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

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

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

#include <ctype.h>

/* For long multiplication, use the O(N**2) school algorithm unless
 * both operands contain more than KARATSUBA_CUTOFF digits (this
 * being an internal Python long digit, in base PyLong_BASE).
 */
#define KARATSUBA_CUTOFF 70
#define KARATSUBA_SQUARE_CUTOFF (2 * KARATSUBA_CUTOFF)

/* For exponentiation, use the binary left-to-right algorithm
 * unless the exponent contains more than FIVEARY_CUTOFF digits.
 * In that case, do 5 bits at a time.  The potential drawback is that
 * a table of 2**5 intermediate results is computed.
 */
#define FIVEARY_CUTOFF 8

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

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

/* Forward */
static PyLongObject *long_normalize(PyLongObject *);
static PyLongObject *mul1(PyLongObject *, wdigit);
static PyLongObject *muladd1(PyLongObject *, wdigit, wdigit);
static PyLongObject *divrem1(PyLongObject *, digit, digit *);

#define SIGCHECK(PyTryBlock) \
	if (--_Py_Ticker < 0) { \
		_Py_Ticker = _Py_CheckInterval; \
		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(register PyLongObject *v)
{
	Py_ssize_t j = ABS(Py_SIZE(v));
	Py_ssize_t i = j;

	while (i > 0 && v->ob_digit[i-1] == 0)
		--i;
	if (i != j)
		Py_SIZE(v) = (Py_SIZE(v) < 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(Py_ssize_t size)
{
	if (size > PY_SSIZE_T_MAX) {
		PyErr_NoMemory();
		return NULL;
	}
	/* coverity[ampersand_in_size] */
	/* XXX(nnorwitz): This can overflow --
           PyObject_NEW_VAR / _PyObject_VAR_SIZE need to detect overflow */
	return PyObject_NEW_VAR(PyLongObject, &PyLong_Type, size);
}

PyObject *
_PyLong_Copy(PyLongObject *src)
{
	PyLongObject *result;
	Py_ssize_t i;

	assert(src != NULL);
	i = src->ob_size;
	if (i < 0)
		i = -(i);
	result = _PyLong_New(i);
	if (result != NULL) {
		result->ob_size = src->ob_size;
		while (--i >= 0)
			result->ob_digit[i] = src->ob_digit[i];
	}
	return (PyObject *)result;
}

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

PyObject *
PyLong_FromLong(long ival)
{
	PyLongObject *v;
        unsigned long abs_ival;
	unsigned long t;  /* unsigned so >> doesn't propagate sign bit */
	int ndigits = 0;
	int negative = 0;

	if (ival < 0) {
		/* if LONG_MIN == -LONG_MAX-1 (true on most platforms) then
		   ANSI C says that the result of -ival is undefined when ival
		   == LONG_MIN.  Hence the following workaround. */
		abs_ival = (unsigned long)(-1-ival) + 1;
		negative = 1;
	}
	else {
		abs_ival = (unsigned long)ival;
	}

	/* Count the number of Python digits.
	   We used to pick 5 ("big enough for anything"), but that's a
	   waste of time and space given that 5*15 = 75 bits are rarely
	   needed. */
	t = abs_ival;
	while (t) {
		++ndigits;
		t >>= PyLong_SHIFT;
	}
	v = _PyLong_New(ndigits);
	if (v != NULL) {
		digit *p = v->ob_digit;
		v->ob_size = negative ? -ndigits : ndigits;
		t = abs_ival;
		while (t) {
			*p++ = (digit)(t & PyLong_MASK);
			t >>= PyLong_SHIFT;
		}
	}
	return (PyObject *)v;
}

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

PyObject *
PyLong_FromUnsignedLong(unsigned long ival)
{
	PyLongObject *v;
	unsigned long t;
	int ndigits = 0;

	/* Count the number of Python digits. */
	t = (unsigned long)ival;
	while (t) {
		++ndigits;
		t >>= PyLong_SHIFT;
	}
	v = _PyLong_New(ndigits);
	if (v != NULL) {
		digit *p = v->ob_digit;
		Py_SIZE(v) = ndigits;
		while (ival) {
			*p++ = (digit)(ival & PyLong_MASK);
			ival >>= PyLong_SHIFT;
		}
	}
	return (PyObject *)v;
}

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

PyObject *
PyLong_FromDouble(double dval)
{
	PyLongObject *v;
	double frac;
	int i, ndig, expo, neg;
	neg = 0;
	if (Py_IS_INFINITY(dval)) {
		PyErr_SetString(PyExc_OverflowError,
			"cannot convert float infinity to integer");
		return NULL;
	}
	if (Py_IS_NAN(dval)) {
		PyErr_SetString(PyExc_ValueError,
			"cannot convert float NaN to integer");
		return NULL;
	}
	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) / PyLong_SHIFT + 1; /* Number of 'digits' in result */
	v = _PyLong_New(ndig);
	if (v == NULL)
		return NULL;
	frac = ldexp(frac, (expo-1) % PyLong_SHIFT + 1);
	for (i = ndig; --i >= 0; ) {
		long bits = (long)frac;
		v->ob_digit[i] = (digit) bits;
		frac = frac - (double)bits;
		frac = ldexp(frac, PyLong_SHIFT);
	}
	if (neg)
		Py_SIZE(v) = -(Py_SIZE(v));
	return (PyObject *)v;
}

/* Checking for overflow in PyLong_AsLong is a PITA since C doesn't define
 * anything about what happens when a signed integer operation overflows,
 * and some compilers think they're doing you a favor by being "clever"
 * then.  The bit pattern for the largest postive signed long is
 * (unsigned long)LONG_MAX, and for the smallest negative signed long
 * it is abs(LONG_MIN), which we could write -(unsigned long)LONG_MIN.
 * However, some other compilers warn about applying unary minus to an
 * unsigned operand.  Hence the weird "0-".
 */
#define PY_ABS_LONG_MIN		(0-(unsigned long)LONG_MIN)
#define PY_ABS_SSIZE_T_MIN	(0-(size_t)PY_SSIZE_T_MIN)

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

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

	if (vv == NULL || !PyLong_Check(vv)) {
		if (vv != NULL && PyInt_Check(vv))
			return PyInt_AsLong(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 << PyLong_SHIFT) + v->ob_digit[i];
		if ((x >> PyLong_SHIFT) != prev)
			goto overflow;
	}
	/* Haven't lost any bits, but casting to long requires extra care
	 * (see comment above).
         */
	if (x <= (unsigned long)LONG_MAX) {
		return (long)x * sign;
	}
	else if (sign < 0 && x == PY_ABS_LONG_MIN) {
		return LONG_MIN;
	}
	/* else overflow */

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

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

Py_ssize_t
PyLong_AsSsize_t(PyObject *vv) {
	register PyLongObject *v;
	size_t x, prev;
	Py_ssize_t i;
	int 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 << PyLong_SHIFT) + v->ob_digit[i];
		if ((x >> PyLong_SHIFT) != prev)
			goto overflow;
	}
	/* Haven't lost any bits, but casting to a signed type requires
	 * extra care (see comment above).
	 */
	if (x <= (size_t)PY_SSIZE_T_MAX) {
		return (Py_ssize_t)x * sign;
	}
	else if (sign < 0 && x == PY_ABS_SSIZE_T_MIN) {
		return PY_SSIZE_T_MIN;
	}
	/* else overflow */

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

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

unsigned long
PyLong_AsUnsignedLong(PyObject *vv)
{
	register PyLongObject *v;
	unsigned long x, prev;
	Py_ssize_t i;

	if (vv == NULL || !PyLong_Check(vv)) {
		if (vv != NULL && PyInt_Check(vv)) {
			long val = PyInt_AsLong(vv);
			if (val < 0) {
				PyErr_SetString(PyExc_OverflowError,
				"can't convert negative value to unsigned long");
				return (unsigned long) -1;
			}
			return val;
		}
		PyErr_BadInternalCall();
		return (unsigned long) -1;
	}
	v = (PyLongObject *)vv;
	i = Py_SIZE(v);
	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 << PyLong_SHIFT) + v->ob_digit[i];
		if ((x >> PyLong_SHIFT) != prev) {
			PyErr_SetString(PyExc_OverflowError,
				"long int too large to convert");
			return (unsigned long) -1;
		}
	}
	return x;
}

/* Get a C unsigned long int from a long int object, ignoring the high bits.
   Returns -1 and sets an error condition if an error occurs. */

unsigned long
PyLong_AsUnsignedLongMask(PyObject *vv)
{
	register PyLongObject *v;
	unsigned long x;
	Py_ssize_t i;
	int sign;

	if (vv == NULL || !PyLong_Check(vv)) {
		if (vv != NULL && PyInt_Check(vv))
			return PyInt_AsUnsignedLongMask(vv);
		PyErr_BadInternalCall();
		return (unsigned long) -1;
	}
	v = (PyLongObject *)vv;
	i = v->ob_size;
	sign = 1;
	x = 0;
	if (i < 0) {
		sign = -1;
		i = -i;
	}
	while (--i >= 0) {
		x = (x << PyLong_SHIFT) + v->ob_digit[i];
	}
	return x * sign;
}

int
_PyLong_Sign(PyObject *vv)
{
	PyLongObject *v = (PyLongObject *)vv;

	assert(v != NULL);
	assert(PyLong_Check(v));

	return Py_SIZE(v) == 0 ? 0 : (Py_SIZE(v) < 0 ? -1 : 1);
}

size_t
_PyLong_NumBits(PyObject *vv)
{
	PyLongObject *v = (PyLongObject *)vv;
	size_t result = 0;
	Py_ssize_t ndigits;

	assert(v != NULL);
	assert(PyLong_Check(v));
	ndigits = ABS(Py_SIZE(v));
	assert(ndigits == 0 || v->ob_digit[ndigits - 1] != 0);
	if (ndigits > 0) {
		digit msd = v->ob_digit[ndigits - 1];

		result = (ndigits - 1) * PyLong_SHIFT;
		if (result / PyLong_SHIFT != (size_t)(ndigits - 1))
			goto Overflow;
		do {
			++result;
			if (result == 0)
				goto Overflow;
			msd >>= 1;
		} while (msd);
	}
	return result;

Overflow:
	PyErr_SetString(PyExc_OverflowError, "long has too many bits "
			"to express in a platform size_t");
	return (size_t)-1;
}

PyObject *
_PyLong_FromByteArray(const unsigned char* bytes, size_t n,
		      int little_endian, int is_signed)
{
	const unsigned char* pstartbyte;/* LSB of bytes */
	int incr;			/* direction to move pstartbyte */
	const unsigned char* pendbyte;	/* MSB of bytes */
	size_t numsignificantbytes;	/* number of bytes that matter */
	size_t ndigits;			/* number of Python long digits */
	PyLongObject* v;		/* result */
	int idigit = 0;  		/* next free index in v->ob_digit */

	if (n == 0)
		return PyLong_FromLong(0L);

	if (little_endian) {
		pstartbyte = bytes;
		pendbyte = bytes + n - 1;
		incr = 1;
	}
	else {
		pstartbyte = bytes + n - 1;
		pendbyte = bytes;
		incr = -1;
	}

	if (is_signed)
		is_signed = *pendbyte >= 0x80;

	/* Compute numsignificantbytes.  This consists of finding the most
	   significant byte.  Leading 0 bytes are insignficant if the number
	   is positive, and leading 0xff bytes if negative. */
	{
		size_t i;
		const unsigned char* p = pendbyte;
		const int pincr = -incr;  /* search MSB to LSB */
		const unsigned char insignficant = is_signed ? 0xff : 0x00;

		for (i = 0; i < n; ++i, p += pincr) {
			if (*p != insignficant)
				break;
		}
		numsignificantbytes = n - i;
		/* 2's-comp is a bit tricky here, e.g. 0xff00 == -0x0100, so
		   actually has 2 significant bytes.  OTOH, 0xff0001 ==
		   -0x00ffff, so we wouldn't *need* to bump it there; but we
		   do for 0xffff = -0x0001.  To be safe without bothering to
		   check every case, bump it regardless. */
		if (is_signed && numsignificantbytes < n)
			++numsignificantbytes;
	}

	/* How many Python long digits do we need?  We have
	   8*numsignificantbytes bits, and each Python long digit has PyLong_SHIFT
	   bits, so it's the ceiling of the quotient. */
	ndigits = (numsignificantbytes * 8 + PyLong_SHIFT - 1) / PyLong_SHIFT;
	if (ndigits > (size_t)INT_MAX)
		return PyErr_NoMemory();
	v = _PyLong_New((int)ndigits);
	if (v == NULL)
		return NULL;

	/* Copy the bits over.  The tricky parts are computing 2's-comp on
	   the fly for signed numbers, and dealing with the mismatch between
	   8-bit bytes and (probably) 15-bit Python digits.*/
	{
		size_t i;
		twodigits carry = 1;		/* for 2's-comp calculation */
		twodigits accum = 0;		/* sliding register */
		unsigned int accumbits = 0; 	/* number of bits in accum */
		const unsigned char* p = pstartbyte;

		for (i = 0; i < numsignificantbytes; ++i, p += incr) {
			twodigits thisbyte = *p;
			/* Compute correction for 2's comp, if needed. */
			if (is_signed) {
				thisbyte = (0xff ^ thisbyte) + carry;
				carry = thisbyte >> 8;
				thisbyte &= 0xff;
			}
			/* Because we're going LSB to MSB, thisbyte is
			   more significant than what's already in accum,
			   so needs to be prepended to accum. */
			accum |= (twodigits)thisbyte << accumbits;
			accumbits += 8;
			if (accumbits >= PyLong_SHIFT) {
				/* There's enough to fill a Python digit. */
				assert(idigit < (int)ndigits);
				v->ob_digit[idigit] = (digit)(accum & PyLong_MASK);
				++idigit;
				accum >>= PyLong_SHIFT;
				accumbits -= PyLong_SHIFT;
				assert(accumbits < PyLong_SHIFT);
			}
		}
		assert(accumbits < PyLong_SHIFT);
		if (accumbits) {
			assert(idigit < (int)ndigits);
			v->ob_digit[idigit] = (digit)accum;
			++idigit;
		}
	}

	Py_SIZE(v) = is_signed ? -idigit : idigit;
	return (PyObject *)long_normalize(v);
}

int
_PyLong_AsByteArray(PyLongObject* v,
		    unsigned char* bytes, size_t n,
		    int little_endian, int is_signed)
{
	Py_ssize_t i;		/* index into v->ob_digit */
	Py_ssize_t ndigits;		/* |v->ob_size| */
	twodigits accum;	/* sliding register */
	unsigned int accumbits; /* # bits in accum */
	int do_twos_comp;	/* store 2's-comp?  is_signed and v < 0 */
	digit carry;		/* for computing 2's-comp */
	size_t j;		/* # bytes filled */
	unsigned char* p;	/* pointer to next byte in bytes */
	int pincr;		/* direction to move p */

	assert(v != NULL && PyLong_Check(v));

	if (Py_SIZE(v) < 0) {
		ndigits = -(Py_SIZE(v));
		if (!is_signed) {
			PyErr_SetString(PyExc_TypeError,
				"can't convert negative long to unsigned");
			return -1;
		}
		do_twos_comp = 1;
	}
	else {
		ndigits = Py_SIZE(v);
		do_twos_comp = 0;
	}

	if (little_endian) {
		p = bytes;
		pincr = 1;
	}
	else {
		p = bytes + n - 1;
		pincr = -1;
	}

	/* Copy over all the Python digits.
	   It's crucial that every Python digit except for the MSD contribute
	   exactly PyLong_SHIFT bits to the total, so first assert that the long is
	   normalized. */
	assert(ndigits == 0 || v->ob_digit[ndigits - 1] != 0);
	j = 0;
	accum = 0;
	accumbits = 0;
	carry = do_twos_comp ? 1 : 0;
	for (i = 0; i < ndigits; ++i) {
		digit thisdigit = v->ob_digit[i];
		if (do_twos_comp) {
			thisdigit = (thisdigit ^ PyLong_MASK) + carry;
			carry = thisdigit >> PyLong_SHIFT;
			thisdigit &= PyLong_MASK;
		}
		/* Because we're going LSB to MSB, thisdigit is more
		   significant than what's already in accum, so needs to be
		   prepended to accum. */
		accum |= (twodigits)thisdigit << accumbits;

		/* The most-significant digit may be (probably is) at least
		   partly empty. */
		if (i == ndigits - 1) {
			/* Count # of sign bits -- they needn't be stored,
			 * although for signed conversion we need later to
			 * make sure at least one sign bit gets stored. */
			digit s = do_twos_comp ? thisdigit ^ PyLong_MASK :
				                thisdigit;
			while (s != 0) {
				s >>= 1;
				accumbits++;
			}
		}
		else
			accumbits += PyLong_SHIFT;

		/* Store as many bytes as possible. */
		while (accumbits >= 8) {
			if (j >= n)
				goto Overflow;
			++j;
			*p = (unsigned char)(accum & 0xff);
			p += pincr;
			accumbits -= 8;
			accum >>= 8;
		}
	}

	/* Store the straggler (if any). */
	assert(accumbits < 8);
	assert(carry == 0);  /* else do_twos_comp and *every* digit was 0 */
	if (accumbits > 0) {
		if (j >= n)
			goto Overflow;
		++j;
		if (do_twos_comp) {
			/* Fill leading bits of the byte with sign bits
			   (appropriately pretending that the long had an
			   infinite supply of sign bits). */
			accum |= (~(twodigits)0) << accumbits;
		}
		*p = (unsigned char)(accum & 0xff);
		p += pincr;
	}
	else if (j == n && n > 0 && is_signed) {
		/* The main loop filled the byte array exactly, so the code
		   just above didn't get to ensure there's a sign bit, and the
		   loop below wouldn't add one either.  Make sure a sign bit
		   exists. */
		unsigned char msb = *(p - pincr);
		int sign_bit_set = msb >= 0x80;
		assert(accumbits == 0);
		if (sign_bit_set == do_twos_comp)
			return 0;
		else
			goto Overflow;
	}

	/* Fill remaining bytes with copies of the sign bit. */
	{
		unsigned char signbyte = do_twos_comp ? 0xffU : 0U;
		for ( ; j < n; ++j, p += pincr)
			*p = signbyte;
	}

	return 0;

Overflow:
	PyErr_SetString(PyExc_OverflowError, "long too big to convert");
	return -1;

}

double
_PyLong_AsScaledDouble(PyObject *vv, int *exponent)
{
/* NBITS_WANTED should be > the number of bits in a double's precision,
   but small enough so that 2**NBITS_WANTED is within the normal double
   range.  nbitsneeded is set to 1 less than that because the most-significant
   Python digit contains at least 1 significant bit, but we don't want to
   bother counting them (catering to the worst case cheaply).

   57 is one more than VAX-D double precision; I (Tim) don't know of a double
   format with more precision than that; it's 1 larger so that we add in at
   least one round bit to stand in for the ignored least-significant bits.
*/
#define NBITS_WANTED 57
	PyLongObject *v;
	double x;
	const double multiplier = (double)(1L << PyLong_SHIFT);
	Py_ssize_t i;
	int sign;
	int nbitsneeded;

	if (vv == NULL || !PyLong_Check(vv)) {
		PyErr_BadInternalCall();
		return -1;
	}
	v = (PyLongObject *)vv;
	i = Py_SIZE(v);
	sign = 1;
	if (i < 0) {
		sign = -1;
		i = -(i);
	}
	else if (i == 0) {
		*exponent = 0;
		return 0.0;
	}
	--i;
	x = (double)v->ob_digit[i];
	nbitsneeded = NBITS_WANTED - 1;
	/* Invariant:  i Python digits remain unaccounted for. */
	while (i > 0 && nbitsneeded > 0) {
		--i;
		x = x * multiplier + (double)v->ob_digit[i];
		nbitsneeded -= PyLong_SHIFT;
	}
	/* There are i digits we didn't shift in.  Pretending they're all
	   zeroes, the true value is x * 2**(i*PyLong_SHIFT). */
	*exponent = i;
	assert(x > 0.0);
	return x * sign;
#undef NBITS_WANTED
}

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

double
PyLong_AsDouble(PyObject *vv)
{
	int e = -1;
	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:
	PyErr_SetString(PyExc_OverflowError,
		"long int too large to convert to float");
	return -1.0;
}

/* Create a new long (or int) object from a C pointer */

PyObject *
PyLong_FromVoidPtr(void *p)
{
#if SIZEOF_VOID_P <= SIZEOF_LONG
	if ((long)p < 0)
		return PyLong_FromUnsignedLong((unsigned long)p);
	return PyInt_FromLong((long)p);
#else

#ifndef HAVE_LONG_LONG
#   error "PyLong_FromVoidPtr: sizeof(void*) > sizeof(long), but no long long"
#endif
#if SIZEOF_LONG_LONG < SIZEOF_VOID_P
#   error "PyLong_FromVoidPtr: sizeof(PY_LONG_LONG) < sizeof(void*)"
#endif
	/* optimize null pointers */
	if (p == NULL)
		return PyInt_FromLong(0);
	return PyLong_FromUnsignedLongLong((unsigned PY_LONG_LONG)p);

#endif /* SIZEOF_VOID_P <= SIZEOF_LONG */
}

/* Get a C pointer from a long object (or an int object in some cases) */

void *
PyLong_AsVoidPtr(PyObject *vv)
{
	/* This function will allow int or long objects. If vv is neither,
	   then the PyLong_AsLong*() functions will raise the exception:
	   PyExc_SystemError, "bad argument to internal function"
	*/
#if SIZEOF_VOID_P <= SIZEOF_LONG
	long x;

	if (PyInt_Check(vv))
		x = PyInt_AS_LONG(vv);
	else if (PyLong_Check(vv) && _PyLong_Sign(vv) < 0)
		x = PyLong_AsLong(vv);
	else
		x = PyLong_AsUnsignedLong(vv);
#else

#ifndef HAVE_LONG_LONG
#   error "PyLong_AsVoidPtr: sizeof(void*) > sizeof(long), but no long long"
#endif
#if SIZEOF_LONG_LONG < SIZEOF_VOID_P
#   error "PyLong_AsVoidPtr: sizeof(PY_LONG_LONG) < sizeof(void*)"
#endif
	PY_LONG_LONG x;

	if (PyInt_Check(vv))
		x = PyInt_AS_LONG(vv);
	else if (PyLong_Check(vv) && _PyLong_Sign(vv) < 0)
		x = PyLong_AsLongLong(vv);
	else
		x = PyLong_AsUnsignedLongLong(vv);

#endif /* SIZEOF_VOID_P <= SIZEOF_LONG */

	if (x == -1 && PyErr_Occurred())
		return NULL;
	return (void *)x;
}

#ifdef HAVE_LONG_LONG

/* Initial PY_LONG_LONG support by Chris Herborth (chrish@qnx.com), later
 * rewritten to use the newer PyLong_{As,From}ByteArray API.
 */

#define IS_LITTLE_ENDIAN (int)*(unsigned char*)&one

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

PyObject *
PyLong_FromLongLong(PY_LONG_LONG ival)
{
	PyLongObject *v;
	unsigned PY_LONG_LONG abs_ival;
	unsigned PY_LONG_LONG t;  /* unsigned so >> doesn't propagate sign bit */
	int ndigits = 0;
	int negative = 0;

	if (ival < 0) {
		/* avoid signed overflow on negation;  see comments
		   in PyLong_FromLong above. */
		abs_ival = (unsigned PY_LONG_LONG)(-1-ival) + 1;
		negative = 1;
	}
	else {
		abs_ival = (unsigned PY_LONG_LONG)ival;
	}

	/* Count the number of Python digits.
	   We used to pick 5 ("big enough for anything"), but that's a
	   waste of time and space given that 5*15 = 75 bits are rarely
	   needed. */
	t = abs_ival;
	while (t) {
		++ndigits;
		t >>= PyLong_SHIFT;
	}
	v = _PyLong_New(ndigits);
	if (v != NULL) {
		digit *p = v->ob_digit;
		Py_SIZE(v) = negative ? -ndigits : ndigits;
		t = abs_ival;
		while (t) {
			*p++ = (digit)(t & PyLong_MASK);
			t >>= PyLong_SHIFT;
		}
	}
	return (PyObject *)v;
}

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

PyObject *
PyLong_FromUnsignedLongLong(unsigned PY_LONG_LONG ival)
{
	PyLongObject *v;
	unsigned PY_LONG_LONG t;
	int ndigits = 0;

	/* Count the number of Python digits. */
	t = (unsigned PY_LONG_LONG)ival;
	while (t) {
		++ndigits;
		t >>= PyLong_SHIFT;
	}
	v = _PyLong_New(ndigits);
	if (v != NULL) {
		digit *p = v->ob_digit;
		Py_SIZE(v) = ndigits;
		while (ival) {
			*p++ = (digit)(ival & PyLong_MASK);
			ival >>= PyLong_SHIFT;
		}
	}
	return (PyObject *)v;
}

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

PyObject *
PyLong_FromSsize_t(Py_ssize_t ival)
{
	Py_ssize_t bytes = ival;
	int one = 1;
	return _PyLong_FromByteArray(
			(unsigned char *)&bytes,
			SIZEOF_SIZE_T, IS_LITTLE_ENDIAN, 1);
}

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

PyObject *
PyLong_FromSize_t(size_t ival)
{
	size_t bytes = ival;
	int one = 1;
	return _PyLong_FromByteArray(
			(unsigned char *)&bytes,
			SIZEOF_SIZE_T, IS_LITTLE_ENDIAN, 0);
}

/* Get a C PY_LONG_LONG int from a long int object.
   Return -1 and set an error if overflow occurs. */

PY_LONG_LONG
PyLong_AsLongLong(PyObject *vv)
{
	PY_LONG_LONG bytes;
	int one = 1;
	int res;

	if (vv == NULL) {
		PyErr_BadInternalCall();
		return -1;
	}
	if (!PyLong_Check(vv)) {
		PyNumberMethods *nb;
		PyObject *io;
		if (PyInt_Check(vv))
			return (PY_LONG_LONG)PyInt_AsLong(vv);
		if ((nb = vv->ob_type->tp_as_number) == NULL ||
		    nb->nb_int == NULL) {
			PyErr_SetString(PyExc_TypeError, "an integer is required");
			return -1;
		}
		io = (*nb->nb_int) (vv);
		if (io == NULL)
			return -1;
		if (PyInt_Check(io)) {
			bytes = PyInt_AsLong(io);
			Py_DECREF(io);
			return bytes;
		}
		if (PyLong_Check(io)) {
			bytes = PyLong_AsLongLong(io);
			Py_DECREF(io);
			return bytes;
		}
		Py_DECREF(io);
		PyErr_SetString(PyExc_TypeError, "integer conversion failed");
		return -1;
	}

	res = _PyLong_AsByteArray(
			(PyLongObject *)vv, (unsigned char *)&bytes,
			SIZEOF_LONG_LONG, IS_LITTLE_ENDIAN, 1);

	/* Plan 9 can't handle PY_LONG_LONG in ? : expressions */
	if (res < 0)
		return (PY_LONG_LONG)-1;
	else
		return bytes;
}

/* Get a C unsigned PY_LONG_LONG int from a long int object.
   Return -1 and set an error if overflow occurs. */

unsigned PY_LONG_LONG
PyLong_AsUnsignedLongLong(PyObject *vv)
{
	unsigned PY_LONG_LONG bytes;
	int one = 1;
	int res;

	if (vv == NULL || !PyLong_Check(vv)) {
		PyErr_BadInternalCall();
		return (unsigned PY_LONG_LONG)-1;
	}

	res = _PyLong_AsByteArray(
			(PyLongObject *)vv, (unsigned char *)&bytes,
			SIZEOF_LONG_LONG, IS_LITTLE_ENDIAN, 0);

	/* Plan 9 can't handle PY_LONG_LONG in ? : expressions */
	if (res < 0)
		return (unsigned PY_LONG_LONG)res;
	else
		return bytes;
}

/* Get a C unsigned long int from a long int object, ignoring the high bits.
   Returns -1 and sets an error condition if an error occurs. */

unsigned PY_LONG_LONG
PyLong_AsUnsignedLongLongMask(PyObject *vv)
{
	register PyLongObject *v;
	unsigned PY_LONG_LONG x;
	Py_ssize_t i;
	int sign;

	if (vv == NULL || !PyLong_Check(vv)) {
		PyErr_BadInternalCall();
		return (unsigned long) -1;
	}
	v = (PyLongObject *)vv;
	i = v->ob_size;
	sign = 1;
	x = 0;
	if (i < 0) {
		sign = -1;
		i = -i;
	}
	while (--i >= 0) {
		x = (x << PyLong_SHIFT) + v->ob_digit[i];
	}
	return x * sign;
}
#undef IS_LITTLE_ENDIAN

#endif /* HAVE_LONG_LONG */


static int
convert_binop(PyObject *v, PyObject *w, PyLongObject **a, PyLongObject **b) {
	if (PyLong_Check(v)) {
		*a = (PyLongObject *) v;
		Py_INCREF(v);
	}
	else if (PyInt_Check(v)) {
		*a = (PyLongObject *) PyLong_FromLong(PyInt_AS_LONG(v));
	}
	else {
		return 0;
	}
	if (PyLong_Check(w)) {
		*b = (PyLongObject *) w;
		Py_INCREF(w);
	}
	else if (PyInt_Check(w)) {
		*b = (PyLongObject *) PyLong_FromLong(PyInt_AS_LONG(w));
	}
	else {
		Py_DECREF(*a);
		return 0;
	}
	return 1;
}

#define CONVERT_BINOP(v, w, a, b) \
	if (!convert_binop(v, w, a, b)) { \
		Py_INCREF(Py_NotImplemented); \
		return Py_NotImplemented; \
	}

/* x[0:m] and y[0:n] are digit vectors, LSD first, m >= n required.  x[0:n]
 * is modified in place, by adding y to it.  Carries are propagated as far as
 * x[m-1], and the remaining carry (0 or 1) is returned.
 */
static digit
v_iadd(digit *x, Py_ssize_t m, digit *y, Py_ssize_t n)
{
	Py_ssize_t i;
	digit carry = 0;

	assert(m >= n);
	for (i = 0; i < n; ++i) {
		carry += x[i] + y[i];
		x[i] = carry & PyLong_MASK;
		carry >>= PyLong_SHIFT;
		assert((carry & 1) == carry);
	}
	for (; carry && i < m; ++i) {
		carry += x[i];
		x[i] = carry & PyLong_MASK;
		carry >>= PyLong_SHIFT;
		assert((carry & 1) == carry);
	}
	return carry;
}

/* x[0:m] and y[0:n] are digit vectors, LSD first, m >= n required.  x[0:n]
 * is modified in place, by subtracting y from it.  Borrows are propagated as
 * far as x[m-1], and the remaining borrow (0 or 1) is returned.
 */
static digit
v_isub(digit *x, Py_ssize_t m, digit *y, Py_ssize_t n)
{
	Py_ssize_t i;
	digit borrow = 0;

	assert(m >= n);
	for (i = 0; i < n; ++i) {
		borrow = x[i] - y[i] - borrow;
		x[i] = borrow & PyLong_MASK;
		borrow >>= PyLong_SHIFT;
		borrow &= 1;	/* keep only 1 sign bit */
	}
	for (; borrow && i < m; ++i) {
		borrow = x[i] - borrow;
		x[i] = borrow & PyLong_MASK;
		borrow >>= PyLong_SHIFT;
		borrow &= 1;
	}
	return borrow;
}

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

static PyLongObject *
mul1(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(PyLongObject *a, wdigit n, wdigit extra)
{
	Py_ssize_t size_a = ABS(Py_SIZE(a));
	PyLongObject *z = _PyLong_New(size_a+1);
	twodigits carry = extra;
	Py_ssize_t 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 & PyLong_MASK);
		carry >>= PyLong_SHIFT;
	}
	z->ob_digit[i] = (digit) carry;
	return long_normalize(z);
}

/* Divide long pin, w/ size digits, by non-zero digit n, storing quotient
   in pout, and returning the remainder.  pin and pout point at the LSD.
   It's OK for pin == pout on entry, which saves oodles of mallocs/frees in
   _PyLong_Format, but that should be done with great care since longs are
   immutable. */

static digit
inplace_divrem1(digit *pout, digit *pin, Py_ssize_t size, digit n)
{
	twodigits rem = 0;

	assert(n > 0 && n <= PyLong_MASK);
	pin += size;
	pout += size;
	while (--size >= 0) {
		digit hi;
		rem = (rem << PyLong_SHIFT) + *--pin;
		*--pout = hi = (digit)(rem / n);
		rem -= (twodigits)hi * n;
	}
	return (digit)rem;
}

/* 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(PyLongObject *a, digit n, digit *prem)
{
	const Py_ssize_t size = ABS(Py_SIZE(a));
	PyLongObject *z;

	assert(n > 0 && n <= PyLong_MASK);
	z = _PyLong_New(size);
	if (z == NULL)
		return NULL;
	*prem = inplace_divrem1(z->ob_digit, a->ob_digit, size, n);
	return long_normalize(z);
}

/* Convert the long to a string object with given base,
   appending a base prefix of 0[box] if base is 2, 8 or 16.
   Add a trailing "L" if addL is non-zero.
   If newstyle is zero, then use the pre-2.6 behavior of octal having
   a leading "0", instead of the prefix "0o" */
PyAPI_FUNC(PyObject *)
_PyLong_Format(PyObject *aa, int base, int addL, int newstyle)
{
	register PyLongObject *a = (PyLongObject *)aa;
	PyStringObject *str;
	Py_ssize_t i, sz;
	Py_ssize_t size_a;
	char *p;
	int bits;
	char sign = '\0';

	if (a == NULL || !PyLong_Check(a)) {
		PyErr_BadInternalCall();
		return NULL;
	}
	assert(base >= 2 && base <= 36);
	size_a = ABS(Py_SIZE(a));

	/* Compute a rough upper bound for the length of the string */
	i = base;
	bits = 0;
	while (i > 1) {
		++bits;
		i >>= 1;
	}
	i = 5 + (addL ? 1 : 0);
	/* ensure we don't get signed overflow in sz calculation */
	if (size_a > (PY_SSIZE_T_MAX - i) / PyLong_SHIFT) {
		PyErr_SetString(PyExc_OverflowError,
				"long is too large to format");
		return NULL;
	}
	sz = i + 1 + (size_a * PyLong_SHIFT - 1) / bits;
	assert(sz >= 0);
	str = (PyStringObject *) PyString_FromStringAndSize((char *)0, sz);
	if (str == NULL)
		return NULL;
	p = PyString_AS_STRING(str) + sz;
	*p = '\0';
	if (addL)
		*--p = 'L';
	if (a->ob_size < 0)
		sign = '-';

	if (a->ob_size == 0) {
		*--p = '0';
	}
	else if ((base & (base - 1)) == 0) {
		/* JRH: special case for power-of-2 bases */
		twodigits accum = 0;
		int accumbits = 0;	/* # of bits in accum */
		int basebits = 1;	/* # of bits in base-1 */
		i = base;
		while ((i >>= 1) > 1)
			++basebits;

		for (i = 0; i < size_a; ++i) {
			accum |= (twodigits)a->ob_digit[i] << accumbits;
			accumbits += PyLong_SHIFT;
			assert(accumbits >= basebits);
			do {
				char cdigit = (char)(accum & (base - 1));
				cdigit += (cdigit < 10) ? '0' : 'a'-10;
				assert(p > PyString_AS_STRING(str));
				*--p = cdigit;
				accumbits -= basebits;
				accum >>= basebits;
			} while (i < size_a-1 ? accumbits >= basebits :
						accum > 0);
		}
	}
	else {
		/* Not 0, and base not a power of 2.  Divide repeatedly by
		   base, but for speed use the highest power of base that
		   fits in a digit. */
		Py_ssize_t size = size_a;
		digit *pin = a->ob_digit;
		PyLongObject *scratch;
		/* powbasw <- largest power of base that fits in a digit. */
		digit powbase = base;  /* powbase == base ** power */
		int power = 1;
		for (;;) {
			unsigned long newpow = powbase * (unsigned long)base;
			if (newpow >> PyLong_SHIFT)
				/* doesn't fit in a digit */
				break;
			powbase = (digit)newpow;
			++power;
		}

		/* Get a scratch area for repeated division. */
		scratch = _PyLong_New(size);
		if (scratch == NULL) {
			Py_DECREF(str);
			return NULL;
		}

		/* Repeatedly divide by powbase. */
		do {
			int ntostore = power;
			digit rem = inplace_divrem1(scratch->ob_digit,
						     pin, size, powbase);
			pin = scratch->ob_digit; /* no need to use a again */
			if (pin[size - 1] == 0)
				--size;
			SIGCHECK({
				Py_DECREF(scratch);
				Py_DECREF(str);
				return NULL;
			})

			/* Break rem into digits. */
			assert(ntostore > 0);
			do {
				digit nextrem = (digit)(rem / base);
				char c = (char)(rem - nextrem * base);
				assert(p > PyString_AS_STRING(str));
				c += (c < 10) ? '0' : 'a'-10;
				*--p = c;
				rem = nextrem;
				--ntostore;
				/* Termination is a bit delicate:  must not
				   store leading zeroes, so must get out if
				   remaining quotient and rem are both 0. */
			} while (ntostore && (size || rem));
		} while (size != 0);
		Py_DECREF(scratch);
	}

	if (base == 2) {
		*--p = 'b';
		*--p = '0';
	}
	else if (base == 8) {
		if (newstyle) {
			*--p = 'o';
			*--p = '0';
		}
		else
			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,
				 (Py_ssize_t) (q - PyString_AS_STRING(str)));
	}
	return (PyObject *)str;
}

/* Table of digit values for 8-bit string -> integer conversion.
 * '0' maps to 0, ..., '9' maps to 9.
 * 'a' and 'A' map to 10, ..., 'z' and 'Z' map to 35.
 * All other indices map to 37.
 * Note that when converting a base B string, a char c is a legitimate
 * base B digit iff _PyLong_DigitValue[Py_CHARMASK(c)] < B.
 */
int _PyLong_DigitValue[256] = {
	37, 37, 37, 37, 37, 37, 37, 37, 37, 37, 37, 37, 37, 37, 37, 37,
	37, 37, 37, 37, 37, 37, 37, 37, 37, 37, 37, 37, 37, 37, 37, 37,
	37, 37, 37, 37, 37, 37, 37, 37, 37, 37, 37, 37, 37, 37, 37, 37,
	0,  1,  2,  3,  4,  5,  6,  7,  8,  9,  37, 37, 37, 37, 37, 37,
	37, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24,
	25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 37, 37, 37, 37, 37,
	37, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24,
	25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 37, 37, 37, 37, 37,
	37, 37, 37, 37, 37, 37, 37, 37, 37, 37, 37, 37, 37, 37, 37, 37,
	37, 37, 37, 37, 37, 37, 37, 37, 37, 37, 37, 37, 37, 37, 37, 37,
	37, 37, 37, 37, 37, 37, 37, 37, 37, 37, 37, 37, 37, 37, 37, 37,
	37, 37, 37, 37, 37, 37, 37, 37, 37, 37, 37, 37, 37, 37, 37, 37,
	37, 37, 37, 37, 37, 37, 37, 37, 37, 37, 37, 37, 37, 37, 37, 37,
	37, 37, 37, 37, 37, 37, 37, 37, 37, 37, 37, 37, 37, 37, 37, 37,
	37, 37, 37, 37, 37, 37, 37, 37, 37, 37, 37, 37, 37, 37, 37, 37,
	37, 37, 37, 37, 37, 37, 37, 37, 37, 37, 37, 37, 37, 37, 37, 37,
};

/* *str points to the first digit in a string of base `base` digits.  base
 * is a power of 2 (2, 4, 8, 16, or 32).  *str is set to point to the first
 * non-digit (which may be *str!).  A normalized long is returned.
 * The point to this routine is that it takes time linear in the number of
 * string characters.
 */
static PyLongObject *
long_from_binary_base(char **str, int base)
{
	char *p = *str;
	char *start = p;
	int bits_per_char;
	Py_ssize_t n;
	PyLongObject *z;
	twodigits accum;
	int bits_in_accum;
	digit *pdigit;

	assert(base >= 2 && base <= 32 && (base & (base - 1)) == 0);
	n = base;
	for (bits_per_char = -1; n; ++bits_per_char)
		n >>= 1;
	/* n <- total # of bits needed, while setting p to end-of-string */
	while (_PyLong_DigitValue[Py_CHARMASK(*p)] < base)
		++p;
	*str = p;
	/* n <- # of Python digits needed, = ceiling(n/PyLong_SHIFT). */
	n = (p - start) * bits_per_char + PyLong_SHIFT - 1;
	if (n / bits_per_char < p - start) {
		PyErr_SetString(PyExc_ValueError,
				"long string too large to convert");
		return NULL;
	}
	n = n / PyLong_SHIFT;
	z = _PyLong_New(n);
	if (z == NULL)
		return NULL;
	/* Read string from right, and fill in long from left; i.e.,
	 * from least to most significant in both.
	 */
	accum = 0;
	bits_in_accum = 0;
	pdigit = z->ob_digit;
	while (--p >= start) {
		int k = _PyLong_DigitValue[Py_CHARMASK(*p)];
		assert(k >= 0 && k < base);
		accum |= (twodigits)k << bits_in_accum;
		bits_in_accum += bits_per_char;
		if (bits_in_accum >= PyLong_SHIFT) {
			*pdigit++ = (digit)(accum & PyLong_MASK);
			assert(pdigit - z->ob_digit <= n);
			accum >>= PyLong_SHIFT;
			bits_in_accum -= PyLong_SHIFT;
			assert(bits_in_accum < PyLong_SHIFT);
		}
	}
	if (bits_in_accum) {
		assert(bits_in_accum <= PyLong_SHIFT);
		*pdigit++ = (digit)accum;
		assert(pdigit - z->ob_digit <= n);
	}
	while (pdigit - z->ob_digit < n)
		*pdigit++ = 0;
	return long_normalize(z);
}

PyObject *
PyLong_FromString(char *str, char **pend, int base)
{
	int sign = 1;
	char *start, *orig_str = str;
	PyLongObject *z;
	PyObject *strobj, *strrepr;
	Py_ssize_t slen;

	if ((base != 0 && base < 2) || base > 36) {
		PyErr_SetString(PyExc_ValueError,
				"long() arg 2 must be >= 2 and <= 36");
		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) {
		/* No base given.  Deduce the base from the contents
		   of the string */
		if (str[0] != '0')
			base = 10;
		else if (str[1] == 'x' || str[1] == 'X')
			base = 16;
		else if (str[1] == 'o' || str[1] == 'O')
			base = 8;
		else if (str[1] == 'b' || str[1] == 'B')
			base = 2;
		else
			/* "old" (C-style) octal literal, still valid in
			   2.x, although illegal in 3.x */
			base = 8;
	}
	/* Whether or not we were deducing the base, skip leading chars
	   as needed */
	if (str[0] == '0' &&
	    ((base == 16 && (str[1] == 'x' || str[1] == 'X')) ||
	     (base == 8  && (str[1] == 'o' || str[1] == 'O')) ||
	     (base == 2  && (str[1] == 'b' || str[1] == 'B'))))
		str += 2;

	start = str;
	if ((base & (base - 1)) == 0)
		z = long_from_binary_base(&str, base);
	else {
/***
Binary bases can be converted in time linear in the number of digits, because
Python's representation base is binary.  Other bases (including decimal!) use
the simple quadratic-time algorithm below, complicated by some speed tricks.

First some math:  the largest integer that can be expressed in N base-B digits
is B**N-1.  Consequently, if we have an N-digit input in base B, the worst-
case number of Python digits needed to hold it is the smallest integer n s.t.

    PyLong_BASE**n-1 >= B**N-1  [or, adding 1 to both sides]
    PyLong_BASE**n >= B**N      [taking logs to base PyLong_BASE]
    n >= log(B**N)/log(PyLong_BASE) = N * log(B)/log(PyLong_BASE)

The static array log_base_PyLong_BASE[base] == log(base)/log(PyLong_BASE) so we can compute
this quickly.  A Python long with that much space is reserved near the start,
and the result is computed into it.

The input string is actually treated as being in base base**i (i.e., i digits
are processed at a time), where two more static arrays hold:

    convwidth_base[base] = the largest integer i such that base**i <= PyLong_BASE
    convmultmax_base[base] = base ** convwidth_base[base]

The first of these is the largest i such that i consecutive input digits
must fit in a single Python digit.  The second is effectively the input
base we're really using.

Viewing the input as a sequence <c0, c1, ..., c_n-1> of digits in base
convmultmax_base[base], the result is "simply"

   (((c0*B + c1)*B + c2)*B + c3)*B + ... ))) + c_n-1

where B = convmultmax_base[base].

Error analysis:  as above, the number of Python digits `n` needed is worst-
case

    n >= N * log(B)/log(PyLong_BASE)

where `N` is the number of input digits in base `B`.  This is computed via

    size_z = (Py_ssize_t)((scan - str) * log_base_PyLong_BASE[base]) + 1;

below.  Two numeric concerns are how much space this can waste, and whether
the computed result can be too small.  To be concrete, assume PyLong_BASE = 2**15,
which is the default (and it's unlikely anyone changes that).

Waste isn't a problem:  provided the first input digit isn't 0, the difference
between the worst-case input with N digits and the smallest input with N
digits is about a factor of B, but B is small compared to PyLong_BASE so at most
one allocated Python digit can remain unused on that count.  If
N*log(B)/log(PyLong_BASE) is mathematically an exact integer, then truncating that
and adding 1 returns a result 1 larger than necessary.  However, that can't
happen:  whenever B is a power of 2, long_from_binary_base() is called
instead, and it's impossible for B**i to be an integer power of 2**15 when
B is not a power of 2 (i.e., it's impossible for N*log(B)/log(PyLong_BASE) to be
an exact integer when B is not a power of 2, since B**i has a prime factor
other than 2 in that case, but (2**15)**j's only prime factor is 2).

The computed result can be too small if the true value of N*log(B)/log(PyLong_BASE)
is a little bit larger than an exact integer, but due to roundoff errors (in
computing log(B), log(PyLong_BASE), their quotient, and/or multiplying that by N)
yields a numeric result a little less than that integer.  Unfortunately, "how
close can a transcendental function get to an integer over some range?"
questions are generally theoretically intractable.  Computer analysis via
continued fractions is practical:  expand log(B)/log(PyLong_BASE) via continued
fractions, giving a sequence i/j of "the best" rational approximations.  Then
j*log(B)/log(PyLong_BASE) is approximately equal to (the integer) i.  This shows that
we can get very close to being in trouble, but very rarely.  For example,
76573 is a denominator in one of the continued-fraction approximations to
log(10)/log(2**15), and indeed:

    >>> log(10)/log(2**15)*76573
    16958.000000654003

is very close to an integer.  If we were working with IEEE single-precision,
rounding errors could kill us.  Finding worst cases in IEEE double-precision
requires better-than-double-precision log() functions, and Tim didn't bother.
Instead the code checks to see whether the allocated space is enough as each
new Python digit is added, and copies the whole thing to a larger long if not.
This should happen extremely rarely, and in fact I don't have a test case
that triggers it(!).  Instead the code was tested by artificially allocating
just 1 digit at the start, so that the copying code was exercised for every
digit beyond the first.
***/
		register twodigits c;	/* current input character */
		Py_ssize_t size_z;
		int i;
		int convwidth;
		twodigits convmultmax, convmult;
		digit *pz, *pzstop;
		char* scan;

		static double log_base_PyLong_BASE[37] = {0.0e0,};
		static int convwidth_base[37] = {0,};
		static twodigits convmultmax_base[37] = {0,};

		if (log_base_PyLong_BASE[base] == 0.0) {
			twodigits convmax = base;
			int i = 1;

			log_base_PyLong_BASE[base] = log((double)base) /
						log((double)PyLong_BASE);
			for (;;) {
				twodigits next = convmax * base;
				if (next > PyLong_BASE)
					break;
				convmax = next;
				++i;
			}
			convmultmax_base[base] = convmax;
			assert(i > 0);
			convwidth_base[base] = i;
		}

		/* Find length of the string of numeric characters. */
		scan = str;
		while (_PyLong_DigitValue[Py_CHARMASK(*scan)] < base)
			++scan;

		/* Create a long object that can contain the largest possible
		 * integer with this base and length.  Note that there's no
		 * need to initialize z->ob_digit -- no slot is read up before
		 * being stored into.
		 */
		size_z = (Py_ssize_t)((scan - str) * log_base_PyLong_BASE[base]) + 1;
		/* Uncomment next line to test exceedingly rare copy code */
		/* size_z = 1; */
		assert(size_z > 0);
		z = _PyLong_New(size_z);
		if (z == NULL)
			return NULL;
		Py_SIZE(z) = 0;

		/* `convwidth` consecutive input digits are treated as a single
		 * digit in base `convmultmax`.
		 */
		convwidth = convwidth_base[base];
		convmultmax = convmultmax_base[base];

		/* Work ;-) */
		while (str < scan) {
			/* grab up to convwidth digits from the input string */
			c = (digit)_PyLong_DigitValue[Py_CHARMASK(*str++)];
			for (i = 1; i < convwidth && str != scan; ++i, ++str) {
				c = (twodigits)(c *  base +
					_PyLong_DigitValue[Py_CHARMASK(*str)]);
				assert(c < PyLong_BASE);
			}

			convmult = convmultmax;
			/* Calculate the shift only if we couldn't get
			 * convwidth digits.
			 */
			if (i != convwidth) {
				convmult = base;
				for ( ; i > 1; --i)
					convmult *= base;
			}

			/* Multiply z by convmult, and add c. */
			pz = z->ob_digit;
			pzstop = pz + Py_SIZE(z);
			for (; pz < pzstop; ++pz) {
				c += (twodigits)*pz * convmult;
				*pz = (digit)(c & PyLong_MASK);
				c >>= PyLong_SHIFT;
			}
			/* carry off the current end? */
			if (c) {
				assert(c < PyLong_BASE);
				if (Py_SIZE(z) < size_z) {
					*pz = (digit)c;
					++Py_SIZE(z);
				}
				else {
					PyLongObject *tmp;
					/* Extremely rare.  Get more space. */
					assert(Py_SIZE(z) == size_z);
					tmp = _PyLong_New(size_z + 1);
					if (tmp == NULL) {
						Py_DECREF(z);
						return NULL;
					}
					memcpy(tmp->ob_digit,
					       z->ob_digit,
					       sizeof(digit) * size_z);
					Py_DECREF(z);
					z = tmp;
					z->ob_digit[size_z] = (digit)c;
					++size_z;
				}
			}
		}
	}
	if (z == NULL)
		return NULL;
	if (str == start)
		goto onError;
	if (sign < 0)
		Py_SIZE(z) = -(Py_SIZE(z));
	if (*str == 'L' || *str == 'l')
		str++;
	while (*str && isspace(Py_CHARMASK(*str)))
		str++;
	if (*str != '\0')
		goto onError;
	if (pend)
		*pend = str;
	return (PyObject *) z;

 onError:
	Py_XDECREF(z);
	slen = strlen(orig_str) < 200 ? strlen(orig_str) : 200;
	strobj = PyString_FromStringAndSize(orig_str, slen);
	if (strobj == NULL)
		return NULL;
	strrepr = PyObject_Repr(strobj);
	Py_DECREF(strobj);
	if (strrepr == NULL)
		return NULL;
	PyErr_Format(PyExc_ValueError,
		     "invalid literal for long() with base %d: %s",
		     base, PyString_AS_STRING(strrepr));
	Py_DECREF(strrepr);
	return NULL;
}

#ifdef Py_USING_UNICODE
PyObject *
PyLong_FromUnicode(Py_UNICODE *u, Py_ssize_t length, int base)
{
	PyObject *result;
	char *buffer = (char *)PyMem_MALLOC(length+1);

	if (buffer == NULL)
		return NULL;

	if (PyUnicode_EncodeDecimal(u, length, buffer, NULL)) {
		PyMem_FREE(buffer);
		return NULL;
	}
	result = PyLong_FromString(buffer, NULL, base);
	PyMem_FREE(buffer);
	return result;
}
#endif

/* forward */
static PyLongObject *x_divrem
	(PyLongObject *, PyLongObject *, PyLongObject **);
static PyObject *long_long(PyObject *v);
static int long_divrem(PyLongObject *, PyLongObject *,
	PyLongObject **, PyLongObject **);

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

static int
long_divrem(PyLongObject *a, PyLongObject *b,
	    PyLongObject **pdiv, PyLongObject **prem)
{
	Py_ssize_t size_a = ABS(Py_SIZE(a)), size_b = ABS(Py_SIZE(b));
	PyLongObject *z;

	if (size_b == 0) {
		PyErr_SetString(PyExc_ZeroDivisionError,
				"long division or modulo by zero");
		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);
		if (*pdiv == NULL)
			return -1;
		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);
		if (*prem == NULL) {
			Py_DECREF(z);
			return -1;
		}
	}
	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(PyLongObject *v1, PyLongObject *w1, PyLongObject **prem)
{
	Py_ssize_t size_v = ABS(Py_SIZE(v1)), size_w = ABS(Py_SIZE(w1));
	digit d = (digit) ((twodigits)PyLong_BASE / (w1->ob_digit[size_w-1] + 1));
	PyLongObject *v = mul1(v1, d);
	PyLongObject *w = mul1(w1, d);
	PyLongObject *a;
	Py_ssize_t 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(Py_REFCNT(v) == 1); /* Since v will be used as accumulator! */
	assert(size_w == ABS(Py_SIZE(w))); /* That's how d was calculated */

	size_v = ABS(Py_SIZE(v));
	k = size_v - size_w;
	a = _PyLong_New(k + 1);

	for (j = size_v; a != NULL && k >= 0; --j, --k) {
		digit vj = (j >= size_v) ? 0 : v->ob_digit[j];
		twodigits q;
		stwodigits carry = 0;
		Py_ssize_t i;

		SIGCHECK({
			Py_DECREF(a);
			a = NULL;
			break;
		})
		if (vj == w->ob_digit[size_w-1])
			q = PyLong_MASK;
		else
			q = (((twodigits)vj << PyLong_SHIFT) + v->ob_digit[j-1]) /
				w->ob_digit[size_w-1];

		while (w->ob_digit[size_w-2]*q >
				((
					((twodigits)vj << PyLong_SHIFT)
					+ v->ob_digit[j-1]
					- q*w->ob_digit[size_w-1]
								) << PyLong_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 >> PyLong_SHIFT);
			carry += v->ob_digit[i+k] - z
				+ ((twodigits)zz << PyLong_SHIFT);
			v->ob_digit[i+k] = (digit)(carry & PyLong_MASK);
			carry = Py_ARITHMETIC_RIGHT_SHIFT(BASE_TWODIGITS_TYPE,
							  carry, PyLong_SHIFT);
			carry -= 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] = (digit)(carry & PyLong_MASK);
				carry = Py_ARITHMETIC_RIGHT_SHIFT(
						BASE_TWODIGITS_TYPE,
						carry, PyLong_SHIFT);
			}
		}
	} /* for j, k */

	if (a == NULL)
		*prem = NULL;
	els…