bignum.c

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

  bignum.c -

  $Author$
  created at: Fri Jun 10 00:48:55 JST 1994

  Copyright (C) 1993-2007 Yukihiro Matsumoto

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

#include "ruby/ruby.h"
#include "ruby/thread.h"
#include "ruby/util.h"
#include "internal.h"

#ifdef HAVE_STRINGS_H
#include <strings.h>
#endif
#include <math.h>
#include <float.h>
#include <ctype.h>
#ifdef HAVE_IEEEFP_H
#include <ieeefp.h>
#endif
#include <assert.h>

VALUE rb_cBignum;

static VALUE big_three = Qnil;

#if defined __MINGW32__
#define USHORT _USHORT
#endif

#define BDIGITS(x) (RBIGNUM_DIGITS(x))
#define BITSPERDIG (SIZEOF_BDIGITS*CHAR_BIT)
#define BIGRAD ((BDIGIT_DBL)1 << BITSPERDIG)
#define BIGRAD_HALF ((BDIGIT)(BIGRAD >> 1))
#define DIGSPERLONG (SIZEOF_LONG/SIZEOF_BDIGITS)
#if HAVE_LONG_LONG
# define DIGSPERLL (SIZEOF_LONG_LONG/SIZEOF_BDIGITS)
#endif
#define BIGUP(x) ((BDIGIT_DBL)(x) << BITSPERDIG)
#define BIGDN(x) RSHIFT((x),BITSPERDIG)
#define BIGLO(x) ((BDIGIT)((x) & (BIGRAD-1)))
#define BDIGMAX ((BDIGIT)-1)

#define BIGZEROP(x) (RBIGNUM_LEN(x) == 0 || \
		     (BDIGITS(x)[0] == 0 && \
		      (RBIGNUM_LEN(x) == 1 || bigzero_p(x))))

#define BIGNUM_DEBUG 0
#if BIGNUM_DEBUG
#define ON_DEBUG(x) do { x; } while (0)
static void
dump_bignum(VALUE x)
{
    long i;
    printf("%c0x0", RBIGNUM_SIGN(x) ? '+' : '-');
    for (i = RBIGNUM_LEN(x); i--; ) {
	printf("_%08"PRIxBDIGIT, BDIGITS(x)[i]);
    }
    printf(", len=%lu", RBIGNUM_LEN(x));
    puts("");
}

static VALUE
rb_big_dump(VALUE x)
{
    dump_bignum(x);
    return x;
}
#else
#define ON_DEBUG(x)
#endif

static int
bigzero_p(VALUE x)
{
    long i;
    BDIGIT *ds = BDIGITS(x);

    for (i = RBIGNUM_LEN(x) - 1; 0 <= i; i--) {
	if (ds[i]) return 0;
    }
    return 1;
}

int
rb_bigzero_p(VALUE x)
{
    return BIGZEROP(x);
}

int
rb_cmpint(VALUE val, VALUE a, VALUE b)
{
    if (NIL_P(val)) {
	rb_cmperr(a, b);
    }
    if (FIXNUM_P(val)) {
        long l = FIX2LONG(val);
        if (l > 0) return 1;
        if (l < 0) return -1;
        return 0;
    }
    if (RB_TYPE_P(val, T_BIGNUM)) {
	if (BIGZEROP(val)) return 0;
	if (RBIGNUM_SIGN(val)) return 1;
	return -1;
    }
    if (RTEST(rb_funcall(val, '>', 1, INT2FIX(0)))) return 1;
    if (RTEST(rb_funcall(val, '<', 1, INT2FIX(0)))) return -1;
    return 0;
}

#define RBIGNUM_SET_LEN(b,l) \
    ((RBASIC(b)->flags & RBIGNUM_EMBED_FLAG) ? \
     (void)(RBASIC(b)->flags = \
	    (RBASIC(b)->flags & ~RBIGNUM_EMBED_LEN_MASK) | \
	    ((l) << RBIGNUM_EMBED_LEN_SHIFT)) : \
     (void)(RBIGNUM(b)->as.heap.len = (l)))

static void
rb_big_realloc(VALUE big, long len)
{
    BDIGIT *ds;
    if (RBASIC(big)->flags & RBIGNUM_EMBED_FLAG) {
	if (RBIGNUM_EMBED_LEN_MAX < len) {
	    ds = ALLOC_N(BDIGIT, len);
	    MEMCPY(ds, RBIGNUM(big)->as.ary, BDIGIT, RBIGNUM_EMBED_LEN_MAX);
	    RBIGNUM(big)->as.heap.len = RBIGNUM_LEN(big);
	    RBIGNUM(big)->as.heap.digits = ds;
	    RBASIC(big)->flags &= ~RBIGNUM_EMBED_FLAG;
	}
    }
    else {
	if (len <= RBIGNUM_EMBED_LEN_MAX) {
	    ds = RBIGNUM(big)->as.heap.digits;
	    RBASIC(big)->flags |= RBIGNUM_EMBED_FLAG;
	    RBIGNUM_SET_LEN(big, len);
	    if (ds) {
		MEMCPY(RBIGNUM(big)->as.ary, ds, BDIGIT, len);
		xfree(ds);
	    }
	}
	else {
	    if (RBIGNUM_LEN(big) == 0) {
		RBIGNUM(big)->as.heap.digits = ALLOC_N(BDIGIT, len);
	    }
	    else {
		REALLOC_N(RBIGNUM(big)->as.heap.digits, BDIGIT, len);
	    }
	}
    }
}

void
rb_big_resize(VALUE big, long len)
{
    rb_big_realloc(big, len);
    RBIGNUM_SET_LEN(big, len);
}

static VALUE
bignew_1(VALUE klass, long len, int sign)
{
    NEWOBJ_OF(big, struct RBignum, klass, T_BIGNUM);
    RBIGNUM_SET_SIGN(big, sign?1:0);
    if (len <= RBIGNUM_EMBED_LEN_MAX) {
	RBASIC(big)->flags |= RBIGNUM_EMBED_FLAG;
	RBIGNUM_SET_LEN(big, len);
    }
    else {
	RBIGNUM(big)->as.heap.digits = ALLOC_N(BDIGIT, len);
	RBIGNUM(big)->as.heap.len = len;
    }
    OBJ_FREEZE(big);
    return (VALUE)big;
}

#define bignew(len,sign) bignew_1(rb_cBignum,(len),(sign))

VALUE
rb_big_new(long len, int sign)
{
    return bignew(len, sign != 0);
}

VALUE
rb_big_clone(VALUE x)
{
    long len = RBIGNUM_LEN(x);
    VALUE z = bignew_1(CLASS_OF(x), len, RBIGNUM_SIGN(x));

    MEMCPY(BDIGITS(z), BDIGITS(x), BDIGIT, len);
    return z;
}

/* modify a bignum by 2's complement */
static void
get2comp(VALUE x)
{
    long i = RBIGNUM_LEN(x);
    BDIGIT *ds = BDIGITS(x);
    BDIGIT_DBL num;

    if (!i) return;
    while (i--) ds[i] = ~ds[i];
    i = 0; num = 1;
    do {
	num += ds[i];
	ds[i++] = BIGLO(num);
	num = BIGDN(num);
    } while (i < RBIGNUM_LEN(x));
    if (num != 0) {
	rb_big_resize(x, RBIGNUM_LEN(x)+1);
	ds = BDIGITS(x);
	ds[RBIGNUM_LEN(x)-1] = 1;
    }
}

void
rb_big_2comp(VALUE x)			/* get 2's complement */
{
    get2comp(x);
}

static inline VALUE
bigtrunc(VALUE x)
{
    long len = RBIGNUM_LEN(x);
    BDIGIT *ds = BDIGITS(x);

    if (len == 0) return x;
    while (--len && !ds[len]);
    if (RBIGNUM_LEN(x) > len+1) {
	rb_big_resize(x, len+1);
    }
    return x;
}

static inline VALUE
bigfixize(VALUE x)
{
    long len = RBIGNUM_LEN(x);
    BDIGIT *ds = BDIGITS(x);

    if (len == 0) return INT2FIX(0);
    if ((size_t)(len*SIZEOF_BDIGITS) <= sizeof(long)) {
	long num = 0;
#if 2*SIZEOF_BDIGITS > SIZEOF_LONG
	num = (long)ds[0];
#else
	while (len--) {
	    num = (long)(BIGUP(num) + ds[len]);
	}
#endif
	if (num >= 0) {
	    if (RBIGNUM_SIGN(x)) {
		if (POSFIXABLE(num)) return LONG2FIX(num);
	    }
	    else {
		if (NEGFIXABLE(-num)) return LONG2FIX(-num);
	    }
	}
    }
    return x;
}

static VALUE
bignorm(VALUE x)
{
    if (RB_TYPE_P(x, T_BIGNUM)) {
	x = bigfixize(bigtrunc(x));
    }
    return x;
}

VALUE
rb_big_norm(VALUE x)
{
    return bignorm(x);
}

VALUE
rb_uint2big(VALUE n)
{
    BDIGIT_DBL num = n;
    long i = 0;
    BDIGIT *digits;
    VALUE big;

    big = bignew(DIGSPERLONG, 1);
    digits = BDIGITS(big);
    while (i < DIGSPERLONG) {
	digits[i++] = BIGLO(num);
	num = BIGDN(num);
    }

    i = DIGSPERLONG;
    while (--i && !digits[i]) ;
    RBIGNUM_SET_LEN(big, i+1);
    return big;
}

VALUE
rb_int2big(SIGNED_VALUE n)
{
    long neg = 0;
    VALUE big;

    if (n < 0) {
	n = -n;
	neg = 1;
    }
    big = rb_uint2big(n);
    if (neg) {
	RBIGNUM_SET_SIGN(big, 0);
    }
    return big;
}

VALUE
rb_uint2inum(VALUE n)
{
    if (POSFIXABLE(n)) return LONG2FIX(n);
    return rb_uint2big(n);
}

VALUE
rb_int2inum(SIGNED_VALUE n)
{
    if (FIXABLE(n)) return LONG2FIX(n);
    return rb_int2big(n);
}

#if SIZEOF_LONG % SIZEOF_BDIGITS != 0
# error unexpected SIZEOF_LONG : SIZEOF_BDIGITS ratio
#endif

/*
 * buf is an array of long integers.
 * buf is ordered from least significant word to most significant word.
 * buf[0] is the least significant word and
 * buf[num_longs-1] is the most significant word.
 * This means words in buf is little endian.
 * However each word in buf is native endian.
 * (buf[i]&1) is the least significant bit and
 * (buf[i]&(1<<(SIZEOF_LONG*CHAR_BIT-1))) is the most significant bit
 * for each 0 <= i < num_longs.
 * So buf is little endian at whole on a little endian machine.
 * But buf is mixed endian on a big endian machine.
 *
 * The buf represents negative integers as two's complement.
 * So, the most significant bit of the most significant word,
 * (buf[num_longs-1]>>(SIZEOF_LONG*CHAR_BIT-1)),
 * is the sign bit: 1 means negative and 0 means zero or positive.
 *
 * If given size of buf (num_longs) is not enough to represent val,
 * higier words (including a sign bit) are ignored.
 */
void
rb_big_pack(VALUE val, unsigned long *buf, long num_longs)
{
    val = rb_to_int(val);
    if (num_longs == 0)
        return;
    if (FIXNUM_P(val)) {
        long i;
        long tmp = FIX2LONG(val);
        buf[0] = (unsigned long)tmp;
        tmp = tmp < 0 ? ~0L : 0;
        for (i = 1; i < num_longs; i++)
            buf[i] = (unsigned long)tmp;
        return;
    }
    else {
        long len = RBIGNUM_LEN(val);
        BDIGIT *ds = BDIGITS(val), *dend = ds + len;
        long i, j;
        for (i = 0; i < num_longs && ds < dend; i++) {
            unsigned long l = 0;
            for (j = 0; j < DIGSPERLONG && ds < dend; j++, ds++) {
                l |= ((unsigned long)*ds << (j * BITSPERDIG));
            }
            buf[i] = l;
        }
        for (; i < num_longs; i++)
            buf[i] = 0;
        if (RBIGNUM_NEGATIVE_P(val)) {
            for (i = 0; i < num_longs; i++) {
                buf[i] = ~buf[i];
            }
            for (i = 0; i < num_longs; i++) {
                buf[i]++;
                if (buf[i] != 0)
                    return;
            }
        }
    }
}

/* See rb_big_pack comment for endianness and sign of buf. */
VALUE
rb_big_unpack(unsigned long *buf, long num_longs)
{
    while (2 <= num_longs) {
        if (buf[num_longs-1] == 0 && (long)buf[num_longs-2] >= 0)
            num_longs--;
        else if (buf[num_longs-1] == ~0UL && (long)buf[num_longs-2] < 0)
            num_longs--;
        else
            break;
    }
    if (num_longs == 0)
        return INT2FIX(0);
    else if (num_longs == 1)
        return LONG2NUM((long)buf[0]);
    else {
        VALUE big;
        BDIGIT *ds;
        long len = num_longs * DIGSPERLONG;
        long i;
        big = bignew(len, 1);
        ds = BDIGITS(big);
        for (i = 0; i < num_longs; i++) {
            unsigned long d = buf[i];
#if SIZEOF_LONG == SIZEOF_BDIGITS
            *ds++ = d;
#else
            int j;
            for (j = 0; j < DIGSPERLONG; j++) {
                *ds++ = BIGLO(d);
                d = BIGDN(d);
            }
#endif
        }
        if ((long)buf[num_longs-1] < 0) {
            get2comp(big);
            RBIGNUM_SET_SIGN(big, 0);
        }
        return bignorm(big);
    }
}

#define QUAD_SIZE 8

#if SIZEOF_LONG_LONG == QUAD_SIZE && SIZEOF_BDIGITS*2 == SIZEOF_LONG_LONG

void
rb_quad_pack(char *buf, VALUE val)
{
    LONG_LONG q;

    val = rb_to_int(val);
    if (FIXNUM_P(val)) {
	q = FIX2LONG(val);
    }
    else {
	long len = RBIGNUM_LEN(val);
	BDIGIT *ds;

	if (len > SIZEOF_LONG_LONG/SIZEOF_BDIGITS) {
	    len = SIZEOF_LONG_LONG/SIZEOF_BDIGITS;
	}
	ds = BDIGITS(val);
	q = 0;
	while (len--) {
	    q = BIGUP(q);
	    q += ds[len];
	}
	if (!RBIGNUM_SIGN(val)) q = -q;
    }
    memcpy(buf, (char*)&q, SIZEOF_LONG_LONG);
}

VALUE
rb_quad_unpack(const char *buf, int sign)
{
    unsigned LONG_LONG q;
    long neg = 0;
    long i;
    BDIGIT *digits;
    VALUE big;

    memcpy(&q, buf, SIZEOF_LONG_LONG);
    if (sign) {
	if (FIXABLE((LONG_LONG)q)) return LONG2FIX((LONG_LONG)q);
	if ((LONG_LONG)q < 0) {
	    q = -(LONG_LONG)q;
	    neg = 1;
	}
    }
    else {
	if (POSFIXABLE(q)) return LONG2FIX(q);
    }

    i = 0;
    big = bignew(DIGSPERLL, 1);
    digits = BDIGITS(big);
    while (i < DIGSPERLL) {
	digits[i++] = BIGLO(q);
	q = BIGDN(q);
    }

    i = DIGSPERLL;
    while (i-- && !digits[i]) ;
    RBIGNUM_SET_LEN(big, i+1);

    if (neg) {
	RBIGNUM_SET_SIGN(big, 0);
    }
    return bignorm(big);
}

#else

static int
quad_buf_complement(char *buf, size_t len)
{
    size_t i;
    for (i = 0; i < len; i++)
        buf[i] = ~buf[i];
    for (i = 0; i < len; i++) {
        buf[i]++;
        if (buf[i] != 0)
            return 0;
    }
    return 1;
}

void
rb_quad_pack(char *buf, VALUE val)
{
    long len;

    memset(buf, 0, QUAD_SIZE);
    val = rb_to_int(val);
    if (FIXNUM_P(val)) {
	val = rb_int2big(FIX2LONG(val));
    }
    len = RBIGNUM_LEN(val) * SIZEOF_BDIGITS;
    if (len > QUAD_SIZE) {
        len = QUAD_SIZE;
    }
    memcpy(buf, (char*)BDIGITS(val), len);
    if (RBIGNUM_NEGATIVE_P(val)) {
        quad_buf_complement(buf, QUAD_SIZE);
    }
}

#define BNEG(b) (RSHIFT(((BDIGIT*)(b))[QUAD_SIZE/SIZEOF_BDIGITS-1],BITSPERDIG-1) != 0)

VALUE
rb_quad_unpack(const char *buf, int sign)
{
    VALUE big = bignew(QUAD_SIZE/SIZEOF_BDIGITS, 1);

    memcpy((char*)BDIGITS(big), buf, QUAD_SIZE);
    if (sign && BNEG(buf)) {
	char *tmp = (char*)BDIGITS(big);

	RBIGNUM_SET_SIGN(big, 0);
        quad_buf_complement(tmp, QUAD_SIZE);
    }

    return bignorm(big);
}

#endif

VALUE
rb_cstr_to_inum(const char *str, int base, int badcheck)
{
    const char *s = str;
    char *end;
    char sign = 1, nondigit = 0;
    int c;
    BDIGIT_DBL num;
    long len, blen = 1;
    long i;
    VALUE z;
    BDIGIT *zds;

#undef ISDIGIT
#define ISDIGIT(c) ('0' <= (c) && (c) <= '9')
#define conv_digit(c) \
    (!ISASCII(c) ? -1 : \
     ISDIGIT(c) ? ((c) - '0') : \
     ISLOWER(c) ? ((c) - 'a' + 10) : \
     ISUPPER(c) ? ((c) - 'A' + 10) : \
     -1)

    if (!str) {
	if (badcheck) goto bad;
	return INT2FIX(0);
    }
    while (ISSPACE(*str)) str++;

    if (str[0] == '+') {
	str++;
    }
    else if (str[0] == '-') {
	str++;
	sign = 0;
    }
    if (str[0] == '+' || str[0] == '-') {
	if (badcheck) goto bad;
	return INT2FIX(0);
    }
    if (base <= 0) {
	if (str[0] == '0') {
	    switch (str[1]) {
	      case 'x': case 'X':
		base = 16;
		break;
	      case 'b': case 'B':
		base = 2;
		break;
	      case 'o': case 'O':
		base = 8;
		break;
	      case 'd': case 'D':
		base = 10;
		break;
	      default:
		base = 8;
	    }
	}
	else if (base < -1) {
	    base = -base;
	}
	else {
	    base = 10;
	}
    }
    switch (base) {
      case 2:
	len = 1;
	if (str[0] == '0' && (str[1] == 'b'||str[1] == 'B')) {
	    str += 2;
	}
	break;
      case 3:
	len = 2;
	break;
      case 8:
	if (str[0] == '0' && (str[1] == 'o'||str[1] == 'O')) {
	    str += 2;
	}
      case 4: case 5: case 6: case 7:
	len = 3;
	break;
      case 10:
	if (str[0] == '0' && (str[1] == 'd'||str[1] == 'D')) {
	    str += 2;
	}
      case 9: case 11: case 12: case 13: case 14: case 15:
	len = 4;
	break;
      case 16:
	len = 4;
	if (str[0] == '0' && (str[1] == 'x'||str[1] == 'X')) {
	    str += 2;
	}
	break;
      default:
	if (base < 2 || 36 < base) {
	    rb_raise(rb_eArgError, "invalid radix %d", base);
	}
	if (base <= 32) {
	    len = 5;
	}
	else {
	    len = 6;
	}
	break;
    }
    if (*str == '0') {		/* squeeze preceding 0s */
	int us = 0;
	while ((c = *++str) == '0' || c == '_') {
	    if (c == '_') {
		if (++us >= 2)
		    break;
	    } else
		us = 0;
	}
	if (!(c = *str) || ISSPACE(c)) --str;
    }
    c = *str;
    c = conv_digit(c);
    if (c < 0 || c >= base) {
	if (badcheck) goto bad;
	return INT2FIX(0);
    }
    len *= strlen(str)*sizeof(char);

    if ((size_t)len <= (sizeof(long)*CHAR_BIT)) {
	unsigned long val = STRTOUL(str, &end, base);

	if (str < end && *end == '_') goto bigparse;
	if (badcheck) {
	    if (end == str) goto bad; /* no number */
	    while (*end && ISSPACE(*end)) end++;
	    if (*end) goto bad;	      /* trailing garbage */
	}

	if (POSFIXABLE(val)) {
	    if (sign) return LONG2FIX(val);
	    else {
		long result = -(long)val;
		return LONG2FIX(result);
	    }
	}
	else {
	    VALUE big = rb_uint2big(val);
	    RBIGNUM_SET_SIGN(big, sign);
	    return bignorm(big);
	}
    }
  bigparse:
    len = (len/BITSPERDIG)+1;
    if (badcheck && *str == '_') goto bad;

    z = bignew(len, sign);
    zds = BDIGITS(z);
    for (i=len;i--;) zds[i]=0;
    while ((c = *str++) != 0) {
	if (c == '_') {
	    if (nondigit) {
		if (badcheck) goto bad;
		break;
	    }
	    nondigit = (char) c;
	    continue;
	}
	else if ((c = conv_digit(c)) < 0) {
	    break;
	}
	if (c >= base) break;
	nondigit = 0;
	i = 0;
	num = c;
	for (;;) {
	    while (i<blen) {
		num += (BDIGIT_DBL)zds[i]*base;
		zds[i++] = BIGLO(num);
		num = BIGDN(num);
	    }
	    if (num) {
		blen++;
		continue;
	    }
	    break;
	}
    }
    if (badcheck) {
	str--;
	if (s+1 < str && str[-1] == '_') goto bad;
	while (*str && ISSPACE(*str)) str++;
	if (*str) {
	  bad:
	    rb_invalid_str(s, "Integer()");
	}
    }

    return bignorm(z);
}

VALUE
rb_str_to_inum(VALUE str, int base, int badcheck)
{
    char *s;
    long len;
    VALUE v = 0;
    VALUE ret;

    StringValue(str);
    rb_must_asciicompat(str);
    if (badcheck) {
	s = StringValueCStr(str);
    }
    else {
	s = RSTRING_PTR(str);
    }
    if (s) {
	len = RSTRING_LEN(str);
	if (s[len]) {		/* no sentinel somehow */
	    char *p = ALLOCV(v, len+1);

	    MEMCPY(p, s, char, len);
	    p[len] = '\0';
	    s = p;
	}
    }
    ret = rb_cstr_to_inum(s, base, badcheck);
    if (v)
	ALLOCV_END(v);
    return ret;
}

#if HAVE_LONG_LONG

static VALUE
rb_ull2big(unsigned LONG_LONG n)
{
    BDIGIT_DBL num = n;
    long i = 0;
    BDIGIT *digits;
    VALUE big;

    big = bignew(DIGSPERLL, 1);
    digits = BDIGITS(big);
    while (i < DIGSPERLL) {
	digits[i++] = BIGLO(num);
	num = BIGDN(num);
    }

    i = DIGSPERLL;
    while (i-- && !digits[i]) ;
    RBIGNUM_SET_LEN(big, i+1);
    return big;
}

static VALUE
rb_ll2big(LONG_LONG n)
{
    long neg = 0;
    VALUE big;

    if (n < 0) {
	n = -n;
	neg = 1;
    }
    big = rb_ull2big(n);
    if (neg) {
	RBIGNUM_SET_SIGN(big, 0);
    }
    return big;
}

VALUE
rb_ull2inum(unsigned LONG_LONG n)
{
    if (POSFIXABLE(n)) return LONG2FIX(n);
    return rb_ull2big(n);
}

VALUE
rb_ll2inum(LONG_LONG n)
{
    if (FIXABLE(n)) return LONG2FIX(n);
    return rb_ll2big(n);
}

#endif  /* HAVE_LONG_LONG */

VALUE
rb_cstr2inum(const char *str, int base)
{
    return rb_cstr_to_inum(str, base, base==0);
}

VALUE
rb_str2inum(VALUE str, int base)
{
    return rb_str_to_inum(str, base, base==0);
}

const char ruby_digitmap[] = "0123456789abcdefghijklmnopqrstuvwxyz";

static VALUE bigsqr(VALUE x);
static void bigdivmod(VALUE x, VALUE y, volatile VALUE *divp, volatile VALUE *modp);

#define POW2_P(x) (((x)&((x)-1))==0)

static inline int
ones(register unsigned long x)
{
#if SIZEOF_LONG == 8
# define MASK_55 0x5555555555555555UL
# define MASK_33 0x3333333333333333UL
# define MASK_0f 0x0f0f0f0f0f0f0f0fUL
#else
# define MASK_55 0x55555555UL
# define MASK_33 0x33333333UL
# define MASK_0f 0x0f0f0f0fUL
#endif
    x -= (x >> 1) & MASK_55;
    x = ((x >> 2) & MASK_33) + (x & MASK_33);
    x = ((x >> 4) + x) & MASK_0f;
    x += (x >> 8);
    x += (x >> 16);
#if SIZEOF_LONG == 8
    x += (x >> 32);
#endif
    return (int)(x & 0x7f);
#undef MASK_0f
#undef MASK_33
#undef MASK_55
}

static inline unsigned long
next_pow2(register unsigned long x)
{
    x |= x >> 1;
    x |= x >> 2;
    x |= x >> 4;
    x |= x >> 8;
    x |= x >> 16;
#if SIZEOF_LONG == 8
    x |= x >> 32;
#endif
    return x + 1;
}

static inline int
floor_log2(register unsigned long x)
{
    x |= x >> 1;
    x |= x >> 2;
    x |= x >> 4;
    x |= x >> 8;
    x |= x >> 16;
#if SIZEOF_LONG == 8
    x |= x >> 32;
#endif
    return (int)ones(x) - 1;
}

static inline int
ceil_log2(register unsigned long x)
{
    return floor_log2(x) + !POW2_P(x);
}

#define LOG2_KARATSUBA_DIGITS 7
#define KARATSUBA_DIGITS (1L<<LOG2_KARATSUBA_DIGITS)
#define MAX_BIG2STR_TABLE_ENTRIES 64

static VALUE big2str_power_cache[35][MAX_BIG2STR_TABLE_ENTRIES];

static void
power_cache_init(void)
{
    int i, j;
    for (i = 0; i < 35; ++i) {
	for (j = 0; j < MAX_BIG2STR_TABLE_ENTRIES; ++j) {
	    big2str_power_cache[i][j] = Qnil;
	}
    }
}

static inline VALUE
power_cache_get_power0(int base, int i)
{
    if (NIL_P(big2str_power_cache[base - 2][i])) {
	big2str_power_cache[base - 2][i] =
	    i == 0 ? rb_big_pow(rb_int2big(base), INT2FIX(KARATSUBA_DIGITS))
		   : bigsqr(power_cache_get_power0(base, i - 1));
	rb_gc_register_mark_object(big2str_power_cache[base - 2][i]);
    }
    return big2str_power_cache[base - 2][i];
}

static VALUE
power_cache_get_power(int base, long n1, long* m1)
{
    int i, m;
    long j;
    VALUE t;

    if (n1 <= KARATSUBA_DIGITS)
	rb_bug("n1 > KARATSUBA_DIGITS");

    m = ceil_log2(n1);
    if (m1) *m1 = 1 << m;
    i = m - LOG2_KARATSUBA_DIGITS;
    if (i >= MAX_BIG2STR_TABLE_ENTRIES)
	i = MAX_BIG2STR_TABLE_ENTRIES - 1;
    t = power_cache_get_power0(base, i);

    j = KARATSUBA_DIGITS*(1 << i);
    while (n1 > j) {
	t = bigsqr(t);
	j *= 2;
    }
    return t;
}

/* big2str_muraken_find_n1
 *
 * Let a natural number x is given by:
 * x = 2^0 * x_0 + 2^1 * x_1 + ... + 2^(B*n_0 - 1) * x_{B*n_0 - 1},
 * where B is BITSPERDIG (i.e. BDIGITS*CHAR_BIT) and n_0 is
 * RBIGNUM_LEN(x).
 *
 * Now, we assume n_1 = min_n \{ n | 2^(B*n_0/2) <= b_1^(n_1) \}, so
 * it is realized that 2^(B*n_0) <= {b_1}^{2*n_1}, where b_1 is a
 * given radix number. And then, we have n_1 <= (B*n_0) /
 * (2*log_2(b_1)), therefore n_1 is given by ceil((B*n_0) /
 * (2*log_2(b_1))).
 */
static long
big2str_find_n1(VALUE x, int base)
{
    static const double log_2[] = {
	1.0,              1.58496250072116, 2.0,
	2.32192809488736, 2.58496250072116, 2.8073549220576,
	3.0,              3.16992500144231, 3.32192809488736,
	3.4594316186373,  3.58496250072116, 3.70043971814109,
	3.8073549220576,  3.90689059560852, 4.0,
	4.08746284125034, 4.16992500144231, 4.24792751344359,
	4.32192809488736, 4.39231742277876, 4.4594316186373,
	4.52356195605701, 4.58496250072116, 4.64385618977472,
	4.70043971814109, 4.75488750216347, 4.8073549220576,
	4.85798099512757, 4.90689059560852, 4.95419631038688,
	5.0,              5.04439411935845, 5.08746284125034,
	5.12928301694497, 5.16992500144231
    };
    long bits;

    if (base < 2 || 36 < base)
	rb_bug("invalid radix %d", base);

    if (FIXNUM_P(x)) {
	bits = (SIZEOF_LONG*CHAR_BIT - 1)/2 + 1;
    }
    else if (BIGZEROP(x)) {
	return 0;
    }
    else if (RBIGNUM_LEN(x) >= LONG_MAX/BITSPERDIG) {
	rb_raise(rb_eRangeError, "bignum too big to convert into `string'");
    }
    else {
	bits = BITSPERDIG*RBIGNUM_LEN(x);
    }

    /* @shyouhei note: vvvvvvvvvvvvv this cast is suspicious.  But I believe it is OK, because if that cast loses data, this x value is too big, and should have raised RangeError. */
    return (long)ceil(((double)bits)/log_2[base - 2]);
}

static long
big2str_orig(VALUE x, int base, char* ptr, long len, long hbase, int trim)
{
    long i = RBIGNUM_LEN(x), j = len;
    BDIGIT* ds = BDIGITS(x);

    while (i && j > 0) {
	long k = i;
	BDIGIT_DBL num = 0;

	while (k--) {               /* x / hbase */
	    num = BIGUP(num) + ds[k];
	    ds[k] = (BDIGIT)(num / hbase);
	    num %= hbase;
	}
	if (trim && ds[i-1] == 0) i--;
	k = SIZEOF_BDIGITS;
	while (k--) {
	    ptr[--j] = ruby_digitmap[num % base];
	    num /= base;
	    if (j <= 0) break;
	    if (trim && i == 0 && num == 0) break;
	}
    }
    if (trim) {
	while (j < len && ptr[j] == '0') j++;
	MEMMOVE(ptr, ptr + j, char, len - j);
	len -= j;
    }
    return len;
}

static long
big2str_karatsuba(VALUE x, int base, char* ptr,
		  long n1, long len, long hbase, int trim)
{
    long lh, ll, m1;
    VALUE b, q, r;

    if (BIGZEROP(x)) {
	if (trim) return 0;
	else {
	    memset(ptr, '0', len);
	    return len;
	}
    }

    if (n1 <= KARATSUBA_DIGITS) {
	return big2str_orig(x, base, ptr, len, hbase, trim);
    }

    b = power_cache_get_power(base, n1, &m1);
    bigdivmod(x, b, &q, &r);
    lh = big2str_karatsuba(q, base, ptr, (len - m1)/2,
			   len - m1, hbase, trim);
    rb_big_resize(q, 0);
    ll = big2str_karatsuba(r, base, ptr + lh, m1/2,
			   m1, hbase, !lh && trim);
    rb_big_resize(r, 0);

    return lh + ll;
}

VALUE
rb_big2str0(VALUE x, int base, int trim)
{
    int off;
    VALUE ss, xx;
    long n1, n2, len, hbase;
    char* ptr;

    if (FIXNUM_P(x)) {
	return rb_fix2str(x, base);
    }
    if (BIGZEROP(x)) {
	return rb_usascii_str_new2("0");
    }

    if (base < 2 || 36 < base)
	rb_raise(rb_eArgError, "invalid radix %d", base);

    n2 = big2str_find_n1(x, base);
    n1 = (n2 + 1) / 2;
    ss = rb_usascii_str_new(0, n2 + 1); /* plus one for sign */
    ptr = RSTRING_PTR(ss);
    ptr[0] = RBIGNUM_SIGN(x) ? '+' : '-';

    hbase = base*base;
#if SIZEOF_BDIGITS > 2
    hbase *= hbase;
#endif
    off = !(trim && RBIGNUM_SIGN(x)); /* erase plus sign if trim */
    xx = rb_big_clone(x);
    RBIGNUM_SET_SIGN(xx, 1);
    if (n1 <= KARATSUBA_DIGITS) {
	len = off + big2str_orig(xx, base, ptr + off, n2, hbase, trim);
    }
    else {
	len = off + big2str_karatsuba(xx, base, ptr + off, n1,
				      n2, hbase, trim);
    }
    rb_big_resize(xx, 0);

    ptr[len] = '\0';
    rb_str_resize(ss, len);

    return ss;
}

VALUE
rb_big2str(VALUE x, int base)
{
    return rb_big2str0(x, base, 1);
}

/*
 *  call-seq:
 *     big.to_s(base=10)   ->  string
 *
 *  Returns a string containing the representation of <i>big</i> radix
 *  <i>base</i> (2 through 36).
 *
 *     12345654321.to_s         #=> "12345654321"
 *     12345654321.to_s(2)      #=> "1011011111110110111011110000110001"
 *     12345654321.to_s(8)      #=> "133766736061"
 *     12345654321.to_s(16)     #=> "2dfdbbc31"
 *     78546939656932.to_s(36)  #=> "rubyrules"
 */

static VALUE
rb_big_to_s(int argc, VALUE *argv, VALUE x)
{
    int base;

    if (argc == 0) base = 10;
    else {
	VALUE b;

	rb_scan_args(argc, argv, "01", &b);
	base = NUM2INT(b);
    }
    return rb_big2str(x, base);
}

static VALUE
big2ulong(VALUE x, const char *type, int check)
{
    long len = RBIGNUM_LEN(x);
    BDIGIT_DBL num;
    BDIGIT *ds;

    if (len > DIGSPERLONG) {
	if (check)
	    rb_raise(rb_eRangeError, "bignum too big to convert into `%s'", type);
	len = DIGSPERLONG;
    }
    ds = BDIGITS(x);
    num = 0;
    while (len--) {
	num = BIGUP(num);
	num += ds[len];
    }
    return (VALUE)num;
}

VALUE
rb_big2ulong_pack(VALUE x)
{
    VALUE num = big2ulong(x, "unsigned long", FALSE);
    if (!RBIGNUM_SIGN(x)) {
	return (VALUE)(-(SIGNED_VALUE)num);
    }
    return num;
}

VALUE
rb_big2ulong(VALUE x)
{
    VALUE num = big2ulong(x, "unsigned long", TRUE);

    if (!RBIGNUM_SIGN(x)) {
	unsigned long v = (unsigned long)(-(long)num);

	if (v <= LONG_MAX)
	    rb_raise(rb_eRangeError, "bignum out of range of unsigned long");
	return (VALUE)v;
    }
    return num;
}

SIGNED_VALUE
rb_big2long(VALUE x)
{
    VALUE num = big2ulong(x, "long", TRUE);

    if ((long)num < 0 &&
	(RBIGNUM_SIGN(x) || (long)num != LONG_MIN)) {
	rb_raise(rb_eRangeError, "bignum too big to convert into `long'");
    }
    if (!RBIGNUM_SIGN(x)) return -(SIGNED_VALUE)num;
    return num;
}

#if HAVE_LONG_LONG

static unsigned LONG_LONG
big2ull(VALUE x, const char *type)
{
    long len = RBIGNUM_LEN(x);
    BDIGIT_DBL num;
    BDIGIT *ds;

    if (len > SIZEOF_LONG_LONG/SIZEOF_BDIGITS)
	rb_raise(rb_eRangeError, "bignum too big to convert into `%s'", type);
    ds = BDIGITS(x);
    num = 0;
    while (len--) {
	num = BIGUP(num);
	num += ds[len];
    }
    return num;
}

unsigned LONG_LONG
rb_big2ull(VALUE x)
{
    unsigned LONG_LONG num = big2ull(x, "unsigned long long");

    if (!RBIGNUM_SIGN(x)) {
	LONG_LONG v = -(LONG_LONG)num;

	/* FIXNUM_MIN-1 .. LLONG_MIN mapped into 0xbfffffffffffffff .. LONG_MAX+1 */
	if ((unsigned LONG_LONG)v <= LLONG_MAX)
	    rb_raise(rb_eRangeError, "bignum out of range of unsigned long long");
	return v;
    }
    return num;
}

LONG_LONG
rb_big2ll(VALUE x)
{
    unsigned LONG_LONG num = big2ull(x, "long long");

    if ((LONG_LONG)num < 0 && (RBIGNUM_SIGN(x)
			       || (LONG_LONG)num != LLONG_MIN)) {
	rb_raise(rb_eRangeError, "bignum too big to convert into `long long'");
    }
    if (!RBIGNUM_SIGN(x)) return -(LONG_LONG)num;
    return num;
}

#endif  /* HAVE_LONG_LONG */

static VALUE
dbl2big(double d)
{
    long i = 0;
    BDIGIT c;
    BDIGIT *digits;
    VALUE z;
    double u = (d < 0)?-d:d;

    if (isinf(d)) {
	rb_raise(rb_eFloatDomainError, d < 0 ? "-Infinity" : "Infinity");
    }
    if (isnan(d)) {
	rb_raise(rb_eFloatDomainError, "NaN");
    }

    while (!POSFIXABLE(u) || 0 != (long)u) {
	u /= (double)(BIGRAD);
	i++;
    }
    z = bignew(i, d>=0);
    digits = BDIGITS(z);
    while (i--) {
	u *= BIGRAD;
	c = (BDIGIT)u;
	u -= c;
	digits[i] = c;
    }

    return z;
}

VALUE
rb_dbl2big(double d)
{
    return bignorm(dbl2big(d));
}

static int
nlz(BDIGIT x)
{
    BDIGIT y;
    int n = BITSPERDIG;
#if BITSPERDIG > 64
    y = x >> 64; if (y) {n -= 64; x = y;}
#endif
#if BITSPERDIG > 32
    y = x >> 32; if (y) {n -= 32; x = y;}
#endif
#if BITSPERDIG > 16
    y = x >> 16; if (y) {n -= 16; x = y;}
#endif
    y = x >>  8; if (y) {n -=  8; x = y;}
    y = x >>  4; if (y) {n -=  4; x = y;}
    y = x >>  2; if (y) {n -=  2; x = y;}
    y = x >>  1; if (y) {return n - 2;}
    return n - x;
}

static double
big2dbl(VALUE x)
{
    double d = 0.0;
    long i = (bigtrunc(x), RBIGNUM_LEN(x)), lo = 0, bits;
    BDIGIT *ds = BDIGITS(x), dl;

    if (i) {
	bits = i * BITSPERDIG - nlz(ds[i-1]);
	if (bits > DBL_MANT_DIG+DBL_MAX_EXP) {
	    d = HUGE_VAL;
	}
	else {
	    if (bits > DBL_MANT_DIG+1)
		lo = (bits -= DBL_MANT_DIG+1) / BITSPERDIG;
	    else
		bits = 0;
	    while (--i > lo) {
		d = ds[i] + BIGRAD*d;
	    }
	    dl = ds[i];
	    if (bits && (dl & (1UL << (bits %= BITSPERDIG)))) {
		int carry = dl & ~(~(BDIGIT)0 << bits);
		if (!carry) {
		    while (i-- > 0) {
			if ((carry = ds[i]) != 0) break;
		    }
		}
		if (carry) {
		    dl &= (BDIGIT)~0 << bits;
		    dl += (BDIGIT)1 << bits;
		    if (!dl) d += 1;
		}
	    }
	    d = dl + BIGRAD*d;
	    if (lo) {
		if (lo > INT_MAX / BITSPERDIG)
		    d = HUGE_VAL;
		else if (lo < INT_MIN / BITSPERDIG)
		    d = 0.0;
		else
		    d = ldexp(d, (int)(lo * BITSPERDIG));
	    }
	}
    }
    if (!RBIGNUM_SIGN(x)) d = -d;
    return d;
}

double
rb_big2dbl(VALUE x)
{
    double d = big2dbl(x);

    if (isinf(d)) {
	rb_warning("Bignum out of Float range");
	if (d < 0.0)
	    d = -HUGE_VAL;
	else
	    d = HUGE_VAL;
    }
    return d;
}

/*
 *  call-seq:
 *     big.to_f -> float
 *
 *  Converts <i>big</i> to a <code>Float</code>. If <i>big</i> doesn't
 *  fit in a <code>Float</code>, the result is infinity.
 *
 */

static VALUE
rb_big_to_f(VALUE x)
{
    return DBL2NUM(rb_big2dbl(x));
}

VALUE
rb_integer_float_cmp(VALUE x, VALUE y)
{
    double yd = RFLOAT_VALUE(y);
    double yi, yf;
    VALUE rel;

    if (isnan(yd))
        return Qnil;
    if (isinf(yd)) {
        if (yd > 0.0) return INT2FIX(-1);
        else return INT2FIX(1);
    }
    yf = modf(yd, &yi);
    if (FIXNUM_P(x)) {
#if SIZEOF_LONG * CHAR_BIT < DBL_MANT_DIG /* assume FLT_RADIX == 2 */
        double xd = (double)FIX2LONG(x);
        if (xd < yd)
            return INT2FIX(-1);
        if (xd > yd)
            return INT2FIX(1);
        return INT2FIX(0);
#else
        long xl, yl;
        if (yi < FIXNUM_MIN)
            return INT2FIX(1);
        if (FIXNUM_MAX+1 <= yi)
            return INT2FIX(-1);
        xl = FIX2LONG(x);
        yl = (long)yi;
        if (xl < yl)
            return INT2FIX(-1);
        if (xl > yl)
            return INT2FIX(1);
        if (yf < 0.0)
            return INT2FIX(1);
        if (0.0 < yf)
            return INT2FIX(-1);
        return INT2FIX(0);
#endif
    }
    y = rb_dbl2big(yi);
    rel = rb_big_cmp(x, y);
    if (yf == 0.0 || rel != INT2FIX(0))
        return rel;
    if (yf < 0.0)
        return INT2FIX(1);
    return INT2FIX(-1);
}

VALUE
rb_integer_float_eq(VALUE x, VALUE y)
{
    double yd = RFLOAT_VALUE(y);
    double yi, yf;

    if (isnan(yd) || isinf(yd))
        return Qfalse;
    yf = modf(yd, &yi);
    if (yf != 0)
        return Qfalse;
    if (FIXNUM_P(x)) {
#if SIZEOF_LONG * CHAR_BIT < DBL_MANT_DIG /* assume FLT_RADIX == 2 */
        double xd = (double)FIX2LONG(x);
        if (xd != yd)
            return Qfalse;
        return Qtrue;
#else
        long xl, yl;
        if (yi < LONG_MIN || LONG_MAX < yi)
            return Qfalse;
        xl = FIX2LONG(x);
        yl = (long)yi;
        if (xl != yl)
            return Qfalse;
        return Qtrue;
#endif
    }
    y = rb_dbl2big(yi);
    return rb_big_eq(x, y);
}

/*
 *  call-seq:
 *     big <=> numeric   -> -1, 0, +1 or nil
 *
 *  Comparison---Returns -1, 0, or +1 depending on whether +big+ is
 *  less than, equal to, or greater than +numeric+. This is the
 *  basis for the tests in Comparable.
 *
 *  +nil+ is returned if the two values are incomparable.
 *
 */

VALUE
rb_big_cmp(VALUE x, VALUE y)
{
    long xlen = RBIGNUM_LEN(x);
    BDIGIT *xds, *yds;

    switch (TYPE(y)) {
      case T_FIXNUM:
	y = rb_int2big(FIX2LONG(y));
	break;

      case T_BIGNUM:
	break;

      case T_FLOAT:
        return rb_integer_float_cmp(x, y);

      default:
	return rb_num_coerce_cmp(x, y, rb_intern("<=>"));
    }

    if (RBIGNUM_SIGN(x) > RBIGNUM_SIGN(y)) return INT2FIX(1);
    if (RBIGNUM_SIGN(x) < RBIGNUM_SIGN(y)) return INT2FIX(-1);
    if (xlen < RBIGNUM_LEN(y))
	return (RBIGNUM_SIGN(x)) ? INT2FIX(-1) : INT2FIX(1);
    if (xlen > RBIGNUM_LEN(y))
	return (RBIGNUM_SIGN(x)) ? INT2FIX(1) : INT2FIX(-1);

    xds = BDIGITS(x);
    yds = BDIGITS(y);

    while (xlen-- && (xds[xlen]==yds[xlen]));
    if (-1 == xlen) return INT2FIX(0);
    return (xds[xlen] > yds[xlen]) ?
	(RBIGNUM_SIGN(x) ? INT2FIX(1) : INT2FIX(-1)) :
	    (RBIGNUM_SIGN(x) ? INT2FIX(-1) : INT2FIX(1));
}

enum big_op_t {
    big_op_gt,
    big_op_ge,
    big_op_lt,
    big_op_le
};

static VALUE
big_op(VALUE x, VALUE y, enum big_op_t op)
{
    VALUE rel;
    int n;

    switch (TYPE(y)) {
      case T_FIXNUM:
      case T_BIGNUM:
	rel = rb_big_cmp(x, y);
	break;

      case T_FLOAT:
        rel = rb_integer_float_cmp(x, y);
        break;

      default:
	{
	    ID id = 0;
	    switch (op) {
		case big_op_gt: id = '>'; break;
		case big_op_ge: id = rb_intern(">="); break;
		case big_op_lt: id = '<'; break;
		case big_op_le: id = rb_intern("<="); break;
	    }
	    return rb_num_coerce_relop(x, y, id);
	}
    }

    if (NIL_P(rel)) return Qfalse;
    n = FIX2INT(rel);

    switch (op) {
	case big_op_gt: return n >  0 ? Qtrue : Qfalse;
	case big_op_ge: return n >= 0 ? Qtrue : Qfalse;
	case big_op_lt: return n <  0 ? Qtrue : Qfalse;
	case big_op_le: return n <= 0 ? Qtrue : Qfalse;
    }
    return Qundef;
}

/*
 * call-seq:
 *   big > real  ->  true or false
 *
 * Returns <code>true</code> if the value of <code>big</code> is
 * greater than that of <code>real</code>.
 */

static VALUE
big_gt(VALUE x, VALUE y)
{
    return big_op(x, y, big_op_gt);
}

/*
 * call-seq:
 *   big >= real  ->  true or false
 *
 * Returns <code>true</code> if the value of <code>big</code> is
 * greater than or equal to that of <code>real</code>.
 */

static VALUE
big_ge(VALUE x, VALUE y)
{
    return big_op(x, y, big_op_ge);
}

/*
 * call-seq:
 *   big < real  ->  true or false
 *
 * Returns <code>true</code> if the value of <code>big</code> is
 * less than that of <code>real</code>.
 */

static VALUE
big_lt(VALUE x, VALUE y)
{
    return big_op(x, y, big_op_lt);
}

/*
 * call-seq:
 *   big <= real  ->  true or false
 *
 * Returns <code>true</code> if the value of <code>big</code> is
 * less than or equal to that of <code>real</code>.
 */

static VALUE
big_le(VALUE x, VALUE y)
{
    return big_op(x, y, big_op_le);
}

/*
 *  call-seq:
 *     big == obj  -> true or false
 *
 *  Returns <code>true</code> only if <i>obj</i> has the same value
 *  as <i>big</i>. Contrast this with <code>Bignum#eql?</code>, which
 *  requires <i>obj</i> to be a <code>Bignum</code>.
 *
 *     68719476736 == 68719476736.0   #=> true
 */

VALUE
rb_big_eq(VALUE x, VALUE y)
{
    switch (TYPE(y)) {
      case T_FIXNUM:
	y = rb_int2big(FIX2LONG(y));
	break;
      case T_BIGNUM:
	break;
      case T_FLOAT:
        return rb_integer_float_eq(x, y);
      default:
	return rb_equal(y, x);
    }
    if (RBIGNUM_SIGN(x) != RBIGNUM_SIGN(y)) return Qfalse;
    if (RBIGNUM_LEN(x) != RBIGNUM_LEN(y)) return Qfalse;
    if (MEMCMP(BDIGITS(x),BDIGITS(y),BDIGIT,RBIGNUM_LEN(y)) != 0) return Qfalse;
    return Qtrue;
}

/*
 *  call-seq:
 *     big.eql?(obj)   -> true or false
 *
 *  Returns <code>true</code> only if <i>obj</i> is a
 *  <code>Bignum</code> with the same value as <i>big</i>. Contrast this
 *  with <code>Bignum#==</code>, which performs type conversions.
 *
 *     68719476736.eql?(68719476736.0)   #=> false
 */

VALUE
rb_big_eql(VALUE x, VALUE y)
{
    if (!RB_TYPE_P(y, T_BIGNUM)) return Qfalse;
    if (RBIGNUM_SIGN(x) != RBIGNUM_SIGN(y)) return Qfalse;
    if (RBIGNUM_LEN(x) != RBIGNUM_LEN(y)) return Qfalse;
    if (MEMCMP(BDIGITS(x),BDIGITS(y),BDIGIT,RBIGNUM_LEN(y)) != 0) return Qfalse;
    return Qtrue;
}

/*
 * call-seq:
 *    -big   ->  integer
 *
 * Unary minus (returns an integer whose value is 0-big)
 */

VALUE
rb_big_uminus(VALUE x)
{
    VALUE z = rb_big_clone(x);

    RBIGNUM_SET_SIGN(z, !RBIGNUM_SIGN(x));

    return bignorm(z);
}

/*
 * call-seq:
 *     ~big  ->  integer
 *
 * Inverts the bits in big. As Bignums are conceptually infinite
 * length, the result acts as if it had an infinite number of one
 * bits to the left. In hex representations, this is displayed
 * as two periods to the left of the digits.
 *
 *   sprintf("%X", ~0x1122334455)    #=> "..FEEDDCCBBAA"
 */

static VALUE
rb_big_neg(VALUE x)
{
    VALUE z = rb_big_clone(x);
    BDIGIT *ds;
    long i;

    if (!RBIGNUM_SIGN(x)) get2comp(z);
    ds = BDIGITS(z);
    i = RBIGNUM_LEN(x);
    if (!i) return INT2FIX(~(SIGNED_VALUE)0);
    while (i--) {
	ds[i] = ~ds[i];
    }
    RBIGNUM_SET_SIGN(z, !RBIGNUM_SIGN(z));
    if (RBIGNUM_SIGN(x)) get2comp(z);

    return bignorm(z);
}

static void
bigsub_core(BDIGIT *xds, long xn, BDIGIT *yds, long yn, BDIGIT *zds, long zn)
{
    BDIGIT_DBL_SIGNED num;
    long i;

    for (i = 0, num = 0; i < yn; i++) {
	num += (BDIGIT_DBL_SIGNED)xds[i] - yds[i];
	zds[i] = BIGLO(num);
	num = BIGDN(num);
    }
    while (num && i < xn) {
	num += xds[i];
	zds[i++] = BIGLO(num);
	num = BIGDN(num);
    }
    while (i < xn) {
	zds[i] = xds[i];
	i++;
    }
    assert(i <= zn);
    while (i < zn) {
	zds[i++] = 0;
    }
}

static VALUE
bigsub(VALUE x, VALUE y)
{
    VALUE z = 0;
    long i = RBIGNUM_LEN(x);
    BDIGIT *xds, *yds;

    /* if x is smaller than y, swap */
    if (RBIGNUM_LEN(x) < RBIGNUM_LEN(y)) {
	z = x; x = y; y = z;	/* swap x y */
    }
    else if (RBIGNUM_LEN(x) == RBIGNUM_LEN(y)) {
	xds = BDIGITS(x);
	yds = BDIGITS(y);
	while (i > 0) {
	    i--;
	    if (xds[i] > yds[i]) {
		break;
	    }
	    if (xds[i] < yds[i]) {
		z = x; x = y; y = z;	/* swap x y */
		break;
	    }
	}
    }

    z = bignew(RBIGNUM_LEN(x), z==0);
    bigsub_core(BDIGITS(x), RBIGNUM_LEN(x),
		BDIGITS(y), RBIGNUM_LEN(y),
		BDIGITS(z), RBIGNUM_LEN(z));

    return z;
}

static VALUE bigadd_int(VALUE x, long y);

static VALUE
bigsub_int(VALUE x, long y0)
{
    VALUE z;
    BDIGIT *xds, *zds;
    long xn;
    BDIGIT_DBL_SIGNED num;
    long i, y;

    y = y0;
    xds = BDIGITS(x);
    xn = RBIGNUM_LEN(x);

    z = bignew(xn, RBIGNUM_SIGN(x));
    zds = BDIGITS(z);

#if SIZEOF_BDIGITS == SIZEOF_LONG
    num = (BDIGIT_DBL_SIGNED)xds[0] - y;
    if (xn == 1 && num < 0) {
	RBIGNUM_SET_SIGN(z, !RBIGNUM_SIGN(x));
	zds[0] = (BDIGIT)-num;
	RB_GC_GUARD(x);
	return bignorm(z);
    }
    zds[0] = BIGLO(num);
    num = BIGDN(num);
    i = 1;
#else
    num = 0;
    for (i=0; i<(int)(sizeof(y)/sizeof(BDIGIT)); i++) {
	num += (BDIGIT_DBL_SIGNED)xds[i] - BIGLO(y);
	zds[i] = BIGLO(num);
	num = BIGDN(num);
	y = BIGDN(y);
    }
#endif
    while (num && i < xn) {
	num += xds[i];
	zds[i++] = BIGLO(num);
	num = BIGDN(num);
    }
    while (i < xn) {
	zds[i] = xds[i];
	i++;
    }
    if (num < 0) {
	z = bigsub(x, rb_int2big(y0));
    }
    RB_GC_GUARD(x);
    return bignorm(z);
}

static VALUE
bigadd_int(VALUE x, long y)
{
    VALUE z;
    BDIGIT *xds, *zds;
    long xn, zn;
    BDIGIT_DBL num;
    long i;

    xds = BDIGITS(x);
    xn = RBIGNUM_LEN(x);

    if (xn < 2) {
	zn = 3;
    }
    else {
	zn = xn + 1;
    }
    z = bignew(zn, RBIGNUM_SIGN(x));
    zds = BDIGITS(z);

#if SIZEOF_BDIGITS == SIZEOF_LONG
    num = (BDIGIT_DBL)xds[0] + y;
    zds[0] = BIGLO(num);
    num = BIGDN(num);
    i = 1;
#else
    num = 0;
    for (i=0; i<(int)(sizeof(y)/sizeof(BDIGIT)); i++) {
	num += (BDIGIT_DBL)xds[i] + BIGLO(y);
	zds[i] = BIGLO(num);
	num = BIGDN(num);
	y = BIGDN(y);
    }
#endif
    while (num && i < xn) {
	num += xds[i];
	zds[i++] = BIGLO(num);
	num = BIGDN(num);
    }
    if (num) zds[i++] = (BDIGIT)num;
    else while (i < xn) {
	zds[i] = xds[i];
	i++;
    }
    assert(i <= zn);
    while (i < zn) {
	zds[i++] = 0;
    }
    RB_GC_GUARD(x);
    return bignorm(z);
}

static void
bigadd_core(BDIGIT *xds, long xn, BDIGIT *yds, long yn, BDIGIT *zds, long zn)
{
    BDIGIT_DBL num = 0;
    long i;

    if (xn > yn) {
	BDIGIT *tds;
	tds = xds; xds = yds; yds = tds;
	i = xn; xn = yn; yn = i;
    }

    i = 0;
    while (i < xn) {
	num += (BDIGIT_DBL)xds[i] + yds[i];
	zds[i++] = BIGLO(num);
	num = BIGDN(num);
    }
    while (num && i < yn) {
	num += yds[i];
	zds[i++] = BIGLO(num);
	num = BIGDN(num);
    }
    while (i < yn) {
	zds[i] = yds[i];
	i++;
    }
    if (num) zds[i++] = (BDIGIT)num;
    assert(i <= zn);
    while (i < zn) {
	zds[i++] = 0;
    }
}

static VALUE
bigadd(VALUE x, VALUE y, int sign)
{
    VALUE z;
    long len;

    sign = (sign == RBIGNUM_SIGN(y));
    if (RBIGNUM_SIGN(x) != sign) {
	if (sign) return bigsub(y, x);
	return bigsub(x, y);
    }

    if (RBIGNUM_LEN(x) > RBIGNUM_LEN(y)) {
	len = RBIGNUM_LEN(x) + 1;
    }
    else {
	len = RBIGNUM_LEN(y) + 1;
    }
    z = bignew(len, sign);

    bigadd_core(BDIGITS(x), RBIGNUM_LEN(x),
		BDIGITS(y), RBIGNUM_LEN(y),
		BDIGITS(z), RBIGNUM_LEN(z));

    return z;
}

/*
 *  call-seq:
 *     big + other  -> Numeric
 *
 *  Adds big and other, returning the result.
 */

VALUE
rb_big_plus(VALUE x, VALUE y)
{
    long n;

    switch (TYPE(y)) {
      case T_FIXNUM:
	n = FIX2LONG(y);
	if ((n > 0) != RBIGNUM_SIGN(x)) {
	    if (n < 0) {
		n = -n;
	    }
	    return bigsub_int(x, n);
	}
	if (n < 0) {
	    n = -n;
	}
	return bigadd_int(x, n);

      case T_BIGNUM:
	return bignorm(bigadd(x, y, 1));

      case T_FLOAT:
	return DBL2NUM(rb_big2dbl(x) + RFLOAT_VALUE(y));

      default:
	return rb_num_coerce_bin(x, y, '+');
    }
}

/*
 *  call-seq:
 *     big - other  -> Numeric
 *
 *  Subtracts other from big, returning the result.
 */

VALUE
rb_big_minus(VALUE x, VALUE y)
{
    long n;

    switch (TYPE(y)) {
      case T_FIXNUM:
	n = FIX2LONG(y);
	if ((n > 0) != RBIGNUM_SIGN(x)) {
	    if (n < 0) {
		n = -n;
	    }
	    return bigadd_int(x, n);
	}
	if (n < 0) {
	    n = -n;
	}
	return bigsub_int(x, n);

      case T_BIGNUM:
	return bignorm(bigadd(x, y, 0));

      case T_FLOAT:
	return DBL2NUM(rb_big2dbl(x) - RFLOAT_VALUE(y));

      default:
	return rb_num_coerce_bin(x, y, '-');
    }
}

static long
big_real_len(VALUE x)
{
    long i = RBIGNUM_LEN(x);
    BDIGIT *xds = BDIGITS(x);
    while (--i && !xds[i]);
    return i + 1;
}

static VALUE
bigmul1_single(VALUE x, VALUE y)
{
    BDIGIT_DBL n;
    VALUE z = bignew(2, RBIGNUM_SIGN(x)==RBIGNUM_SIGN(y));
    BDIGIT *xds, *yds, *zds;

    xds = BDIGITS(x);
    yds = BDIGITS(y);
    zds = BDIGITS(z);

    n = (BDIGIT_DBL)xds[0] * yds[0];
    zds[0] = BIGLO(n);
    zds[1] = (BDIGIT)BIGDN(n);

    return z;
}

static VALUE
bigmul1_normal(VALUE x, VALUE y)
{
    long xl = RBIGNUM_LEN(x), yl = RBIGNUM_LEN(y), i, j = xl + yl + 1;
    BDIGIT_DBL n = 0;
    VALUE z = bignew(j, RBIGNUM_SIGN(x)==RBIGNUM_SIGN(y));
    BDIGIT *xds, *yds, *zds;

    xds = BDIGITS(x);
    yds = BDIGITS(y);
    zds = BDIGITS(z);
    while (j--) zds[j] = 0;
    for (i = 0; i < xl; i++) {
	BDIGIT_DBL dd;
	dd = xds[i];
	if (dd == 0) continue;
	n = 0;
	for (j = 0; j < yl; j++) {
	    BDIGIT_DBL ee = n + (BDIGIT_DBL)dd * yds[j];
	    n = zds[i + j] + ee;
	    if (ee) zds[i + j] = BIGLO(n);
	    n = BIGDN(n);
	}
	if (n) {
	    zds[i + j] = (BDIGIT)n;
	}
    }
    rb_thread_check_ints();
    return z;
}

static VALUE bigmul0(VALUE x, VALUE y);

/* balancing multiplication by slicing larger argument */
static VALUE
bigmul1_balance(VALUE x, VALUE y)
{
    VALUE z, t1, t2;
    long i, xn, yn, r, n;
    BDIGIT *yds, *zds, *t1ds;

    xn = RBIGNUM_LEN(x);
    yn = RBIGNUM_LEN(y);
    assert(2 * xn <= yn || 3 * xn <= 2*(yn+2));

    z = bignew(xn + yn, RBIGNUM_SIGN(x)==RBIGNUM_SIGN(y));
    t1 = bignew(xn, 1);

    yds = BDIGITS(y);
    zds = BDIGITS(z);
    t1ds = BDIGITS(t1);

    for (i = 0; i < xn + yn; i++) zds[i] = 0;

    n = 0;
    while (yn > 0) {
	r = xn > yn ? yn : xn;
	MEMCPY(t1ds, yds + n, BDIGIT, r);
	RBIGNUM_SET_LEN(t1, r);
	t2 = bigmul0(x, t1);
	bigadd_core(zds + n, RBIGNUM_LEN(z) - n,
		    BDIGITS(t2), big_real_len(t2),
		    zds + n, RBIGNUM_LEN(z) - n);
	yn -= r;
	n += r;
    }

    return z;
}

/* split a bignum into high and low bignums */
static void
big_split(VALUE v, long n, volatile VALUE *ph, volatile VALUE *pl)
{
    long hn = 0, ln = RBIGNUM_LEN(v);
    VALUE h, l;
    BDIGIT *vds = BDIGITS(v);

    if (ln > n) {
	hn = ln - n;
	ln = n;
    }

    if (!hn) {
	h = rb_uint2big(0);
    }
    else {
	while (--hn && !vds[hn + ln]);
	h = bignew(hn += 2, 1);
	MEMCPY(BDIGITS(h), vds + ln, BDIGIT, hn - 1);
	BDIGITS(h)[hn - 1] = 0; /* margin for carry */
    }

    while (--ln && !vds[ln]);
    l = bignew(ln += 2, 1);
    MEMCPY(BDIGITS(l), vds, BDIGIT, ln - 1);
    BDIGITS(l)[ln - 1] = 0; /* margin for carry */

    *pl = l;
    *ph = h;
}

/* multiplication by karatsuba method */
static VALUE
bigmul1_karatsuba(VALUE x, VALUE y)
{
    long i, n, xn, yn, t1n, t2n;
    VALUE xh, xl, yh, yl, z, t1, t2, t3;
    BDIGIT *zds;

    xn = RBIGNUM_LEN(x);
    yn = RBIGNUM_LEN(y);
    n = yn / 2;
    big_split(x, n, &xh, &xl);
    if (x == y) {
	yh = xh; yl = xl;
    }
    else big_split(y, n, &yh, &yl);

    /* x = xh * b + xl
     * y = yh * b + yl
     *
     * Karatsuba method:
     *   x * y = z2 * b^2 + z1 * b + z0
     *   where
     *     z2 = xh * yh
     *     z0 = xl * yl
     *     z1 = (xh + xl) * (yh + yl) - z2 - z0
     *
     *  ref: http://en.wikipedia.org/wiki/Karatsuba_algorithm
     */

    /* allocate a result bignum */
    z = bignew(xn + yn, RBIGNUM_SIGN(x)==RBIGNUM_SIGN(y));
    zds = BDIGITS(z);

    /* t1 <- xh * yh */
    t1 = bigmul0(xh, yh);
    t1n = big_real_len(t1);

    /* copy t1 into high bytes of the result (z2) */
    MEMCPY(zds + 2 * n, BDIGITS(t1), BDIGIT, t1n);
    for (i = 2 * n + t1n; i < xn + yn; i++) zds[i] = 0;

    if (!BIGZEROP(xl) && !BIGZEROP(yl)) {
	/* t2 <- xl * yl */
	t2 = bigmul0(xl, yl);
	t2n = big_real_len(t2);

	/* copy t2 into low bytes of the result (z0) */
	MEMCPY(zds, BDIGITS(t2), BDIGIT, t2n);
	for (i = t2n; i < 2 * n; i++) zds[i] = 0;
    }
    else {
	t2 = Qundef;
	t2n = 0;

	/* copy 0 into low bytes of the result (z0) */
	for (i = 0; i < 2 * n; i++) zds[i] = 0;
    }

    /* xh <- xh + xl */
    if (RBIGNUM_LEN(xl) > RBIGNUM_LEN(xh)) {
	t3 = xl; xl = xh; xh = t3;
    }
    /* xh has a margin for carry */
    bigadd_core(BDIGITS(xh), RBIGNUM_LEN(xh),
		BDIGITS(xl), RBIGNUM_LEN(xl),
		BDIGITS(xh), RBIGNUM_LEN(xh));

    /* yh <- yh + yl */
    if (x != y) {
	if (RBIGNUM_LEN(yl) > RBIGNUM_LEN(yh)) {
	    t3 = yl; yl = yh; yh = t3;
	}
	/* yh has a margin for carry */
	bigadd_core(BDIGITS(yh), RBIGNUM_LEN(yh),
		    BDIGITS(yl), RBIGNUM_LEN(yl),
		    BDIGITS(yh), RBIGNUM_LEN(yh));
    }
    else yh = xh;

    /* t3 <- xh * yh */
    t3 = bigmul0(xh, yh);

    i = xn + yn - n;
    /* subtract t1 from t3 */
    bigsub_core(BDIGITS(t3), big_real_len(t3), BDIGITS(t1), t1n, BDIGITS(t3), big_real_len(t3));

    /* subtract t2 from t3; t3 is now the middle term of the product */
    if (t2 != Qundef) bigsub_core(BDIGITS(t3), big_real_len(t3), BDIGITS(t2), t2n, BDIGITS(t3), big_real_len(t3));

    /* add t3 to middle bytes of the result (z1) */
    bigadd_core(zds + n, i, BDIGITS(t3), big_real_len(t3), zds + n, i);

    return z;
}

static void
biglsh_bang(BDIGIT *xds, long xn, unsigned long shift)
{
    long const s1 = shift/BITSPERDIG;
    int const s2 = (int)(shift%BITSPERDIG);
    int const s3 = BITSPERDIG-s2;
    BDIGIT* zds;
    BDIGIT num;
    long i;
    if (s1 >= xn) {
	MEMZERO(xds, BDIGIT, xn);
	return;
    }
    zds = xds + xn - 1;
    xn -= s1 + 1;
    num = xds[xn]<<s2;
    do {
	*zds-- = num | xds[--xn]>>s3;
	num = xds[xn]<<s2;
    }
    while (xn > 0);
    *zds = num;
    for (i = s1; i > 0; --i)
	*zds-- = 0;
}

static void
bigrsh_bang(BDIGIT* xds, long xn, unsigned long shift)
{
    long s1 = shift/BITSPERDIG;
    int s2 = (int)(shift%BITSPERDIG);
    int s3 = BITSPERDIG - s2;
    int i;
    BDIGIT num;
    BDIGIT* zds;
    if (s1 >= xn) {
	MEMZERO(xds, BDIGIT, xn);
	return;
    }

    i = 0;
    zds = xds + s1;
    num = *zds++>>s2;
    do {
	xds[i++] = (BDIGIT)(*zds<<s3) | num;
	num = *zds++>>s2;
    }
    while (i < xn - s1 - 1);
    xds[i] = num;
    MEMZERO(xds + xn - s1, BDIGIT, s1);
}

static void
big_split3(VALUE v, long n, volatile VALUE* p0, volatile VALUE* p1, volatile VALUE* p2)
{
    VALUE v0, v12, v1, v2;

    big_split(v, n, &v12, &v0);
    big_split(v12, n, &v2, &v1);

    *p0 = bigtrunc(v0);
    *p1 = bigtrunc(v1);
    *p2 = bigtrunc(v2);
}

static VALUE big_lshift(VALUE, unsigned long);
static VALUE big_rshift(VALUE, unsigned long);
static VALUE bigdivrem(VALUE, VALUE, volatile VALUE*, volatile VALUE*);

static VALUE
bigmul1_toom3(VALUE x, VALUE y)
{
    long n, xn, yn, zn;
    VALUE x0, x1, x2, y0, y1, y2;
    VALUE u0, u1, u2, u3, u4, v1, v2, v3;
    VALUE z0, z1, z2, z3, z4, z, t;
    BDIGIT* zds;

    xn = RBIGNUM_LEN(x);
    yn = RBIGNUM_LEN(y);
    assert(xn <= yn);  /* assume y >= x */

    n = (yn + 2) / 3;
    big_split3(x, n, &x0, &x1, &x2);
    if (x == y) {
	y0 = x0; y1 = x1; y2 = x2;
    }
    else big_split3(y, n, &y0, &y1, &y2);

    /*
     * ref. http://en.wikipedia.org/wiki/Toom%E2%80%93Cook_multiplication
     *
     * x(b) = x0 * b^0 + x1 * b^1 + x2 * b^2
     * y(b) = y0 * b^0 + y1 * b^1 + y2 * b^2
     *
     * z(b) = x(b) * y(b)
     * z(b) = z0 * b^0 + z1 * b^1 + z2 * b^2 + z3 * b^3 + z4 * b^4
     * where:
     *   z0 = x0 * y0
     *   z1 = x0 * y1 + x1 * y0
     *   z2 = x0 * y2 + x1 * y1 + x2 * y0
     *   z3 = x1 * y2 + x2 * y1
     *   z4 = x2 * y2
     *
     * Toom3 method (a.k.a. Toom-Cook method):
     * (Step1) calculating 5 points z(b0), z(b1), z(b2), z(b3), z(b4),
     * where:
     *   b0 = 0, b1 = 1, b2 = -1, b3 = -2, b4 = inf,
     *   z(0)   = x(0)   * y(0)   = x0 * y0
     *   z(1)   = x(1)   * y(1)   = (x0 + x1 + x2) * (y0 + y1 + y2)
     *   z(-1)  = x(-1)  * y(-1)  = (x0 - x1 + x2) * (y0 - y1 + y2)
     *   z(-2)  = x(-2)  * y(-2)  = (x0 - 2 * (x1 - 2 * x2)) * (y0 - 2 * (y1 - 2 * y2))
     *   z(inf) = x(inf) * y(inf) = x2 * y2
     *
     * (Step2) interpolating z0, z1, z2, z3, z4, and z5.
     *
     * (Step3) Substituting base value into b of the polynomial z(b),
     */

    /*
     * [Step1] calculating 5 points z(b0), z(b1), z(b2), z(b3), z(b4)
     */

    /* u1 <- x0 + x2 */
    u1 = bigtrunc(bigadd(x0, x2, 1));

    /* x(-1) : u2 <- u1 - x1 = x0 - x1 + x2 */
    u2 = bigtrunc(bigsub(u1, x1));

    /* x(1) : u1 <- u1 + x1 = x0 + x1 + x2 */
    u1 = bigtrunc(bigadd(u1, x1, 1));

    /* x(-2) : u3 <- 2 * (u2 + x2) - x0 = x0 - 2 * (x1 - 2 * x2) */
    u3 = bigadd(u2, x2, 1);
    if (BDIGITS(u3)[RBIGNUM_LEN(u3)-1] & BIGRAD_HALF) {
	rb_big_resize(u3, RBIGNUM_LEN(u3) + 1);
	BDIGITS(u3)[RBIGNUM_LEN(u3)-1] = 0;
    }
    biglsh_bang(BDIGITS(u3), RBIGNUM_LEN(u3), 1);
    u3 = bigtrunc(bigadd(bigtrunc(u3), x0, 0));

    if (x == y) {
	v1 = u1; v2 = u2; v3 = u3;
    }
    else {
	/* v1 <- y0 + y2 */
	v1 = bigtrunc(bigadd(y0, y2, 1));

	/* y(-1) : v2 <- v1 - y1 = y0 - y1 + y2 */
	v2 = bigtrunc(bigsub(v1, y1));

	/* y(1) : v1 <- v1 + y1 = y0 + y1 + y2 */
	v1 = bigtrunc(bigadd(v1, y1, 1));

	/* y(-2) : v3 <- 2 * (v2 + y2) - y0 = y0 - 2 * (y1 - 2 * y2) */
	v3 = bigadd(v2, y2, 1);
	if (BDIGITS(v3)[RBIGNUM_LEN(v3)-1] & BIGRAD_HALF) {
	    rb_big_resize(v3, RBIGNUM_LEN(v3) + 1);
	    BDIGITS(v3)[RBIGNUM_LEN(v3)-1] = 0;
	}
	biglsh_bang(BDIGITS(v3), RBIGNUM_LEN(v3), 1);
	v3 = bigtrunc(bigadd(bigtrunc(v3), y0, 0));
    }

    /* z(0) : u0 <- x0 * y0 */
    u0 = bigtrunc(bigmul0(x0, y0));

    /* z(1) : u1 <- u1 * v1 */
    u1 = bigtrunc(bigmul0(u1, v1));

    /* z(-1) : u2 <- u2 * v2 */
    u2 = bigtrunc(bigmul0(u2, v2));

    /* z(-2) : u3 <- u3 * v3 */
    u3 = bigtrunc(bigmul0(u3, v3));

    /* z(inf) : u4 <- x2 * y2 */
    u4 = bigtrunc(bigmul0(x2, y2));

    /* for GC */
    v1 = v2 = v3 = Qnil;

    /*
     * [Step2] interpolating z0, z1, z2, z3, z4, and z5.
     */

    /* z0 <- z(0) == u0 */
    z0 = u0;

    /* z4 <- z(inf) == u4 */
    z4 = u4;

    /* z3 <- (z(-2) - z(1)) / 3 == (u3 - u1) / 3 */
    z3 = bigadd(u3, u1, 0);
    bigdivrem(z3, big_three, &z3, NULL); /* TODO: optimize */
    bigtrunc(z3);

    /* z1 <- (z(1) - z(-1)) / 2 == (u1 - u2) / 2 */
    z1 = bigtrunc(bigadd(u1, u2, 0));
    bigrsh_bang(BDIGITS(z1), RBIGNUM_LEN(z1), 1);

    /* z2 <- z(-1) - z(0) == u2 - u0 */
    z2 = bigtrunc(bigadd(u2, u0, 0));

    /* z3 <- (z2 - z3) / 2 + 2 * z(inf) == (z2 - z3) / 2 + 2 * u4 */
    z3 = bigtrunc(bigadd(z2, z3, 0));
    bigrsh_bang(BDIGITS(z3), RBIGNUM_LEN(z3), 1);
    t = big_lshift(u4, 1); /* TODO: combining with next addition */
    z3 = bigtrunc(bigadd(z3, t, 1));

    /* z2 <- z2 + z1 - z(inf) == z2 + z1 - u4 */
    z2 = bigtrunc(bigadd(z2, z1, 1));
    z2 = bigtrunc(bigadd(z2, u4, 0));

    /* z1 <- z1 - z3 */
    z1 = bigtrunc(bigadd(z1, z3, 0));

    /*
     * [Step3] Substituting base value into b of the polynomial z(b),
     */

    zn = 6*n + 1;
    z = bignew(zn, RBIGNUM_SIGN(x)==RBIGNUM_SIGN(y));
    zds = BDIGITS(z);
    MEMCPY(zds, BDIGITS(z0), BDIGIT, RBIGNUM_LEN(z0));
    MEMZERO(zds + RBIGNUM_LEN(z0), BDIGIT, zn - RBIGNUM_LEN(z0));
    bigadd_core(zds +   n, zn -   n, BDIGITS(z1), big_real_len(z1), zds +   n, zn -   n);
    bigadd_core(zds + 2*n, zn - 2*n, BDIGITS(z2), big_real_len(z2), zds + 2*n, zn - 2*n);
    bigadd_core(zds + 3*n, zn - 3*n, BDIGITS(z3), big_real_len(z3), zds + 3*n, zn - 3*n);
    bigadd_core(zds + 4*n, zn - 4*n, BDIGITS(z4), big_real_len(z4), zds + 4*n, zn - 4*n);
    z = bignorm(z);

    return bignorm(z);
}

/* efficient squaring (2 times faster than normal multiplication)
 * ref: Handbook of Applied Cryptography, Algorithm 14.16
 *      http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf
 */
static VALUE
bigsqr_fast(VALUE x)
{
    long len = RBIGNUM_LEN(x), i, j;
    VALUE z = bignew(2 * len + 1, 1);
    BDIGIT *xds = BDIGITS(x), *zds = BDIGITS(z);
    BDIGIT_DBL c, v, w;

    for (i = 2 * len + 1; i--; ) zds[i] = 0;
    for (i = 0; i < len; i++) {
	v = (BDIGIT_DBL)xds[i];
	if (!v) continue;
	c = (BDIGIT_DBL)zds[i + i] + v * v;
	zds[i + i] = BIGLO(c);
	c = BIGDN(c);
	v *= 2;
	for (j = i + 1; j < len; j++) {
	    w = (BDIGIT_DBL)xds[j];
	    c += (BDIGIT_DBL)zds[i + j] + BIGLO(v) * w;
	    zds[i + j] = BIGLO(c);
	    c = BIGDN(c);
	    if (BIGDN(v)) c += w;
	}
	if (c) {
	    c += (BDIGIT_DBL)zds[i + len];
	    zds[i + len] = BIGLO(c);
	    c = BIGDN(c);
	}
	if (c) zds[i + len + 1] += (BDIGIT)c;
    }
    return z;
}

#define KARATSUBA_MUL_DIGITS 70
#define TOOM3_MUL_DIGITS 150


/* determine whether a bignum is sparse or not by random sampling */
static inline VALUE
big_sparse_p(VALUE x)
{
    long c = 0, n = RBIGNUM_LEN(x);

    if (          BDIGITS(x)[rb_genrand_ulong_limited(n / 2) + n / 4]) c++;
    if (c <= 1 && BDIGITS(x)[rb_genrand_ulong_limited(n / 2) + n / 4]) c++;
    if (c <= 1 && BDIGITS(x)[rb_genrand_ulong_limited(n / 2) + n / 4]) c++;

    return (c <= 1) ? Qtrue : Qfalse;
}

static VALUE
bigmul0(VALUE x, VALUE y)
{
    long xn, yn;

    xn = RBIGNUM_LEN(x);
    yn = RBIGNUM_LEN(y);

    /* make sure that y is longer than x */
    if (xn > yn) {
	VALUE t;
	long tn;
	t = x; x = y; y = t;
	tn = xn; xn = yn; yn = tn;
    }
    assert(xn <= yn);

    /* normal multiplication when x is small */
    if (xn < KARATSUBA_MUL_DIGITS) {
      normal:
	if (x == y) return bigsqr_fast(x);
	if (xn == 1 && yn == 1) return bigmul1_single(x, y);
	return bigmul1_normal(x, y);
    }

    /* normal multiplication when x or y is a sparse bignum */
    if (big_sparse_p(x)) goto normal;
    if (big_sparse_p(y)) return bigmul1_normal(y, x);

    /* balance multiplication by slicing y when x is much smaller than y */
    if (2 * xn <= yn) return bigmul1_balance(x, y);

    if (xn < TOOM3_MUL_DIGITS) {
	/* multiplication by karatsuba method */
	return bigmul1_karatsuba(x, y);
    }
    else if (3*xn <= 2*(yn + 2))
	return bigmul1_balance(x, y);
    return bigmul1_toom3(x, y);
}

/*
 *  call-seq:
 *     big * other  -> Numeric
 *
 *  Multiplies big and other, returning the result.
 */

VALUE
rb_big_mul(VALUE x, VALUE y)
{
    switch (TYPE(y)) {
      case T_FIXNUM:
	y = rb_int2big(FIX2LONG(y));
	break;

      case T_BIGNUM:
	break;

      case T_FLOAT:
	return DBL2NUM(rb_big2dbl(x) * RFLOAT_VALUE(y));

      default:
	return rb_num_coerce_bin(x, y, '*');
    }

    return bignorm(bigmul0(x, y));
}

struct big_div_struct {
    long nx, ny, j, nyzero;
    BDIGIT *yds, *zds;
    volatile VALUE stop;
};

static void *
bigdivrem1(void *ptr)
{
    struct big_div_struct *bds = (struct big_div_struct*)ptr;
    long ny = bds->ny;
    long i, j;
    BDIGIT *yds = bds->yds, *zds = bds->zds;
    BDIGIT_DBL t2;
    BDIGIT_DBL_SIGNED num;
    BDIGIT q;

    j = bds->j;
    do {
	if (bds->stop) {
	    bds->j = j;
	    return 0;
        }
	if (zds[j] ==  yds[ny-1]) q = (BDIGIT)BIGRAD-1;
	else q = (BDIGIT)((BIGUP(zds[j]) + zds[j-1])/yds[ny-1]);
	if (q) {
           i = bds->nyzero; num = 0; t2 = 0;
	    do {			/* multiply and subtract */
		BDIGIT_DBL ee;
		t2 += (BDIGIT_DBL)yds[i] * q;
		ee = num - BIGLO(t2);
		num = (BDIGIT_DBL)zds[j - ny + i] + ee;
		if (ee) zds[j - ny + i] = BIGLO(num);
		num = BIGDN(num);
		t2 = BIGDN(t2);
	    } while (++i < ny);
	    num += zds[j - ny + i] - t2;/* borrow from high digit; don't update */
	    while (num) {		/* "add back" required */
		i = 0; num = 0; q--;
		do {
		    BDIGIT_DBL ee = num + yds[i];
		    num = (BDIGIT_DBL)zds[j - ny + i] + ee;
		    if (ee) zds[j - ny + i] = BIGLO(num);
		    num = BIGDN(num);
		} while (++i < ny);
		num--;
	    }
	}
	zds[j] = q;
    } while (--j >= ny);
    return 0;
}

static void
rb_big_stop(void *ptr)
{
    struct big_div_struct *bds = ptr;
    bds->stop = Qtrue;
}

static VALUE
bigdivrem(VALUE x, VALUE y, volatile VALUE *divp, volatile VALUE *modp)
{
    struct big_div_struct bds;
    long nx = RBIGNUM_LEN(x), ny = RBIGNUM_LEN(y);
    long i, j;
    VALUE z, yy, zz;
    BDIGIT *xds, *yds, *zds, *tds;
    BDIGIT_DBL t2;
    BDIGIT dd, q;

    if (BIGZEROP(y)) rb_num_zerodiv();
    xds = BDIGITS(x);
    yds = BDIGITS(y);
    if (nx < ny || (nx == ny && xds[nx - 1] < yds[ny - 1])) {
	if (divp) *divp = rb_int2big(0);
	if (modp) *modp = x;
	return Qnil;
    }
    if (ny == 1) {
	dd = yds[0];
	z = rb_big_clone(x);
	zds = BDIGITS(z);
	t2 = 0; i = nx;
	while (i--) {
	    t2 = BIGUP(t2) + zds[i];
	    zds[i] = (BDIGIT)(t2 / dd);
	    t2 %= dd;
	}
	RBIGNUM_SET_SIGN(z, RBIGNUM_SIGN(x)==RBIGNUM_SIGN(y));
	if (modp) {
	    *modp = rb_uint2big((VALUE)t2);
	    RBIGNUM_SET_SIGN(*modp, RBIGNUM_SIGN(x));
	}
	if (divp) *divp = z;
	return Qnil;
    }

    z = bignew(nx==ny?nx+2:nx+1, RBIGNUM_SIGN(x)==RBIGNUM_SIGN(y));
    zds = BDIGITS(z);
    if (nx==ny) zds[nx+1] = 0;
    while (!yds[ny-1]) ny--;

    dd = 0;
    q = yds[ny-1];
    while ((q & (BDIGIT)(1UL<<(BITSPERDIG-1))) == 0) {
	q <<= 1UL;
	dd++;
    }
    if (dd) {
	yy = rb_big_clone(y);
	tds = BDIGITS(yy);
	j = 0;
	t2 = 0;
	while (j<ny) {
	    t2 += (BDIGIT_DBL)yds[j]<<dd;
	    tds[j++] = BIGLO(t2);
	    t2 = BIGDN(t2);
	}
	yds = tds;
	RB_GC_GUARD(y) = yy;
	j = 0;
	t2 = 0;
	while (j<nx) {
	    t2 += (BDIGIT_DBL)xds[j]<<dd;
	    zds[j++] = BIGLO(t2);
	    t2 = BIGDN(t2);
	}
	zds[j] = (BDIGIT)t2;
    }
    else {
	zds[nx] = 0;
	j = nx;
	while (j--) zds[j] = xds[j];
    }

    bds.nx = nx;
    bds.ny = ny;
    bds.zds = zds;
    bds.yds = yds;
    bds.stop = Qfalse;
    bds.j = nx==ny?nx+1:nx;
    for (bds.nyzero = 0; !yds[bds.nyzero]; bds.nyzero++);
    if (nx > 10000 || ny > 10000) {
      retry:
	bds.stop = Qfalse;
	rb_thread_call_without_gvl(bigdivrem1, &bds, rb_big_stop, &bds);

	if (bds.stop == Qtrue) {
	    /* execute trap handler, but exception was not raised. */
	    goto retry;
	}
    }
    else {
	bigdivrem1(&bds);
    }

    if (divp) {			/* move quotient down in z */
	*divp = zz = rb_big_clone(z);
	zds = BDIGITS(zz);
	j = (nx==ny ? nx+2 : nx+1) - ny;
	for (i = 0;i < j;i++) zds[i] = zds[i+ny];
	if (!zds[i-1]) i--;
	RBIGNUM_SET_LEN(zz, i);
    }
    if (modp) {			/* normalize remainder */
	*modp = zz = rb_big_clone(z);
	zds = BDIGITS(zz);
	while (ny > 1 && !zds[ny-1]) --ny;
	if (dd) {
	    t2 = 0; i = ny;
	    while (i--) {
		t2 = (t2 | zds[i]) >> dd;
		q = zds[i];
		zds[i] = BIGLO(t2);
		t2 = BIGUP(q);
	    }
	}
	if (!zds[ny-1]) ny--;
	RBIGNUM_SET_LEN(zz, ny);
	RBIGNUM_SET_SIGN(zz, RBIGNUM_SIGN(x));
    }
    return z;
}

static void
bigdivmod(VALUE x, VALUE y, volatile VALUE *divp, volatile VALUE *modp)
{
    VALUE mod;

    bigdivrem(x, y, divp, &mod);
    if (RBIGNUM_SIGN(x) != RBIGNUM_SIGN(y) && !BIGZEROP(mod)) {
	if (divp) *divp = bigadd(*divp, rb_int2big(1), 0);
	if (modp) *modp = bigadd(mod, y, 1);
    }
    else if (modp) {
	*modp = mod;
    }
}


static VALUE
rb_big_divide(VALUE x, VALUE y, ID op)
{
    VALUE z;

    switch (TYPE(y)) {
      case T_FIXNUM:
	y = rb_int2big(FIX2LONG(y));
	break;

      case T_BIGNUM:
	break;

      case T_FLOAT:
	{
	    if (op == '/') {
		return DBL2NUM(rb_big2dbl(x) / RFLOAT_VALUE(y));
	    }
	    else {
		double dy = RFLOAT_VALUE(y);
		if (dy == 0.0) rb_num_zerodiv();
		return rb_dbl2big(rb_big2dbl(x) / dy);
	    }
	}

      default:
	return rb_num_coerce_bin(x, y, op);
    }
    bigdivmod(x, y, &z, 0);

    return bignorm(z);
}

/*
 *  call-seq:
 *     big / other     -> Numeric
 *
 * Performs division: the class of the resulting object depends on
 * the class of <code>numeric</code> and on the magnitude of the
 * result.
 */

VALUE
rb_big_div(VALUE x, VALUE y)
{
    return rb_big_divide(x, y, '/');
}

/*
 *  call-seq:
 *     big.div(other)  -> integer
 *
 * Performs integer division: returns integer value.
 */

VALUE
rb_big_idiv(VALUE x, VALUE y)
{
    return rb_big_divide(x, y, rb_intern("div"));
}

/*
 *  call-seq:
 *     big % other         -> Numeric
 *     big.modulo(other)   -> Numeric
 *
 *  Returns big modulo other. See Numeric.divmod for more
 *  information.
 */

VALUE
rb_big_modulo(VALUE x, VALUE y)
{
    VALUE z;

    switch (TYPE(y)) {
      case T_FIXNUM:
	y = rb_int2big(FIX2LONG(y));
	break;

      case T_BIGNUM:
	break;

      default:
	return rb_num_coerce_bin(x, y, '%');
    }
    bigdivmod(x, y, 0, &z);

    return bignorm(z);
}

/*
 *  call-seq:
 *     big.remainder(numeric)    -> number
 *
 *  Returns the remainder after dividing <i>big</i> by <i>numeric</i>.
 *
 *     -1234567890987654321.remainder(13731)      #=> -6966
 *     -1234567890987654321.remainder(13731.24)   #=> -9906.22531493148
 */
static VALUE
rb_big_remainder(VALUE x, VALUE y)
{
    VALUE z;

    switch (TYPE(y)) {
      case T_FIXNUM:
	y = rb_int2big(FIX2LONG(y));
	break;

      case T_BIGNUM:
	break;

      default:
	return rb_num_coerce_bin(x, y, rb_intern("remainder"));
    }
    bigdivrem(x, y, 0, &z);

    return bignorm(z);
}

/*
 *  call-seq:
 *     big.divmod(numeric)   -> array
 *
 *  See <code>Numeric#divmod</code>.
 *
 */
VALUE
rb_big_divmod(VALUE x, VALUE y)
{
    VALUE div, mod;

    switch (TYPE(y)) {
      case T_FIXNUM:
	y = rb_int2big(FIX2LONG(y));
	break;

      case T_BIGNUM:
	break;

      default:
	return rb_num_coerce_bin(x, y, rb_intern("divmod"));
    }
    bigdivmod(x, y, &div, &mod);

    return rb_assoc_new(bignorm(div), bignorm(mod));
}

static int
bdigbitsize(BDIGIT x)
{
    int size = 1;
    int nb = BITSPERDIG / 2;
    BDIGIT bits = (~0 << nb);

    if (!x) return 0;
    while (x > 1) {
	if (x & bits) {
	    size += nb;
	    x >>= nb;
	}
	x &= ~bits;
	nb /= 2;
	bits >>= nb;
    }

    return size;
}

static VALUE big_lshift(VALUE, unsigned long);
static VALUE big_rshift(VALUE, unsigned long);

static VALUE
big_shift(VALUE x, long n)
{
    if (n < 0)
	return big_lshift(x, (unsigned long)-n);
    else if (n > 0)
	return big_rshift(x, (unsigned long)n);
    return x;
}

static VALUE
big_fdiv(VALUE x, VALUE y)
{
#define DBL_BIGDIG ((DBL_MANT_DIG + BITSPERDIG) / BITSPERDIG)
    VALUE z;
    long l, ex, ey;
    int i;

    bigtrunc(x);
    l = RBIGNUM_LEN(x) - 1;
    ex = l * BITSPERDIG;
    ex += bdigbitsize(BDIGITS(x)[l]);
    ex -= 2 * DBL_BIGDIG * BITSPERDIG;
    if (ex) x = big_shift(x, ex);

    switch (TYPE(y)) {
      case T_FIXNUM:
	y = rb_int2big(FIX2LONG(y));
      case T_BIGNUM:
	bigtrunc(y);
	l = RBIGNUM_LEN(y) - 1;
	ey = l * BITSPERDIG;
	ey += bdigbitsize(BDIGITS(y)[l]);
	ey -= DBL_BIGDIG * BITSPERDIG;
	if (ey) y = big_shift(y, ey);
	break;
      case T_FLOAT:
	y = dbl2big(ldexp(frexp(RFLOAT_VALUE(y), &i), DBL_MANT_DIG));
	ey = i - DBL_MANT_DIG;
	break;
      default:
	rb_bug("big_fdiv");
    }
    bigdivrem(x, y, &z, 0);
    l = ex - ey;
#if SIZEOF_LONG > SIZEOF_INT
    {
	/* Visual C++ can't be here */
	if (l > INT_MAX) return DBL2NUM(INFINITY);
	if (l < INT_MIN) return DBL2NUM(0.0);
    }
#endif
    return DBL2NUM(ldexp(big2dbl(z), (int)l));
}

/*
 *  call-seq:
  *     big.fdiv(numeric) -> float
 *
 *  Returns the floating point result of dividing <i>big</i> by
 *  <i>numeric</i>.
 *
 *     -1234567890987654321.fdiv(13731)      #=> -89910996357705.5
 *     -1234567890987654321.fdiv(13731.24)   #=> -89909424858035.7
 *
 */


VALUE
rb_big_fdiv(VALUE x, VALUE y)
{
    double dx, dy;

    dx = big2dbl(x);
    switch (TYPE(y)) {
      case T_FIXNUM:
	dy = (double)FIX2LONG(y);
	if (isinf(dx))
	    return big_fdiv(x, y);
	break;

      case T_BIGNUM:
	dy = rb_big2dbl(y);
	if (isinf(dx) || isinf(dy))
	    return big_fdiv(x, y);
	break;

      case T_FLOAT:
	dy = RFLOAT_VALUE(y);
	if (isnan(dy))
	    return y;
	if (isinf(dx))
	    return big_fdiv(x, y);
	break;

      default:
	return rb_num_coerce_bin(x, y, rb_intern("fdiv"));
    }
    return DBL2NUM(dx / dy);
}

static VALUE
bigsqr(VALUE x)
{
    return bigtrunc(bigmul0(x, x));
}

/*
 *  call-seq:
 *     big ** exponent   -> numeric
 *
 *  Raises _big_ to the _exponent_ power (which may be an integer, float,
 *  or anything that will coerce to a number). The result may be
 *  a Fixnum, Bignum, or Float
 *
 *    123456789 ** 2      #=> 15241578750190521
 *    123456789 ** 1.2    #=> 5126464716.09932
 *    123456789 ** -2     #=> 6.5610001194102e-17
 */

VALUE
rb_big_pow(VALUE x, VALUE y)
{
    double d;
    SIGNED_VALUE yy;

    if (y == INT2FIX(0)) return INT2FIX(1);
    switch (TYPE(y)) {
      case T_FLOAT:
	d = RFLOAT_VALUE(y);
	if ((!RBIGNUM_SIGN(x) && !BIGZEROP(x)) && d != round(d))
	    return rb_funcall(rb_complex_raw1(x), rb_intern("**"), 1, y);
	break;

      case T_BIGNUM:
	rb_warn("in a**b, b may be too big");
	d = rb_big2dbl(y);
	break;

      case T_FIXNUM:
	yy = FIX2LONG(y);

	if (yy < 0)
	    return rb_funcall(rb_rational_raw1(x), rb_intern("**"), 1, y);
	else {
	    VALUE z = 0;
	    SIGNED_VALUE mask;
	    const long xlen = RBIGNUM_LEN(x) - 1;
	    const long xbits = ffs(RBIGNUM_DIGITS(x)[xlen]) + SIZEOF_BDIGITS*BITSPERDIG*xlen;
	    const long BIGLEN_LIMIT = BITSPERDIG*1024*1024;

	    if ((xbits > BIGLEN_LIMIT) || (xbits * yy > BIGLEN_LIMIT)) {
		rb_warn("in a**b, b may be too big");
		d = (double)yy;
		break;
	    }
	    for (mask = FIXNUM_MAX + 1; mask; mask >>= 1) {
		if (z) z = bigsqr(z);
		if (yy & mask) {
		    z = z ? bigtrunc(bigmul0(z, x)) : x;
		}
	    }
	    return bignorm(z);
	}
	/* NOTREACHED */
	break;

      default:
	return rb_num_coerce_bin(x, y, rb_intern("**"));
    }
    return DBL2NUM(pow(rb_big2dbl(x), d));
}

static VALUE
bigand_int(VALUE x, long y)
{
    VALUE z;
    BDIGIT *xds, *zds;
    long xn, zn;
    long i;
    char sign;

    if (y == 0) return INT2FIX(0);
    sign = (y > 0);
    xds = BDIGITS(x);
    zn = xn = RBIGNUM_LEN(x);
#if SIZEOF_BDIGITS == SIZEOF_LONG
    if (sign) {
	y &= xds[0];
	return LONG2NUM(y);
    }
#endif

    z = bignew(zn, RBIGNUM_SIGN(x) || sign);
    zds = BDIGITS(z);

#if SIZEOF_BDIGITS == SIZEOF_LONG
    i = 1;
    zds[0] = xds[0] & y;
#else
    {
	BDIGIT_DBL num = y;

	for (i=0; i<(int)(sizeof(y)/sizeof(BDIGIT)); i++) {
	    zds[i] = xds[i] & BIGLO(num);
	    num = BIGDN(num);
	}
    }
#endif
    while (i < xn) {
	zds[i] = sign?0:xds[i];
	i++;
    }
    if (!RBIGNUM_SIGN(z)) get2comp(z);
    return bignorm(z);
}

/*
 * call-seq:
 *     big & numeric   ->  integer
 *
 * Performs bitwise +and+ between _big_ and _numeric_.
 */

VALUE
rb_big_and(VALUE xx, VALUE yy)
{
    volatile VALUE x, y, z;
    BDIGIT *ds1, *ds2, *zds;
    long i, l1, l2;
    char sign;

    if (!FIXNUM_P(yy) && !RB_TYPE_P(yy, T_BIGNUM)) {
	return rb_num_coerce_bit(xx, yy, '&');
    }

    x = xx;
    y = yy;

    if (!RBIGNUM_SIGN(x)) {
	x = rb_big_clone(x);
	get2comp(x);
    }
    if (FIXNUM_P(y)) {
	return bigand_int(x, FIX2LONG(y));
    }
    if (!RBIGNUM_SIGN(y)) {
	y = rb_big_clone(y);
	get2comp(y);
    }
    if (RBIGNUM_LEN(x) > RBIGNUM_LEN(y)) {
	l1 = RBIGNUM_LEN(y);
	l2 = RBIGNUM_LEN(x);
	ds1 = BDIGITS(y);
	ds2 = BDIGITS(x);
	sign = RBIGNUM_SIGN(y);
    }
    else {
	l1 = RBIGNUM_LEN(x);
	l2 = RBIGNUM_LEN(y);
	ds1 = BDIGITS(x);
	ds2 = BDIGITS(y);
	sign = RBIGNUM_SIGN(x);
    }
    z = bignew(l2, RBIGNUM_SIGN(x) || RBIGNUM_SIGN(y));
    zds = BDIGITS(z);

    for (i=0; i<l1; i++) {
	zds[i] = ds1[i] & ds2[i];
    }
    for (; i<l2; i++) {
	zds[i] = sign?0:ds2[i];
    }
    if (!RBIGNUM_SIGN(z)) get2comp(z);
    return bignorm(z);
}

static VALUE
bigor_int(VALUE x, long y)
{
    VALUE z;
    BDIGIT *xds, *zds;
    long xn, zn;
    long i;
    char sign;

    sign = (y >= 0);
    xds = BDIGITS(x);
    zn = xn = RBIGNUM_LEN(x);
    z = bignew(zn, RBIGNUM_SIGN(x) && sign);
    zds = BDIGITS(z);

#if SIZEOF_BDIGITS == SIZEOF_LONG
    i = 1;
    zds[0] = xds[0] | y;
#else
    {
	BDIGIT_DBL num = y;

	for (i=0; i<(int)(sizeof(y)/sizeof(BDIGIT)); i++) {
	    zds[i] = xds[i] | BIGLO(num);
	    num = BIGDN(num);
	}
    }
#endif
    while (i < xn) {
	zds[i] = sign?xds[i]:(BDIGIT)(BIGRAD-1);
	i++;
    }
    if (!RBIGNUM_SIGN(z)) get2comp(z);
    return bignorm(z);
}

/*
 * call-seq:
 *     big | numeric   ->  integer
 *
 * Performs bitwise +or+ between _big_ and _numeric_.
 */

VALUE
rb_big_or(VALUE xx, VALUE yy)
{
    volatile VALUE x, y, z;
    BDIGIT *ds1, *ds2, *zds;
    long i, l1, l2;
    char sign;

    if (!FIXNUM_P(yy) && !RB_TYPE_P(yy, T_BIGNUM)) {
	return rb_num_coerce_bit(xx, yy, '|');
    }

    x = xx;
    y = yy;

    if (!RBIGNUM_SIGN(x)) {
	x = rb_big_clone(x);
	get2comp(x);
    }
    if (FIXNUM_P(y)) {
	return bigor_int(x, FIX2LONG(y));
    }
    if (!RBIGNUM_SIGN(y)) {
	y = rb_big_clone(y);
	get2comp(y);
    }
    if (RBIGNUM_LEN(x) > RBIGNUM_LEN(y)) {
	l1 = RBIGNUM_LEN(y);
	l2 = RBIGNUM_LEN(x);
	ds1 = BDIGITS(y);
	ds2 = BDIGITS(x);
	sign = RBIGNUM_SIGN(y);
    }
    else {
	l1 = RBIGNUM_LEN(x);
	l2 = RBIGNUM_LEN(y);
	ds1 = BDIGITS(x);
	ds2 = BDIGITS(y);
	sign = RBIGNUM_SIGN(x);
    }
    z = bignew(l2, RBIGNUM_SIGN(x) && RBIGNUM_SIGN(y));
    zds = BDIGITS(z);

    for (i=0; i<l1; i++) {
	zds[i] = ds1[i] | ds2[i];
    }
    for (; i<l2; i++) {
	zds[i] = sign?ds2[i]:(BDIGIT)(BIGRAD-1);
    }
    if (!RBIGNUM_SIGN(z)) get2comp(z);
    return bignorm(z);
}

static VALUE
bigxor_int(VALUE x, long y)
{
    VALUE z;
    BDIGIT *xds, *zds;
    long xn, zn;
    long i;
    char sign;

    sign = (y >= 0) ? 1 : 0;
    xds = BDIGITS(x);
    zn = xn = RBIGNUM_LEN(x);
    z = bignew(zn, !(RBIGNUM_SIGN(x) ^ sign));
    zds = BDIGITS(z);

#if SIZEOF_BDIGITS == SIZEOF_LONG
    i = 1;
    zds[0] = xds[0] ^ y;
#else
    {
	BDIGIT_DBL num = y;

	for (i=0; i<(int)(sizeof(y)/sizeof(BDIGIT)); i++) {
	    zds[i] = xds[i] ^ BIGLO(num);
	    num = BIGDN(num);
	}
    }
#endif
    while (i < xn) {
	zds[i] = sign?xds[i]:~xds[i];
	i++;
    }
    if (!RBIGNUM_SIGN(z)) get2comp(z);
    return bignorm(z);
}
/*
 * call-seq:
 *     big ^ numeric   ->  integer
 *
 * Performs bitwise +exclusive or+ between _big_ and _numeric_.
 */

VALUE
rb_big_xor(VALUE xx, VALUE yy)
{
    volatile VALUE x, y;
    VALUE z;
    BDIGIT *ds1, *ds2, *zds;
    long i, l1, l2;
    char sign;

    if (!FIXNUM_P(yy) && !RB_TYPE_P(yy, T_BIGNUM)) {
	return rb_num_coerce_bit(xx, yy, '^');
    }

    x = xx;
    y = yy;

    if (!RBIGNUM_SIGN(x)) {
	x = rb_big_clone(x);
	get2comp(x);
    }
    if (FIXNUM_P(y)) {
	return bigxor_int(x, FIX2LONG(y));
    }
    if (!RBIGNUM_SIGN(y)) {
	y = rb_big_clone(y);
	get2comp(y);
    }
    if (RBIGNUM_LEN(x) > RBIGNUM_LEN(y)) {
	l1 = RBIGNUM_LEN(y);
	l2 = RBIGNUM_LEN(x);
	ds1 = BDIGITS(y);
	ds2 = BDIGITS(x);
	sign = RBIGNUM_SIGN(y);
    }
    else {
	l1 = RBIGNUM_LEN(x);
	l2 = RBIGNUM_LEN(y);
	ds1 = BDIGITS(x);
	ds2 = BDIGITS(y);
	sign = RBIGNUM_SIGN(x);
    }
    RBIGNUM_SET_SIGN(x, RBIGNUM_SIGN(x)?1:0);
    RBIGNUM_SET_SIGN(y, RBIGNUM_SIGN(y)?1:0);
    z = bignew(l2, !(RBIGNUM_SIGN(x) ^ RBIGNUM_SIGN(y)));
    zds = BDIGITS(z);

    for (i=0; i<l1; i++) {
	zds[i] = ds1[i] ^ ds2[i];
    }
    for (; i<l2; i++) {
	zds[i] = sign?ds2[i]:~ds2[i];
    }
    if (!RBIGNUM_SIGN(z)) get2comp(z);

    return bignorm(z);
}

static VALUE
check_shiftdown(VALUE y, VALUE x)
{
    if (!RBIGNUM_LEN(x)) return INT2FIX(0);
    if (RBIGNUM_LEN(y) > SIZEOF_LONG / SIZEOF_BDIGITS) {
	return RBIGNUM_SIGN(x) ? INT2FIX(0) : INT2FIX(-1);
    }
    return Qnil;
}

/*
 * call-seq:
 *     big << numeric   ->  integer
 *
 * Shifts big left _numeric_ positions (right if _numeric_ is negative).
 */

VALUE
rb_big_lshift(VALUE x, VALUE y)
{
    long shift;
    int neg = 0;

    for (;;) {
	if (FIXNUM_P(y)) {
	    shift = FIX2LONG(y);
	    if (shift < 0) {
		neg = 1;
		shift = -shift;
	    }
	    break;
	}
	else if (RB_TYPE_P(y, T_BIGNUM)) {
	    if (!RBIGNUM_SIGN(y)) {
		VALUE t = check_shiftdown(y, x);
		if (!NIL_P(t)) return t;
		neg = 1;
	    }
	    shift = big2ulong(y, "long", TRUE);
	    break;
	}
	y = rb_to_int(y);
    }

    x = neg ? big_rshift(x, shift) : big_lshift(x, shift);
    return bignorm(x);
}

static VALUE
big_lshift(VALUE x, unsigned long shift)
{
    BDIGIT *xds, *zds;
    long s1 = shift/BITSPERDIG;
    int s2 = (int)(shift%BITSPERDIG);
    VALUE z;
    BDIGIT_DBL num = 0;
    long len, i;

    len = RBIGNUM_LEN(x);
    z = bignew(len+s1+1, RBIGNUM_SIGN(x));
    zds = BDIGITS(z);
    for (i=0; i<s1; i++) {
	*zds++ = 0;
    }
    xds = BDIGITS(x);
    for (i=0; i<len; i++) {
	num = num | (BDIGIT_DBL)*xds++<<s2;
	*zds++ = BIGLO(num);
	num = BIGDN(num);
    }
    *zds = BIGLO(num);
    return z;
}

/*
 * call-seq:
 *     big >> numeric   ->  integer
 *
 * Shifts big right _numeric_ positions (left if _numeric_ is negative).
 */

VALUE
rb_big_rshift(VALUE x, VALUE y)
{
    long shift;
    int neg = 0;

    for (;;) {
	if (FIXNUM_P(y)) {
	    shift = FIX2LONG(y);
	    if (shift < 0) {
		neg = 1;
		shift = -shift;
	    }
	    break;
	}
	else if (RB_TYPE_P(y, T_BIGNUM)) {
	    if (RBIGNUM_SIGN(y)) {
		VALUE t = check_shiftdown(y, x);
		if (!NIL_P(t)) return t;
	    }
	    else {
		neg = 1;
	    }
	    shift = big2ulong(y, "long", TRUE);
	    break;
	}
	y = rb_to_int(y);
    }

    x = neg ? big_lshift(x, shift) : big_rshift(x, shift);
    return bignorm(x);
}

static VALUE
big_rshift(VALUE x, unsigned long shift)
{
    BDIGIT *xds, *zds;
    long s1 = shift/BITSPERDIG;
    int s2 = (int)(shift%BITSPERDIG);
    VALUE z;
    BDIGIT_DBL num = 0;
    long i, j;
    volatile VALUE save_x;

    if (s1 > RBIGNUM_LEN(x)) {
	if (RBIGNUM_SIGN(x))
	    return INT2FIX(0);
	else
	    return INT2FIX(-1);
    }
    if (!RBIGNUM_SIGN(x)) {
	x = rb_big_clone(x);
	get2comp(x);
    }
    save_x = x;
    xds = BDIGITS(x);
    i = RBIGNUM_LEN(x); j = i - s1;
    if (j == 0) {
	if (RBIGNUM_SIGN(x)) return INT2FIX(0);
	else return INT2FIX(-1);
    }
    z = bignew(j, RBIGNUM_SIGN(x));
    if (!RBIGNUM_SIGN(x)) {
	num = ((BDIGIT_DBL)~0) << BITSPERDIG;
    }
    zds = BDIGITS(z);
    while (i--, j--) {
	num = (num | xds[i]) >> s2;
	zds[j] = BIGLO(num);
	num = BIGUP(xds[i]);
    }
    if (!RBIGNUM_SIGN(x)) {
	get2comp(z);
    }
    RB_GC_GUARD(save_x);
    return z;
}

/*
 *  call-seq:
 *     big[n] -> 0, 1
 *
 *  Bit Reference---Returns the <em>n</em>th bit in the (assumed) binary
 *  representation of <i>big</i>, where <i>big</i>[0] is the least
 *  significant bit.
 *
 *     a = 9**15
 *     50.downto(0) do |n|
 *       print a[n]
 *     end
 *
 *  <em>produces:</em>
 *
 *     000101110110100000111000011110010100111100010111001
 *
 */

static VALUE
rb_big_aref(VALUE x, VALUE y)
{
    BDIGIT *xds;
    BDIGIT_DBL num;
    VALUE shift;
    long i, s1, s2;

    if (RB_TYPE_P(y, T_BIGNUM)) {
	if (!RBIGNUM_SIGN(y))
	    return INT2FIX(0);
	bigtrunc(y);
	if (RBIGNUM_LEN(y) > DIGSPERLONG) {
	  out_of_range:
	    return RBIGNUM_SIGN(x) ? INT2FIX(0) : INT2FIX(1);
	}
	shift = big2ulong(y, "long", FALSE);
    }
    else {
	i = NUM2LONG(y);
	if (i < 0) return INT2FIX(0);
	shift = (VALUE)i;
    }
    s1 = shift/BITSPERDIG;
    s2 = shift%BITSPERDIG;

    if (s1 >= RBIGNUM_LEN(x)) goto out_of_range;
    if (!RBIGNUM_SIGN(x)) {
	xds = BDIGITS(x);
	i = 0; num = 1;
	while (num += ~xds[i], ++i <= s1) {
	    num = BIGDN(num);
	}
    }
    else {
	num = BDIGITS(x)[s1];
    }
    if (num & ((BDIGIT_DBL)1<<s2))
	return INT2FIX(1);
    return INT2FIX(0);
}

/*
 * call-seq:
 *   big.hash   -> fixnum
 *
 * Compute a hash based on the value of _big_.
 */

static VALUE
rb_big_hash(VALUE x)
{
    st_index_t hash;

    hash = rb_memhash(BDIGITS(x), sizeof(BDIGIT)*RBIGNUM_LEN(x)) ^ RBIGNUM_SIGN(x);
    return INT2FIX(hash);
}

/*
 * MISSING: documentation
 */

static VALUE
rb_big_coerce(VALUE x, VALUE y)
{
    if (FIXNUM_P(y)) {
	y = rb_int2big(FIX2LONG(y));
    }
    else if (!RB_TYPE_P(y, T_BIGNUM)) {
	rb_raise(rb_eTypeError, "can't coerce %s to Bignum",
		 rb_obj_classname(y));
    }
    return rb_assoc_new(y, x);
}

/*
 *  call-seq:
 *     big.abs -> aBignum
 *
 *  Returns the absolute value of <i>big</i>.
 *
 *     -1234567890987654321.abs   #=> 1234567890987654321
 */

static VALUE
rb_big_abs(VALUE x)
{
    if (!RBIGNUM_SIGN(x)) {
	x = rb_big_clone(x);
	RBIGNUM_SET_SIGN(x, 1);
    }
    return x;
}

/*
 *  call-seq:
 *     big.size -> integer
 *
 *  Returns the number of bytes in the machine representation of
 *  <i>big</i>.
 *
 *     (256**10 - 1).size   #=> 12
 *     (256**20 - 1).size   #=> 20
 *     (256**40 - 1).size   #=> 40
 */

static VALUE
rb_big_size(VALUE big)
{
    return LONG2FIX(RBIGNUM_LEN(big)*SIZEOF_BDIGITS);
}

/*
 *  call-seq:
 *     big.odd? -> true or false
 *
 *  Returns <code>true</code> if <i>big</i> is an odd number.
 */

static VALUE
rb_big_odd_p(VALUE num)
{
    if (BDIGITS(num)[0] & 1) {
	return Qtrue;
    }
    return Qfalse;
}

/*
 *  call-seq:
 *     big.even? -> true or false
 *
 *  Returns <code>true</code> if <i>big</i> is an even number.
 */

static VALUE
rb_big_even_p(VALUE num)
{
    if (BDIGITS(num)[0] & 1) {
	return Qfalse;
    }
    return Qtrue;
}

/*
 *  Bignum objects hold integers outside the range of
 *  Fixnum. Bignum objects are created
 *  automatically when integer calculations would otherwise overflow a
 *  Fixnum. When a calculation involving
 *  Bignum objects returns a result that will fit in a
 *  Fixnum, the result is automatically converted.
 *
 *  For the purposes of the bitwise operations and <code>[]</code>, a
 *  Bignum is treated as if it were an infinite-length
 *  bitstring with 2's complement representation.
 *
 *  While Fixnum values are immediate, Bignum
 *  objects are not---assignment and parameter passing work with
 *  references to objects, not the objects themselves.
 *
 */

void
Init_Bignum(void)
{
    rb_cBignum = rb_define_class("Bignum", rb_cInteger);

    rb_define_method(rb_cBignum, "to_s", rb_big_to_s, -1);
    rb_define_alias(rb_cBignum, "inspect", "to_s");
    rb_define_method(rb_cBignum, "coerce", rb_big_coerce, 1);
    rb_define_method(rb_cBignum, "-@", rb_big_uminus, 0);
    rb_define_method(rb_cBignum, "+", rb_big_plus, 1);
    rb_define_method(rb_cBignum, "-", rb_big_minus, 1);
    rb_define_method(rb_cBignum, "*", rb_big_mul, 1);
    rb_define_method(rb_cBignum, "/", rb_big_div, 1);
    rb_define_method(rb_cBignum, "%", rb_big_modulo, 1);
    rb_define_method(rb_cBignum, "div", rb_big_idiv, 1);
    rb_define_method(rb_cBignum, "divmod", rb_big_divmod, 1);
    rb_define_method(rb_cBignum, "modulo", rb_big_modulo, 1);
    rb_define_method(rb_cBignum, "remainder", rb_big_remainder, 1);
    rb_define_method(rb_cBignum, "fdiv", rb_big_fdiv, 1);
    rb_define_method(rb_cBignum, "**", rb_big_pow, 1);
    rb_define_method(rb_cBignum, "&", rb_big_and, 1);
    rb_define_method(rb_cBignum, "|", rb_big_or, 1);
    rb_define_method(rb_cBignum, "^", rb_big_xor, 1);
    rb_define_method(rb_cBignum, "~", rb_big_neg, 0);
    rb_define_method(rb_cBignum, "<<", rb_big_lshift, 1);
    rb_define_method(rb_cBignum, ">>", rb_big_rshift, 1);
    rb_define_method(rb_cBignum, "[]", rb_big_aref, 1);

    rb_define_method(rb_cBignum, "<=>", rb_big_cmp, 1);
    rb_define_method(rb_cBignum, "==", rb_big_eq, 1);
    rb_define_method(rb_cBignum, ">", big_gt, 1);
    rb_define_method(rb_cBignum, ">=", big_ge, 1);
    rb_define_method(rb_cBignum, "<", big_lt, 1);
    rb_define_method(rb_cBignum, "<=", big_le, 1);
    rb_define_method(rb_cBignum, "===", rb_big_eq, 1);
    rb_define_method(rb_cBignum, "eql?", rb_big_eql, 1);
    rb_define_method(rb_cBignum, "hash", rb_big_hash, 0);
    rb_define_method(rb_cBignum, "to_f", rb_big_to_f, 0);
    rb_define_method(rb_cBignum, "abs", rb_big_abs, 0);
    rb_define_method(rb_cBignum, "magnitude", rb_big_abs, 0);
    rb_define_method(rb_cBignum, "size", rb_big_size, 0);
    rb_define_method(rb_cBignum, "odd?", rb_big_odd_p, 0);
    rb_define_method(rb_cBignum, "even?", rb_big_even_p, 0);

    power_cache_init();

    big_three = rb_uint2big(3);
    rb_gc_register_mark_object(big_three);
}