from rpython.rlib.rarithmetic import LONG_BIT, intmask, longlongmask, r_uint, r_ulonglong
from rpython.rlib.rarithmetic import ovfcheck, r_longlong, widen
from rpython.rlib.rarithmetic import most_neg_value_of_same_type
from rpython.rlib.rarithmetic import check_support_int128
from rpython.rlib.rstring import StringBuilder
from rpython.rlib.debug import make_sure_not_resized, check_regular_int
from rpython.rlib.objectmodel import we_are_translated, specialize, \
        not_rpython, newlist_hint, always_inline
from rpython.rlib import jit
from rpython.rtyper.lltypesystem import lltype, rffi
from rpython.rtyper import extregistry

import math, sys

SUPPORT_INT128 = check_support_int128()
BYTEORDER = sys.byteorder

# note about digit sizes:
# In division, the native integer type must be able to hold
# a sign bit plus two digits plus 1 overflow bit.

#SHIFT = (LONG_BIT // 2) - 1
if SUPPORT_INT128:
    SHIFT = 63
    UDIGIT_TYPE = r_ulonglong
    if LONG_BIT >= 64:
        UDIGIT_MASK = intmask
    else:
        UDIGIT_MASK = longlongmask
    LONG_TYPE = rffi.__INT128_T
    ULONG_TYPE = rffi.__UINT128_T
    if LONG_BIT > SHIFT:
        STORE_TYPE = lltype.Signed
        UNSIGNED_TYPE = lltype.Unsigned
    else:
        STORE_TYPE = rffi.LONGLONG
        UNSIGNED_TYPE = rffi.ULONGLONG
else:
    SHIFT = 31
    UDIGIT_TYPE = r_uint
    UDIGIT_MASK = intmask
    STORE_TYPE = lltype.Signed
    UNSIGNED_TYPE = lltype.Unsigned
    LONG_TYPE = rffi.LONGLONG
    ULONG_TYPE = rffi.ULONGLONG

    # TODO if LONG_BIT >= 64, it would be best to use r_uint32, but
    #      int32 and uint32 ops are unimplemented

MASK = int((1 << SHIFT) - 1)
FLOAT_MULTIPLIER = float(1 << SHIFT)

# For BIGINT and INT mix.
#
# The VALID range of an int is different than a valid range of a bigint of length one.
# -1 << LONG_BIT is actually TWO digits, because they are stored without the sign.
if SHIFT == LONG_BIT - 1:
    MIN_INT_VALUE = -1 << SHIFT
    def int_in_valid_range(x):
        if x == MIN_INT_VALUE:
            return False
        return True
else:
    # Means we don't have INT128 on 64bit.
    def int_in_valid_range(x):
        if x > MASK or x < -MASK:
            return False
        return True

int_in_valid_range._always_inline_ = True

# Debugging digit array access.
#
# False == no checking at all
# True == check 0 <= value <= MASK


# 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 BASE).

# Karatsuba is O(N**1.585)
USE_KARATSUBA = True # set to False for comparison

if SHIFT > 31:
    KARATSUBA_CUTOFF = 19
else:
    KARATSUBA_CUTOFF = 38

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.

FIVEARY_CUTOFF = 8

@specialize.argtype(0)
def _mask_digit(x):
    return UDIGIT_MASK(x & MASK)

def _widen_digit(x):
    return rffi.cast(LONG_TYPE, x)

def _unsigned_widen_digit(x):
    return rffi.cast(ULONG_TYPE, x)

@specialize.argtype(0)
def _store_digit(x):
    return rffi.cast(STORE_TYPE, x)

def _load_unsigned_digit(x):
    return rffi.cast(UNSIGNED_TYPE, x)

_load_unsigned_digit._always_inline_ = True

NULLDIGIT = _store_digit(0)
ONEDIGIT = _store_digit(1)
NULLDIGITS = [NULLDIGIT]

def _check_digits(l):
    for x in l:
        assert type(x) is type(NULLDIGIT)
        assert UDIGIT_MASK(x) & MASK == UDIGIT_MASK(x)

class InvalidEndiannessError(Exception):
    pass

class InvalidSignednessError(Exception):
    pass

class MaxIntError(Exception):
    pass


class Entry(extregistry.ExtRegistryEntry):
    _about_ = _check_digits

    def compute_result_annotation(self, s_list):
        from rpython.annotator import model as annmodel
        assert isinstance(s_list, annmodel.SomeList)
        s_DIGIT = self.bookkeeper.valueoftype(type(NULLDIGIT))
        assert s_DIGIT.contains(s_list.listdef.listitem.s_value)

    def specialize_call(self, hop):
        hop.exception_cannot_occur()

def intsign(i):
    if i == 0:
        return 0
    return -1 if i < 0 else 1

class rbigint(object):
    """This is a reimplementation of longs using a list of digits."""
    _immutable_ = True
    _immutable_fields_ = ["_digits[*]", "_size"]

    def __init__(self, digits=NULLDIGITS, sign=0, size=0):
        if not we_are_translated():
            _check_digits(digits)
        make_sure_not_resized(digits)
        self._digits = digits

        assert size >= 0
        self._size = (size or len(digits)) * sign

    def get_sign(self):
        return intsign(self._size)

    def _set_sign(self, sign):
        self._size = abs(self._size) * sign

    # __eq__ and __ne__ method exist for testing only, they are not RPython!
    @not_rpython
    def __eq__(self, other):
        if not isinstance(other, rbigint):
            return NotImplemented
        return self.eq(other)

    @not_rpython
    def __ne__(self, other):
        return not (self == other)

    @specialize.argtype(1)
    def digit(self, x):
        """Return the x'th digit, as an int."""
        return self._digits[x]
    digit._always_inline_ = True

    def widedigit(self, x):
        """Return the x'th digit, as a long long int if needed
        to have enough room to contain two digits."""
        return _widen_digit(self._digits[x])
    widedigit._always_inline_ = True

    def uwidedigit(self, x):
        """Return the x'th digit, as a long long int if needed
        to have enough room to contain two digits."""
        return _unsigned_widen_digit(self._digits[x])
    uwidedigit._always_inline_ = True

    def udigit(self, x):
        """Return the x'th digit, as an unsigned int."""
        return _load_unsigned_digit(self._digits[x])
    udigit._always_inline_ = True

    @specialize.argtype(2)
    def setdigit(self, x, val):
        val = _mask_digit(val)
        assert val >= 0
        self._digits[x] = _store_digit(val)
    setdigit._always_inline_ = True

    def numdigits(self):
        w = abs(self._size)
        if not w:
            w = 1
        assert w > 0
        return w
    numdigits._always_inline_ = True

    @staticmethod
    @jit.elidable
    def fromint(intval):
        # This function is marked as pure, so you must not call it and
        # then modify the result.

        # for hypothesis testing, we want to be able to set SHIFT to a small
        # number to hit edge cases more easily. so use a slower path if SHIFT
        # is a nonstandard value
        if SHIFT != 63 and SHIFT != 31:
            return rbigint.fromrarith_int(intval)
        check_regular_int(intval)

        if intval < 0:
            sign = -1
            ival = -r_uint(intval)
            carry = ival >> SHIFT
        elif intval > 0:
            sign = 1
            ival = r_uint(intval)
            carry = 0
        else:
            return NULLRBIGINT

        if SHIFT != LONG_BIT - 1:
            # Means we don't have INT128 on 64bit.
            if intval > 0:
                carry = ival >> SHIFT

            if carry > 0:
                carry2 = carry >> SHIFT
            else:
                carry2 = 0

            if carry2:
                return rbigint([_store_digit(ival & MASK),
                                _store_digit(carry & MASK),
                                _store_digit(carry2)], sign, 3)

        if carry:
            return rbigint([_store_digit(ival & MASK),
                _store_digit(carry)], sign, 2)
        else:
            return rbigint([_store_digit(ival & MASK)], sign, 1)

    @staticmethod
    @jit.elidable
    def frombool(b):
        # You must not call this function and then modify the result.
        if b:
            return ONERBIGINT
        return NULLRBIGINT

    @staticmethod
    @not_rpython
    def fromlong(l):
        return rbigint(*args_from_long(l))

    @staticmethod
    @jit.elidable
    def fromfloat(dval):
        """ Create a new bigint object from a float """
        # This function is not marked as pure because it can raise
        if math.isinf(dval):
            raise OverflowError("cannot convert float infinity to integer")
        if math.isnan(dval):
            raise ValueError("cannot convert float NaN to integer")
        return rbigint._fromfloat_finite(dval)

    @staticmethod
    @jit.elidable
    def _fromfloat_finite(dval):
        sign = 1
        if dval < 0.0:
            sign = -1
            dval = -dval
        frac, expo = math.frexp(dval) # dval = frac*2**expo; 0.0 <= frac < 1.0
        if expo <= 0:
            return NULLRBIGINT
        ndig = (expo-1) // SHIFT + 1 # Number of 'digits' in result
        v = rbigint([NULLDIGIT] * ndig, sign, ndig)
        frac = math.ldexp(frac, (expo-1) % SHIFT + 1)
        for i in range(ndig-1, -1, -1):
            # use int(int(frac)) as a workaround for a CPython bug:
            # with frac == 2147483647.0, int(frac) == 2147483647L
            bits = int(int(frac))
            v.setdigit(i, bits)
            frac -= float(bits)
            frac = math.ldexp(frac, SHIFT)
        return v

    @staticmethod
    @jit.elidable
    @specialize.argtype(0)
    def fromrarith_int(i):
        # This function is marked as pure, so you must not call it and
        # then modify the result.
        return rbigint(*args_from_rarith_int(i))

    @staticmethod
    @jit.elidable
    def fromdecimalstr(s):
        # This function is marked as elidable, so you must not call it and
        # then modify the result.
        return _decimalstr_to_bigint(s)

    @staticmethod
    @jit.elidable
    def fromstr(s, base=0, allow_underscores=False):
        """As string_to_int(), but ignores an optional 'l' or 'L' suffix
        and returns an rbigint."""
        from rpython.rlib.rstring import NumberStringParser, \
            strip_spaces
        s = literal = strip_spaces(s) # XXX could get rid of this slice
        end = len(s)
        if (s.endswith('l') or s.endswith('L')) and base < 22:
            # in base 22 and above, 'L' is a valid digit!  try: long('L',22)
            end -= 1
        parser = NumberStringParser(s, literal, base, 'long',
                                    allow_underscores=allow_underscores,
                                    end=end)
        return rbigint._from_numberstring_parser(parser)

    @staticmethod
    def _from_numberstring_parser(parser):
        return parse_digit_string(parser)

    @staticmethod
    @jit.elidable
    def frombytes(s, byteorder, signed):
        if byteorder not in ('big', 'little'):
            raise InvalidEndiannessError()
        if not s:
            return NULLRBIGINT

        if byteorder == 'big':
            msb = ord(s[0])
            itr = range(len(s)-1, -1, -1)
        else:
            msb = ord(s[-1])
            itr = range(0, len(s))

        sign = -1 if msb >= 0x80 and signed else 1
        accum = _widen_digit(0)
        accumbits = 0
        digits = newlist_hint(len(s) * 8 // LONG_BIT + 1)
        carry = 1

        for i in itr:
            c = _widen_digit(ord(s[i]))
            if sign == -1:
                c = (0xFF ^ c) + carry
                carry = c >> 8
                c &= 0xFF

            accum |= c << accumbits
            accumbits += 8
            if accumbits >= SHIFT:
                digits.append(_store_digit(intmask(accum & MASK)))
                accum >>= SHIFT
                accumbits -= SHIFT

        if accumbits:
            digits.append(_store_digit(intmask(accum)))
        result = rbigint(digits[:], sign)
        result._normalize()
        return result

    @jit.elidable
    def tobytes(self, nbytes, byteorder, signed):
        if byteorder not in ('big', 'little'):
            raise InvalidEndiannessError()
        if not signed and self.get_sign() == -1:
            raise InvalidSignednessError()

        bswap = byteorder == 'big'
        d = _widen_digit(0)
        j = 0
        imax = self.numdigits()
        accum = _widen_digit(0)
        accumbits = 0
        result = StringBuilder(nbytes)
        carry = 1

        for i in range(0, imax):
            d = self.widedigit(i)
            if self.get_sign() == -1:
                d = (d ^ MASK) + carry
                carry = d >> SHIFT
                d &= MASK

            accum |= d << accumbits
            if i == imax - 1:
                # Avoid bogus 0's
                s = d ^ MASK if self.get_sign() == -1 else d
                while s:
                    s >>= 1
                    accumbits += 1
            else:
                accumbits += SHIFT

            while accumbits >= 8:
                if j >= nbytes:
                    raise OverflowError()
                j += 1

                result.append(chr(accum & 0xFF))
                accum >>= 8
                accumbits -= 8

        if accumbits:
            if j >= nbytes:
                raise OverflowError()
            j += 1

            if self.get_sign() == -1:
                # Add a sign bit
                accum |= (~_widen_digit(0)) << accumbits

            result.append(chr(accum & 0xFF))

        if j < nbytes:
            signbyte = 0xFF if self.get_sign() == -1 else 0
            result.append_multiple_char(chr(signbyte), nbytes - j)

        digits = result.build()

        if j == nbytes and nbytes > 0 and signed:
            # If not already set, we cannot contain the sign bit
            msb = digits[-1]
            if (self.get_sign() == -1) != (ord(msb) >= 0x80):
                raise OverflowError()

        if bswap:
            # Bah, this is very inefficient. At least it's not
            # quadratic.
            length = len(digits)
            if length >= 0:
                digits = ''.join([digits[i] for i in range(length-1, -1, -1)])
        return digits

    @jit.elidable
    def toint(self):
        """
        Get an integer from a bigint object.
        Raises OverflowError if overflow occurs.
        """
        if self.numdigits() > MAX_DIGITS_THAT_CAN_FIT_IN_INT:
            raise OverflowError
        return self._toint_helper()

    @jit.elidable
    def _toint_helper(self):
        x = self._touint_helper()
        # Haven't lost any bits so far
        if self.get_sign() >= 0:
            res = intmask(x)
            if res < 0:
                raise OverflowError
        else:
            # Use "-" on the unsigned number, not on the signed number.
            # This is needed to produce valid C code.
            res = intmask(-x)
            if res >= 0:
                raise OverflowError
        return res

    def fits_int(self):
        n = self.numdigits()
        if n < MAX_DIGITS_THAT_CAN_FIT_IN_INT:
            return True
        if n > MAX_DIGITS_THAT_CAN_FIT_IN_INT:
            return False
        try:
            x = self._touint_helper()
        except OverflowError:
            return False
        if self.get_sign() >= 0:
            res = intmask(x)
            return res >= 0
        else:
            res = intmask(-x)
            return res < 0

    @jit.elidable
    def tolonglong(self):
        return _AsLongLong(self)

    def tobool(self):
        return self.get_sign() != 0

    @jit.elidable
    def touint(self):
        if self.get_sign() == -1:
            raise ValueError("cannot convert negative integer to unsigned int")
        return self._touint_helper()

    @jit.elidable
    def _touint_helper(self):
        x = r_uint(0)
        i = self.numdigits() - 1
        while i >= 0:
            prev = x
            x = (x << SHIFT) + self.udigit(i)
            if (x >> SHIFT) != prev:
                raise OverflowError("long int too large to convert to unsigned int")
            i -= 1
        return x

    @jit.elidable
    def toulonglong(self):
        if self.get_sign() == -1:
            raise ValueError("cannot convert negative integer to unsigned int")
        return _AsULonglong_ignore_sign(self)

    @jit.elidable
    def uintmask(self):
        return _AsUInt_mask(self)

    @jit.elidable
    def ulonglongmask(self):
        """Return r_ulonglong(self), truncating."""
        return _AsULonglong_mask(self)

    @jit.elidable
    def tofloat(self):
        return _AsDouble(self)

    @jit.elidable
    def format(self, digits, prefix='', suffix='', max_str_digits=0):
        # 'digits' is a string whose length is the base to use,
        # and where each character is the corresponding digit.
        return _format(self, digits, prefix, suffix, max_str_digits)

    @jit.elidable
    def repr(self):
        try:
            x = self.toint()
        except OverflowError:
            return self.format(BASE10, suffix="L")
        return str(x) + "L"

    @jit.elidable
    def str(self, max_str_digits=0):
        try:
            x = self.toint()
        except OverflowError:
            return self.format(BASE10, max_str_digits=max_str_digits)
        return str(x)

    @jit.elidable
    def eq(self, other):
        if (self.get_sign() != other.get_sign() or
            self.numdigits() != other.numdigits()):
            return False

        i = 0
        ld = self.numdigits()
        while i < ld:
            if self.digit(i) != other.digit(i):
                return False
            i += 1
        return True

    @jit.elidable
    def int_eq(self, iother):
        """ eq with int """
        if not int_in_valid_range(iother):
            # Fallback to Long.
            return self.eq(rbigint.fromint(iother))

        if self.numdigits() > 1:
            return False

        return (self.get_sign() * self.digit(0)) == iother

    def ne(self, other):
        return not self.eq(other)

    def int_ne(self, iother):
        return not self.int_eq(iother)

    @jit.elidable
    def lt(self, other):
        selfsign = self.get_sign()
        othersign = other.get_sign()
        if selfsign > othersign:
            return False
        if selfsign < othersign:
            return True
        ld1 = self.numdigits()
        ld2 = other.numdigits()
        if ld1 > ld2:
            if othersign > 0:
                return False
            else:
                return True
        elif ld1 < ld2:
            if othersign > 0:
                return True
            else:
                return False
        i = ld1 - 1
        while i >= 0:
            d1 = self.digit(i)
            d2 = other.digit(i)
            if d1 < d2:
                if othersign > 0:
                    return True
                else:
                    return False
            elif d1 > d2:
                if othersign > 0:
                    return False
                else:
                    return True
            i -= 1
        return False

    @jit.elidable
    def int_lt(self, iother):
        """ lt where other is an int """

        if not int_in_valid_range(iother):
            # Fallback to Long.
            return self.lt(rbigint.fromint(iother))

        return _x_int_lt(self, iother, False)

    def le(self, other):
        return not other.lt(self)

    def int_le(self, iother):
        """ le where iother is an int """

        if not int_in_valid_range(iother):
            # Fallback to Long.
            return self.le(rbigint.fromint(iother))

        return _x_int_lt(self, iother, True)

    def gt(self, other):
        return other.lt(self)

    def int_gt(self, iother):
        return not self.int_le(iother)

    def ge(self, other):
        return not self.lt(other)

    def int_ge(self, iother):
        return not self.int_lt(iother)

    @jit.elidable
    def hash(self):
        return _hash(self)

    @jit.elidable
    def add(self, other):
        selfsign = self.get_sign()
        othersign = other.get_sign()
        if selfsign == 0:
            return other
        if othersign == 0:
            return self
        if selfsign == othersign:
            result = _x_add(self, other)
        else:
            result = _x_sub(other, self)
        result._set_sign(result.get_sign() * othersign)
        return result

    @jit.elidable
    def int_add(self, iother):
        selfsign = self.get_sign()
        if not int_in_valid_range(iother):
            # Fallback to long.
            return self.add(rbigint.fromint(iother))
        elif selfsign == 0:
            return rbigint.fromint(iother)
        elif iother == 0:
            return self

        othersign = intsign(iother)
        if selfsign == othersign:
            result = _x_int_add(self, iother)
        else:
            result = _x_int_sub(self, iother)
            result._set_sign(-result.get_sign())
        result._set_sign(result.get_sign() * othersign)
        return result

    @jit.elidable
    def sub(self, other):
        selfsign = self.get_sign()
        othersign = other.get_sign()
        if othersign == 0:
            return self
        elif selfsign == 0:
            return rbigint(other._digits[:other.numdigits()], -othersign, other.numdigits())
        elif selfsign == othersign:
            result = _x_sub(self, other)
        else:
            result = _x_add(self, other)
        result._set_sign(result.get_sign() * selfsign)
        return result

    @jit.elidable
    def int_sub(self, iother):
        selfsign = self.get_sign()
        if not int_in_valid_range(iother):
            # Fallback to long.
            return self.sub(rbigint.fromint(iother))
        elif iother == 0:
            return self
        elif selfsign == 0:
            return rbigint.fromint(-iother)
        elif selfsign == intsign(iother):
            result = _x_int_sub(self, iother)
        else:
            result = _x_int_add(self, iother)
        result._set_sign(result.get_sign() * selfsign)
        return result

    @jit.elidable
    def mul(self, other):
        selfsize = self.numdigits()
        othersize = other.numdigits()
        selfsign = self.get_sign()
        othersign = other.get_sign()

        if selfsize > othersize:
            self, other, selfsize, othersize = other, self, othersize, selfsize

        if selfsign == 0 or othersign == 0:
            return NULLRBIGINT

        if selfsize == 1:
            if self._digits[0] == ONEDIGIT:
                return rbigint(other._digits[:othersize], selfsign * othersign, othersize)
            elif othersize == 1:
                res = other.uwidedigit(0) * self.udigit(0)
                carry = res >> SHIFT
                if carry:
                    return rbigint([_store_digit(res & MASK), _store_digit(carry)], selfsign * othersign, 2)
                else:
                    return rbigint([_store_digit(res & MASK)], selfsign * othersign, 1)

            result = _x_mul(self, other, self.digit(0))
        elif USE_KARATSUBA:
            if self is other:
                i = KARATSUBA_SQUARE_CUTOFF
            else:
                i = KARATSUBA_CUTOFF

            if selfsize <= i:
                result = _x_mul(self, other)
            else:
                result = _k_mul(self, other)
        else:
            result = _x_mul(self, other)

        result._set_sign(selfsign * othersign)
        return result

    @jit.elidable
    def int_mul(self, iother):
        if not int_in_valid_range(iother):
            # Fallback to long.
            return self.mul(rbigint.fromint(iother))

        selfsign = self.get_sign()
        if selfsign == 0 or iother == 0:
            return NULLRBIGINT

        asize = self.numdigits()
        digit = abs(iother)

        othersign = intsign(iother)

        if digit == 1:
            if othersign == 1:
                return self
            return rbigint(self._digits[:asize], selfsign * othersign, asize)
        elif asize == 1:
            udigit = r_uint(digit)
            res = self.uwidedigit(0) * udigit
            carry = res >> SHIFT
            if carry:
                return rbigint([_store_digit(res & MASK), _store_digit(carry)], selfsign * othersign, 2)
            else:
                return rbigint([_store_digit(res & MASK)], selfsign * othersign, 1)
        elif digit & (digit - 1) == 0:
            result = self.lqshift(ptwotable[digit])
        else:
            result = _muladd1(self, digit)

        result._set_sign(selfsign * othersign)
        return result

    @staticmethod
    @jit.elidable
    def mul_int_int_bigint_result(iself, iother):
        if not SUPPORT_INT128 or SHIFT != 63 or not int_in_valid_range(iself):
            return rbigint.fromint(iself).int_mul(iother)
        if iself == 0 or iother == 0:
            return NULLRBIGINT
        selfsign = intsign(iself)
        othersign = intsign(iother)
        otherdigit = r_uint(iother)
        if iother < 0: # can use abs because of minint
            otherdigit = -otherdigit
        res = _unsigned_widen_digit(abs(iself)) * otherdigit
        carry = res >> SHIFT
        if carry:
            return rbigint([_store_digit(res & MASK), _store_digit(carry)], selfsign * othersign, 2)
        else:
            return rbigint([_store_digit(res & MASK)], selfsign * othersign, 1)

    @jit.elidable
    def truediv(self, other):
        div = _bigint_true_divide(self, other)
        return div

    def floordiv(self, other):
        div, mod = self.divmod(other)
        return div

    def div(self, other):
        return self.floordiv(other)

    @jit.elidable
    def int_floordiv(self, iother):
        if not int_in_valid_range(iother):
            # Fallback to long.
            return self.floordiv(rbigint.fromint(iother))

        if iother == 0:
            raise ZeroDivisionError("long division by zero")

        digit = abs(iother)
        assert digit > 0

        selfsign = self.get_sign()
        if selfsign == 1 and iother > 0:
            if digit == 1:
                return self
            elif digit & (digit - 1) == 0:
                return self.rqshift(ptwotable[digit])

        div, mod = _divrem1(self, digit)

        othersign = intsign(iother)
        if mod != 0 and selfsign * othersign == -1:
            if div.get_sign() == 0:
                return ONENEGATIVERBIGINT
            div = div.int_add(1)
        div._set_sign(selfsign * othersign)
        div._normalize()
        return div

    def int_div(self, iother):
        return self.int_floordiv(iother)

    def mod(self, other):
        div, mod = self.divmod(other)
        return mod

    @jit.elidable
    def int_mod(self, iother):
        if iother == 0:
            raise ZeroDivisionError("long division or modulo by zero")
        selfsign = self.get_sign()
        if selfsign == 0:
            return NULLRBIGINT

        elif not int_in_valid_range(iother):
            # Fallback to long.
            return self.mod(rbigint.fromint(iother))

        if 1: # preserve indentation to preserve history
            digit = abs(iother)
            if digit == 1:
                return NULLRBIGINT
            elif digit == 2:
                modm = self.digit(0) & 1
                if modm:
                    return ONENEGATIVERBIGINT if iother < 0 else ONERBIGINT
                return NULLRBIGINT
            elif digit & (digit - 1) == 0:
                mod = self.int_and_(digit - 1)
            else:
                rem = _int_rem_core(self, digit)
                if rem == 0:
                    return NULLRBIGINT
                mod = rbigint([rem], -1 if selfsign < 0 else 1, 1)

        if mod.get_sign() * intsign(iother) == -1:
            mod = mod.int_add(iother)
        return mod

    @jit.elidable
    def int_mod_int_result(self, iother):
        selfsign = self.get_sign()
        if iother == 0:
            raise ZeroDivisionError("long division or modulo by zero")
        if selfsign == 0:
            return 0

        elif not int_in_valid_range(iother):
            # Fallback to long.
            return self.mod(rbigint.fromint(iother)).toint() # cannot raise

        assert iother != -sys.maxint-1 # covered by int_in_valid_range above
        digit = abs(iother)
        if digit == 1:
            return 0
        elif digit == 2:
            modm = self.digit(0) & 1
            if modm:
                return -1 if iother < 0 else 1
            return 0
        elif digit & (digit - 1) == 0:
            mod = self.int_and_(digit - 1).toint() # XXX improve
        else:
            mod = _int_rem_core(self, digit) * selfsign
        if intsign(mod) * intsign(iother) == -1:
            mod = mod + iother
        return mod

    @jit.elidable
    def divmod(self, other):
        """
        The / and % operators are now defined in terms of divmod().
        The expression a mod b has the value a - b*floor(a/b).
        The _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.
        """
        selfsign = self.get_sign()
        othersign = other.get_sign()
        if othersign == 0:
            raise ZeroDivisionError("long division or modulo by zero")
        if selfsign == 0:
            return TWO_NULLRBIGINTS
        if other.numdigits() == 1 and not (-1 == othersign != selfsign):
            otherint = other.digit(0) * othersign
            assert int_in_valid_range(otherint)
            return self.int_divmod(otherint)

        if self.numdigits() > 1.2 * other.numdigits() and \
                other.numdigits() > HOLDER.DIV_LIMIT * 2: # * 2 to offset setup cost
            res = divmod_big(self, other)
            # be paranoid: keep the assert here for a bit
            div, mod = res
            assert div.mul(other).add(mod).eq(self)
            return res

        return self._divmod_small(other)

    def _divmod_small(self, other):
        div, mod = _divrem(self, other)
        if mod.get_sign() * other.get_sign() == -1:
            mod = mod.add(other)
            if div.get_sign() == 0:
                return ONENEGATIVERBIGINT, mod
            div = div.int_sub(1)
        return div, mod

    @jit.elidable
    def int_divmod(self, iother):
        """ Divmod with int """

        if iother == 0:
            raise ZeroDivisionError("long division or modulo by zero")

        selfsign = self.get_sign()
        othersign = intsign(iother)
        if not int_in_valid_range(iother) or (othersign == -1 and selfsign != othersign):
            # Just fallback.
            return self.divmod(rbigint.fromint(iother))

        digit = abs(iother)
        assert digit > 0

        div, mod = _divrem1(self, digit)
        # _divrem1 doesn't fix the sign
        if div._size == 0:
            assert div.get_sign() == 0
        else:
            div._set_sign(selfsign * othersign)
        if selfsign < 0:
            mod = -mod
        if mod and selfsign * othersign == -1:
            mod += iother
            if div.get_sign() == 0:
                div = ONENEGATIVERBIGINT
            else:
                div = div.int_sub(1)
        mod = rbigint.fromint(mod)
        return div, mod

    @jit.elidable
    def pow(self, other, modulus=None):
        selfsign = self.get_sign()
        othersign = other.get_sign()
        negativeOutput = False  # if x<0 return negative output

        # 5-ary values.  If the exponent is large enough, table is
        # precomputed so that table[i] == self**i % modulus for i in range(32).
        # python translation: the table is computed when needed.

        if othersign < 0:  # if exponent is negative
            if modulus is not None:
                raise TypeError(
                    "pow() 2nd argument "
                    "cannot be negative when 3rd argument specified")
            raise ValueError("bigint pow() too negative")

        size_b = UDIGIT_TYPE(other.numdigits())

        if modulus is not None:
            modulussign = modulus.get_sign()
            if modulussign == 0:
                raise ValueError("pow() 3rd argument cannot be 0")

            if modulussign < 0:
                negativeOutput = True
                modulus = modulus.neg()

            # if modulus == 1:
            #     return 0
            if modulus.numdigits() == 1 and modulus._digits[0] == ONEDIGIT:
                return NULLRBIGINT

            # Reduce base by modulus in some cases:
            # 1. If base < 0.  Forcing the base non-neg makes things easier.
            # 2. If base is obviously larger than the modulus.  The "small
            #    exponent" case later can multiply directly by base repeatedly,
            #    while the "large exponent" case multiplies directly by base 31
            #    times.  It can be unboundedly faster to multiply by
            #    base % modulus instead.
            # We could _always_ do this reduction, but mod() isn't cheap,
            # so we only do it when it buys something.
            if selfsign < 0 or self.numdigits() > modulus.numdigits():
                self = self.mod(modulus)
        elif othersign == 0:
            return ONERBIGINT
        elif selfsign == 0:
            return NULLRBIGINT
        elif size_b == 1:
            if other._digits[0] == ONEDIGIT:
                return self
            elif self.numdigits() == 1 and modulus is None:
                adigit = self.digit(0)
                digit = other.digit(0)
                if adigit == 1:
                    if selfsign == -1 and digit % 2:
                        return ONENEGATIVERBIGINT
                    return ONERBIGINT
                elif adigit & (adigit - 1) == 0:
                    ret = self.lshift(((digit-1)*(ptwotable[adigit]-1)) + digit-1)
                    if selfsign == -1 and not digit % 2:
                        ret._set_sign(1)
                    return ret

        # At this point self, other, and modulus are guaranteed non-negative UNLESS
        # modulus is NULL, in which case self may be negative. */

        z = ONERBIGINT

        # python adaptation: moved macros REDUCE(X) and MULT(X, Y, result)
        # into helper function result = _help_mult(x, y, c)
        if size_b <= FIVEARY_CUTOFF:
            # Left-to-right binary exponentiation (HAC Algorithm 14.79)
            # http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf

            while size_b > 0:
                size_b -= 1
                bi = other.digit(size_b)
                j = 1 << (SHIFT-1)
                while j != 0:
                    z = _help_mult(z, z, modulus)
                    if bi & j:
                        z = _help_mult(z, self, modulus)
                    j >>= 1


        else:
            # Left-to-right 5-ary exponentiation (HAC Algorithm 14.82)
            # This is only useful in the case where modulus != None.
            # z still holds 1L
            table = [z] * 32
            table[0] = z
            for i in range(1, 32):
                table[i] = _help_mult(table[i-1], self, modulus)

            # Note that here SHIFT is not a multiple of 5.  The difficulty
            # is to extract 5 bits at a time from 'other', starting from the
            # most significant digits, so that at the end of the algorithm
            # it falls exactly to zero.
            # m  = max number of bits = i * SHIFT
            # m+ = m rounded up to the next multiple of 5
            # j  = (m+) % SHIFT = (m+) - (i * SHIFT)
            # (computed without doing "i * SHIFT", which might overflow)
            j = size_b % 5
            j = _jmapping[j]
            if not we_are_translated():
                assert j == (size_b*SHIFT+4)//5*5 - size_b*SHIFT
            #
            accum = r_uint(0)
            while True:
                j -= 5
                if j >= 0:
                    index = (accum >> j) & 0x1f
                else:
                    # 'accum' does not have enough digit.
                    # must get the next digit from 'other' in order to complete
                    if size_b == 0:
                        break # Done

                    size_b -= 1
                    assert size_b >= 0
                    bi = other.udigit(size_b)
                    index = ((accum << (-j)) | (bi >> (j+SHIFT))) & 0x1f
                    accum = bi
                    j += SHIFT
                #
                for k in range(5):
                    z = _help_mult(z, z, modulus)
                if index:
                    z = _help_mult(z, table[index], modulus)
            #
            assert j == -5

        if negativeOutput and z.get_sign() != 0:
            z = z.sub(modulus)
        return z

    @jit.elidable
    def int_pow(self, iother, modulus=None):
        negativeOutput = False  # if x<0 return negative output

        # 5-ary values.  If the exponent is large enough, table is
        # precomputed so that table[i] == self**i % modulus for i in range(32).
        # python translation: the table is computed when needed.

        if iother < 0:  # if exponent is negative
            if modulus is not None:
                raise TypeError(
                    "pow() 2nd argument "
                    "cannot be negative when 3rd argument specified")
            raise ValueError("bigint pow() too negative")

        selfsign = self.get_sign()
        assert iother >= 0
        if modulus is not None:
            modulussign = modulus.get_sign()
            if modulussign == 0:
                raise ValueError("pow() 3rd argument cannot be 0")

            if modulussign < 0:
                negativeOutput = True
                modulus = modulus.neg()

            # if modulus == 1:
            #     return 0
            if modulus.numdigits() == 1 and modulus._digits[0] == ONEDIGIT:
                return NULLRBIGINT

            # Reduce base by modulus in some cases:
            # 1. If base < 0.  Forcing the base non-neg makes things easier.
            # 2. If base is obviously larger than the modulus.  The "small
            #    exponent" case later can multiply directly by base repeatedly,
            #    while the "large exponent" case multiplies directly by base 31
            #    times.  It can be unboundedly faster to multiply by
            #    base % modulus instead.
            # We could _always_ do this reduction, but mod() isn't cheap,
            # so we only do it when it buys something.
            if selfsign < 0 or self.numdigits() > modulus.numdigits():
                self = self.mod(modulus)
        elif iother == 0:
            return ONERBIGINT
        elif selfsign == 0:
            return NULLRBIGINT
        elif iother == 1:
            return self
        elif self.numdigits() == 1:
            adigit = self.digit(0)
            if adigit == 1:
                if selfsign == -1 and iother % 2:
                    return ONENEGATIVERBIGINT
                return ONERBIGINT
            elif adigit & (adigit - 1) == 0:
                ret = self.lshift(((iother-1)*(ptwotable[adigit]-1)) + iother-1)
                if selfsign == -1 and not iother % 2:
                    ret._set_sign(1)
                return ret

        # At this point self, iother, and modulus are guaranteed non-negative UNLESS
        # modulus is NULL, in which case self may be negative. */

        z = ONERBIGINT

        # Left-to-right binary exponentiation (HAC Algorithm 14.79)
        # http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf
        j = 1 << (LONG_BIT-2)

        while j != 0:
            z = _help_mult(z, z, modulus)
            if iother & j:
                z = _help_mult(z, self, modulus)
            j >>= 1

        if negativeOutput and z.get_sign() != 0:
            z = z.sub(modulus)
        return z

    @jit.elidable
    def neg(self):
        return rbigint(self._digits, -self.get_sign(), self.numdigits())

    @jit.elidable
    def abs(self):
        selfsign = self.get_sign()
        if selfsign != -1:
            return self
        return rbigint(self._digits, abs(selfsign), self.numdigits())

    @jit.elidable
    def invert(self): #Implement ~x as -(x + 1)
        if self.get_sign() == 0:
            return ONENEGATIVERBIGINT

        ret = self.int_add(1)
        ret._set_sign(-ret.get_sign())
        return ret

    @jit.elidable
    def lshift(self, int_other):
        selfsign = self.get_sign()
        if int_other < 0:
            raise ValueError("negative shift count")
        elif int_other == 0 or selfsign == 0:
            return self

        # wordshift, remshift = divmod(int_other, SHIFT)
        wordshift = int_other // SHIFT
        remshift = int_other - wordshift * SHIFT
        hishift = SHIFT - remshift

        if not remshift:
            # So we can avoid problems with eq, AND avoid the need for normalize.
            return rbigint([NULLDIGIT] * wordshift + self._digits, selfsign, self.numdigits() + wordshift)

        oldsize = self.numdigits()
        newsize = oldsize + wordshift + 1
        z = rbigint([NULLDIGIT] * newsize, selfsign, newsize)
        j = 0
        prevdigit = NULLDIGIT
        while j < oldsize:
            digit = self.udigit(j)
            newdigit = (digit << remshift) | (prevdigit >> hishift)
            z.setdigit(wordshift, newdigit)
            prevdigit = digit
            wordshift += 1
            j += 1

        newsize -= 1
        assert newsize >= 0
        z.setdigit(newsize, prevdigit >> hishift)

        z._normalize()
        return z
    lshift._always_inline_ = True # It's so fast that it's always beneficial.

    @jit.elidable
    def lqshift(self, int_other):
        " A quicker one with much less checks, int_other is valid and for the most part constant."
        assert int_other > 0

        oldsize = self.numdigits()
        selfsign = self.get_sign()

        z = rbigint([NULLDIGIT] * (oldsize + 1), selfsign, (oldsize + 1))
        i = 0
        hishift = SHIFT - int_other
        prevdigit = NULLDIGIT
        while i < oldsize:
            digit = self.udigit(i)
            newdigit = (digit << int_other) | (prevdigit >> hishift)
            z.setdigit(i, newdigit)
            prevdigit = digit
            i += 1
        z.setdigit(oldsize, prevdigit >> hishift)
        z._normalize()
        return z
    lqshift._always_inline_ = True # It's so fast that it's always beneficial.

    @jit.elidable
    def rshift(self, int_other, dont_invert=False):
        if int_other < 0:
            raise ValueError("negative shift count")
        elif int_other == 0:
            return self
        selfsign = self.get_sign()
        if selfsign == -1 and not dont_invert:
            a = self.invert().rshift(int_other)
            return a.invert()

        wordshift = int_other / SHIFT
        newsize = self.numdigits() - wordshift
        if newsize <= 0:
            return NULLRBIGINT

        loshift = int_other % SHIFT
        hishift = SHIFT - loshift
        z = rbigint([NULLDIGIT] * newsize, selfsign, newsize)
        i = 0
        while i < newsize:
            newdigit = (self.digit(wordshift) >> loshift)
            if i+1 < newsize:
                newdigit |= (self.digit(wordshift+1) << hishift)
            z.setdigit(i, newdigit)
            i += 1
            wordshift += 1
        z._normalize()
        return z
    rshift._always_inline_ = 'try' # It's so fast that it's always benefitial.

    @jit.elidable
    def rqshift(self, int_other):
        wordshift = int_other / SHIFT
        loshift = int_other % SHIFT
        newsize = self.numdigits() - wordshift

        if newsize <= 0:
            return NULLRBIGINT

        hishift = SHIFT - loshift
        selfsign = self.get_sign()
        z = rbigint([NULLDIGIT] * newsize, selfsign, newsize)
        i = 0

        while i < newsize:
            digit = self.udigit(wordshift)
            newdigit = (digit >> loshift)
            if i+1 < newsize:
                newdigit |= (self.udigit(wordshift+1) << hishift)
            z.setdigit(i, newdigit)
            i += 1
            wordshift += 1
        z._normalize()
        return z
    rshift._always_inline_ = 'try' # It's so fast that it's always beneficial.

    @jit.elidable
    def abs_rshift_and_mask(self, bigshiftcount, mask):
        assert isinstance(bigshiftcount, r_ulonglong)
        assert mask >= 0
        wordshift = bigshiftcount / SHIFT
        numdigits = self.numdigits()
        if wordshift >= numdigits:
            return 0
        wordshift = intmask(wordshift)
        loshift = intmask(intmask(bigshiftcount) - intmask(wordshift * SHIFT))
        lastdigit = self.digit(wordshift) >> loshift
        if mask > (MASK >> loshift) and wordshift + 1 < numdigits:
            hishift = SHIFT - loshift
            lastdigit |= self.digit(wordshift+1) << hishift
            if SHIFT != LONG_BIT - 1:
                # Means we don't have INT128 on 64bit.
                if mask > (MASK << (SHIFT - loshift)) and wordshift + 2 < numdigits:
                    hishift = 2*SHIFT - loshift
                    lastdigit |= self.digit(wordshift+2) << hishift
        return lastdigit & mask

    @staticmethod
    def from_list_n_bits(list, nbits):
        if len(list) == 0:
            return NULLRBIGINT

        if nbits == SHIFT:
            z = rbigint(list, 1)
        else:
            if not (1 <= nbits < SHIFT):
                raise ValueError

            lllength = (r_ulonglong(len(list)) * nbits) // SHIFT
            length = intmask(lllength) + 1
            z = rbigint([NULLDIGIT] * length, 1)

            out = 0
            i = 0
            accum = 0
            for input in list:
                accum |= (input << i)
                original_i = i
                i += nbits
                if i > SHIFT:
                    z.setdigit(out, accum)
                    out += 1
                    accum = input >> (SHIFT - original_i)
                    i -= SHIFT
            assert out < length
            z.setdigit(out, accum)

        z._normalize()
        return z

    @jit.elidable
    def and_(self, other):
        return _bitwise(self, '&', other)

    @jit.elidable
    def int_and_(self, iother):
        return _int_bitwise(self, '&', iother)

    @jit.elidable
    def xor(self, other):
        return _bitwise(self, '^', other)

    @jit.elidable
    def int_xor(self, iother):
        return _int_bitwise(self, '^', iother)

    @jit.elidable
    def or_(self, other):
        return _bitwise(self, '|', other)

    @jit.elidable
    def int_or_(self, iother):
        return _int_bitwise(self, '|', iother)

    @jit.elidable
    def oct(self):
        if self.get_sign() == 0:
            return '0L'
        else:
            return _format(self, BASE8, '0', 'L')

    @jit.elidable
    def hex(self):
        return _format(self, BASE16, '0x', 'L')

    @jit.elidable
    def log(self, base):
        # base is supposed to be positive or 0.0, which means we use e
        if base == 10.0:
            return _loghelper(math.log10, self)
        if base == 2.0:
            from rpython.rlib import rfloat
            return _loghelper(rfloat.log2, self)
        ret = _loghelper(math.log, self)
        if base != 0.0:
            ret /= math.log(base)
        return ret

    @not_rpython
    def tolong(self):
        l = 0L
        digits = list(self._digits)
        digits.reverse()
        for d in digits:
            l = l << SHIFT
            l += intmask(d)
        result = l * self.get_sign()
        if result == 0:
            assert self.get_sign() == 0
        return result

    def _normalize(self):
        i = self.numdigits()

        while i > 1 and self._digits[i - 1] == NULLDIGIT:
            i -= 1
        assert i > 0

        self._size = i * self.get_sign()
        if i == 1 and self._digits[0] == NULLDIGIT:
            self._size = 0
            self._digits = NULLDIGITS

    _normalize._always_inline_ = True

    @jit.elidable
    def bit_length(self):
        i = self.numdigits()
        if i == 1 and self._digits[0] == NULLDIGIT:
            return 0
        msd = self.digit(i - 1)
        msd_bits = bits_in_digit(msd)
        # yes, this can overflow: a huge number which fits 3 gigabytes of
        # memory has around 24 gigabits!
        bits = ovfcheck((i-1) * SHIFT) + msd_bits
        return bits

    @jit.elidable
    def bit_count(self):
        res = 0
        for i in range(self.numdigits()):
            res = ovfcheck(res + bit_count_digit(self.digit(i)))
        return res

    def gcd(self, other):
        """ Compute the (always positive) greatest common divisor of self and
        other """
        return gcd_lehmer(self.abs(), other.abs())

    def isqrt(self):
        """ Compute the integer square root of self """
        if self.int_lt(0):
            raise ValueError("isqrt() argument must be nonnegative")
        if self.int_eq(0):
            return NULLRBIGINT
        c = (self.bit_length() - 1) // 2
        a = ONERBIGINT
        d = 0
        for s in range(bits_in_digit(_store_digit(c)) - 1, -1, -1):
            # Loop invariant: (a-1)**2 < (self >> 2*(c - d)) < (a+1)**2
            e = d
            d = c >> s
            a = a.lshift(d - e - 1).add(self.rshift(2*c - e - d + 1).floordiv(a))
        return a.int_sub(a.mul(a).gt(self))

    def __repr__(self):
        return "<rbigint digits=%s, sign=%s, size=%d, len=%d, %s>" % (self._digits,
                                            self.get_sign(), self.numdigits(), len(self._digits),
                                            self.tolong())

ONERBIGINT = rbigint([ONEDIGIT], 1, 1)
ONENEGATIVERBIGINT = rbigint([ONEDIGIT], -1, 1)
NULLRBIGINT = rbigint()

TWO_NULLRBIGINTS = (NULLRBIGINT, NULLRBIGINT)

_jmapping = [(5 * SHIFT) % 5,
             (4 * SHIFT) % 5,
             (3 * SHIFT) % 5,
             (2 * SHIFT) % 5,
             (1 * SHIFT) % 5]



#_________________________________________________________________

# Helper Functions


def _help_mult(x, y, c):
    """
    Multiply two values, then reduce the result:
    result = X*Y % c.  If c is None, skip the mod.
    """
    res = x.mul(y)
    # Perform a modular reduction, X = X % c, but leave X alone if c
    # is NULL.
    if c is not None:
        res = res.mod(c)

    return res

@specialize.argtype(0)
def digits_from_nonneg_long(l):
    digits = []
    while True:
        digits.append(_store_digit(_mask_digit(l & MASK)))
        l = l >> SHIFT
        if not l:
            return digits[:] # to make it non-resizable

@specialize.argtype(0)
def digits_for_most_neg_long(l):
    # This helper only works if 'l' is the most negative integer of its
    # type, which in base 2 looks like: 1000000..0000
    digits = []
    while _mask_digit(l) == 0:
        digits.append(NULLDIGIT)
        l = l >> SHIFT
    # now 'l' looks like: ...111100000
    # turn it into:       ...000100000
    # to drop the extra unwanted 1's introduced by the signed right shift
    l = -intmask(l)
    assert l & MASK == l
    digits.append(_store_digit(l))
    return digits[:] # to make it non-resizable

@specialize.argtype(0)
def args_from_rarith_int1(x):
    if x > 0:
        return digits_from_nonneg_long(x), 1
    elif x == 0:
        return NULLDIGITS, 0
    elif x != most_neg_value_of_same_type(x):
        # normal case
        return digits_from_nonneg_long(-x), -1
    else:
        # the most negative integer! hacks needed...
        return digits_for_most_neg_long(x), -1

@specialize.argtype(0)
def args_from_rarith_int(x):
    return args_from_rarith_int1(widen(x))
# ^^^ specialized by the precise type of 'x', which is typically a r_xxx
#     instance from rlib.rarithmetic

@not_rpython
def args_from_long(x):
    if x >= 0:
        if x == 0:
            return NULLDIGITS, 0
        else:
            return digits_from_nonneg_long(x), 1
    else:
        return digits_from_nonneg_long(-x), -1

def _x_add(a, b):
    """ Add the absolute values of two bigint integers. """
    size_a = a.numdigits()
    size_b = b.numdigits()

    # Ensure a is the larger of the two:
    if size_a < size_b:
        a, b = b, a
        size_a, size_b = size_b, size_a
    z = rbigint([NULLDIGIT] * (size_a + 1), 1)
    i = UDIGIT_TYPE(0)
    carry = UDIGIT_TYPE(0)
    while i < size_b:
        carry += a.udigit(i) + b.udigit(i)
        z.setdigit(i, carry)
        carry >>= SHIFT
        i += 1
    while i < size_a:
        carry += a.udigit(i)
        z.setdigit(i, carry)
        carry >>= SHIFT
        i += 1
    z.setdigit(i, carry)
    z._normalize()
    return z

def _x_int_add(a, b):
    """ Add the absolute values of one bigint and one integer. """
    size_a = a.numdigits()

    z = rbigint([NULLDIGIT] * (size_a + 1), 1)
    i = UDIGIT_TYPE(1)
    carry = a.udigit(0) + abs(b)
    z.setdigit(0, carry)
    carry >>= SHIFT

    while i < size_a:
        carry += a.udigit(i)
        z.setdigit(i, carry)
        carry >>= SHIFT
        i += 1
    z.setdigit(i, carry)
    z._normalize()
    return z

def _x_sub(a, b):
    """ Subtract the absolute values of two integers. """

    size_a = a.numdigits()
    size_b = b.numdigits()
    sign = 1

    # Ensure a is the larger of the two:
    if size_a < size_b:
        sign = -1
        a, b = b, a
        size_a, size_b = size_b, size_a
    elif size_a == size_b:
        # Find highest digit where a and b differ:
        i = size_a - 1
        while i >= 0 and a.digit(i) == b.digit(i):
            i -= 1
        if i < 0:
            return NULLRBIGINT
        if a.digit(i) < b.digit(i):
            sign = -1
            a, b = b, a
        size_a = size_b = i+1

    z = rbigint([NULLDIGIT] * size_a, sign, size_a)
    borrow = UDIGIT_TYPE(0)
    i = _load_unsigned_digit(0)
    while i < size_b:
        # The following assumes unsigned arithmetic
        # works modulo 2**N for some N>SHIFT.
        borrow = a.udigit(i) - b.udigit(i) - borrow
        z.setdigit(i, borrow)
        borrow >>= SHIFT
        borrow &= 1 # Keep only one sign bit
        i += 1
    while i < size_a:
        borrow = a.udigit(i) - borrow
        z.setdigit(i, borrow)
        borrow >>= SHIFT
        borrow &= 1
        i += 1

    assert borrow == 0
    z._normalize()
    return z

def _x_int_sub(a, b):
    """ Subtract the absolute values of two integers. """

    size_a = a.numdigits()

    bdigit = abs(b)

    if size_a == 1:
        # Find highest digit where a and b differ:
        adigit = a.digit(0)

        if adigit == bdigit:
            return NULLRBIGINT

        return rbigint.fromint(adigit - bdigit)

    z = rbigint([NULLDIGIT] * size_a, 1, size_a)
    i = _load_unsigned_digit(1)
    # The following assumes unsigned arithmetic
    # works modulo 2**N for some N>SHIFT.
    borrow = a.udigit(0) - bdigit
    z.setdigit(0, borrow)
    borrow >>= SHIFT
    borrow &= 1 # Keep only one sign bit

    while i < size_a:
        borrow = a.udigit(i) - borrow
        z.setdigit(i, borrow)
        borrow >>= SHIFT
        borrow &= 1
        i += 1

    assert borrow == 0
    z._normalize()
    return z

# A neat little table of power of twos.
ptwotable = {}
for x in range(SHIFT-1):
    ptwotable[r_longlong(2 << x)] = x+1
    ptwotable[r_longlong(-2 << x)] = x+1
del x

def _x_mul(a, b, digit=0):
    """
    Grade school multiplication, ignoring the signs.
    Returns the absolute value of the product, or None if error.
    """

    size_a = a.numdigits()
    size_b = b.numdigits()

    if a is b:
        # Efficient squaring per HAC, Algorithm 14.16:
        # http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf
        # Gives slightly less than a 2x speedup when a == b,
        # via exploiting that each entry in the multiplication
        # pyramid appears twice (except for the size_a squares).
        z = rbigint([NULLDIGIT] * (size_a + size_b), 1)
        i = UDIGIT_TYPE(0)
        while i < size_a:
            f = a.uwidedigit(i)
            pz = i << 1
            pa = i + 1

            carry = z.uwidedigit(pz) + f * f
            z.setdigit(pz, carry)
            pz += 1
            carry >>= SHIFT
            assert carry <= MASK

            # Now f is added in twice in each column of the
            # pyramid it appears.  Same as adding f<<1 once.
            f <<= 1
            while pa < size_a:
                carry += z.uwidedigit(pz) + a.uwidedigit(pa) * f
                pa += 1
                z.setdigit(pz, carry)
                pz += 1
                carry >>= SHIFT
            if carry:
                carry += z.udigit(pz)
                z.setdigit(pz, carry)
                pz += 1
                carry >>= SHIFT
            if carry:
                z.setdigit(pz, z.udigit(pz) + carry)
            assert (carry >> SHIFT) == 0
            i += 1
        z._normalize()
        return z

    elif digit:
        if digit & (digit - 1) == 0:
            return b.lqshift(ptwotable[digit])

        # Even if it's not power of two it can still be useful.
        return _muladd1(b, digit)

    # a is not b
    # use the following identity to reduce the number of operations
    # a * b = a_0*b_0 + sum_{i=1}^n(a_0*b_i + a_1*b_{i-1}) + a_1*b_n
    z = rbigint([NULLDIGIT] * (size_a + size_b), 1)
    i = UDIGIT_TYPE(0)
    size_a1 = UDIGIT_TYPE(size_a - 1)
    size_b1 = UDIGIT_TYPE(size_b - 1)
    while i < size_a1:
        f0 = a.uwidedigit(i)
        f1 = a.uwidedigit(i + 1)
        pz = i
        carry = z.uwidedigit(pz) + b.uwidedigit(0) * f0
        z.setdigit(pz, carry)
        pz += 1
        carry >>= SHIFT
        j = UDIGIT_TYPE(0)
        while j < size_b1:
            # this operation does not overflow using
            # SHIFT = (LONG_BIT // 2) - 1 = B - 1; in fact before it
            # carry and z.widedigit(pz) are less than 2**(B - 1);
            # b.widedigit(j + 1) * f0 < (2**(B-1) - 1)**2; so
            # carry + z.widedigit(pz) + b.widedigit(j + 1) * f0 +
            # b.widedigit(j) * f1 < 2**(2*B - 1) - 2**B < 2**LONG)BIT - 1
            carry += z.uwidedigit(pz) + b.uwidedigit(j + 1) * f0 + \
                     b.uwidedigit(j) * f1
            z.setdigit(pz, carry)
            pz += 1
            carry >>= SHIFT
            j += 1
        # carry < 2**(B + 1) - 2
        carry += z.uwidedigit(pz) + b.uwidedigit(size_b1) * f1
        z.setdigit(pz, carry)
        pz += 1
        carry >>= SHIFT
        # carry < 4
        if carry:
            z.setdigit(pz, carry)
        assert (carry >> SHIFT) == 0
        i += 2
    if size_a & 1:
        pz = size_a1
        f = a.uwidedigit(pz)
        pb = 0
        carry = _unsigned_widen_digit(0)
        while pb < size_b:
            carry += z.uwidedigit(pz) + b.uwidedigit(pb) * f
            pb += 1
            z.setdigit(pz, carry)
            pz += 1
            carry >>= SHIFT
        if carry:
            z.setdigit(pz, z.udigit(pz) + carry)
    z._normalize()
    return z

def _kmul_split(n, size):
    """
    A helper for Karatsuba multiplication (k_mul).
    Takes a bigint "n" and an integer "size" representing the place to
    split, and sets low and high such that abs(n) == (high << size) + low,
    viewing the shift as being by digits.  The sign bit is ignored, and
    the return values are >= 0.
    """
    size_n = n.numdigits()
    size_lo = min(size_n, size)

    # We use "or" her to avoid having a check where list can be empty in _normalize.
    lo = rbigint(n._digits[:size_lo] or NULLDIGITS, 1)
    hi = rbigint(n._digits[size_lo:size_n] or NULLDIGITS, 1)
    lo._normalize()
    hi._normalize()
    return hi, lo

def _k_mul(a, b):
    """
    Karatsuba multiplication.  Ignores the input signs, and returns the
    absolute value of the product (or raises if error).
    See Knuth Vol. 2 Chapter 4.3.3 (Pp. 294-295).
    """
    asize = a.numdigits()
    bsize = b.numdigits()

    # (ah*X+al)(bh*X+bl) = ah*bh*X*X + (ah*bl + al*bh)*X + al*bl
    # Let k = (ah+al)*(bh+bl) = ah*bl + al*bh  + ah*bh + al*bl
    # Then the original product is
    #     ah*bh*X*X + (k - ah*bh - al*bl)*X + al*bl
    # By picking X to be a power of 2, "*X" is just shifting, and it's
    # been reduced to 3 multiplies on numbers half the size.

    # allocate result for both paths, asize + bsize is always enough
    ret = rbigint([NULLDIGIT] * (asize + bsize), 1)

    # Split a & b into hi & lo pieces.
    shift = bsize >> 1
    bh, bl = _kmul_split(b, shift)
    if a is b:
        ah = bh
        al = bl
    elif asize <= shift:
        # a is more than 2x smaller than b. it's important that we still use
        # .mul to get karatsuba for sub-parts. the computation is just:
        # a*(bh*X+bl) = a*bh*X + a*bl

        # multiply lower bits, copy into result
        t1 = a.mul(bl)
        for i in range(t1.numdigits()):
            ret._digits[i] = t1._digits[i]
        t2 = a.mul(bh)
        i = ret.numdigits() - shift  # digits after shift
        carry = _v_iadd(ret, shift, i, t2, t2.numdigits())
        ret._normalize()
        return ret
    else:
        ah, al = _kmul_split(a, shift)

    # The plan:
    # 1. Allocate result space (asize + bsize digits:  that's always
    #    enough).
    # 2. Compute ah*bh, and copy into result at 2*shift.
    # 3. Compute al*bl, and copy into result at 0.  Note that this
    #    can't overlap with #2.
    # 4. Subtract al*bl from the result, starting at shift.  This may
    #    underflow (borrow out of the high digit), but we don't care:
    #    we're effectively doing unsigned arithmetic mod
    #    BASE**(sizea + sizeb), and so long as the *final* result fits,
    #    borrows and carries out of the high digit can be ignored.
    # 5. Subtract ah*bh from the result, starting at shift.
    # 6. Compute (ah+al)*(bh+bl), and add it into the result starting
    #    at shift.

    # 1. Allocate result space. (done, see above)

    # 2. t1 <- ah*bh, and copy into high digits of result.
    t1 = ah.mul(bh)

    assert t1.get_sign() >= 0
    assert 2*shift + t1.numdigits() <= ret.numdigits()
    for i in range(t1.numdigits()):
        ret._digits[2*shift + i] = t1._digits[i]

    # 3. t2 <- al*bl, and copy into the low digits.
    t2 = al.mul(bl)
    assert t2.get_sign() >= 0
    assert t2.numdigits() <= 2*shift # no overlap with high digits
    for i in range(t2.numdigits()):
        ret._digits[i] = t2._digits[i]

    # 4 & 5. Subtract ah*bh (t1) and al*bl (t2).  We do al*bl first
    # because it's fresher in cache.
    i = ret.numdigits() - shift  # # digits after shift
    _v_isub(ret, shift, i, t2, t2.numdigits())
    _v_isub(ret, shift, i, t1, t1.numdigits())

    # 6. t3 <- (ah+al)(bh+bl), and add into result.
    t1 = _x_add(ah, al)

    if a is b:
        t2 = t1
    else:
        t2 = _x_add(bh, bl)

    t3 = t1.mul(t2)
    assert t3.get_sign() >= 0

    # Add t3.  It's not obvious why we can't run out of room here.
    # See the (*) comment after this function.
    _v_iadd(ret, shift, i, t3, t3.numdigits())

    ret._normalize()
    return ret

""" (*) Why adding t3 can't "run out of room" above.

Let f(x) mean the floor of x and c(x) mean the ceiling of x.  Some facts
to start with:

1. For any integer i, i = c(i/2) + f(i/2).  In particular,
   bsize = c(bsize/2) + f(bsize/2).
2. shift = f(bsize/2)
3. asize <= bsize
4. Since we call k_lopsided_mul if asize*2 <= bsize, asize*2 > bsize in this
   routine, so asize > bsize/2 >= f(bsize/2) in this routine.

We allocated asize + bsize result digits, and add t3 into them at an offset
of shift.  This leaves asize+bsize-shift allocated digit positions for t3
to fit into, = (by #1 and #2) asize + f(bsize/2) + c(bsize/2) - f(bsize/2) =
asize + c(bsize/2) available digit positions.

bh has c(bsize/2) digits, and bl at most f(size/2) digits.  So bh+hl has
at most c(bsize/2) digits + 1 bit.

If asize == bsize, ah has c(bsize/2) digits, else ah has at most f(bsize/2)
digits, and al has at most f(bsize/2) digits in any case.  So ah+al has at
most (asize == bsize ? c(bsize/2) : f(bsize/2)) digits + 1 bit.

The product (ah+al)*(bh+bl) therefore has at most

    c(bsize/2) + (asize == bsize ? c(bsize/2) : f(bsize/2)) digits + 2 bits

and we have asize + c(bsize/2) available digit positions.  We need to show
this is always enough.  An instance of c(bsize/2) cancels out in both, so
the question reduces to whether asize digits is enough to hold
(asize == bsize ? c(bsize/2) : f(bsize/2)) digits + 2 bits.  If asize < bsize,
then we're asking whether asize digits >= f(bsize/2) digits + 2 bits.  By #4,
asize is at least f(bsize/2)+1 digits, so this in turn reduces to whether 1
digit is enough to hold 2 bits.  This is so since SHIFT=15 >= 2.  If
asize == bsize, then we're asking whether bsize digits is enough to hold
c(bsize/2) digits + 2 bits, or equivalently (by #1) whether f(bsize/2) digits
is enough to hold 2 bits.  This is so if bsize >= 2, which holds because
bsize >= KARATSUBA_CUTOFF >= 2.

Note that since there's always enough room for (ah+al)*(bh+bl), and that's
clearly >= each of ah*bh and al*bl, there's always enough room to subtract
ah*bh and al*bl too.
"""


def _inplace_divrem1(pout, pin, n):
    """
    Divide bigint pin by non-zero digit n, storing quotient
    in pout, and returning the remainder. It's OK for pin == pout on entry.
    """
    rem = _unsigned_widen_digit(0)
    assert n > 0 and n <= MASK
    size = pin.numdigits() - 1
    while size >= 0:
        rem = (rem << SHIFT) | pin.udigit(size)
        hi = rem // n
        pout.setdigit(size, hi)
        rem -= hi * n
        size -= 1
    return rffi.cast(lltype.Signed, rem)

@jit.elidable
def _divrem1(a, n):
    """
    Divide a bigint integer by a digit, returning both the quotient
    and the remainder as a tuple.
    The sign of a is ignored; n should not be zero.
    """
    assert n > 0 and n <= MASK

    size = a.numdigits()
    z = rbigint([NULLDIGIT] * size, 1, size)
    rem = _inplace_divrem1(z, a, n)
    z._normalize()
    return z, rem

def _int_rem_core(a, digit):
    # digit must be positive
    size = UDIGIT_TYPE(a.numdigits() - 1)

    if size > 0:
        wrem = a.widedigit(size)
        while size > 0:
            size -= 1
            wrem = ((wrem << SHIFT) | a.digit(size)) % digit
        rem = _store_digit(wrem)
    else:
        rem = _store_digit(a.digit(0) % digit)

    return rem

def _v_iadd(x, xofs, m, y, n):
    """
    x and y are rbigints, m >= n required.  x.digits[0:n] is modified in place,
    by adding y.digits[0:m] to it.  Carries are propagated as far as
    x[m-1], and the remaining carry (0 or 1) is returned.
    Python adaptation: x is addressed relative to xofs!
    """
    carry = UDIGIT_TYPE(0)

    assert m >= n
    i = _load_unsigned_digit(xofs)
    iend = xofs + n
    while i < iend:
        carry += x.udigit(i) + y.udigit(i-xofs)
        x.setdigit(i, carry)
        carry >>= SHIFT
        i += 1
    iend = xofs + m
    while carry and i < iend:
        carry += x.udigit(i)
        x.setdigit(i, carry)
        carry >>= SHIFT
        i += 1
    return carry

def _v_isub(x, xofs, m, y, n):
    """
    x and y are rbigints, m >= n required.  x.digits[0:n] is modified in place,
    by substracting y.digits[0:m] to it. Borrows are propagated as
    far as x[m-1], and the remaining borrow (0 or 1) is returned.
    Python adaptation: x is addressed relative to xofs!
    """
    borrow = UDIGIT_TYPE(0)

    assert m >= n
    i = _load_unsigned_digit(xofs)
    iend = xofs + n
    while i < iend:
        borrow = x.udigit(i) - y.udigit(i-xofs) - borrow
        x.setdigit(i, borrow)
        borrow >>= SHIFT
        borrow &= 1    # keep only 1 sign bit
        i += 1
    iend = xofs + m
    while borrow and i < iend:
        borrow = x.udigit(i) - borrow
        x.setdigit(i, borrow)
        borrow >>= SHIFT
        borrow &= 1
        i += 1
    return borrow

@specialize.argtype(2)
def _muladd1(a, n, extra=0):
    """Multiply by a single digit and add a single digit, ignoring the sign.
    """
    assert n > 0

    size_a = a.numdigits()
    z = rbigint([NULLDIGIT] * (size_a+1), 1)
    assert extra & MASK == extra
    carry = _unsigned_widen_digit(extra)
    i = 0
    while i < size_a:
        carry += a.uwidedigit(i) * n
        z.setdigit(i, carry)
        carry >>= SHIFT
        i += 1
    z.setdigit(i, carry)
    z._normalize()
    return z

def _v_lshift(z, a, m, d):
    """ Shift digit vector a[0:m] d bits left, with 0 <= d < SHIFT. Put
        * result in z[0:m], and return the d bits shifted out of the top.
    """

    carry = 0
    #assert 0 <= d and d < SHIFT
    i = 0
    while i < m:
        acc = a.uwidedigit(i) << d | carry
        z.setdigit(i, acc)
        carry = acc >> SHIFT
        i += 1

    return carry

def _v_rshift(z, a, m, d):
    """ Shift digit vector a[0:m] d bits right, with 0 <= d < PyLong_SHIFT. Put
        * result in z[0:m], and return the d bits shifted out of the bottom.
    """

    carry = _unsigned_widen_digit(0)
    acc = _unsigned_widen_digit(0)
    mask = (1 << d) - 1

    #assert 0 <= d and d < SHIFT
    i = m-1
    while i >= 0:
        acc = (carry << SHIFT) | a.udigit(i)
        carry = acc & mask
        z.setdigit(i, acc >> d)
        i -= 1

    return carry

def _x_divrem(v1, w1):
    """ Unsigned bigint division with remainder -- the algorithm """
    size_v = v1.numdigits()
    size_w = w1.numdigits()
    assert size_v >= size_w and size_w > 1

    v = rbigint([NULLDIGIT] * (size_v + 1), 1, size_v + 1)
    w = rbigint([NULLDIGIT] * size_w, 1, size_w)

    """ normalize: shift w1 left so that its top digit is >= PyLong_BASE/2.
        shift v1 left by the same amount. Results go into w and v. """

    d = SHIFT - bits_in_digit(w1.digit(abs(size_w-1)))
    carry = _v_lshift(w, w1, size_w, d)
    assert carry == 0
    carry = _v_lshift(v, v1, size_v, d)
    if carry != 0 or v.digit(abs(size_v-1)) >= w.digit(abs(size_w-1)):
        v.setdigit(size_v, carry)
        size_v += 1

    """ Now v->ob_digit[size_v-1] < w->ob_digit[size_w-1], so quotient has
        at most (and usually exactly) k = size_v - size_w digits. """
    k = size_v - size_w
    if k == 0:
        # We can't use v1, nor NULLRBIGINT here as some function modify the result.
        assert _v_rshift(w, v, size_w, d) == 0
        w._normalize()
        return rbigint([NULLDIGIT]), w

    assert k > 0
    a = rbigint([NULLDIGIT] * k, 1, k)

    wm1 = w.widedigit(abs(size_w-1))
    wm2 = w.widedigit(abs(size_w-2))

    j = size_v - 1
    k -= 1
    while k >= 0:
        assert j >= 0
        """ inner loop: divide vk[0:size_w+1] by w0[0:size_w], giving
            single-digit quotient q, remainder in vk[0:size_w]. """

        # estimate quotient digit q; may overestimate by 1 (rare)
        if j >= size_v:
            vtop = 0
        else:
            vtop = v.widedigit(j)
        assert vtop <= wm1

        vv = (vtop << SHIFT) | v.widedigit(abs(j-1))

        # Hints to make division just as fast as doing it unsigned. But avoids casting to get correct results.
        assert vv >= 0
        assert wm1 >= 1

        q = vv / wm1
        r = vv % wm1 # This seems to be slightly faster on widen digits than vv - wm1 * q.
        vj2 = v.digit(abs(j-2))
        while wm2 * q > ((r << SHIFT) | vj2):
            q -= 1
            r += wm1

        #assert q <= MASK+1, We need to compare to BASE <=, but ehm, it gives a buildin long error. So we ignore this.

        # subtract q*w0[0:size_w] from vk[0:size_w+1]
        zhi = 0
        i = 0
        while i < size_w:
            z = v.widedigit(k+i) + zhi - q * w.widedigit(i)
            v.setdigit(k+i, z)
            zhi = z >> SHIFT
            i += 1

        # add w back if q was too large (this branch taken rarely)
        if vtop + zhi < 0:
            carry = UDIGIT_TYPE(0)
            i = 0
            while i < size_w:
                carry += v.udigit(k+i) + w.udigit(i)
                v.setdigit(k+i, carry)
                carry >>= SHIFT
                i += 1
            q -= 1

        # store quotient digit
        a.setdigit(k, q)
        k -= 1
        j -= 1

    carry = _v_rshift(w, v, size_w, d)
    assert carry == 0

    a._normalize()
    w._normalize()

    return a, w

@jit.elidable
def _divrem(a, b):
    """ Long division with remainder, top-level routine """
    size_a = a.numdigits()
    size_b = b.numdigits()

    if b.get_sign() == 0:
        raise ZeroDivisionError("long division or modulo by zero")

    if (size_a < size_b or
        (size_a == size_b and
         a.digit(abs(size_a-1)) < b.digit(abs(size_b-1)))):
        # |a| < |b|
        return NULLRBIGINT, a# result is 0
    if size_b == 1:
        z, urem = _divrem1(a, b.digit(0))
        rem = rbigint([_store_digit(urem)], int(urem != 0), 1)
    else:
        z, rem = _x_divrem(a, b)
    # 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.get_sign() != b.get_sign():
        z._set_sign(-z.get_sign())
    if a.get_sign() < 0 and rem.get_sign() != 0:
        rem._set_sign(-rem.get_sign())
    return z, rem



class LimitHolder:
    pass

HOLDER = LimitHolder()
HOLDER.DIV_LIMIT = 21
HOLDER.STR2INT_LIMIT = 2048
HOLDER.MINSIZE_STR2INT = 4000


def _extract_digits(a, startindex, numdigits):
    assert startindex >= 0
    if startindex >= a.numdigits():
        return NULLRBIGINT
    stop = min(startindex + numdigits, a.numdigits())
    assert stop >= 0
    digits = a._digits[startindex: stop]
    if not digits:
        return NULLRBIGINT
    r = rbigint(digits, 1)
    r._normalize()
    return r

def div2n1n(a_container, a_startindex, b, n_S):
    """Divide a 2*n_S-digit nonnegative integer a by an n_S-digit positive integer
    b, using a recursive divide-and-conquer algorithm.

    Inputs:
      n_S is a positive integer
      b is a positive rbigint with exactly n_S digits
      a is a nonnegative integer such that a < 2**(n_S * SHIFT) * b

    Output:
      (q, r) such that a = b*q+r and 0 <= r < b.

    a is represented as a slice of a bigger number a_container, 2 * n_S digits
    wide, starting at a_startindex
    """
    if n_S <= HOLDER.DIV_LIMIT:
        a = _extract_digits(a_container, a_startindex, 2 * n_S)
        if a.get_sign() == 0:
            return NULLRBIGINT, NULLRBIGINT
        res = _divrem(a, b)
        return res
    assert n_S & 1 == 0
    half_n_S = n_S >> 1
    # school division: (diagram from Burnikel & Ziegler, p 3)
    #
    #   size half_n_S                                     size n_S
    #    |                                                     |
    #    v                                                     v
    # +----+----+----+----+   +----+----+   +----+----+   +---------+
    # | a1 | a2 | a3 | a4 | / | b1 | b2 | = | q1 | q2 | = |    q    |
    # +====+====+====+====+   +----+----+   +----+----+   +---------+
    # | q1 * b1 |
    # +----+----+----+               <
    #      | q1 * b2 | subtracting  <   first call to div3n2n
    #      +---------+----+          <
    #      |    r1   | a4 |
    #      +---------+----+
    #      | q2 * b1 |
    #      +----+----+----+              <
    #           | q2 * b2 | subtracing  <   second call to div3n2n
    #           +---------+              <
    #           |    r    |
    #           +---------+

    b1, b2 = _extract_digits(b, half_n_S, half_n_S), _extract_digits(b, 0, half_n_S)
    q1, r1 = div3n2n(a_container, a_startindex + n_S, a_container, a_startindex + half_n_S, b, b1, b2, half_n_S)
    q2, r = div3n2n(r1, 0, a_container, a_startindex, b, b1, b2, half_n_S)
    return _full_digits_lshift_then_or(q1, half_n_S, q2), r

def div3n2n(a12_container, a12_startindex, a3_container, a3_startindex, b, b1, b2, n_S):
    """Helper function for div2n1n; not intended to be called directly."""
    q, r = div2n1n(a12_container, a12_startindex, b1, n_S)
    # equivalent to r = _full_digits_lshift_then_or(r, n_S, _extract_digits(a_container, a3_startindex, n_S))
    if r.get_sign() == 0:
        r = _extract_digits(a3_container, a3_startindex, n_S)
    else:
        digits = [NULLDIGIT] * (n_S + r.numdigits())
        index = 0
        for i in range(a3_startindex, min(a3_startindex + n_S, a3_container.numdigits())):
            digits[index] = a3_container._digits[i]
            index += 1
        index = n_S
        for i in range(r.numdigits()):
            digits[index] = r._digits[i]
            index += 1
        r = rbigint(digits, 1)
        r._normalize()
    if q.get_sign() == 0:
        return q, r
    r = r.sub(q.mul(b2))

    # loop runs at most twice
    while r.get_sign() < 0:
        q = q.int_sub(1)
        r = r.add(b)
    return q, r

def _full_digits_lshift_then_or(a, n, b):
    """ equivalent to a.lshift(n * SHIFT).or_(b)
    the size of b must be smaller than n
    """
    if a.get_sign() == 0:
        return b
    bdigits = b.numdigits()
    assert bdigits <= n
    # b._digits + [NULLDIGIT] * (n - bdigits) + a._digits
    digits = [NULLDIGIT] * (a.numdigits() + n)
    for i in range(b.numdigits()):
        digit = b._digits[i]
        digits[i] = digit
    index = n
    for i in range(a.numdigits()):
        digits[index] = a._digits[i]
        index += 1

    return rbigint(digits, 1)

def _divmod_fast_pos(a, b):
    """Divide a positive integer a by a positive integer b, giving
    quotient and remainder."""
    # Use grade-school algorithm in base 2**n, n = nbits(b)
    n = b.bit_length()
    m = a.bit_length()
    if m < n:
        return NULLRBIGINT, a
    # make n of the form SHIFT * HOLDER.DIV_LIMIT * 2 ** x
    new_n = SHIFT * HOLDER.DIV_LIMIT
    while new_n < n:
        new_n <<= 1
    rest_shift = new_n - n
    if rest_shift:
        a = a.lshift(rest_shift)
        b = b.lshift(rest_shift)
        assert b.bit_length() == new_n
    n = new_n

    n_S = n // SHIFT
    r = range(0, a.numdigits(), n_S)
    a_digits_base_two_pow_n = [None] * len(r)
    index = 0
    for i in r:
        assert i >= 0
        stop = i + n_S
        assert stop >= 0
        a_digits_base_two_pow_n[index] = rbigint(a._digits[i: stop], 1)
        index += 1

    a_digits_index = len(a_digits_base_two_pow_n) - 1
    assert a_digits_index >= 0
    if a_digits_base_two_pow_n[a_digits_index].ge(b):
        r = NULLRBIGINT
    else:
        r = a_digits_base_two_pow_n[a_digits_index]
        a_digits_index -= 1
    q_digits = None
    q_index_start = a_digits_index * n_S
    while a_digits_index >= 0:
        arg1 = _full_digits_lshift_then_or(r, n_S, a_digits_base_two_pow_n[a_digits_index])
        q_digitbase_two_pow_n, r = div2n1n(arg1, 0, b, n_S)
        if q_digits is None:
            q_digits = [NULLDIGIT] * (a_digits_index * n_S + q_digitbase_two_pow_n.numdigits())
        for i in range(q_digitbase_two_pow_n.numdigits()):
            q_digits[q_index_start + i] = q_digitbase_two_pow_n._digits[i]
        q_index_start -= n_S
        a_digits_index -= 1
    if rest_shift:
        r = r.rshift(rest_shift)
    if q_digits is None:
        q = NULLRBIGINT
    else:
        q = rbigint(q_digits, 1)
    q._normalize()
    r._normalize()
    return q, r

def divmod_big(a, b):
    # code from Mark Dickinson via https://bugs.python.org/file11060/fast_div.py
    # follows cr.yp.to/bib/1998/burnikel.ps
    if b.eq(NULLRBIGINT):
        raise ZeroDivisionError
    elif b.get_sign() < 0:
        q, r = divmod_big(a.neg(), b.neg())
        return q, r.neg()
    elif a.get_sign() < 0:
        q, r = divmod_big(a.invert(), b)
        return q.invert(), b.add(r.invert())
    elif a.get_sign() == 0:
        return TWO_NULLRBIGINTS
    else:
        return _divmod_fast_pos(a, b)

def _x_int_lt(a, b, eq=False):
    """ Compare bigint a with int b for less than or less than or equal """
    osign = 1
    if b == 0:
        osign = 0
    elif b < 0:
        osign = -1

    if a.get_sign() > osign:
        return False
    elif a.get_sign() < osign:
        return True

    digits = a.numdigits()

    if digits > 1:
        if osign == 1:
            return False
        else:
            return True

    d1 = a.get_sign() * a.digit(0)
    if eq:
        if d1 <= b:
            return True
    else:
        if d1 < b:
            return True
    return False

# ______________ conversions to double _______________

def _AsScaledDouble(v):
    """
    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.
    """
    NBITS_WANTED = 57
    if v.get_sign() == 0:
        return 0.0, 0
    i = v.numdigits() - 1
    sign = v.get_sign()
    x = float(v.digit(i))
    nbitsneeded = NBITS_WANTED - 1
    # Invariant:  i Python digits remain unaccounted for.
    while i > 0 and nbitsneeded > 0:
        i -= 1
        x = x * FLOAT_MULTIPLIER + float(v.digit(i))
        nbitsneeded -= SHIFT
    # There are i digits we didn't shift in.  Pretending they're all
    # zeroes, the true value is x * 2**(i*SHIFT).
    exponent = i
    assert x > 0.0
    return x * sign, exponent

##def ldexp(x, exp):
##    assert type(x) is float
##    lb1 = LONG_BIT - 1
##    multiplier = float(1 << lb1)
##    while exp >= lb1:
##        x *= multiplier
##        exp -= lb1
##    if exp:
##        x *= float(1 << exp)
##    return x

# note that math.ldexp checks for overflows,
# while the C ldexp is not guaranteed to do.
# XXX make sure that we don't ignore this!
# YYY no, we decided to do ignore this!

@jit.dont_look_inside
def _AsDouble(n):
    """ Get a C double from a bigint object. """
    # This is a "correctly-rounded" version from Python 2.7.
    #
    from rpython.rlib import rfloat
    DBL_MANT_DIG = rfloat.DBL_MANT_DIG  # 53 for IEEE 754 binary64
    DBL_MAX_EXP = rfloat.DBL_MAX_EXP    # 1024 for IEEE 754 binary64
    assert DBL_MANT_DIG < r_ulonglong.BITS

    # Reduce to case n positive.
    sign = n.get_sign()
    if sign == 0:
        return 0.0
    elif sign < 0:
        n = n.neg()

    # Find exponent: 2**(exp - 1) <= n < 2**exp
    exp = n.bit_length()

    # Get top DBL_MANT_DIG + 2 significant bits of n, with a 'sticky'
    # last bit: that is, the least significant bit of the result is 1
    # iff any of the shifted-out bits is set.
    shift = DBL_MANT_DIG + 2 - exp
    if shift >= 0:
        q = _AsULonglong_mask(n) << shift
        if not we_are_translated():
            assert q == n.tolong() << shift   # no masking actually done
    else:
        shift = -shift
        n2 = n.rshift(shift)
        q = _AsULonglong_mask(n2)
        if not we_are_translated():
            assert q == n2.tolong()           # no masking actually done
        if not n.eq(n2.lshift(shift)):
            q |= 1

    # Now remove the excess 2 bits, rounding to nearest integer (with
    # ties rounded to even).
    q = (q >> 2) + r_uint((bool(q & 2) and bool(q & 5)))

    if exp > DBL_MAX_EXP or (exp == DBL_MAX_EXP and
                             q == r_ulonglong(1) << DBL_MANT_DIG):
        raise OverflowError("integer too large to convert to float")

    ad = math.ldexp(float(q), exp - DBL_MANT_DIG)
    if sign < 0:
        ad = -ad
    return ad

@specialize.arg(0)
def _loghelper(func, arg):
    """
    A decent logarithm is easy to compute even for huge bigints, but libm can't
    do that by itself -- loghelper can.  func is log or log10.
    Note that overflow isn't possible:  a bigint can contain
    no more than INT_MAX * SHIFT bits, so has value certainly less than
    2**(2**64 * 2**16) == 2**2**80, and log2 of that is 2**80, which is
    small enough to fit in an IEEE single.  log and log10 are even smaller.
    """
    x, e = _AsScaledDouble(arg)
    if x <= 0.0:
        raise ValueError
    # Value is ~= x * 2**(e*SHIFT), so the log ~=
    # log(x) + log(2) * e * SHIFT.
    # CAUTION:  e*SHIFT may overflow using int arithmetic,
    # so force use of double. */
    return func(x) + (e * float(SHIFT) * func(2.0))

# ____________________________________________________________

BASE_AS_FLOAT = float(1 << SHIFT)     # note that it may not fit an int

bit_length_table = [0] * 2**8
for i in range(1, len(bit_length_table)):
    bit_length_table[i] = 1 + bit_length_table[i//2]
bit_length_table = "".join(chr(c) for c in bit_length_table)
del i


def bits_in_digit(d):
    # returns the unique integer k such that 2**(k-1) <= d <
    # 2**k if d is nonzero, else 0.
    assert d >= 0
    d_bits = 0
    while d >= 2 ** 8:
        d_bits += 8
        d >>= 8
    d_bits += ord(bit_length_table[d])
    return d_bits


@jit.elidable
def bit_length_int(val):
    if val < 0:
        # warning, "-val" can overflow here
        val = -((val + 1) >> 1)
        count = 1
    else:
        count = 0
    if LONG_BIT > 32 and val >= 2**32:
        val >>= 32
        count += 32
    if val >= 2**16:
        val >>= 16
        count += 16
    if val >= 2**8:
        val >>= 8
        count += 8
    return count + ord(bit_length_table[val])


def bit_count_digit(val):
    if SHIFT == 63:
        return _bitcount64(r_ulonglong(val))
    count = 0
    while val:
        count += val & 1
        val >>= 1
    return count

BITCOUNT_K1 = r_ulonglong(0x5555555555555555) # Repeating 01
BITCOUNT_K2 = r_ulonglong(0x3333333333333333) # Repeating 0011
BITCOUNT_K4 = r_ulonglong(0x0f0f0f0f0f0f0f0f) # Repeating 00001111
BITCOUNT_KF = r_ulonglong(0x0101010101010101) # Repeating 00000001

@jit.elidable
def _bitcount64(x):
    # takes an r_ulonglong
    return intmask(_bitcount64_ops(x))

@always_inline
def _bitcount64_ops(x, K1=BITCOUNT_K1, K2=BITCOUNT_K2, K4=BITCOUNT_K4, KF=BITCOUNT_KF):
    # this is a separate function for Z3 testing
    x -= (x >> 1) & K1
    x = (x & K2) + ((x >> 2) & K2)
    x = (x + (x >> 4)) & K4 
    return (x * KF) >> 56

def _truediv_result(result, negate):
    if negate:
        result = -result
    return result

def _truediv_overflow():
    raise OverflowError("integer division result too large for a float")

def _bigint_true_divide(a, b):
    # A longish method to obtain the floating-point result with as much
    # precision as theoretically possible.  The code is almost directly
    # copied from CPython.  See there (Objects/longobject.c,
    # long_true_divide) for detailled comments.  Method in a nutshell:
    #
    #    0. reduce to case a, b > 0; filter out obvious underflow/overflow
    #    1. choose a suitable integer 'shift'
    #    2. use integer arithmetic to compute x = floor(2**-shift*a/b)
    #    3. adjust x for correct rounding
    #    4. convert x to a double dx with the same value
    #    5. return ldexp(dx, shift).

    from rpython.rlib import rfloat
    DBL_MANT_DIG = rfloat.DBL_MANT_DIG  # 53 for IEEE 754 binary64
    DBL_MAX_EXP = rfloat.DBL_MAX_EXP    # 1024 for IEEE 754 binary64
    DBL_MIN_EXP = rfloat.DBL_MIN_EXP
    MANT_DIG_DIGITS = DBL_MANT_DIG // SHIFT
    MANT_DIG_BITS = DBL_MANT_DIG % SHIFT

    # Reduce to case where a and b are both positive.
    negate = (a.get_sign() < 0) ^ (b.get_sign() < 0)
    if not b.tobool():
        raise ZeroDivisionError("long division or modulo by zero")
    if not a.tobool():
        return _truediv_result(0.0, negate)

    a_size = a.numdigits()
    b_size = b.numdigits()

    # Fast path for a and b small (exactly representable in a double).
    # Relies on floating-point division being correctly rounded; results
    # may be subject to double rounding on x86 machines that operate with
    # the x87 FPU set to 64-bit precision.
    a_is_small = (a_size <= MANT_DIG_DIGITS or
                  (a_size == MANT_DIG_DIGITS+1 and
                   a.digit(MANT_DIG_DIGITS) >> MANT_DIG_BITS == 0))
    b_is_small = (b_size <= MANT_DIG_DIGITS or
                  (b_size == MANT_DIG_DIGITS+1 and
                   b.digit(MANT_DIG_DIGITS) >> MANT_DIG_BITS == 0))
    if a_is_small and b_is_small:
        a_size -= 1
        da = float(a.digit(a_size))
        while True:
            a_size -= 1
            if a_size < 0:
                break
            da = da * BASE_AS_FLOAT + a.digit(a_size)

        b_size -= 1
        db = float(b.digit(b_size))
        while True:
            b_size -= 1
            if b_size < 0:
                break
            db = db * BASE_AS_FLOAT + b.digit(b_size)

        return _truediv_result(da / db, negate)

    # Catch obvious cases of underflow and overflow
    diff = a_size - b_size
    if diff > sys.maxint/SHIFT - 1:
        return _truediv_overflow()           # Extreme overflow
    elif diff < 1 - sys.maxint/SHIFT:
        return _truediv_result(0.0, negate)  # Extreme underflow
    # Next line is now safe from overflowing integers
    diff = (diff * SHIFT + bits_in_digit(a.digit(a_size - 1)) -
                           bits_in_digit(b.digit(b_size - 1)))
    # Now diff = a_bits - b_bits.
    if diff > DBL_MAX_EXP:
        return _truediv_overflow()
    elif diff < DBL_MIN_EXP - DBL_MANT_DIG - 1:
        return _truediv_result(0.0, negate)

    # Choose value for shift; see comments for step 1 in CPython.
    shift = max(diff, DBL_MIN_EXP) - DBL_MANT_DIG - 2

    inexact = False

    # x = abs(a * 2**-shift)
    if shift <= 0:
        x = a.lshift(-shift)
    else:
        x = a.rshift(shift, dont_invert=True)
        # set inexact if any of the bits shifted out is nonzero
        if not a.eq(x.lshift(shift)):
            inexact = True

    # x //= b. If the remainder is nonzero, set inexact.
    x, rem = _divrem(x, b)
    if rem.tobool():
        inexact = True

    assert x.tobool()    # result of division is never zero
    x_size = x.numdigits()
    x_bits = (x_size-1)*SHIFT + bits_in_digit(x.digit(x_size-1))

    # The number of extra bits that have to be rounded away.
    extra_bits = max(x_bits, DBL_MIN_EXP - shift) - DBL_MANT_DIG
    assert extra_bits == 2 or extra_bits == 3

    # Round by remembering a modified copy of the low digit of x
    mask = r_uint(1 << (extra_bits - 1))
    low = x.udigit(0) | r_uint(inexact)
    if (low & mask) != 0 and (low & (3*mask-1)) != 0:
        low += mask
    x_digit_0 = low & ~(mask-1)

    # Convert x to a double dx; the conversion is exact.
    x_size -= 1
    dx = 0.0
    while x_size > 0:
        dx += x.digit(x_size)
        dx *= BASE_AS_FLOAT
        x_size -= 1
    dx += x_digit_0

    # Check whether ldexp result will overflow a double.
    if (shift + x_bits >= DBL_MAX_EXP and
        (shift + x_bits > DBL_MAX_EXP or dx == math.ldexp(1.0, x_bits))):
        return _truediv_overflow()

    return _truediv_result(math.ldexp(dx, shift), negate)

# ____________________________________________________________

BASE8  = '01234567'
BASE10 = '0123456789'
BASE16 = '0123456789abcdef'

def _format_base2_notzero(a, digits, prefix='', suffix='', max_str_digits=0):
        base = len(digits)
        # JRH: special case for power-of-2 bases
        accum = 0
        accumbits = 0  # # of bits in accum
        basebits = 0
        i = base
        while i > 1:
            basebits += 1
            i >>= 1

        # Compute a rough upper bound for the length of the string
        size_a = a.numdigits()
        i = 5 + len(prefix) + len(suffix) + (size_a*SHIFT + basebits-1) // basebits
        result = [chr(0)] * i
        next_char_index = i
        j = len(suffix)
        while j > 0:
            next_char_index -= 1
            j -= 1
            result[next_char_index] = suffix[j]

        i = 0
        while i < size_a:
            accum |= a.widedigit(i) << accumbits
            accumbits += SHIFT
            assert accumbits >= basebits
            while 1:
                cdigit = intmask(accum & (base - 1))
                next_char_index -= 1
                assert next_char_index >= 0
                result[next_char_index] = digits[cdigit]
                accumbits -= basebits
                accum >>= basebits
                if i < size_a - 1:
                    if accumbits < basebits:
                        break
                else:
                    if accum <= 0:
                        break
            i += 1
        j = len(prefix)
        while j > 0:
            next_char_index -= 1
            j -= 1
            result[next_char_index] = prefix[j]

        if a.get_sign() < 0:
            next_char_index -= 1
            result[next_char_index] = '-'

        assert next_char_index >= 0    # otherwise, buffer overflow (this is also a
                         # hint for the annotator for the slice below)
        return ''.join(result[next_char_index:])


class PartsCacheBase(object):
    def __init__(self, base):
        mindigits = 1
        curr = base
        while 1:
            try:
                next = ovfcheck(curr * base)
            except OverflowError:
                break
            if next >= MASK:
                break
            curr = next
            mindigits += 1
        self.mindigits = mindigits
        part = rbigint.fromint(curr)
        self.lowest_part = curr
        self.parts_cache = [part]

class _PartsCache(object):
    def __init__(self):
        # 36 - 3, because bases 0, 1 make no sense
        # and 2 is handled differently
        self.parts_cache = [None] * 34

    def get_cached_parts(self, base):
        index = base - 3
        res = self.parts_cache[index]
        if res is None:
            res = PartsCacheBase(base)
            self.parts_cache[index] = res
        return res


_parts_cache = _PartsCache()
_parts_cache_10 = _parts_cache.get_cached_parts(10)


def _format_int_general(val, digits):
    base = len(digits)
    out = []
    while val:
        out.append(digits[val % base])
        val //= base
    out.reverse()
    return "".join(out)

def _format_int10(val, digits):
    return str(val)

_format10_table2 = "".join(str(i // 10) + str(i % 10) for i in range(100))

def _format_int10_18digits(val, builder):
    assert 0 <= val < 10**18
    top2 = val // 10**16
    assert top2 < 100
    val = val % 10**16
    a = val // 10**8
    b = val % 10**8

    aa = a // 10**4
    ab = a % 10**4
    ba = b // 10**4
    bb = b % 10**4

    aaa = aa // 10**2
    aab = aa % 10**2
    aba = ab // 10**2
    abb = ab % 10**2
    baa = ba // 10**2
    bab = ba % 10**2
    bba = bb // 10**2
    bbb = bb % 10**2
    builder.append_slice(_format10_table2, 2*top2, 2*top2 + 2)
    builder.append_slice(_format10_table2, 2*aaa, 2*aaa + 2)
    builder.append_slice(_format10_table2, 2*aab, 2*aab + 2)
    builder.append_slice(_format10_table2, 2*aba, 2*aba + 2)
    builder.append_slice(_format10_table2, 2*abb, 2*abb + 2)
    builder.append_slice(_format10_table2, 2*baa, 2*baa + 2)
    builder.append_slice(_format10_table2, 2*bab, 2*bab + 2)
    builder.append_slice(_format10_table2, 2*bba, 2*bba + 2)
    builder.append_slice(_format10_table2, 2*bbb, 2*bbb + 2)

@specialize.arg(6)
def _format_recursive(x, i, output, pcb, digits, size_prefix, _format_int, max_str_digits):
    while i > 0:
        top, x = x.divmod(pcb.parts_cache[i]) # split the number
        if not top.tobool() and output.getlength() == size_prefix:
            # the top half can often be 0, because the number isn't perfectly a
            # power of the base
            pass
        else:
            _format_recursive(top, i-1, output, pcb, digits, size_prefix, _format_int, max_str_digits)
        # do the second recursive call by means of manual tail calling
        i -= 1
    # bottomed out with mindigits sized pieces
    # use str of ints
    mindigits = pcb.mindigits
    curlen = output.getlength()
    # the last divmod is guaranteed to return two ints
    high, low = _format_lowest_level_divmod_int_results(x, pcb.lowest_part)
    # this checks whether any digit has been appended yet
    lowdone = False
    if curlen == size_prefix:
        if high:
            s = _format_int(high, digits)
            output.append(s)
            curlen += len(s)
        else:
            if low:
                s = _format_int(low, digits)
                output.append(s)
                curlen += len(s)
            lowdone = True
    else:
        if SHIFT == 63 and _format_int is _format_int10 and mindigits == 18:
            _format_int10_18digits(high, output)
        else:
            s = _format_int(high, digits)
            output.append_multiple_char(digits[0], mindigits - len(s))
            output.append(s)
        curlen += mindigits
    if not lowdone:
        if SHIFT == 63 and _format_int is _format_int10 and mindigits == 18:
            _format_int10_18digits(low, output)
        else:
            s = _format_int(low, digits)
            output.append_multiple_char(digits[0], mindigits - len(s))
            output.append(s)
        curlen += mindigits
    if max_str_digits > 0 and curlen  - size_prefix > max_str_digits:
        raise MaxIntError("requested output too large")

@always_inline
def _format_lowest_level_divmod_int_results(x, iother):
    # this is only useful in the context of _format_recursive, where we know
    # that at the lowest levels the division leaves a result that fits into an
    # int (and the mod does anyway fit into an int)
    # x must be smaller than iother**2
    # iother must be positive
    if not x.tobool():
        return 0, 0
    assert iother > 0 and iother <= MASK
    size = x.numdigits() - 1
    if size == 1:
        rem = x.uwidedigit(1)
        assert rem < iother # otherwise iother ** 2 >= x
    else:
        rem = _unsigned_widen_digit(0)
    rem = (rem << SHIFT) | x.uwidedigit(0)
    div = rem // iother
    rem -= div * iother
    return rffi.cast(lltype.Signed, div), rffi.cast(lltype.Signed, rem)

def _format(x, digits, prefix='', suffix='', max_str_digits=0):
    if x.get_sign() == 0:
        return prefix + "0" + suffix
    base = len(digits)
    assert base >= 2 and base <= 36
    if (base & (base - 1)) == 0:
        # base is 2, 4, 8, 16, ...
        return _format_base2_notzero(x, digits, prefix, suffix, max_str_digits)
    negative = x.get_sign() < 0
    if negative:
        x = x.neg()

    pcb = _parts_cache.get_cached_parts(base)
    mindigits = pcb.mindigits
    stringsize = mindigits
    startindex = 0
    pts = pcb.parts_cache
    for startindex, part in enumerate(pts):
        if not part.lt(x):
            break
        stringsize *= 2 # XXX can this overflow on 32 bit?
    else:
        # not enough parts computed yet
        while pts[-1].lt(x):
            pts.append(pts[-1].int_pow(2))
            stringsize *= 2

        startindex = len(pts) - 1

    # remove first base**2**i greater than x
    startindex -= 1

    stringsize += len(prefix) + len(suffix) + negative
    output = StringBuilder(stringsize)
    if negative:
        output.append('-')
    output.append(prefix)
    if startindex < 0:
        if digits == BASE10:
            s = _format_int10(x.toint(), digits)
        else:
            s = _format_int_general(x.toint(), digits)
        output.append(s)
    else:
        if digits == BASE10:
            _format_recursive(
                x, startindex, output, pcb, digits, output.getlength(),
                _format_int10, max_str_digits)
        else:
            _format_recursive(
                x, startindex, output, pcb, digits, output.getlength(),
                _format_int_general, max_str_digits)

    output.append(suffix)
    return output.build()


@specialize.arg(1)
def _bitwise(a, op, b): # '&', '|', '^'
    """ Bitwise and/or/xor operations """

    if a.get_sign() < 0:
        a = a.invert()
        maska = MASK
    else:
        maska = 0
    if b.get_sign() < 0:
        b = b.invert()
        maskb = MASK
    else:
        maskb = 0

    negz = 0
    if op == '^':
        if maska != maskb:
            maska ^= MASK
            negz = -1
    elif op == '&':
        if maska and maskb:
            op = '|'
            maska ^= MASK
            maskb ^= MASK
            negz = -1
    elif op == '|':
        if maska or maskb:
            op = '&'
            maska ^= MASK
            maskb ^= MASK
            negz = -1
    else:
        assert 0, "unreachable"

    # JRH: The original logic here was to allocate the result value (z)
    # as the longer of the two operands.  However, there are some cases
    # where the result is guaranteed to be shorter than that: AND of two
    # positives, OR of two negatives: use the shorter number.  AND with
    # mixed signs: use the positive number.  OR with mixed signs: use the
    # negative number.  After the transformations above, op will be '&'
    # iff one of these cases applies, and mask will be non-0 for operands
    # whose length should be ignored.

    size_a = a.numdigits()
    size_b = b.numdigits()
    if op == '&':
        if maska:
            size_z = size_b
        else:
            if maskb:
                size_z = size_a
            else:
                size_z = min(size_a, size_b)
    else:
        size_z = max(size_a, size_b)

    z = rbigint([NULLDIGIT] * size_z, 1, size_z)
    i = 0
    while i < size_z:
        if i < size_a:
            diga = a.digit(i) ^ maska
        else:
            diga = maska
        if i < size_b:
            digb = b.digit(i) ^ maskb
        else:
            digb = maskb

        if op == '&':
            z.setdigit(i, diga & digb)
        elif op == '|':
            z.setdigit(i, diga | digb)
        elif op == '^':
            z.setdigit(i, diga ^ digb)
        i += 1

    z._normalize()
    if negz == 0:
        return z

    return z.invert()

@specialize.arg(1)
def _int_bitwise(a, op, b): # '&', '|', '^'
    """ Bitwise and/or/xor operations """

    if not int_in_valid_range(b):
        # Fallback to long.
        return _bitwise(a, op, rbigint.fromint(b))

    if a.get_sign() < 0:
        a = a.invert()
        maska = MASK
    else:
        maska = 0
    if b < 0:
        b = ~b
        maskb = MASK
    else:
        maskb = 0

    negz = 0
    if op == '^':
        if maska != maskb:
            maska ^= MASK
            negz = -1
    elif op == '&':
        if maska and maskb:
            op = '|'
            maska ^= MASK
            maskb ^= MASK
            negz = -1
    elif op == '|':
        if maska or maskb:
            op = '&'
            maska ^= MASK
            maskb ^= MASK
            negz = -1

    # JRH: The original logic here was to allocate the result value (z)
    # as the longer of the two operands.  However, there are some cases
    # where the result is guaranteed to be shorter than that: AND of two
    # positives, OR of two negatives: use the shorter number.  AND with
    # mixed signs: use the positive number.  OR with mixed signs: use the
    # negative number.  After the transformations above, op will be '&'
    # iff one of these cases applies, and mask will be non-0 for operands
    # whose length should be ignored.

    size_a = a.numdigits()
    if op == '&':
        if maska:
            size_z = 1
        else:
            if maskb:
                size_z = size_a
            else:
                size_z = 1
    else:
        size_z = size_a

    z = rbigint([NULLDIGIT] * size_z, 1, size_z)
    i = 0
    while i < size_z:
        if i < size_a:
            diga = a.digit(i) ^ maska
        else:
            diga = maska
        if i == 0:
            digb = b ^ maskb
        else:
            digb = maskb

        if op == '&':
            z.setdigit(i, diga & digb)
        elif op == '|':
            z.setdigit(i, diga | digb)
        elif op == '^':
            z.setdigit(i, diga ^ digb)
        i += 1

    z._normalize()
    if negz == 0:
        return z

    return z.invert()

ULONGLONG_BOUND = r_ulonglong(1L << (r_longlong.BITS-1))
LONGLONG_MIN = r_longlong(-(1L << (r_longlong.BITS-1)))

def _AsLongLong(v):
    """
    Get a r_longlong integer from a bigint object.
    Raises OverflowError if overflow occurs.
    """
    x = _AsULonglong_ignore_sign(v)
    # grr grr grr
    if x >= ULONGLONG_BOUND:
        if x == ULONGLONG_BOUND and v.get_sign() < 0:
            x = LONGLONG_MIN
        else:
            raise OverflowError
    else:
        x = r_longlong(x)
        if v.get_sign() < 0:
            x = -x
    return x

def _AsULonglong_ignore_sign(v):
    x = r_ulonglong(0)
    i = v.numdigits() - 1
    while i >= 0:
        prev = x
        x = (x << SHIFT) + r_ulonglong(v.widedigit(i))
        if (x >> SHIFT) != prev:
                raise OverflowError(
                    "long int too large to convert to unsigned long long int")
        i -= 1
    return x

def make_unsigned_mask_conversion(T):
    def _As_unsigned_mask(v):
        x = T(0)
        i = v.numdigits() - 1
        while i >= 0:
            x = (x << SHIFT) + T(v.digit(i))
            i -= 1
        if v.get_sign() < 0:
            x = -x
        return x
    return _As_unsigned_mask

_AsULonglong_mask = make_unsigned_mask_conversion(r_ulonglong)
_AsUInt_mask = make_unsigned_mask_conversion(r_uint)

def _hash(v):
    # 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.  Moreover, purely
    # to please decimal.py, we return a hash that satisfies
    # hash(x) == hash(x % ULONG_MAX).  In particular, this
    # implies that hash(x) == hash(x % (2**64-1)).
    i = v.numdigits() - 1
    sign = v.get_sign()
    x = r_uint(0)
    LONG_BIT_SHIFT = LONG_BIT - SHIFT
    while i >= 0:
        # Force a native long #-bits (32 or 64) circular shift
        x = (x << SHIFT) | (x >> LONG_BIT_SHIFT)
        x += v.udigit(i)
        # If the addition above overflowed we compensate by
        # incrementing.  This preserves the value modulo
        # ULONG_MAX.
        if x < v.udigit(i):
            x += 1
        i -= 1
    res = intmask(intmask(x) * sign)
    return res

#_________________________________________________________________

# a few internal helpers

def digits_max_for_base(base):
    dec_per_digit = 1
    while base ** dec_per_digit < MASK:
        dec_per_digit += 1
    dec_per_digit -= 1
    return base ** dec_per_digit

BASE_MAX = [0, 1] + [digits_max_for_base(_base) for _base in range(2, 37)]
DEC_MAX = digits_max_for_base(10)
assert DEC_MAX == BASE_MAX[10]

def _decimalstr_to_bigint(s, start=0, lim=-1):
    # a string that has been already parsed to be decimal and valid,
    # is turned into a bigint
    p = start
    if lim < 0:
        lim = len(s)
    sign = False
    if s[p] == '-':
        sign = True
        p += 1
    elif s[p] == '+':
        p += 1

    a = NULLRBIGINT
    tens = 1
    dig = 0
    ord0 = ord('0')
    while p < lim:
        dig = dig * 10 + ord(s[p]) - ord0
        p += 1
        tens *= 10
        if tens == DEC_MAX or p == lim:
            if a is not None:
                a = _muladd1(a, tens, dig)
            else:
                assert dig & MASK == dig
                a = rbigint([_store_digit(dig)], int(dig != 0))
            tens = 1
            dig = 0
    if sign and a.get_sign() == 1:
        a._set_sign(-1)
    return a

def parse_digit_string(parser):
    # helper for fromstr
    base = parser.base
    if (base & (base - 1)) == 0 and base >= 2:
        return parse_string_from_binary_base(parser)
    if base == 10 and (parser.end - parser.start) > HOLDER.MINSIZE_STR2INT:
        # check for errors and potentially remove underscores
        s, start, end = parser._all_digits10()
        a = _str_to_int_big_base10(s, start, end, HOLDER.STR2INT_LIMIT)
        a._set_sign(a.get_sign() * parser.sign)
        return a
    a = NULLRBIGINT
    digitmax = BASE_MAX[base]
    baseexp, dig = 1, 0
    while True:
        digit = parser.next_digit()
        if baseexp == digitmax or digit < 0:
            if a is not None:
                a = _muladd1(a, baseexp, dig)
            else:
                assert dig & MASK == dig
                a = rbigint([_store_digit(dig)], int(dig != 0))
            if digit < 0:
                break
            dig = digit
            baseexp = base
        else:
            dig = dig * base + digit
            baseexp *= base
    a._set_sign(a.get_sign() * parser.sign)
    return a


FIVERBIGINT = rbigint.fromint(5)

def _str_to_int_big_w5pow(w, mem, limit):
    """Return 5**w and store the result.
    Also possibly save some intermediate results. In context, these
    are likely to be reused across various levels of the conversion
    to 'int'.
    """
    result = mem.get(w, None)
    if result is None:
        if w <= limit:
            result = FIVERBIGINT.int_pow(w)
        elif w - 1 in mem:
            result = mem[w - 1].int_mul(5)
        else:
            w2 = w >> 1
            # If w happens to be odd, w-w2 is one larger then w2
            # now. Recurse on the smaller first (w2), so that it's
            # in the cache and the larger (w-w2) can be handled by
            # the cheaper `w-1 in mem` branch instead.
            result = _str_to_int_big_w5pow(w2, mem, limit).mul(
                    _str_to_int_big_w5pow(w - w2, mem, limit))
        mem[w] = result
    return result

def _str_to_int_big_inner10(s, a, b, mem, limit):
    diff = b - a
    if diff <= limit:
        return _decimalstr_to_bigint(s, a, b)
    # choose the midpoint rounding up, as that yields slightly fewer entries in
    # mem, see comment in _str_to_int_big_w5pow too
    mid = a + (diff + 1) // 2
    right = _str_to_int_big_inner10(s, mid, b, mem, limit)
    left = _str_to_int_big_inner10(s, a, mid, mem, limit)
    left = left.mul(_str_to_int_big_w5pow(b - mid, mem, limit)).lshift(b - mid)
    return right.add(left)

def _str_to_int_big_base10(s, start, end, limit=20):
    """Asymptotically fast conversion of a 'str' to an 'int'."""

    # Function due to Bjorn Martinsson.  See GH issue #90716 for details.
    # https://github.com/python/cpython/issues/90716
    #
    # The implementation in longobject.c of base conversion algorithms
    # between power-of-2 and non-power-of-2 bases are quadratic time.
    # This function implements a divide-and-conquer algorithm making use
    # of Python's built in big int multiplication. Since Python uses the
    # Karatsuba algorithm for multiplication, the time complexity
    # of this function is O(len(s)**1.58).

    mem = {}
    result = _str_to_int_big_inner10(s, start, end, mem, limit)
    return result

def parse_string_from_binary_base(parser):
    # The point to this routine is that it takes time linear in the number of
    # string characters.
    from rpython.rlib.rstring import ParseStringError

    base = parser.base
    if   base ==  2: bits_per_char = 1
    elif base ==  4: bits_per_char = 2
    elif base ==  8: bits_per_char = 3
    elif base == 16: bits_per_char = 4
    elif base == 32: bits_per_char = 5
    else:
        raise AssertionError

    # n <- total number of bits needed, while moving 'parser' to the end
    n = 0
    while parser.next_digit() >= 0:
        n += 1

    # b <- number of Python digits needed, = ceiling(n/SHIFT). */
    try:
        b = ovfcheck(n * bits_per_char)
        b = ovfcheck(b + (SHIFT - 1))
    except OverflowError:
        raise ParseStringError("long string too large to convert")
    b = (b // SHIFT) or 1
    z = rbigint([NULLDIGIT] * b, sign=parser.sign)

    # Read string from right, and fill in long from left; i.e.,
    # from least to most significant in both.
    accum = _widen_digit(0)
    bits_in_accum = 0
    pdigit = 0
    for _ in range(n):
        k = parser.prev_digit()
        accum |= _widen_digit(k) << bits_in_accum
        bits_in_accum += bits_per_char
        if bits_in_accum >= SHIFT:
            z.setdigit(pdigit, accum)
            pdigit += 1
            assert pdigit <= b
            accum >>= SHIFT
            bits_in_accum -= SHIFT

    if bits_in_accum:
        z.setdigit(pdigit, accum)
    z._normalize()
    return z

@jit.elidable
def gcd_binary(a, b):
    """ Compute the greatest common divisor of non-negative integers a and b
    using the binary GCD algorithm. Raises ValueError on negative input. """
    if a < 0 or b < 0:
        raise ValueError

    if a == 0:
        return b

    if b == 0:
        return a

    shift = 0
    while (a | b) & 1 == 0:
        a >>= 1
        b >>= 1
        shift += 1

    while a & 1 == 0:
        a >>= 1

    while b & 1 == 0:
        b >>= 1

    while a != b:
        a, b = abs(a - b), min(a, b)
        while a & 1 == 0:
            a >>= 1

    return a << shift

def lehmer_xgcd(a, b):
    s_old, s_new = 1, 0
    t_old, t_new = 0, 1
    while b >> (SHIFT >> 1):
        q, r = a // b, a % b

        a, b = b, r
        s_old, s_new = s_new, s_old - q * s_new
        t_old, t_new = t_new, t_old - q * t_new

    return s_old, t_old, s_new, t_new

@jit.elidable
def gcd_lehmer(a, b):
    if a.lt(b):
        a, b = b, a

    while b.numdigits() > 1:
        a_ms = a.digit(abs(a.numdigits()-1))

        x = 0
        while a_ms & (0xFF << SHIFT-8) == 0:
            a_ms <<= 8
            x += 8

        while a_ms & (1 << SHIFT-1) == 0:
            a_ms <<= 1
            x += 1

        a_ms |= a.digit(abs(a.numdigits()-2)) >> SHIFT-x

        if a.numdigits() == b.numdigits():
            b_ms = (b.digit(abs(b.numdigits()-1)) << x) | (b.digit(abs(b.numdigits()-2)) >> SHIFT-x)
        elif a.numdigits() == b.numdigits()+1:
            b_ms = b.digit(abs(b.numdigits()-1)) >> SHIFT-x
        else:
            b_ms = 0

        if b_ms >> (SHIFT+1 >> 1) == 0:
            a, b = b, a.mod(b)
            continue

        s_old, t_old, s_new, t_new = lehmer_xgcd(a_ms, b_ms)

        n_a = a.int_mul(s_new).add(b.int_mul(t_new)).abs()
        b = a.int_mul(s_old).add(b.int_mul(t_old)).abs()
        a = n_a

        if a.lt(b):
            a, b = b, a

    if not b.tobool():
        return a

    a = a.mod(b)
    return rbigint.fromint(gcd_binary(b.toint(), a.toint()))


# if the bigint has more digits than this, it cannot fit into an int
# Also, if it has less digits than this, then it must be <=sys.maxint in
# absolute value and so it must fit an int.
MAX_DIGITS_THAT_CAN_FIT_IN_INT = rbigint.fromint(-sys.maxint - 1).numdigits()


# _________________________________________________________________
# tobytes and frombytes for integers

@jit.elidable
def frombytes_int(s, byteorder, signed):
    """ like rbigint.frombytes but returns an int. raises OverflowError if not
    possible """
    if byteorder not in ('big', 'little'):
        raise InvalidEndiannessError()
    if not s:
        return 0

    if byteorder == 'big':
        msb = ord(s[0])
        itr = range(len(s)-1, -1, -1)
    else:
        msb = ord(s[-1])
        itr = range(0, len(s))

    sign = -1 if msb >= 0x80 and signed else 1
    result = r_uint(0)
    bitpos = 0
    pad_byte = 0xdead

    for i in itr:
        c = r_uint(ord(s[i]))
        if bitpos == LONG_BIT - 8:
            if signed and c & (1 << 7): # signed and negative
                pad_byte = 0xff
            else:
                pad_byte = 0x00

        if bitpos >= LONG_BIT:
            # Extra bytes are only OK if they're padding (0x00 or 0xff)
            if c != pad_byte:
                raise OverflowError("value does not fit in int")
        else:
            result |= c << bitpos
        bitpos += 8
    if signed and bitpos <= LONG_BIT:
        # sign extend
        m = r_uint(1) << (bitpos - 1)
        result = (result ^ m) - m

    result = intmask(result)

    if sign == -1:
        assert result < 0
    elif not signed and result < 0:
        raise OverflowError("value does not fit in int")

    return result


@jit.elidable
def tobytes_int(intval, nbytes, byteorder, signed):
    """ like rbigint.tobytes, but starts from an int """
    if byteorder not in ('big', 'little'):
        raise InvalidEndiannessError()
    if not signed and intval < 0:
        raise InvalidSignednessError()

    bswap = byteorder == 'big'
    result = StringBuilder(nbytes)

    currval = intval
    byte = 0
    for i in range(nbytes):
        byte = currval & 0xff
        result.append(chr(byte))
        currval >>= 8
    if currval != 0 and currval != -1:
        raise OverflowError

    digits = result.build()
    if nbytes > 0 and signed:
        # If not already set, we cannot contain the sign bit
        if (intval < 0) != (byte >= 0x80):
            raise OverflowError()

    if bswap:
        # Bah, this is very inefficient. At least it's not
        # quadratic.
        length = len(digits)
        if length >= 0:
            digits = ''.join([digits[i] for i in range(length-1, -1, -1)])
    return digits

