b;
{
	int size_a;
	int size_b;
	PyLongObject *z;
	int i;
	
	size_a = ABS(a->ob_size);
	size_b = ABS(b->ob_size);
	z = _PyLong_New(size_a + size_b);
	if (z == NULL)
		return NULL;
	for (i = 0; i < z->ob_size; ++i)
		z->ob_digit[i] = 0;
	for (i = 0; i < size_a; ++i) {
		twodigits carry = 0;
		twodigits f = a->ob_digit[i];
		int j;
		
		SIGCHECK({
			Py_DECREF(z);
			return NULL;
		})
		for (j = 0; j < size_b; ++j) {
			carry += z->ob_digit[i+j] + b->ob_digit[j] * f;
			z->ob_digit[i+j] = (digit) (carry & MASK);
			carry >>= SHIFT;
		}
		for (; carry != 0; ++j) {
			assert(i+j < z->ob_size);
			carry += z->ob_digit[i+j];
			z->ob_digit[i+j] = (digit) (carry & MASK);
			carry >>= SHIFT;
		}
	}
	if (a->ob_size < 0)
		z->ob_size = -(z->ob_size);
	if (b->ob_size < 0)
		z->ob_size = -(z->ob_size);
	return (PyObject *) long_normalize(z);
}

/* The / and % operators are now defined in terms of divmod().
   The expression a mod b has the value a - b*floor(a/b).
   The long_divrem function gives the remainder after division of
   |a| by |b|, with the sign of a.  This is also expressed
   as a - b*trunc(a/b), if trunc truncates towards zero.
   Some examples:
   	 a	 b	a rem b		a mod b
   	 13	 10	 3		 3
   	-13	 10	-3		 7
   	 13	-10	 3		-7
   	-13	-10	-3		-3
   So, to get from rem to mod, we have to add b if a and b
   have different signs.  We then subtract one from the 'div'
   part of the outcome to keep the invariant intact. */

static int l_divmod Py_PROTO((PyLongObject *, PyLongObject *,
	PyLongObject **, PyLongObject **));
static int
l_divmod(v, w, pdiv, pmod)
	PyLongObject *v;
	PyLongObject *w;
	PyLongObject **pdiv;
	PyLongObject **pmod;
{
	PyLongObject *div, *mod;
	
	if (long_divrem(v, w, &div, &mod) < 0)
		return -1;
	if ((mod->ob_size < 0 && w->ob_size > 0) ||
	    (mod->ob_size > 0 && w->ob_size < 0)) {
		PyLongObject *temp;
		PyLongObject *one;
		temp = (PyLongObject *) long_add(mod, w);
		Py_DECREF(mod);
		mod = temp;
		if (mod == NULL) {
			Py_DECREF(div);
			return -1;
		}
		one = (PyLongObject *) PyLong_FromLong(1L);
		if (one == NULL ||
		    (temp = (PyLongObject *) long_sub(div, one)) == NULL) {
			Py_DECREF(mod);
			Py_DECREF(div);
			Py_XDECREF(one);
			return -1;
		}
		Py_DECREF(one);
		Py_DECREF(div);
		div = temp;
	}
	*pdiv = div;
	*pmod = mod;
	return 0;
}

static PyObject *
long_div(v, w)
	PyLongObject *v;
	PyLongObject *w;
{
	PyLongObject *div, *mod;
	if (l_divmod(v, w, &div, &mod) < 0)
		return NULL;
	Py_DECREF(mod);
	return (PyObject *)div;
}

static PyObject *
long_mod(v, w)
	PyLongObject *v;
	PyLongObject *w;
{
	PyLongObject *div, *mod;
	if (l_divmod(v, w, &div, &mod) < 0)
		return NULL;
	Py_DECREF(div);
	return (PyObject *)mod;
}

static PyObject *
long_divmod(v, w)
	PyLongObject *v;
	PyLongObject *w;
{
	PyObject *z;
	PyLongObject *div, *mod;
	if (l_divmod(v, w, &div, &mod) < 0)
		return NULL;
	z = PyTuple_New(2);
	if (z != NULL) {
		PyTuple_SetItem(z, 0, (PyObject *) div);
		PyTuple_SetItem(z, 1, (PyObject *) mod);
	}
	else {
		Py_DECREF(div);
		Py_DECREF(mod);
	}
	return z;
}

static PyObject *
long_pow(a, b, c)
	PyLongObject *a;
	PyLongObject *b;
	PyLongObject *c;
{
	PyLongObject *z, *div, *mod;
	int size_b, i;
	
	size_b = b->ob_size;
	if (size_b < 0) {
		PyErr_SetString(PyExc_ValueError,
				"long integer to the negative power");
		return NULL;
	}
	z = (PyLongObject *)PyLong_FromLong(1L);
	Py_INCREF(a);
	for (i = 0; i < size_b; ++i) {
		digit bi = b->ob_digit[i];
		int j;
	
		for (j = 0; j < SHIFT; ++j) {
			PyLongObject *temp;
		
			if (bi & 1) {
				temp = (PyLongObject *)long_mul(z, a);
				Py_DECREF(z);
			 	if ((PyObject*)c!=Py_None && temp!=NULL) {
			 		l_divmod(temp, c, &div, &mod);
				 	Py_XDECREF(div);
				 	Py_DECREF(temp);
				 	temp = mod;
				}
			 	z = temp;
				if (z == NULL)
					break;
			}
			bi >>= 1;
			if (bi == 0 && i+1 == size_b)
				break;
			temp = (PyLongObject *)long_mul(a, a);
			Py_DECREF(a);
		 	if ((PyObject*)c!=Py_None && temp!=NULL) {
			 	l_divmod(temp, c, &div, &mod);
			 	Py_XDECREF(div);
			 	Py_DECREF(temp);
			 	temp = mod;
			}
			a = temp;
			if (a == NULL) {
				Py_DECREF(z);
				z = NULL;
				break;
			}
		}
		if (a == NULL || z == NULL)
			break;
	}
	Py_XDECREF(a);
	if ((PyObject*)c!=Py_None && z!=NULL) {
			l_divmod(z, c, &div, &mod);
			Py_XDECREF(div);
			Py_DECREF(z);
			z=mod;
	}
	return (PyObject *)z;
}

static PyObject *
long_invert(v)
	PyLongObject *v;
{
	/* Implement ~x as -(x+1) */
	PyLongObject *x;
	PyLongObject *w;
	w = (PyLongObject *)PyLong_FromLong(1L);
	if (w == NULL)
		return NULL;
	x = (PyLongObject *) long_add(v, w);
	Py_DECREF(w);
	if (x == NULL)
		return NULL;
	if (x->ob_size != 0)
		x->ob_size = -(x->ob_size);
	return (PyObject *)x;
}

static PyObject *
long_pos(v)
	PyLongObject *v;
{
	Py_INCREF(v);
	return (PyObject *)v;
}

static PyObject *
long_neg(v)
	PyLongObject *v;
{
	PyLongObject *z;
	int i, n;
	n = ABS(v->ob_size);
	if (n == 0) {
		/* -0 == 0 */
		Py_INCREF(v);
		return (PyObject *) v;
	}
	z = _PyLong_New(ABS(n));
	if (z == NULL)
		return NULL;
	for (i = 0; i < n; i++)
		z->ob_digit[i] = v->ob_digit[i];
	z->ob_size = -(v->ob_size);
	return (PyObject *)z;
}

static PyObject *
long_abs(v)
	PyLongObject *v;
{
	if (v->ob_size < 0)
		return long_neg(v);
	else {
		Py_INCREF(v);
		return (PyObject *)v;
	}
}

static int
long_nonzero(v)
	PyLongObject *v;
{
	return ABS(v->ob_size) != 0;
}

static PyObject *
long_rshift(a, b)
	PyLongObject *a;
	PyLongObject *b;
{
	PyLongObject *z;
	long shiftby;
	int newsize, wordshift, loshift, hishift, i, j;
	digit lomask, himask;
	
	if (a->ob_size < 0) {
		/* Right shifting negative numbers is harder */
		PyLongObject *a1, *a2, *a3;
		a1 = (PyLongObject *) long_invert(a);
		if (a1 == NULL) return NULL;
		a2 = (PyLongObject *) long_rshift(a1, b);
		Py_DECREF(a1);
		if (a2 == NULL) return NULL;
		a3 = (PyLongObject *) long_invert(a2);
		Py_DECREF(a2);
		return (PyObject *) a3;
	}
	
	shiftby = PyLong_AsLong((PyObject *)b);
	if (shiftby == -1L && PyErr_Occurred())
		return NULL;
	if (shiftby < 0) {
		PyErr_SetString(PyExc_ValueError, "negative shift count");
		return NULL;
	}
	wordshift = shiftby / SHIFT;
	newsize = ABS(a->ob_size) - wordshift;
	if (newsize <= 0) {
		z = _PyLong_New(0);
		return (PyObject *)z;
	}
	loshift = shiftby % SHIFT;
	hishift = SHIFT - loshift;
	lomask = ((digit)1 << hishift) - 1;
	himask = MASK ^ lomask;
	z = _PyLong_New(newsize);
	if (z == NULL)
		return NULL;
	if (a->ob_size < 0)
		z->ob_size = -(z->ob_size);
	for (i = 0, j = wordshift; i < newsize; i++, j++) {
		z->ob_digit[i] = (a->ob_digit[j] >> loshift) & lomask;
		if (i+1 < newsize)
			z->ob_digit[i] |=
			  (a->ob_digit[j+1] << hishift) & himask;
	}
	return (PyObject *) long_normalize(z);
}

static PyObject *
long_lshift(a, b)
	PyLongObject *a;
	PyLongObject *b;
{
	/* This version due to Tim Peters */
	PyLongObject *z;
	long shiftby;
	int oldsize, newsize, wordshift, remshift, i, j;
	twodigits accum;
	
	shiftby = PyLong_AsLong((PyObject *)b);
	if (shiftby == -1L && PyErr_Occurred())
		return NULL;
	if (shiftby < 0) {
		PyErr_SetString(PyExc_ValueError, "negative shift count");
		return NULL;
	}
	if ((long)(int)shiftby != shiftby) {
		PyErr_SetString(PyExc_ValueError,
				"outrageous left shift count");
		return NULL;
	}
	/* wordshift, remshift = divmod(shiftby, SHIFT) */
	wordshift = (int)shiftby / SHIFT;
	remshift  = (int)shiftby - wordshift * SHIFT;

	oldsize = ABS(a->ob_size);
	newsize = oldsize + wordshift;
	if (remshift)
		++newsize;
	z = _PyLong_New(newsize);
	if (z == NULL)
		return NULL;
	if (a->ob_size < 0)
		z->ob_size = -(z->ob_size);
	for (i = 0; i < wordshift; i++)
		z->ob_digit[i] = 0;
	accum = 0;	
	for (i = wordshift, j = 0; j < oldsize; i++, j++) {
		accum |= a->ob_digit[j] << remshift;
		z->ob_digit[i] = (digit)(accum & MASK);
		accum >>= SHIFT;
	}
	if (remshift)
		z->ob_digit[newsize-1] = (digit)accum;
	else	
		assert(!accum);
	return (PyObject *) long_normalize(z);
}


/* Bitwise and/xor/or operations */

#define MAX(x, y) ((x) < (y) ? (y) : (x))
#define MIN(x, y) ((x) > (y) ? (y) : (x))

static PyObject *long_bitwise Py_PROTO((PyLongObject *, int, PyLongObject *));
static PyObject *
long_bitwise(a, op, b)
	PyLongObject *a;
	int op; /* '&', '|', '^' */
	PyLongObject *b;
{
	digit maska, maskb; /* 0 or MASK */
	int negz;
	int size_a, size_b, size_z;
	PyLongObject *z;
	int i;
	digit diga, digb;
	PyObject *v;
	
	if (a->ob_size < 0) {
		a = (PyLongObject *) long_invert(a);
		maska = MASK;
	}
	else {
		Py_INCREF(a);
		maska = 0;
	}
	if (b->ob_size < 0) {
		b = (PyLongObject *) long_invert(b);
		maskb = MASK;
	}
	else {
		Py_INCREF(b);
		maskb = 0;
	}
	
	size_a = a->ob_size;
	size_b = b->ob_size;
	size_z = MAX(size_a, size_b);
	z = _PyLong_New(size_z);
	if (a == NULL || b == NULL || z == NULL) {
		Py_XDECREF(a);
		Py_XDECREF(b);
		Py_XDECREF(z);
		return NULL;
	}
	
	negz = 0;
	switch (op) {
	case '^':
		if (maska != maskb) {
			maska ^= MASK;
			negz = -1;
		}
		break;
	case '&':
		if (maska && maskb) {
			op = '|';
			maska ^= MASK;
			maskb ^= MASK;
			negz = -1;
		}
		break;
	case '|':
		if (maska || maskb) {
			op = '&';
			maska ^= MASK;
			maskb ^= MASK;
			negz = -1;
		}
		break;
	}
	
	for (i = 0; i < size_z; ++i) {
		diga = (i < size_a ? a->ob_digit[i] : 0) ^ maska;
		digb = (i < size_b ? b->ob_digit[i] : 0) ^ maskb;
		switch (op) {
		case '&': z->ob_digit[i] = diga & digb; break;
		case '|': z->ob_digit[i] = diga | digb; break;
		case '^': z->ob_digit[i] = diga ^ digb; break;
		}
	}
	
	Py_DECREF(a);
	Py_DECREF(b);
	z = long_normalize(z);
	if (negz == 0)
		return (PyObject *) z;
	v = long_invert(z);
	Py_DECREF(z);
	return v;
}

static PyObject *
long_and(a, b)
	PyLongObject *a;
	PyLongObject *b;
{
	return long_bitwise(a, '&', b);
}

static PyObject *
long_xor(a, b)
	PyLongObject *a;
	PyLongObject *b;
{
	return long_bitwise(a, '^', b);
}

static PyObject *
long_or(a, b)
	PyLongObject *a;
	PyLongObject *b;
{
	return long_bitwise(a, '|', b);
}

static int
long_coerce(pv, pw)
	PyObject **pv;
	PyObject **pw;
{
	if (PyInt_Check(*pw)) {
		*pw = PyLong_FromLong(PyInt_AsLong(*pw));
		Py_INCREF(*pv);
		return 0;
	}
	return 1; /* Can't do it */
}

static PyObject *
long_int(v)
	PyObject *v;
{
	long x;
	x = PyLong_AsLong(v);
	if (PyErr_Occurred())
		return NULL;
	return PyInt_FromLong(x);
}

static PyObject *
long_long(v)
	PyObject *v;
{
	Py_INCREF(v);
	return v;
}

static PyObject *
long_float(v)
	PyObject *v;
{
	double result;
	PyFPE_START_PROTECT("long_float", return 0)
	result = PyLong_AsDouble(v);
	PyFPE_END_PROTECT(result)
	return PyFloat_FromDouble(result);
}

static PyObject *
long_oct(v)
	PyObject *v;
{
	return long_format(v, 8);
}

static PyObject *
long_hex(v)
	PyObject *v;
{
	return long_format(v, 16);
}


#define UF (unaryfunc)
#define BF (binaryfunc)
#define TF (ternaryfunc)
#define IF (inquiry)

static PyNumberMethods long_as_number = {
	BF long_add,	/*nb_add*/
	BF long_sub,	/*nb_subtract*/
	BF long_mul,	/*nb_multiply*/
	BF long_div,	/*nb_divide*/
	BF long_mod,	/*nb_remainder*/
	BF long_divmod,	/*nb_divmod*/
	TF long_pow,	/*nb_power*/
	UF long_neg,	/*nb_negative*/
	UF long_pos,	/*tp_positive*/
	UF long_abs,	/*tp_absolute*/
	IF long_nonzero,/*tp_nonzero*/
	UF long_invert,	/*nb_invert*/
	BF long_lshift,	/*nb_lshift*/
	BF long_rshift,	/*nb_rshift*/
	BF long_and,	/*nb_and*/
	BF long_xor,	/*nb_xor*/
	BF long_or,	/*nb_or*/
	(int (*) Py_FPROTO((PyObject **, PyObject **)))
	(coercion)long_coerce, /*nb_coerce*/
	UF long_int,	/*nb_int*/
	UF long_long,	/*nb_long*/
	UF long_float,	/*nb_float*/
	UF long_oct,	/*nb_oct*/
	UF long_hex,	/*nb_hex*/
};

PyTypeObject PyLong_Type = {
	PyObject_HEAD_INIT(&PyType_Type)
	0,
	"long int",
	sizeof(PyLongObject) - sizeof(digit),
	sizeof(digit),
	(destructor)long_dealloc, /*tp_dealloc*/
	0,		/*tp_print*/
	0,		/*tp_getattr*/
	0,		/*tp_setattr*/
	(int (*) Py_FPROTO((PyObject *, PyObject *)))
	(cmpfunc)long_compare, /*tp_compare*/
	(reprfunc)long_repr, /*tp_repr*/
	&long_as_number,/*tp_as_number*/
	0,		/*tp_as_sequence*/
	0,		/*tp_as_mapping*/
	(long (*) Py_FPROTO((PyObject *)))
	(hashfunc)long_hash, /*tp_hash*/
};