Quellverzeichnis integer.c Sprache: C

/****************************************************************************
**
** This file is part of GAP, a system for computational discrete algebra.
**
** Copyright of GAP belongs to its developers, whose names are too numerous
** to list here. Please refer to the COPYRIGHT file for details.
**
** SPDX-License-Identifier: GPL-2.0-or-later
**
** This file implements the functions handling GMP integers.
**
** There are three integer types in GAP: 'T_INT', 'T_INTPOS' and 'T_INTNEG'.
** Each integer has a unique representation, e.g., an integer that can be
** represented as 'T_INT' is never represented as 'T_INTPOS' or 'T_INTNEG'.
**
** In the following, let 'N' be the number of bits in an UInt (so 32 or
** 64, depending on the system). 'T_INT' is the type of those integers small
** enough to fit into N-3 bits. Therefore the value range of this small
** integers is $-2^{N-4}...2^{N-4}-1$. Only these small integers can be used
** as index expression into sequences.
**
** Small integers are represented by an immediate integer handle, containing
** the value instead of pointing to it, which has the following form:
**
** +-------+-------+-------+-------+- - - -+-------+-------+-------+
** | guard | sign | bit | bit | | bit | tag | tag |
** | bit | bit | N-5 | N-6 | | 0 | = 0 | = 1 |
** +-------+-------+-------+-------+- - - -+-------+-------+-------+
**
** Immediate integers handles carry the tag 'T_INT', i.e. the last bit is 1.
** This distinguishes immediate integers from other handles which point to
** structures aligned on even boundaries and therefore have last bit zero.
** (The second bit is reserved as tag to allow extensions of this scheme.)
** Using immediates as pointers and dereferencing them gives address errors.
**
** To aid overflow check the most significant two bits must always be equal,
** that is to say that the sign bit of immediate integers has a guard bit.
**
** The macros 'INTOBJ_INT' and 'INT_INTOBJ' should be used to convert between
** a small integer value and its representation as immediate integer handle.
**
** 'T_INTPOS' and 'T_INTNEG' are the types of positive respectively negative
** integer values that cannot be represented by immediate integers.
**
** These large integers values are represented as low-level GMP integer
** objects, that is, in base 2^N. That means that the bag of a large integer
** has the following form, where the "digits" are also more commonly referred
** to as "limbs".:
**
** +-------+-------+-------+-------+- - - -+-------+-------+-------+
** | digit | digit | digit | digit | | digit | digit | digit |
** | 0 | 1 | 2 | 3 | | <n>-2 | <n>-1 | <n> |
** +-------+-------+-------+-------+- - - -+-------+-------+-------+
**
** The value of this is: $d0 + d1 2^N + d2 (2^N)^2 + ... + d_n (2^N)^n$,
** respectively the negative of this if the object type is 'T_INTNEG'.
**
** Each digit resp. limb is stored as a N bit wide unsigned integer.
**
** Note that we require that all large integers be normalized (that is, they
** must not contain leading zero limbs) and reduced (they do not fit into a
** small integer). Internally, a large integer may temporarily not be
** normalized or not be reduced, but all kernel functions must make sure
** that they eventually return normalized and reduced values. The function
** GMP_NORMALIZE can be used to ensure this.
*/

#include "integer.h"

#include "ariths.h"
#include "bool.h"
#include "calls.h"
#include "error.h"
#include "intfuncs.h"
#include "io.h"
#include "modules.h"
#include "opers.h"
#include "saveload.h"
#include "stats.h"
#include "stringobj.h"

#include "config.h"

// TODO: Remove after Ward2
#ifndef WARD_ENABLED

#include <gmp.h>

#if GMP_NAIL_BITS != 0
#error Aborting compile: GAP does not support non-zero GMP nail size
#endif
#if !defined(__GNU_MP_RELEASE)
#if __GMP_MP_RELEASE < 50002
#error Aborting compile: GAP requires GMP 5.0.2 or newer
#endif
#endif

GAP_STATIC_ASSERT(GMP_LIMB_BITS == 8 * sizeof(UInt),
 "GMP_LIMB_BITS != 8 * sizeof(UInt)");
GAP_STATIC_ASSERT(sizeof(mp_limb_t) == sizeof(UInt),
 "sizeof(mp_limb_t) != sizeof(UInt)");

static Obj ObjInt_UIntInv( UInt i );

// debugging
#ifdef GAP_KERNEL_DEBUG
#define DEBUG_GMP 1
#else
#define DEBUG_GMP 0
#endif

#if DEBUG_GMP

#define CHECK_INT(op) IS_NORMALIZED_AND_REDUCED(op, __func__, __LINE__)
#else
#define CHECK_INT(op) do { } while(0);
#endif

// macros to save typing later :)
#define VAL_LIMB0(obj) (*CONST_ADDR_INT(obj))
#define SET_VAL_LIMB0(obj,val) do { *ADDR_INT(obj) = val; } while(0)
#define IS_INTPOS(obj) (TNUM_OBJ(obj) == T_INTPOS)
#define IS_INTNEG(obj) (TNUM_OBJ(obj) == T_INTNEG)

#define SIZE_INT_OR_INTOBJ(obj) (IS_INTOBJ(obj) ? 1 : SIZE_INT(obj))

#define RequireNonzero(funcname, op, argname) \
 do { \
 if (op == INTOBJ_INT(0)) { \
 RequireArgumentEx(funcname, op, "<" argname ">", \
 "must be a nonzero integer"); \
 } \
 } while (0)

GAP_STATIC_ASSERT( sizeof(mp_limb_t) == sizeof(UInt), "gmp limb size incompatible with GAP word size");

/* This ensures that all memory underlying a bag is actually committed
** to physical memory and can be written to.
** This is a workaround to a bug specific to Cygwin 64-bit and bad
** interaction with GMP, so this is only needed specifically for new
** bags created in this module to hold the outputs of GMP routines.
**
** Thus, any time NewBag is called, it is also necessary to call
** ENSURE_BAG(bag) on the newly created bag if some GMP function will be
** the first place that bag's data is written to.
**
** To give a counter-example, ENSURE_BAG is *not* needed in ObjInt_Int,
** because it just creates a bag to hold a single mp_limb_t, and
** immediately assigns it a value.
**
** The bug this works around is explained more in
** https://github.com/gap-system/gap/issues/3434
*/
static inline void ENSURE_BAG(Bag bag)
{
// Note: This workaround is only required with the original GMP and not with
// MPIR
#if defined(SYS_IS_CYGWIN32) && defined(SYS_IS_64_BIT) && \
 !defined(__MPIR_VERSION)
 memset(PTR_BAG(bag), 0, SIZE_BAG(bag));
#endif
}

// for fallbacks to library
static Obj String;
static Obj IsIntFilt;

/****************************************************************************
**
*F TypeInt(<val>) . . . . . . . . . . . . . . . . . . . . . type of integer
**
** 'TypeInt' returns the type of the integer <val>.
**
** 'TypeInt' is the function in 'TypeObjFuncs' for integers.
*/
static Obj TYPE_INT_SMALL_ZERO;
static Obj TYPE_INT_SMALL_POS;
static Obj TYPE_INT_SMALL_NEG;
static Obj TYPE_INT_LARGE_POS;
static Obj TYPE_INT_LARGE_NEG;

static Obj TypeIntSmall(Obj val)
{
 if ( 0 == INT_INTOBJ(val) ) {
 return TYPE_INT_SMALL_ZERO;
 }
 else if ( 0 < INT_INTOBJ(val) ) {
 return TYPE_INT_SMALL_POS;
 }
 else {
 return TYPE_INT_SMALL_NEG;
 }
}

static Obj TypeIntLargePos(Obj val)
{
 return TYPE_INT_LARGE_POS;
}

static Obj TypeIntLargeNeg(Obj val)
{
 return TYPE_INT_LARGE_NEG;
}

/****************************************************************************
**
*F FiltIS_INT( <self>, <val> ) . . . . . . . . . . internal function 'IsInt'
**
** 'FiltIS_INT' implements the internal filter 'IsInt'.
**
** 'IsInt( <val> )'
**
** 'IsInt' returns 'true' if the value <val> is a small integer or a
** large int, and 'false' otherwise.
*/
static Obj FiltIS_INT(Obj self, Obj val)
{
 if ( IS_INT(val) ) {
 return True;
 }
 else if ( TNUM_OBJ(val) < FIRST_EXTERNAL_TNUM ) {
 return False;
 }
 else {
 return DoFilter( self, val );
 }
}

/****************************************************************************
**
*F SaveInt( <op> )
**
**
*/
#ifdef GAP_ENABLE_SAVELOAD
static void SaveInt(Obj op)
{
 const UInt * ptr = CONST_ADDR_INT(op);
 for (UInt i = 0; i < SIZE_INT(op); i++)
 SaveUInt(*ptr++);
 return;
}
#endif

/****************************************************************************
**
*F LoadInt( <op> )
**
**
*/
#ifdef GAP_ENABLE_SAVELOAD
static void LoadInt(Obj op)
{
 UInt * ptr = ADDR_INT(op);
 for (UInt i = 0; i < SIZE_INT(op); i++)
 *ptr++ = LoadUInt();
 return;
}
#endif

/****************************************************************************
**
** In order to use the high-level GMP mpz_* functions conveniently while
** retaining a low overhead, we introduce fake_mpz_t, which can be thought
** of as a "subclass" of mpz_t, extending it by temporary storage for
** single limb of data, as well as a reference to a corresponding GAP
** object.
**
** For an example on how to correctly use this, see GcdInt.
*/
typedef struct {
 mpz_t v;
 mp_limb_t tmp;
 Obj obj;
} fake_mpz_t[1];

/****************************************************************************
**
*F NEW_FAKEMPZ( <fake>, <size> )
**
** Setup a fake mpz_t object for capturing the output of a GMP mpz_ function,
** with space for up to <size> limbs allocated.
*/
static void NEW_FAKEMPZ( fake_mpz_t fake, UInt size )
{
 fake->v->_mp_alloc = size;
 fake->v->_mp_size = 0;
 if (size == 1) {
 fake->obj = 0;
 }
 else {
 fake->obj = NewBag( T_INTPOS, size * sizeof(mp_limb_t) );
 ENSURE_BAG(fake->obj);
 }
}

/****************************************************************************
**
*F FAKEMPZ_GMPorINTOBJ( <fake>, <op> )
**
** Initialize <fake> to reference the content of <op>. For this, <op>
** must be an integer, either small or large, but this is *not* checked.
** The calling code is responsible for any verification.
*/
static void FAKEMPZ_GMPorINTOBJ( fake_mpz_t fake, Obj op )
{
 if (IS_INTOBJ(op)) {
 fake->obj = 0;
 fake->v->_mp_alloc = 1;
 const Int i = INT_INTOBJ(op);
 if ( i >= 0 ) {
 fake->tmp = i;
 fake->v->_mp_size = i ? 1 : 0;
 }
 else {
 fake->tmp = -i;
 fake->v->_mp_size = -1;
 }
 }
 else {
 fake->obj = op;
 fake->v->_mp_alloc = SIZE_INT(op);
 fake->v->_mp_size = IS_INTPOS(op) ? SIZE_INT(op) : -SIZE_INT(op);
 }
}

/****************************************************************************
**
*F GMPorINTOBJ_FAKEMPZ( <fake> )
**
** This function converts a fake mpz_t into a GAP integer object.
*/
static Obj GMPorINTOBJ_FAKEMPZ( fake_mpz_t fake )
{
 Obj obj = fake->obj;
 if ( fake->v->_mp_size == 0 ) {
 obj = INTOBJ_INT(0);
 }
 else if ( obj != 0 ) {
 if ( fake->v->_mp_size < 0 ) {
 /* Warning: changing the bag type is only correct if the object was
 not yet visible to the outside world. Thus, it is safe to use
 with an fake_mpz_t initialized with NEW_FAKEMPZ, but not with
 one that was setup by FAKEMPZ_GMPorINTOBJ. */
 RetypeBag( obj, T_INTNEG );
 }
 obj = GMP_NORMALIZE( obj );
 }
 else {
 if ( fake->v->_mp_size == 1 )
 obj = ObjInt_UInt( fake->tmp );
 else
 obj = ObjInt_UIntInv( fake->tmp );
 }
 return obj;
}

/****************************************************************************
**
*F GMPorINTOBJ_MPZ( <fake> )
**
** This function converts an mpz_t into a GAP integer object.
*/
static Obj GMPorINTOBJ_MPZ( mpz_t v )
{
 return MakeObjInt((const UInt *)v->_mp_d, v->_mp_size);
}

/****************************************************************************
**
** MPZ_FAKEMPZ( <fake> )
**
** This converts a fake_mpz_t into an mpz_t. As a side effect, it updates
** fake->v->_mp_d. This allows us to use SWAP on fake_mpz_t objects, and
** also protects against garbage collection moving around data.
*/
#define MPZ_FAKEMPZ(fake) (UPDATE_FAKEMPZ(fake), fake->v)

// UPDATE_FAKEMPZ is a helper function for the MPZ_FAKEMPZ macro
static inline void UPDATE_FAKEMPZ( fake_mpz_t fake )
{
 fake->v->_mp_d = fake->obj ? (mp_ptr)ADDR_INT(fake->obj) : &fake->tmp;
}

// some extra debugging tools for FAKMPZ objects
#if DEBUG_GMP
#define CHECK_FAKEMPZ(fake) \
 assert( ((fake)->v->_mp_d == ((fake)->obj ? (mp_ptr)ADDR_INT((fake)->obj) : &(fake)->tmp )) \
 && (fake->v->_mp_alloc == ((fake)->obj ? SIZE_INT((fake)->obj) : 1 )) )
#else
#define CHECK_FAKEMPZ(fake) do { } while(0);
#endif

/****************************************************************************
**
*F GMP_NORMALIZE( <op> ) . . . . . . . remove leading zeros from a GMP bag
**
** 'GMP_NORMALIZE' removes any leading zeros from a large integer object
** and returns a small int or resizes the bag if possible.
**
*/
Obj GMP_NORMALIZE(Obj op)
{
 mp_size_t size;
 if (IS_INTOBJ(op)) {
 return op;
 }
 for (size = SIZE_INT(op); size != (mp_size_t)1; size--) {
 if (CONST_ADDR_INT(op)[(size - 1)] != 0) {
 break;
 }
 }
 if (size < SIZE_INT(op)) {
 ResizeBag(op, size * sizeof(mp_limb_t));
 }

 if (SIZE_INT(op) == 1) {
 if ((VAL_LIMB0(op) <= INT_INTOBJ_MAX) ||
 (IS_INTNEG(op) && VAL_LIMB0(op) == -INT_INTOBJ_MIN)) {
 if (IS_INTNEG(op)) {
 return INTOBJ_INT(-(Int)VAL_LIMB0(op));
 }
 else {
 return INTOBJ_INT((Int)VAL_LIMB0(op));
 }
 }
 }
 return op;
}

/****************************************************************************
**
** This is a helper function for the CHECK_INT macro, which checks that
** the given integer object <op> is normalized and reduced.
**
*/
#if DEBUG_GMP
static BOOL IS_NORMALIZED_AND_REDUCED(Obj op, const char * func, int line)
{
 mp_size_t size;
 if ( IS_INTOBJ( op ) ) {
 return TRUE;
 }
 if ( !IS_LARGEINT( op ) ) {
 // ignore non-integers
 return FALSE;
 }
 for ( size = SIZE_INT(op); size != (mp_size_t)1; size-- ) {
 if ( CONST_ADDR_INT(op)[(size - 1)] != 0 ) {
 break;
 }
 }
 if ( size < SIZE_INT(op) ) {
 Pr("WARNING: non-normalized gmp value (%s:%d)\n",(Int)func,line);
 }
 if ( SIZE_INT(op) == 1) {
 if ( ( VAL_LIMB0(op) <= INT_INTOBJ_MAX ) ||
 ( IS_INTNEG(op) && VAL_LIMB0(op) == -INT_INTOBJ_MIN ) ) {
 if ( IS_INTNEG(op) ) {
 Pr("WARNING: non-reduced negative gmp value (%s:%d)\n",(Int)func,line);
 return FALSE;
 }
 else {
 Pr("WARNING: non-reduced positive gmp value (%s:%d)\n",(Int)func,line);
 return FALSE;
 }
 }
 }
 return TRUE;
}
#endif

/****************************************************************************
**
*F ObjInt_Int( <cint> ) . . . . . . . . . . convert c int to integer object
**
** 'ObjInt_Int' takes the C integer <cint> and returns the equivalent
** GMP obj or int obj, according to the value of <cint>.
**
*/
Obj ObjInt_Int( Int i )
{
 Obj gmp;

 if (INT_INTOBJ_MIN <= i && i <= INT_INTOBJ_MAX) {
 return INTOBJ_INT(i);
 }
 else if (i < 0 ) {
 gmp = NewBag( T_INTNEG, sizeof(mp_limb_t) );
 i = -i;
 }
 else {
 gmp = NewBag( T_INTPOS, sizeof(mp_limb_t) );
 }
 SET_VAL_LIMB0( gmp, i );
 return gmp;
}

Obj ObjInt_UInt( UInt i )
{
 Obj gmp;
 if (i <= INT_INTOBJ_MAX) {
 return INTOBJ_INT(i);
 }
 else {
 gmp = NewBag( T_INTPOS, sizeof(mp_limb_t) );
 SET_VAL_LIMB0( gmp, i );
 return gmp;
 }
}

// initialize to -i
Obj ObjInt_UIntInv( UInt i )
{
 Obj gmp;
 // we need to test INT_INTOBJ_MIN <= -i; to express this with unsigned
 // values, we must avoid all negative terms, which leads to this equivalent
 // check:
 if (i <= -INT_INTOBJ_MIN) {
 return INTOBJ_INT(-i);
 }
 else {
 gmp = NewBag( T_INTNEG, sizeof(mp_limb_t) );
 SET_VAL_LIMB0( gmp, i );
 return gmp;
 }
}

Obj ObjInt_Int8( Int8 i )
{
#ifdef SYS_IS_64_BIT
 return ObjInt_Int(i);
#else
 if (i == (Int4)i) {
 return ObjInt_Int((Int4)i);
 }

 // we need two limbs to store this integer
 assert( sizeof(mp_limb_t) == 4 );
 Obj gmp;
 if (i >= 0) {
 gmp = NewBag( T_INTPOS, 2 * sizeof(mp_limb_t) );
 } else {
 gmp = NewBag( T_INTNEG, 2 * sizeof(mp_limb_t) );
 i = -i;
 }

 UInt *ptr = ADDR_INT(gmp);
 ptr[0] = (UInt4)i;
 ptr[1] = ((UInt8)i) >> 32;
 return gmp;
#endif
}

Obj ObjInt_UInt8( UInt8 i )
{
#ifdef SYS_IS_64_BIT
 return ObjInt_UInt(i);
#else
 if (i == (UInt4)i) {
 return ObjInt_UInt((UInt4)i);
 }

 // we need two limbs to store this integer
 assert( sizeof(mp_limb_t) == 4 );
 Obj gmp = NewBag( T_INTPOS, 2 * sizeof(mp_limb_t) );
 UInt *ptr = ADDR_INT(gmp);
 ptr[0] = (UInt4)i;
 ptr[1] = ((UInt8)i) >> 32;
 return gmp;
#endif
}

/****************************************************************************
**
** Convert GAP Integers to various C types -- see header file
*/
Int Int_ObjInt(Obj i)
{
 UInt sign = 0;
 if (IS_INTOBJ(i))
 return INT_INTOBJ(i);
 // must be a single limb
 if (TNUM_OBJ(i) == T_INTPOS)
 sign = 0;
 else if (TNUM_OBJ(i) == T_INTNEG)
 sign = 1;
 else
 RequireArgument("Conversion error", i, "must be an integer");
 if (SIZE_BAG(i) != sizeof(mp_limb_t))
 ErrorMayQuit("Conversion error: integer too large", 0, 0);

 // now check if val is small enough to fit in the signed Int type
 // that has a range from -2^N to 2^N-1 so we need to check both ends
 // Since -2^N is the same bit pattern as the UInt 2^N (N is 31 or 63)
 // we can do it as below which avoids some compiler warnings
 UInt val = VAL_LIMB0(i);
#ifdef SYS_IS_64_BIT
 if ((!sign && (val > INT64_MAX)) || (sign && (val > (UInt)INT64_MIN)))
#else
 if ((!sign && (val > INT32_MAX)) || (sign && (val > (UInt)INT32_MIN)))
#endif
 ErrorMayQuit("Conversion error: integer too large", 0, 0);
 return sign ? -(Int)val : (Int)val;
}

UInt UInt_ObjInt(Obj i)
{
 if (IS_NEG_INT(i))
 ErrorMayQuit("Conversion error: cannot convert negative integer to unsigned type", 0, 0);
 if (IS_INTOBJ(i))
 return (UInt)INT_INTOBJ(i);
 if (TNUM_OBJ(i) != T_INTPOS)
 RequireArgument("Conversion error", i, "must be a non-negative integer");

 // must be a single limb
 if (SIZE_INT(i) != 1)
 ErrorMayQuit("Conversion error: integer too large", 0, 0);
 return VAL_LIMB0(i);
}

Int8 Int8_ObjInt(Obj i)
{
#ifdef SYS_IS_64_BIT
 // in this case Int8 is Int
 return Int_ObjInt(i);
#else
 if (IS_INTOBJ(i))
 return (Int8)INT_INTOBJ(i);

 UInt sign = 0;
 if (TNUM_OBJ(i) == T_INTPOS)
 sign = 0;
 else if (TNUM_OBJ(i) == T_INTNEG)
 sign = 1;
 else
 RequireArgument("Conversion error", i, "must be an integer");

 // must be at most two limbs
 if (SIZE_INT(i) > 2)
 ErrorMayQuit("Conversion error: integer too large", 0, 0);
 UInt vall = VAL_LIMB0(i);
 UInt valh = (SIZE_INT(i) == 1) ? 0 : CONST_ADDR_INT(i)[1];
 UInt8 val = (UInt8)vall + ((UInt8)valh << 32);
 // now check if val is small enough to fit in the signed Int8 type
 // that has a range from -2^63 to 2^63-1 so we need to check both ends
 // Since -2^63 is the same bit pattern as the UInt8 2^63 we can do it
 // this way which avoids some compiler warnings
 if ((!sign && (val > INT64_MAX)) || (sign && (val > (UInt8)INT64_MIN)))
 ErrorMayQuit("Conversion error: integer too large", 0, 0);
 return sign ? -(Int8)val : (Int8)val;
#endif
}

UInt8 UInt8_ObjInt(Obj i)
{
#ifdef SYS_IS_64_BIT
 // in this case UInt8 is UInt
 return UInt_ObjInt(i);
#else
 if (IS_NEG_INT(i))
 ErrorMayQuit("Conversion error: cannot convert negative integer to unsigned type", 0, 0);
 if (IS_INTOBJ(i))
 return (UInt8)INT_INTOBJ(i);
 if (TNUM_OBJ(i) != T_INTPOS)
 RequireArgument("Conversion error", i, "must be a non-negative integer");
 if (SIZE_INT(i) > 2)
 ErrorMayQuit("Conversion error: integer too large", 0, 0);
 UInt vall = VAL_LIMB0(i);
 UInt valh = (SIZE_INT(i) == 1) ? 0 : CONST_ADDR_INT(i)[1];
 return (UInt8)vall + ((UInt8)valh << 32);
#endif
}

/****************************************************************************
**
*F MakeObjInt(<limbs>, <size>) . . . . . . . . . create a new integer object
**
*/
Obj MakeObjInt(const UInt * limbs, int size)
{
 Obj obj;
 if (size == 0)
 obj = INTOBJ_INT(0);
 else if (size == 1)
 obj = ObjInt_UInt(limbs[0]);
 else if (size == -1)
 obj = ObjInt_UIntInv(limbs[0]);
 else {
 UInt tnum = (size > 0 ? T_INTPOS : T_INTNEG);
 if (size < 0) size = -size;
 obj = NewBag(tnum, size * sizeof(mp_limb_t));
 memcpy(ADDR_INT(obj), limbs, size * sizeof(mp_limb_t));

 obj = GMP_NORMALIZE(obj);
 }
 return obj;
}

// This function returns an immediate integer, or
// an integer object with exactly one limb, and returns
// its absolute value as an unsigned integer.
static inline UInt AbsOfSmallInt(Obj x)
{
 if (!IS_INTOBJ(x)) {
 GAP_ASSERT(SIZE_INT(x) == 1);
 return VAL_LIMB0(x);
 }
 Int val = INT_INTOBJ(x);
 return val > 0 ? val : -val;
}

/****************************************************************************
**
*F PrintInt( <op> ) . . . . . . . . . . . . . . . . print an integer object
**
** 'PrintInt' prints the integer <op> in the usual decimal notation.
*/
void PrintInt ( Obj op )
{
 // print a small integer
 if ( IS_INTOBJ(op) ) {
 Pr("%>%d%<", INT_INTOBJ(op), 0);
 }

 // print a large integer
 else if ( SIZE_INT(op) < 1000 ) {
 CHECK_INT(op);

 /* Use GMP to print the integer to a buffer. We are looking at an
 integer with less than 1000 limbs, hence in decimal notation, it
 will take up at most LogInt( 2^(1000 * GMP_LIMB_BITS), 10)
 digits. Since 1000*Log(2)/Log(10) = 301.03, we get the following
 estimate for the buffer size (the overestimate is big enough to
 include space for a sign and a null terminator). */
 Char buf[302 * GMP_LIMB_BITS];
 mpz_t v;
 v->_mp_alloc = SIZE_INT(op);
 v->_mp_size = IS_INTPOS(op) ? v->_mp_alloc : -v->_mp_alloc;
 v->_mp_d = (mp_ptr)ADDR_INT(op);
 mpz_get_str(buf, 10, v);

 // print the buffer, %> means insert '\' before a linebreak
 Pr("%>%s%<",(Int)buf, 0);
 }
 else {
 Obj str = CALL_1ARGS( String, op );
 Pr("%>", 0, 0);
 PrintString1(str);
 Pr("%<", 0, 0);
 /* for a long time Print of large ints did not follow the general idea
 * that Print should produce something that can be read back into GAP:
 Pr("<<an integer too large to be printed>>", 0, 0); */
 }
}

/****************************************************************************
**
*F StringIntBase( <op>, <base> )
**
** Convert the integer <op> to a string relative to the given base <base>.
** Here, base may range from 2 to 36.
*/
static Obj StringIntBase(Obj op, int base)
{
 int len;
 Obj res;
 fake_mpz_t v;

 GAP_ASSERT(IS_INT(op));
 CHECK_INT(op);
 GAP_ASSERT(2 <= base && base <= 36);

 // 0 is special
 if (op == INTOBJ_INT(0)) {
 res = NEW_STRING(1);
 CHARS_STRING(res)[0] = '0';
 return res;
 }

 // convert integer to fake_mpz_t
 FAKEMPZ_GMPorINTOBJ(v, op);

 // allocate the result string
 len = mpz_sizeinbase( MPZ_FAKEMPZ(v), base ) + 2;
 res = NEW_STRING( len );

 // ask GMP to perform the actual conversion
 mpz_get_str( CSTR_STRING( res ), -base, MPZ_FAKEMPZ(v) );

 // we may have to shrink the string
 int real_len = strlen( CONST_CSTR_STRING(res) );
 if ( real_len != GET_LEN_STRING(res) ) {
 SET_LEN_STRING(res, real_len);
 }

 return res;
}

/****************************************************************************
**
*F FuncHexStringInt( <self>, <n> ) . . . . . . . . . hex string for GAP int
*F FuncIntHexString( <self>, <string> ) . . . . . . GAP int from hex string
**
** The function `FuncHexStringInt' constructs from a GAP integer the
** corresponding string in hexadecimal notation. It has a leading '-'
** for negative numbers and the digits 10..15 are written as A..F.
**
** The function `FuncIntHexString' does the converse, but here the
** letters a..f are also allowed in <string> instead of A..F.
**
*/
static Obj FuncHexStringInt(Obj self, Obj n)
{
 RequireInt(SELF_NAME, n);
 return StringIntBase(n, 16);
}

/****************************************************************************
**
** This helper function for IntHexString reads <len> bytes in the string
** and parses them as a hexadecimal. The resulting integer is returned.
** This function does not check for overflow, so make sure that len*4 does
** not exceed the number of bits in mp_limb_t.
**/
static mp_limb_t hexstr2int( const UInt1 *p, UInt len )
{
 mp_limb_t n = 0;
 UInt1 a;
 while (len--) {
 a = *p++;
 if (a >= 'a')
 a -= 'a' - 10;
 else if ( a>= 'A')
 a -= 'A' - 10;
 else
 a -= '0';
 if (a > 15)
 ErrorMayQuit("IntHexString: invalid character in hex-string", 0, 0);
 n = (n << 4) + a;
 }
 return n;
}

static Obj FuncIntHexString(Obj self, Obj str)
{
 RequireStringRep(SELF_NAME, str);
 return IntHexString(str);
}

Obj IntHexString(Obj str)
{
 Obj res;
 Int i, len, sign, nd;
 mp_limb_t n;
 const UInt1 *p;
 UInt *limbs;

 GAP_ASSERT(IS_STRING_REP(str));

 len = GET_LEN_STRING(str);
 if (len == 0) {
 res = INTOBJ_INT(0);
 return res;
 }
 p = CONST_CHARS_STRING(str);
 if (*p == '-') {
 sign = -1;
 i = 1;
 }
 else {
 sign = 1;
 i = 0;
 }

 while (p[i] == '0' && i < len)
 i++;
 len -= i;

 if (len*4 <= NR_SMALL_INT_BITS) {
 n = hexstr2int( p + i, len );
 res = INTOBJ_INT(sign * n);
 return res;
 }

 else {
 /* Each hex digit corresponds to 4 bits, and each GMP limb has sizeof(UInt)
 bytes, thus 2*sizeof(UInt) hex digits fit into one limb. We use this
 to compute the number of limbs minus 1: */
 nd = (len - 1) / (2*sizeof(UInt));
 res = NewBag( (sign == 1) ? T_INTPOS : T_INTNEG, (nd + 1) * sizeof(mp_limb_t) );

 // update pointer, in case a garbage collection happened
 p = CONST_CHARS_STRING(str) + i;
 limbs = ADDR_INT(res);

 // if len is not divisible by 2*sizeof(UInt), then take care of the extra bytes
 UInt diff = len - nd * (2*sizeof(UInt));
 if ( diff ) {
 n = hexstr2int( p, diff );
 p += diff;
 len -= diff;
 limbs[nd--] = n;
 }

 while ( len ) {
 n = hexstr2int( p, 2*sizeof(UInt) );
 p += 2*sizeof(UInt);
 len -= 2*sizeof(UInt);
 limbs[nd--] = n;
 }

 res = GMP_NORMALIZE(res);
 return res;
 }
}

/****************************************************************************
**
** Implementation of Log2Int for C integers.
**
** When available, we try to use GCC builtins. Otherwise, fall back to code
** based on
** https://graphics.stanford.edu/~seander/bithacks.html#IntegerLogLookup
** On a test machine with x86 64bit, the builtins are about 4 times faster
** than the generic code.
**
*/

static Int CLog2UInt(UInt a)
{
#if SIZEOF_VOID_P == SIZEOF_INT && defined(HAVE___BUILTIN_CLZ)
 return GMP_LIMB_BITS - 1 - __builtin_clz(a);
#elif SIZEOF_VOID_P == SIZEOF_LONG && defined(HAVE___BUILTIN_CLZL)
 return GMP_LIMB_BITS - 1 - __builtin_clzl(a);
#elif SIZEOF_VOID_P == SIZEOF_LONG_LONG && defined(HAVE___BUILTIN_CLZLL)
 return GMP_LIMB_BITS - 1 - __builtin_clzll(a);
#else
 static const char LogTable256[256] = {
 -1, 0, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3,
 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4,
 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5,
 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5,
 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6,
 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6,
 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6,
 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6,
 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7,
 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7,
 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7,
 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7,
 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7,
 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7,
 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7,
 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7
 };

 Int res = 0;
 UInt b;
 b = a >> 32; if (b) { res+=32; a=b; }
 b = a >> 16; if (b) { res+=16; a=b; }
 b = a >> 8; if (b) { res+= 8; a=b; }
 return res + LogTable256[a];
#endif
}

Int CLog2Int(Int a)
{
 if (a == 0) return -1;
 if (a < 0) a = -a;
 return CLog2UInt(a);
}

/****************************************************************************
**
*F FuncLog2Int( <self>, <n> ) . . . . . . . . . . . nr of bits of a GAP int
**
** Given to GAP-Level as "Log2Int".
*/
static Obj FuncLog2Int(Obj self, Obj n)
{
 RequireInt(SELF_NAME, n);

 if (IS_INTOBJ(n)) {
 return INTOBJ_INT(CLog2Int(INT_INTOBJ(n)));
 }

 UInt len = SIZE_INT(n) - 1;
 UInt a = CLog2UInt(CONST_ADDR_INT(n)[len]);

 CHECK_INT(n);

#ifdef SYS_IS_64_BIT
 return INTOBJ_INT(len * GMP_LIMB_BITS + a);
#else
 /* The final result is len * GMP_LIMB_BITS - d, which may not
 fit into an immediate integer (at least on a 32bit system) */
 return SumInt(ProdInt(INTOBJ_INT(len), INTOBJ_INT(GMP_LIMB_BITS)),
 INTOBJ_INT(a));
#endif
}

/****************************************************************************
**
*F FuncSTRING_INT( <self>, <n> ) . . . . . . convert an integer to a string
**
** `FuncSTRING_INT' returns an immutable string representing the integer <n>
**
*/
static Obj FuncSTRING_INT(Obj self, Obj n)
{
 RequireInt(SELF_NAME, n);
 return StringIntBase(n, 10);
}

Obj IntStringInternal(Obj string, const Char *str)
{
 Obj val; // value = <upp> * <pow> + <low>
 Obj upp; // upper part
 Int pow; // power
 Int low; // lower part
 Int sign; // is the integer negative
 UInt i; // loop variable

 // if <string> is given, then we ignore <str>
 if (string)
 str = CONST_CSTR_STRING(string);

 // get the sign, if any
 sign = 1;
 i = 0;
 if (str[i] == '-') {
 sign = -sign;
 i++;
 }

 // collect the digits in groups of 8, for improved performance
 // note that 2^26 < 10^8 < 2^27, so the intermediate
 // values always fit into an immediate integer
 low = 0;
 pow = 1;
 upp = INTOBJ_INT(0);
 while (str[i] != '\0') {
 if (str[i] < '0' || str[i] > '9') {
 return Fail;
 }
 low = 10 * low + str[i] - '0';
 pow = 10 * pow;
 if (pow == 100000000L) {
 upp = ProdInt(upp, INTOBJ_INT(pow));
 upp = SumInt(upp, INTOBJ_INT(sign*low));
 // refresh 'str', in case the arithmetic operations triggered
 // a garbage collection
 if (string)
 str = CONST_CSTR_STRING(string);
 pow = 1;
 low = 0;
 }
 i++;
 }

 // check if 0 char does not mark the end of the string
 if (string && i < GET_LEN_STRING(string))
 return Fail;

 // compose the integer value
 if (upp == INTOBJ_INT(0)) {
 val = INTOBJ_INT(sign * low);
 }
 else if (pow == 1) {
 val = upp;
 }
 else {
 upp = ProdInt(upp, INTOBJ_INT(pow));
 val = SumInt(upp, INTOBJ_INT(sign * low));
 }

 // return the integer value
 return val;
}

/****************************************************************************
**
*F FuncINT_STRING( <self>, <string> ) . . . . convert a string to an integer
**
** `FuncINT_STRING' returns an integer representing the string, or
** fail if the string is not a valid integer.
**
*/
static Obj FuncINT_STRING(Obj self, Obj string)
{
 if( !IS_STRING(string) ) {
 return Fail;
 }

 if( !IS_STRING_REP(string) ) {
 string = CopyToStringRep(string);
 }

 return IntStringInternal(string, 0);
}

/****************************************************************************
**
*F EqInt( <opL>, <opR> ) . . . . . . . . . . test if two integers are equal
**
** 'EqInt' returns 1 if the two integer arguments <opL> and <opR> are equal
** and 0 otherwise.
*/
Int EqInt(Obj opL, Obj opR)
{
 CHECK_INT(opL);
 CHECK_INT(opR);

 // if at least one input is a small int, a naive equality test suffices
 if (IS_INTOBJ(opL) || IS_INTOBJ(opR))
 return opL == opR;

 // compare the sign and size
 // note: at this point we know that opL and opR are proper bags, so
 // we can use TNUM_BAG instead of TNUM_OBJ
 if (TNUM_BAG(opL) != TNUM_BAG(opR) || SIZE_INT(opL) != SIZE_INT(opR))
 return 0;

 return !mpn_cmp((mp_srcptr)CONST_ADDR_INT(opL),
 (mp_srcptr)CONST_ADDR_INT(opR), SIZE_INT(opL));
}

/****************************************************************************
**
*F LtInt( <opL>, <opR> ) . . . . . . test if an integer is less than another
**
** 'LtInt' returns 1 if the integer <opL> is strictly less than the
** integer <opR> and 0 otherwise.
*/
Int LtInt(Obj opL, Obj opR)
{
 Int res;

 CHECK_INT(opL);
 CHECK_INT(opR);

 // compare two small integers
 if (ARE_INTOBJS(opL, opR))
 return (Int)opL < (Int)opR;

 // a small int is always less than a positive large int,
 // and always more than a negative large int
 if (IS_INTOBJ(opL))
 return IS_INTPOS(opR);
 if (IS_INTOBJ(opR))
 return IS_INTNEG(opL);

 // at this point, both inputs are large integers, and we compare their
 // signs first
 if (TNUM_OBJ(opL) != TNUM_OBJ(opR))
 return IS_INTNEG(opL);

 // signs are equal; compare sizes and absolute values
 if (SIZE_INT(opL) < SIZE_INT(opR))
 res = -1;
 else if (SIZE_INT(opL) > SIZE_INT(opR))
 res = +1;
 else
 res = mpn_cmp((mp_srcptr)CONST_ADDR_INT(opL),
 (mp_srcptr)CONST_ADDR_INT(opR), SIZE_INT(opL));

 // if both arguments are negative, flip the result
 if (IS_INTNEG(opL))
 res = -res;

 return res < 0;
}

/****************************************************************************
**
*F SumOrDiffInt( <opL>, <opR>, <sign> ) . . . . sum or diff of two integers
**
** 'SumOrDiffInt' returns the sum or difference of the two integer arguments
** <opL> and <opR>, depending on whether sign is +1 or -1.
**
** 'SumOrDiffInt' is a little bit tricky since there are many different
** cases to handle: each operand can be positive or negative, small or large
** integer.
*/
static Obj SumOrDiffInt(Obj opL, Obj opR, Int sign)
{
 UInt sizeL, sizeR;
 fake_mpz_t mpzL, mpzR, mpzResult;
 Obj result;

 CHECK_INT(opL);
 CHECK_INT(opR);

 // handle trivial cases first
 if (opR == INTOBJ_INT(0))
 return opL;
 if (opL == INTOBJ_INT(0)) {
 if (sign == 1)
 return opR;
 else
 return AInvInt(opR);
 }

 sizeL = SIZE_INT_OR_INTOBJ(opL);
 sizeR = SIZE_INT_OR_INTOBJ(opR);

 NEW_FAKEMPZ( mpzResult, sizeL > sizeR ? sizeL+1 : sizeR+1 );
 FAKEMPZ_GMPorINTOBJ(mpzL, opL);
 FAKEMPZ_GMPorINTOBJ(mpzR, opR);

 // add or subtract
 if (sign == 1)
 mpz_add( MPZ_FAKEMPZ(mpzResult), MPZ_FAKEMPZ(mpzL), MPZ_FAKEMPZ(mpzR) );
 else
 mpz_sub( MPZ_FAKEMPZ(mpzResult), MPZ_FAKEMPZ(mpzL), MPZ_FAKEMPZ(mpzR) );

 // convert result to GAP object and return it
 CHECK_FAKEMPZ(mpzResult);
 CHECK_FAKEMPZ(mpzL);
 CHECK_FAKEMPZ(mpzR);
 result = GMPorINTOBJ_FAKEMPZ( mpzResult );
 CHECK_INT(result);
 return result;
}

/****************************************************************************
**
*F SumInt( <opL>, <opR> ) . . . . . . . . . . . . . . . sum of two integers
**
*/
inline Obj SumInt(Obj opL, Obj opR)
{
 Obj sum;

 if (!ARE_INTOBJS(opL, opR) || !SUM_INTOBJS(sum, opL, opR))
 sum = SumOrDiffInt(opL, opR, +1);

 CHECK_INT(sum);
 return sum;
}

/****************************************************************************
**
*F DiffInt( <opL>, <opR> ) . . . . . . . . . . . difference of two integers
**
*/
inline Obj DiffInt(Obj opL, Obj opR)
{
 Obj dif;

 if (!ARE_INTOBJS(opL, opR) || !DIFF_INTOBJS(dif, opL, opR))
 dif = SumOrDiffInt(opL, opR, -1);

 CHECK_INT(dif);
 return dif;
}

/****************************************************************************
**
*F ZeroInt(<op>) . . . . . . . . . . . . . . . . . . . . . zero of integers
*/
static Obj ZeroInt(Obj op)
{
 return INTOBJ_INT(0);
}

/****************************************************************************
**
*F AInvInt( <op> ) . . . . . . . . . . . . . additive inverse of an integer
*/
Obj AInvInt(Obj op)
{
 Obj inv;

 CHECK_INT(op);

 // handle small integer
 if (IS_INTOBJ(op)) {

 // special case (ugh)
 if (op == INTOBJ_MIN) {
 inv = NewBag( T_INTPOS, sizeof(mp_limb_t) );
 SET_VAL_LIMB0( inv, -INT_INTOBJ_MIN );
 }

 // general case
 else {
 inv = INTOBJ_INT(-INT_INTOBJ(op));
 }

 }

 else {
 if (IS_INTPOS(op)) {
 // special case
 if (SIZE_INT(op) == 1 && VAL_LIMB0(op) == -INT_INTOBJ_MIN) {
 return INTOBJ_MIN;
 }
 else {
 inv = NewBag( T_INTNEG, SIZE_OBJ(op) );
 }
 }

 else {
 inv = NewBag( T_INTPOS, SIZE_OBJ(op) );
 }

 memcpy( ADDR_INT(inv), CONST_ADDR_INT(op), SIZE_OBJ(op) );
 }

 CHECK_INT(inv);
 return inv;

}

/****************************************************************************
**
*F AbsInt(<op>) . . . . . . . . . . . . . . . . absolute value of an integer
*/
Obj AbsInt( Obj op )
{
 Obj a;
 if ( IS_INTOBJ(op) ) {
 if ( ((Int)op) > 0 ) // non-negative?
 return op;
 else if ( op == INTOBJ_MIN ) {
 a = NewBag( T_INTPOS, sizeof(mp_limb_t) );
 SET_VAL_LIMB0( a, -INT_INTOBJ_MIN );
 return a;
 } else
 return (Obj)( 2 - (Int)op );
 }
 CHECK_INT(op);
 if ( IS_INTPOS(op) ) {
 return op;
 } else if ( IS_INTNEG(op) ) {
 a = NewBag( T_INTPOS, SIZE_OBJ(op) );
 memcpy( ADDR_INT(a), CONST_ADDR_INT(op), SIZE_OBJ(op) );
 return a;
 }
 return Fail;
}

static Obj FuncABS_INT(Obj self, Obj n)
{
 RequireInt(SELF_NAME, n);
 Obj res = AbsInt(n);
 CHECK_INT(res);
 return res;
}

/****************************************************************************
**
*F SignInt(<op>) . . . . . . . . . . . . . . . . . . . . sign of an integer
*/
Obj SignInt( Obj op )
{
 if ( IS_INTOBJ(op) ) {
 if ( op == INTOBJ_INT(0) )
 return INTOBJ_INT(0);
 else if ( ((Int)op) > (Int)INTOBJ_INT(0) )
 return INTOBJ_INT(1);
 else
 return INTOBJ_INT(-1);
 }
 CHECK_INT(op);
 if ( IS_INTPOS(op) ) {
 return INTOBJ_INT(1);
 } else if ( IS_INTNEG(op) ) {
 return INTOBJ_INT(-1);
 }
 return Fail;
}

static Obj FuncSIGN_INT(Obj self, Obj n)
{
 RequireInt(SELF_NAME, n);
 Obj res = SignInt(n);
 CHECK_INT(res);
 return res;
}

/****************************************************************************
**
*F ProdInt( <opL>, <opR> ) . . . . . . . . . . . . . product of two integers
**
** 'ProdInt' returns the product of the two integer arguments <opL> and
** <opR>.
*/
Obj ProdInt(Obj opL, Obj opR)
{
 Obj prd; // handle of the result bag
 UInt sizeL, sizeR;
 fake_mpz_t mpzL, mpzR, mpzResult;

 CHECK_INT(opL);
 CHECK_INT(opR);

 // multiplying two small integers
 if ( ARE_INTOBJS( opL, opR ) ) {

 // multiply two small integers with a small product
 if ( PROD_INTOBJS( prd, opL, opR ) ) {
 CHECK_INT(prd);
 return prd;
 }
 }

 // handle trivial cases first
 if ( opL == INTOBJ_INT(0) || opR == INTOBJ_INT(1) )
 return opL;
 if ( opR == INTOBJ_INT(0) || opL == INTOBJ_INT(1) )
 return opR;
 if ( opR == INTOBJ_INT(-1) )
 return AInvInt( opL );
 if ( opL == INTOBJ_INT(-1) )
 return AInvInt( opR );

 sizeL = SIZE_INT_OR_INTOBJ(opL);
 sizeR = SIZE_INT_OR_INTOBJ(opR);

 NEW_FAKEMPZ( mpzResult, sizeL + sizeR );
 FAKEMPZ_GMPorINTOBJ( mpzL, opL );
 FAKEMPZ_GMPorINTOBJ( mpzR, opR );

 // multiply
 mpz_mul( MPZ_FAKEMPZ(mpzResult), MPZ_FAKEMPZ(mpzL), MPZ_FAKEMPZ(mpzR) );

 // convert result to GAP object and return it
 CHECK_FAKEMPZ(mpzResult);
 CHECK_FAKEMPZ(mpzL);
 CHECK_FAKEMPZ(mpzR);
 prd = GMPorINTOBJ_FAKEMPZ( mpzResult );
 CHECK_INT(prd);
 return prd;
}

/****************************************************************************
**
*F ProdIntObj(<n>,<op>) . . . . . . . . product of an integer and an object
*/
static Obj ProdIntObj ( Obj n, Obj op )
{
 Obj res = 0; // result
 UInt i, k; // loop variables
 mp_limb_t l; // loop variable

 CHECK_INT(n);

 // if the integer is zero, return the neutral element of the operand
 if ( n == INTOBJ_INT(0) ) {
 res = ZERO_SAMEMUT( op );
 }

 /* if the integer is one, return the object if immutable -
 if mutable, add the object to its ZeroSameMutability to
 ensure correct mutability propagation */
 else if ( n == INTOBJ_INT(1) ) {
 if (IS_MUTABLE_OBJ(op))
 res = SUM(ZERO_SAMEMUT(op),op);
 else
 res = op;
 }

 // if the integer is minus one, return the inverse of the operand
 else if ( n == INTOBJ_INT(-1) ) {
 res = AINV_SAMEMUT( op );
 }

 // if the integer is negative, invert the operand and the integer
 else if ( IS_NEG_INT(n) ) {
 res = AINV_SAMEMUT( op );
 if ( res == Fail ) {
 ErrorMayQuit("Operations: must have an additive inverse", 0, 0);
 }
 res = PROD(AINV_SAMEMUT(n), res);
 }

 // if the integer is small, compute the product by repeated doubling
 // the loop invariant is <result> = <k>*<res> + <l>*<op>, <l> < <k>
 // <res> = 0 means that <res> is the neutral element
 else if ( IS_INTOBJ(n) && INT_INTOBJ(n) > 1 ) {
 res = 0;
 k = (Int)1 << NR_SMALL_INT_BITS;
 l = INT_INTOBJ(n);
 while ( 0 < k ) {
 res = (res == 0 ? res : SUM( res, res ));
 if ( k <= l ) {
 res = (res == 0 ? op : SUM( res, op ));
 l = l - k;
 }
 k = k / 2;
 }
 }

 // if the integer is large, compute the product by repeated doubling
 else if ( IS_INTPOS(n) ) {
 res = 0;
 for ( i = SIZE_INT(n); 0 < i; i-- ) {
 k = 8*sizeof(mp_limb_t);
 l = CONST_ADDR_INT(n)[i-1];
 while ( 0 < k ) {
 res = (res == 0 ? res : SUM( res, res ));
 k--;
 if ( (l >> k) & 1 ) {
 res = (res == 0 ? op : SUM( res, op ));
 }
 }
 }
 }

 return res;
}

static Obj FuncPROD_INT_OBJ(Obj self, Obj opL, Obj opR)
{
 return ProdIntObj( opL, opR );
}

/****************************************************************************
**
*F OneInt(<op>) . . . . . . . . . . . . . . . . . . . . . one of an integer
*/
static Obj OneInt(Obj op)
{
 return INTOBJ_INT( 1 );
}

/****************************************************************************
**
*F PowInt( <opL>, <opR> ) . . . . . . . . . . . . . . . power of an integer
**
** 'PowInt' returns the <opR>-th (an integer) power of the integer <opL>.
*/
Obj PowInt ( Obj opL, Obj opR )
{
 Int i;
 Obj pow;

 CHECK_INT(opL);
 CHECK_INT(opR);

 if ( opR == INTOBJ_INT(0) ) {
 pow = INTOBJ_INT(1);
 }
 else if ( opL == INTOBJ_INT(0) ) {
 if ( IS_NEG_INT( opR ) ) {
 ErrorMayQuit("Integer operands: must not be zero", 0, 0);
 }
 pow = INTOBJ_INT(0);
 }
 else if ( opL == INTOBJ_INT(1) ) {
 pow = INTOBJ_INT(1);
 }
 else if ( opL == INTOBJ_INT(-1) ) {
 pow = IS_EVEN_INT(opR) ? INTOBJ_INT(1) : INTOBJ_INT(-1);
 }

 // power with a large exponent
 else if ( ! IS_INTOBJ(opR) ) {
 ErrorMayQuit("Integer operands: is too large", 0, 0);
 }

 // power with a negative exponent
 else if ( INT_INTOBJ(opR) < 0 ) {
 pow = QUO( INTOBJ_INT(1),
 PowInt( opL, INTOBJ_INT( -INT_INTOBJ(opR)) ) );
 }

 // findme - can we use the gmp function mpz_n_pow_ui?

 // power with a small positive exponent, do it by a repeated squaring
 else {
 pow = INTOBJ_INT(1);
 i = INT_INTOBJ(opR);
 while ( i != 0 ) {
 if ( i % 2 == 1 ) pow = ProdInt( pow, opL );
 if ( i > 1 ) opL = ProdInt( opL, opL );
 TakeInterrupt();
 i = i / 2;
 }
 }

 // return the power
 CHECK_INT(pow);
 return pow;
}

/****************************************************************************
**
*F PowObjInt(<op>,<n>) . . . . . . . . . . power of an object and an integer
*/
static Obj PowObjInt(Obj op, Obj n)
{
 Obj res = 0; // result
 UInt i, k; // loop variables
 mp_limb_t l; // loop variable

 CHECK_INT(n);

 // if the integer is zero, return the neutral element of the operand
 if ( n == INTOBJ_INT(0) ) {
 return ONE_SAMEMUT( op );
 }

 // if the integer is one, return a copy of the operand
 else if ( n == INTOBJ_INT(1) ) {
 res = CopyObj( op, 1 );
 }

 // if the integer is minus one, return the inverse of the operand
 else if ( n == INTOBJ_INT(-1) ) {
 res = INV_SAMEMUT( op );
 }

 // if the integer is negative, invert the operand and the integer
 else if ( IS_NEG_INT(n) ) {
 res = INV_SAMEMUT( op );
 if ( res == Fail ) {
 ErrorMayQuit("Operations: must have an inverse", 0, 0);
 }
 res = POW(res, AINV_SAMEMUT(n));
 }

 // if the integer is small, compute the power by repeated squaring
 // the loop invariant is <result> = <res>^<k> * <op>^<l>, <l> < <k>
 // <res> = 0 means that <res> is the neutral element
 else if ( IS_INTOBJ(n) && INT_INTOBJ(n) > 0 ) {
 res = 0;
 k = (Int)1 << NR_SMALL_INT_BITS;
 l = INT_INTOBJ(n);
 while ( 0 < k ) {
 res = (res == 0 ? res : PROD( res, res ));
 if ( k <= l ) {
 res = (res == 0 ? op : PROD( res, op ));
 l = l - k;
 }
 k = k / 2;
 }
 }

 // if the integer is large, compute the power by repeated squaring
 else if ( IS_INTPOS(n) ) {
 res = 0;
 for ( i = SIZE_INT(n); 0 < i; i-- ) {
 k = 8*sizeof(mp_limb_t);
 l = CONST_ADDR_INT(n)[i-1];
 while ( 0 < k ) {
 res = (res == 0 ? res : PROD( res, res ));
 k--;
 if ( (l >> k) & 1 ) {
 res = (res == 0 ? op : PROD( res, op ));
 }
 }
 }
 }

 return res;
}

static Obj FuncPOW_OBJ_INT(Obj self, Obj opL, Obj opR)
{
 return PowObjInt( opL, opR );
}

/****************************************************************************
**
*F ModInt( <opL>, <opR> ) . . representative of residue class of an integer
**
** 'ModInt' returns the smallest positive representative of the residue
** class of the integer <opL> modulo the integer <opR>.
*/
Obj ModInt(Obj opL, Obj opR)
{
 Int i; // loop count, value for small int
 Int k; // loop count, value for small int
 UInt c; // product of two digits
 Obj mod; // handle of the remainder bag
 Obj quo; // handle of the quotient bag

 CHECK_INT(opL);
 CHECK_INT(opR);

 // pathological case first
 RequireNonzero("Integer operations", opR, "divisor");

 // compute the remainder of two small integers
 if ( ARE_INTOBJS( opL, opR ) ) {

 // get the integer values
 i = INT_INTOBJ(opL);
 k = INT_INTOBJ(opR);

 // compute the remainder, make sure we divide only positive numbers
 i %= k;
 if (i < 0)
 i += k > 0 ? k : -k;
 mod = INTOBJ_INT(i);
 }

 // compute the remainder of a small integer by a large integer
 else if ( IS_INTOBJ(opL) ) {

 // the small int -(1<<28) mod the large int (1<<28) is 0
 if ( opL == INTOBJ_MIN
 && ( IS_INTPOS(opR) )
 && ( SIZE_INT(opR) == 1 )
 && ( VAL_LIMB0(opR) == -INT_INTOBJ_MIN ) )
 mod = INTOBJ_INT(0);

 // in all other cases the remainder is equal the left operand
 else if ( 0 <= INT_INTOBJ(opL) )
 mod = opL;
 else if ( IS_INTPOS(opR) )
 mod = SumOrDiffInt( opL, opR, 1 );
 else
 mod = SumOrDiffInt( opL, opR, -1 );
 }

 // compute the remainder of a large integer by a small integer
 else if ( IS_INTOBJ(opR) ) {

 // get the integer value, make positive
 i = INT_INTOBJ(opR); if ( i < 0 ) i = -i;

 // check whether right operand is a small power of 2
 if ( !(i & (i-1)) ) {
 c = VAL_LIMB0(opL) & (i-1);
 }

 // otherwise use the gmp function to divide
 else {
 c = mpn_mod_1( (mp_srcptr)CONST_ADDR_INT(opL), SIZE_INT(opL), i );
 }

 // now c is the absolute value of the actual result. Thus, if the left
 // operand is negative, and c is non-zero, we have to adjust it.
 if (IS_INTPOS(opL) || c == 0)
 mod = INTOBJ_INT( c );
 else
 // even if opR is INT_INTOBJ_MIN, and hence i is INT_INTOBJ_MAX+1, we
 // have 0 <= i-c <= INT_INTOBJ_MAX, so i-c fits into a small integer
 mod = INTOBJ_INT( i - (Int)c );

 }

 // compute the remainder of a large integer modulo a large integer
 else {

 // trivial case first
 if ( SIZE_INT(opL) < SIZE_INT(opR) ) {
 if ( IS_INTPOS(opL) )
 return opL;
 else if ( IS_INTPOS(opR) )
 mod = SumOrDiffInt( opL, opR, 1 );
 else
 mod = SumOrDiffInt( opL, opR, -1 );
#if DEBUG_GMP
 assert( !IS_NEG_INT(mod) );
#endif
 CHECK_INT(mod);
 return mod;
 }

 mod = NewBag( TNUM_OBJ(opL), (SIZE_INT(opL)+1)*sizeof(mp_limb_t) );
 ENSURE_BAG(mod);

 quo = NewBag( T_INTPOS,
 (SIZE_INT(opL)-SIZE_INT(opR)+1)*sizeof(mp_limb_t) );
 ENSURE_BAG(quo);

 // and let gmp do the work
 mpn_tdiv_qr( (mp_ptr)ADDR_INT(quo), (mp_ptr)ADDR_INT(mod), 0,
 (mp_srcptr)CONST_ADDR_INT(opL), SIZE_INT(opL),
 (mp_srcptr)CONST_ADDR_INT(opR), SIZE_INT(opR) );

 // reduce to small integer if possible, otherwise shrink bag
 mod = GMP_NORMALIZE( mod );

 // make the representative positive
 if ( IS_NEG_INT(mod) ) {
 if ( IS_INTPOS(opR) )
 mod = SumOrDiffInt( mod, opR, 1 );
 else
 mod = SumOrDiffInt( mod, opR, -1 );
 }

 }

#if DEBUG_GMP
 assert( !IS_NEG_INT(mod) );
#endif
 CHECK_INT(mod);
 return mod;
}

/****************************************************************************
**
*F QuoInt( <opL>, <opR> ) . . . . . . . . . . . . . quotient of two integers
**
** 'QuoInt' returns the integer part of the two integers <opL> and <opR>.
**
** Note that this routine is not called from 'EvalQuo', the division of two
** integers yields a rational and is therefore performed in 'QuoRat'. This
** operation is however available through the internal function 'Quo'.
*/
Obj QuoInt(Obj opL, Obj opR)
{
 Int i; // loop count, value for small int
 Int k; // loop count, value for small int
 Obj quo; // handle of the result bag
 Obj rem; // handle of the remainder bag

 CHECK_INT(opL);
 CHECK_INT(opR);

 // pathological case first
 RequireNonzero("Integer operations", opR, "divisor");

 // divide two small integers
 if ( ARE_INTOBJS( opL, opR ) ) {

 // the small int -(1<<28) divided by -1 is the large int (1<<28)
 if ( opL == INTOBJ_MIN && opR == INTOBJ_INT(-1) ) {
 quo = NewBag( T_INTPOS, sizeof(mp_limb_t) );
 SET_VAL_LIMB0( quo, -INT_INTOBJ_MIN );
 return quo;
 }

 // get the integer values
 i = INT_INTOBJ(opL);
 k = INT_INTOBJ(opR);

 // divide, truncated towards zero; this is also what section 6.5.5 of the
 // C99 standard guarantees for integer division
 quo = INTOBJ_INT(i / k);
 }

 // divide a small integer by a large one
 else if ( IS_INTOBJ(opL) ) {

 // the small int -(1<<28) divided by the large int (1<<28) is -1
 if ( opL == INTOBJ_MIN
 && IS_INTPOS(opR) && SIZE_INT(opR) == 1
 && VAL_LIMB0(opR) == -INT_INTOBJ_MIN )
 quo = INTOBJ_INT(-1);

 // in all other cases the quotient is of course zero
 else
 quo = INTOBJ_INT(0);

 }

 // divide a large integer by a small integer
 else if ( IS_INTOBJ(opR) ) {

 k = INT_INTOBJ(opR);

 // allocate a bag for the result and set up the pointers
 if ( IS_INTNEG(opL) == ( k < 0 ) )
 quo = NewBag( T_INTPOS, SIZE_OBJ(opL) );
 else
 quo = NewBag( T_INTNEG, SIZE_OBJ(opL) );

 ENSURE_BAG(quo);

 if ( k < 0 ) k = -k;

 // use gmp function for dividing by a 1-limb number
 mpn_divrem_1( (mp_ptr)ADDR_INT(quo), 0,
 (mp_srcptr)CONST_ADDR_INT(opL), SIZE_INT(opL),
 k );
 }

 // divide a large integer by a large integer
 else {

 // trivial case first
 if ( SIZE_INT(opL) < SIZE_INT(opR) )
 return INTOBJ_INT(0);

 // create a new bag for the remainder
 rem = NewBag( TNUM_OBJ(opL), (SIZE_INT(opL)+1)*sizeof(mp_limb_t) );
 ENSURE_BAG(rem);

 // allocate a bag for the quotient
 if ( TNUM_OBJ(opL) == TNUM_OBJ(opR) )
 quo = NewBag( T_INTPOS,
 (SIZE_INT(opL)-SIZE_INT(opR)+1)*sizeof(mp_limb_t) );
 else
 quo = NewBag( T_INTNEG,
 (SIZE_INT(opL)-SIZE_INT(opR)+1)*sizeof(mp_limb_t) );
 ENSURE_BAG(quo);

 mpn_tdiv_qr( (mp_ptr)ADDR_INT(quo), (mp_ptr)ADDR_INT(rem), 0,
 (mp_srcptr)CONST_ADDR_INT(opL), SIZE_INT(opL),
 (mp_srcptr)CONST_ADDR_INT(opR), SIZE_INT(opR) );
 }

 quo = GMP_NORMALIZE(quo);
 return quo;
}

/****************************************************************************
**
*F FuncQUO_INT( <self>, <a>, ) . . . . . . . internal function 'QuoInt'
**
** 'FuncQUO_INT' implements the internal function 'QuoInt'.
**
** 'QuoInt( <a>, )'
**
** 'Quo' returns the integer part of the quotient of its integer operands.
** If <a> and are positive 'Quo( <a>, )' is the largest positive
** integer <q> such that '<q> * \<= <a>'. If <a> or or both are
** negative we define 'Abs( Quo(<a>,) ) = Quo( Abs(<a>), Abs() )' and
** 'Sign( Quo(<a>,) ) = Sign(<a>) * Sign()'. Dividing by 0 causes an
** error. 'Rem' (see "Rem") can be used to compute the remainder.
*/
static Obj FuncQUO_INT(Obj self, Obj a, Obj b)
{
 RequireInt(SELF_NAME, a);
 RequireInt(SELF_NAME, b);
 return QuoInt(a, b);
}

/****************************************************************************
**
*F RemInt( <opL>, <opR> ) . . . . . . . . . . . . remainder of two integers
**
** 'RemInt' returns the remainder of the quotient of the integers <opL>
** and <opR>.
**
** Note that the remainder is different from the value returned by the 'mod'
** operator which is always positive, while the result of 'RemInt' has
** the same sign as <opL>.
*/
Obj RemInt(Obj opL, Obj opR)
{
 Int i; // loop count, value for small int
 Int k; // loop count, value for small int
 UInt c;
 Obj rem; // handle of the remainder bag
 Obj quo; // handle of the quotient bag

 CHECK_INT(opL);
 CHECK_INT(opR);

 // pathological case first
 RequireNonzero("Integer operations", opR, "divisor");

 // compute the remainder of two small integers
 if ( ARE_INTOBJS( opL, opR ) ) {

 // get the integer values
 i = INT_INTOBJ(opL);
 k = INT_INTOBJ(opR);

 // compute the remainder with sign matching that of the dividend i;
 // this matches the sign of i % k as specified in section 6.5.5 of
 // the C99 standard, which indicates division truncates towards zero,
 // and the invariant i%k == i - (i/k)*k holds
 rem = INTOBJ_INT(i % k);
 }

 // compute the remainder of a small integer by a large integer
 else if ( IS_INTOBJ(opL) ) {

 // the small int -(1<<28) rem the large int (1<<28) is 0
 if ( opL == INTOBJ_MIN
 && IS_INTPOS(opR) && SIZE_INT(opR) == 1
 && VAL_LIMB0(opR) == -INT_INTOBJ_MIN )
 rem = INTOBJ_INT(0);

 // in all other cases the remainder is equal the left operand
 else
 rem = opL;
 }

 // compute the remainder of a large integer by a small integer
 else if ( IS_INTOBJ(opR) ) {

 // get the integer value, make positive
 i = INT_INTOBJ(opR); if ( i < 0 ) i = -i;

 // check whether right operand is a small power of 2
 if ( !(i & (i-1)) ) {
 c = VAL_LIMB0(opL) & (i-1);
 }

 // otherwise use the gmp function to divide
 else {
 c = mpn_mod_1( (mp_srcptr)CONST_ADDR_INT(opL), SIZE_INT(opL), i );
 }

 // adjust c for the sign of the left operand
 if ( IS_INTPOS(opL) )
 rem = INTOBJ_INT( c );
 else
 rem = INTOBJ_INT( -(Int)c );

 }

 // compute the remainder of a large integer modulo a large integer
 else {

 // trivial case first
 if ( SIZE_INT(opL) < SIZE_INT(opR) )
 return opL;

 rem = NewBag( TNUM_OBJ(opL), (SIZE_INT(opL)+1)*sizeof(mp_limb_t) );
 ENSURE_BAG(rem);

 quo = NewBag( T_INTPOS,
 (SIZE_INT(opL)-SIZE_INT(opR)+1)*sizeof(mp_limb_t) );
 ENSURE_BAG(quo);

 // and let gmp do the work
 mpn_tdiv_qr( (mp_ptr)ADDR_INT(quo), (mp_ptr)ADDR_INT(rem), 0,
 (mp_srcptr)CONST_ADDR_INT(opL), SIZE_INT(opL),
 (mp_srcptr)CONST_ADDR_INT(opR), SIZE_INT(opR) );

 // reduce to small integer if possible, otherwise shrink bag
 rem = GMP_NORMALIZE( rem );
 }

 CHECK_INT(rem);
 return rem;
}

/****************************************************************************
**
*F FuncREM_INT( <self>, <a>, ) . . . . . . . internal function 'RemInt'
**
** 'FuncREM_INT' implements the internal function 'RemInt'.
**
** 'RemInt( <a>, )'
**
** 'RemInt' returns the remainder of its two integer operands, i.e., if <k>
** is not equal to zero 'Rem( , <k> ) = - <k> * Quo( , <k> )'.
** Note that the rules given for 'Quo' imply that 'Rem( , <k> )' has the
** same sign as and its absolute value is strictly less than the
** absolute value of <k>. Dividing by 0 causes an error.
*/
static Obj FuncREM_INT(Obj self, Obj a, Obj b)
{
 RequireInt(SELF_NAME, a);
 RequireInt(SELF_NAME, b);
 return RemInt(a, b);
}

/****************************************************************************
**
*F GcdInt( <opL>, <opR> ) . . . . . . . . . . . . . . . gcd of two integers
**
** 'GcdInt' returns the gcd of the two integers <opL> and <opR>.
**
** It is called from 'FuncGCD_INT' and from the rational package.
*/
Obj GcdInt ( Obj opL, Obj opR )
{
 UInt sizeL, sizeR;
 fake_mpz_t mpzL, mpzR, mpzResult;
 Obj result;

 CHECK_INT(opL);
 CHECK_INT(opR);

 if (opL == INTOBJ_INT(0)) return AbsInt(opR);
 if (opR == INTOBJ_INT(0)) return AbsInt(opL);

 sizeL = SIZE_INT_OR_INTOBJ(opL);
 sizeR = SIZE_INT_OR_INTOBJ(opR);

 // for small inputs, run Euclid directly
 if (sizeL == 1 || sizeR == 1) {
 if (sizeR != 1) {
 SWAP(Obj, opL, opR);
 }
 UInt r = AbsOfSmallInt(opR);
 FAKEMPZ_GMPorINTOBJ(mpzL, opL);
 r = mpz_gcd_ui(0, MPZ_FAKEMPZ(mpzL), r);
 CHECK_FAKEMPZ(mpzL);
 return ObjInt_UInt(r);
 }

 NEW_FAKEMPZ( mpzResult, sizeL < sizeR ? sizeL : sizeR );
 FAKEMPZ_GMPorINTOBJ( mpzL, opL );
 FAKEMPZ_GMPorINTOBJ( mpzR, opR );

 // compute the gcd
 mpz_gcd( MPZ_FAKEMPZ(mpzResult), MPZ_FAKEMPZ(mpzL), MPZ_FAKEMPZ(mpzR) );

 // convert result to GAP object and return it
 CHECK_FAKEMPZ(mpzResult);
 CHECK_FAKEMPZ(mpzL);
 CHECK_FAKEMPZ(mpzR);
 result = GMPorINTOBJ_FAKEMPZ( mpzResult );
 CHECK_INT(result);
 return result;
}

/****************************************************************************
**
*F FuncGCD_INT(<self>,<a>,) . . . . . . . internal function 'GcdInt'
**
** 'FuncGCD_INT' implements the internal function 'GcdInt'.
**
** 'GcdInt( <a>, )'
**
** 'Gcd' returns the greatest common divisor of the two integers <a> and
** , i.e., the greatest integer that divides both <a> and . The
** greatest common divisor is never negative, even if the arguments are. We
** define $gcd( a, 0 ) = gcd( 0, a ) = abs( a )$ and $gcd( 0, 0 ) = 0$.
*/
static Obj FuncGCD_INT(Obj self, Obj a, Obj b)
{
 RequireInt(SELF_NAME, a);
 RequireInt(SELF_NAME, b);
 return GcdInt(a, b);
}

/****************************************************************************
**
*F LcmInt( <opL>, <opR> ) . . . . . . . . . . . . . . . lcm of two integers
**
** 'LcmInt' returns the lcm of the two integers <opL> and <opR>.
*/
Obj LcmInt(Obj opL, Obj opR)
{
 UInt sizeL, sizeR;
 fake_mpz_t mpzL, mpzR, mpzResult;
 Obj result;

 CHECK_INT(opL);
 CHECK_INT(opR);

 if (opL == INTOBJ_INT(0) || opR == INTOBJ_INT(0))
 return INTOBJ_INT(0);

 if (IS_INTOBJ(opL) || IS_INTOBJ(opR)) {
 if (!IS_INTOBJ(opR)) {
 SWAP(Obj, opL, opR);
 }
 Obj gcd = GcdInt(opL, opR);
 opR = QuoInt(opR, gcd);
 return AbsInt(ProdInt(opL, opR));
 }

 sizeL = SIZE_INT_OR_INTOBJ(opL);
 sizeR = SIZE_INT_OR_INTOBJ(opR);

 NEW_FAKEMPZ(mpzResult, sizeL + sizeR);
 FAKEMPZ_GMPorINTOBJ(mpzL, opL);
 FAKEMPZ_GMPorINTOBJ(mpzR, opR);

 // compute the gcd
 mpz_lcm(MPZ_FAKEMPZ(mpzResult), MPZ_FAKEMPZ(mpzL), MPZ_FAKEMPZ(mpzR));

 // convert result to GAP object and return it
 CHECK_FAKEMPZ(mpzResult);
 CHECK_FAKEMPZ(mpzL);
 CHECK_FAKEMPZ(mpzR);
 result = GMPorINTOBJ_FAKEMPZ(mpzResult);
 CHECK_INT(result);
 return result;
}

static Obj FuncLCM_INT(Obj self, Obj a, Obj b)
{
 RequireInt(SELF_NAME, a);
 RequireInt(SELF_NAME, b);
 return LcmInt(a, b);
}

/****************************************************************************
**
*/
static Obj FuncFACTORIAL_INT(Obj self, Obj n)
{
 RequireNonnegativeSmallInt(SELF_NAME, n);

 mpz_t mpzResult;
 mpz_init(mpzResult);
 mpz_fac_ui(mpzResult, INT_INTOBJ(n));

 // convert mpzResult into a GAP integer object.
 Obj result = GMPorINTOBJ_MPZ(mpzResult);

 // free mpzResult
 mpz_clear(mpzResult);

 return result;
}

/****************************************************************************
**
*/
Obj BinomialInt(Obj n, Obj k)
{
 Int negate_result = 0;

 // deal with k <= 1
 if (k == INTOBJ_INT(0))
 return INTOBJ_INT(1);
 if (k == INTOBJ_INT(1))
 return n;
 if (IS_NEG_INT(k))
 return INTOBJ_INT(0);

 // deal with n < 0
 if (IS_NEG_INT(n)) {
 // use the identity Binomial(n,k) = (-1)^k * Binomial(-n+k-1, k)
 negate_result = IS_ODD_INT(k);
 n = DiffInt(DiffInt(k,n), INTOBJ_INT(1));
 }

 // deal with n <= k
 if (n == k)
 return negate_result ? INTOBJ_INT(-1) : INTOBJ_INT(1);
 if (LtInt(n, k))
 return INTOBJ_INT(0);

 // deal with n-k < k <=> n < 2k
 Obj k2 = DiffInt(n, k);
 if (LtInt(k2, k))
 k = k2;

 // From here on, we only support single limb integers for k. Anything else
 // would lead to output too big for storage anyway, at least on a 64 bit
 // system. To be specific, in that case n >= 2k and k >= 2^60. Thus, in
 // the *best* case, we are trying to compute the central binomial
 // coefficient Binomial(2k k) for k = 2^60. This value is approximately
 // (4^k / sqrt(pi*k)); taking the logarithm and dividing by 8 yields that
 // we need about k/4 = 2^58 bytes to store this value. No computer in the
 // foreseeable future will be able to store the result (and much less
 // compute it in a reasonable time).
 //
 // On 32 bit systems, the limit is k = 2^28, and then the central binomial
 // coefficient Binomial(2k,k) takes up about 64 MB, so that would still be
 // feasible. However, GMP does not support computing binomials when k is
 // larger than a single limb, so we'd have to implement this on our own.
 //
 // Since GAP previously was effectively unable to compute such binomial
 // coefficients (unless you were willing to wait for a few days or so), we
 // simply do not implement this on 32bit systems, and instead return Fail,
 // jut as we do on 64 bit. If somebody complains about this, we can still
 // look into implementing this (and in the meantime, tell the user to use
 // the old GAP version of this function).

 if (SIZE_INT_OR_INTOBJ(k) > 1)
 return Fail;

 UInt K = IS_INTOBJ(k) ? INT_INTOBJ(k) : VAL_LIMB0(k);
 mpz_t mpzResult;
 mpz_init( mpzResult );

 if (SIZE_INT_OR_INTOBJ(n) == 1) {
 UInt N = IS_INTOBJ(n) ? INT_INTOBJ(n) : VAL_LIMB0(n);
 mpz_bin_uiui(mpzResult, N, K);
 } else {
 fake_mpz_t mpzN;
 FAKEMPZ_GMPorINTOBJ( mpzN, n );
 mpz_bin_ui(mpzResult, MPZ_FAKEMPZ(mpzN), K);
 }

 // adjust sign of result
 if (negate_result)
 mpzResult->_mp_size = -mpzResult->_mp_size;

 // convert mpzResult into a GAP integer object.
 Obj result = GMPorINTOBJ_MPZ(mpzResult);

 // free mpzResult
 mpz_clear( mpzResult );

 return result;
}

static Obj FuncBINOMIAL_INT(Obj self, Obj n, Obj k)
{
 RequireInt(SELF_NAME, n);
 RequireInt(SELF_NAME, k);
 return BinomialInt(n, k);
}

/****************************************************************************
**
*/
--> --------------------

--> maximum size reached

--> --------------------

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