void
mpz_sqrt (mpz_ptr root, mpz_srcptr op)
{
mp_size_t op_size, root_size;
mp_ptr root_ptr, op_ptr;
mp_ptr free_me = NULL;
mp_size_t free_me_size;
TMP_DECL (marker);
TMP_MARK (marker);
op_size = op->_mp_size;
if (op_size <= 0)
{
if (op_size < 0)
SQRT_OF_NEGATIVE;
SIZ(root) = 0;
return;
}
/* The size of the root is accurate after this simple calculation. */
root_size = (op_size + 1) / 2;
root_ptr = root->_mp_d;
op_ptr = op->_mp_d;
if (root->_mp_alloc < root_size)
{
if (root_ptr == op_ptr)
{
free_me = root_ptr;
free_me_size = root->_mp_alloc;
}
else
(*__gmp_free_func) (root_ptr, root->_mp_alloc * BYTES_PER_MP_LIMB);
root->_mp_alloc = root_size;
root_ptr = (mp_ptr) (*__gmp_allocate_func) (root_size * BYTES_PER_MP_LIMB);
root->_mp_d = root_ptr;
}
else
{
/* Make OP not overlap with ROOT. */
if (root_ptr == op_ptr)
{
/* ROOT and OP are identical. Allocate temporary space for OP. */
op_ptr = (mp_ptr) TMP_ALLOC (op_size * BYTES_PER_MP_LIMB);
/* Copy to the temporary space. Hack: Avoid temporary variable
by using ROOT_PTR. */
MPN_COPY (op_ptr, root_ptr, op_size);
}
}
mpn_sqrtrem (root_ptr, NULL, op_ptr, op_size);
root->_mp_size = root_size;
if (free_me != NULL)
(*__gmp_free_func) (free_me, free_me_size * BYTES_PER_MP_LIMB);
TMP_FREE (marker);
}
void
gmp_wrap_isqrt(mp_limb_t *res, mp_size_t l2, mp_limb_t *n1, mp_size_t l1)
{
if (n1[l1 - 1] == 0) {
l1--;
res[l2 - 1] = 0;
}
mpn_sqrtrem(res, 0, n1, l1);
}
void
gmp_sb_isqrt(char * n1l, char * resl)
{
WORD_PTR_TYPE n1 = TO_WORD_PTR(n1l);
mp_size_t l1 = BIGNUM_LENGTH(n1);
WORD_PTR_TYPE res = TO_WORD_PTR(resl);
unsigned long l2 = BIGNUM_LENGTH(res);
if (n1[l1 - 1] == 0) {
l1--;
res[l2 - 1] = 0;
}
mpn_sqrtrem(res, 0, n1, l1);
}
void
mpf_sqrt (mpf_ptr r, mpf_srcptr u)
{
mp_size_t usize;
mp_ptr up, tp;
mp_size_t prec, tsize;
mp_exp_t uexp, expodd;
TMP_DECL;
usize = u->_mp_size;
if (usize <= 0)
{
if (usize < 0)
SQRT_OF_NEGATIVE;
r->_mp_size = 0;
r->_mp_exp = 0;
return;
}
TMP_MARK;
uexp = u->_mp_exp;
prec = r->_mp_prec;
up = u->_mp_d;
expodd = (uexp & 1);
tsize = 2 * prec - expodd;
r->_mp_size = prec;
r->_mp_exp = (uexp + expodd) / 2; /* ceil(uexp/2) */
/* root size is ceil(tsize/2), this will be our desired "prec" limbs */
ASSERT ((tsize + 1) / 2 == prec);
tp = (mp_ptr) TMP_ALLOC (tsize * BYTES_PER_MP_LIMB);
if (usize > tsize)
{
up += usize - tsize;
usize = tsize;
MPN_COPY (tp, up, tsize);
}
else
{
MPN_ZERO (tp, tsize - usize);
MPN_COPY (tp + (tsize - usize), up, usize);
}
mpn_sqrtrem (r->_mp_d, NULL, tp, tsize);
TMP_FREE;
}
void fmpz_sqrtrem(fmpz_t sqrt, fmpz_t rem, fmpz_t n)
{
long size = n[0];
if (size < 0L)
{
printf("Cannot take the square root of a negative number!\n");
abort();
}
if (!size)
{
fmpz_set_ui(sqrt, 0L);
fmpz_set_ui(rem, 0L);
return;
}
rem[0] = mpn_sqrtrem(sqrt+1, rem+1, n+1, size);
sqrt[0] = (size+1)/2;
}
void
mpf_sqrt_ui (mpf_ptr r, unsigned long int u)
{
mp_size_t rsize, zeros;
mp_ptr tp;
mp_size_t prec;
TMP_DECL;
if (UNLIKELY (u == 0))
{
r->_mp_size = 0;
r->_mp_exp = 0;
return;
}
TMP_MARK;
prec = r->_mp_prec;
zeros = 2 * prec - 2;
rsize = zeros + 1 + U2;
tp = TMP_ALLOC_LIMBS (rsize);
MPN_ZERO (tp, zeros);
tp[zeros] = u & GMP_NUMB_MASK;
#if U2
{
mp_limb_t uhigh = u >> GMP_NUMB_BITS;
tp[zeros + 1] = uhigh;
rsize -= (uhigh == 0);
}
#endif
mpn_sqrtrem (r->_mp_d, NULL, tp, rsize);
r->_mp_size = prec;
r->_mp_exp = 1;
TMP_FREE;
}
mp_limb_t _ll_factor_SQUFOF(mp_limb_t n_hi, mp_limb_t n_lo, ulong max_iters)
{
mp_limb_t n[2];
mp_limb_t sqrt[2];
mp_limb_t rem[2];
mp_size_t num, sqroot, p, q;
mp_limb_t l, l2, iq, pnext;
mp_limb_t qarr[50];
mp_limb_t qupto, qlast, t, r = 0;
ulong i, j;
n[0] = n_lo;
n[1] = n_hi;
if (n_hi) num = mpn_sqrtrem(sqrt, rem, n, 2);
else num = ((sqrt[0] = n_sqrtrem(rem, n_lo)) != 0UL);
sqroot = sqrt[0];
p = sqroot;
q = rem[0];
if ((q == 0) || (num == 0))
{
return sqroot;
}
l = 1 + 2*n_sqrt(2*p);
l2 = l/2;
qupto = 0;
qlast = 1;
for (i = 0; i < max_iters; i++)
{
iq = (sqroot + p)/q;
pnext = iq*q - p;
if (q <= l)
{
if ((q & 1UL) == 0UL)
{
qarr[qupto] = q/2;
qupto++;
if (qupto >= 50UL) return 0UL;
} else if (q <= l2)
{
qarr[qupto] = q;
qupto++;
if (qupto >= 50UL) return 0UL;
}
}
t = qlast + iq*(p - pnext);
qlast = q;
q = t;
p = pnext;
if ((i & 1) == 1) continue;
if (!n_is_square(q)) continue;
r = n_sqrt(q);
if (qupto == 0UL) break;
for (j = 0; j < qupto; j++)
if (r == qarr[j]) goto cont;
break;
cont: ;
if (r == 1UL) return 0UL;
}
if (i == max_iters) return 0UL; /* taken too long, give up */
qlast = r;
p = p + r*((sqroot - p)/r);
umul_ppmm(rem[1], rem[0], p, p);
sub_ddmmss(sqrt[1], sqrt[0], n[1], n[0], rem[1], rem[0]);
if (sqrt[1])
{
int norm;
count_leading_zeros(norm, qlast);
udiv_qrnnd(q, rem[0], (sqrt[1] << norm) + r_shift(sqrt[0], FLINT_BITS - norm), sqrt[0] << norm, qlast << norm);
rem[0] >>= norm;
}
else
{
int
mpfr_sqrt (mpfr_ptr r, mpfr_srcptr u, mpfr_rnd_t rnd_mode)
{
mp_size_t rsize; /* number of limbs of r (plus 1 if exact limb multiple) */
mp_size_t rrsize;
mp_size_t usize; /* number of limbs of u */
mp_size_t tsize; /* number of limbs of the sqrtrem remainder */
mp_size_t k;
mp_size_t l;
mpfr_limb_ptr rp, rp0;
mpfr_limb_ptr up;
mpfr_limb_ptr sp;
mp_limb_t sticky0; /* truncated part of input */
mp_limb_t sticky1; /* truncated part of rp[0] */
mp_limb_t sticky;
int odd_exp;
int sh; /* number of extra bits in rp[0] */
int inexact; /* return ternary flag */
mpfr_exp_t expr;
MPFR_TMP_DECL(marker);
MPFR_LOG_FUNC
(("x[%Pu]=%.*Rg rnd=%d", mpfr_get_prec (u), mpfr_log_prec, u, rnd_mode),
("y[%Pu]=%.*Rg inexact=%d",
mpfr_get_prec (r), mpfr_log_prec, r, inexact));
if (MPFR_UNLIKELY(MPFR_IS_SINGULAR(u)))
{
if (MPFR_IS_NAN(u))
{
MPFR_SET_NAN(r);
MPFR_RET_NAN;
}
else if (MPFR_IS_ZERO(u))
{
/* 0+ or 0- */
MPFR_SET_SAME_SIGN(r, u);
MPFR_SET_ZERO(r);
MPFR_RET(0); /* zero is exact */
}
else
{
MPFR_ASSERTD(MPFR_IS_INF(u));
/* sqrt(-Inf) = NAN */
if (MPFR_IS_NEG(u))
{
MPFR_SET_NAN(r);
MPFR_RET_NAN;
}
MPFR_SET_POS(r);
MPFR_SET_INF(r);
MPFR_RET(0);
}
}
if (MPFR_UNLIKELY(MPFR_IS_NEG(u)))
{
MPFR_SET_NAN(r);
MPFR_RET_NAN;
}
MPFR_SET_POS(r);
MPFR_TMP_MARK (marker);
MPFR_UNSIGNED_MINUS_MODULO(sh,MPFR_PREC(r));
if (sh == 0 && rnd_mode == MPFR_RNDN)
sh = GMP_NUMB_BITS; /* ugly case */
rsize = MPFR_LIMB_SIZE(r) + (sh == GMP_NUMB_BITS);
/* rsize is the number of limbs of r + 1 if exact limb multiple and rounding
to nearest, this is the number of wanted limbs for the square root */
rrsize = rsize + rsize;
usize = MPFR_LIMB_SIZE(u); /* number of limbs of u */
rp0 = MPFR_MANT(r);
rp = (sh < GMP_NUMB_BITS) ? rp0 : MPFR_TMP_LIMBS_ALLOC (rsize);
up = MPFR_MANT(u);
sticky0 = MPFR_LIMB_ZERO; /* truncated part of input */
sticky1 = MPFR_LIMB_ZERO; /* truncated part of rp[0] */
odd_exp = (unsigned int) MPFR_GET_EXP (u) & 1;
inexact = -1; /* return ternary flag */
sp = MPFR_TMP_LIMBS_ALLOC (rrsize);
/* copy the most significant limbs of u to {sp, rrsize} */
if (MPFR_LIKELY(usize <= rrsize)) /* in case r and u have the same precision,
we have indeed rrsize = 2 * usize */
{
k = rrsize - usize;
if (MPFR_LIKELY(k))
MPN_ZERO (sp, k);
if (odd_exp)
{
if (MPFR_LIKELY(k))
sp[k - 1] = mpn_rshift (sp + k, up, usize, 1);
else
sticky0 = mpn_rshift (sp, up, usize, 1);
}
else
MPN_COPY (sp + rrsize - usize, up, usize);
}
else /* usize > rrsize: truncate the input */
{
k = usize - rrsize;
if (odd_exp)
sticky0 = mpn_rshift (sp, up + k, rrsize, 1);
else
MPN_COPY (sp, up + k, rrsize);
l = k;
while (sticky0 == MPFR_LIMB_ZERO && l != 0)
sticky0 = up[--l];
}
/* sticky0 is non-zero iff the truncated part of the input is non-zero */
/* mpn_rootrem with NULL 2nd argument is faster than mpn_sqrtrem, thus use
it if available and if the user asked to use GMP internal functions */
#if defined(WANT_GMP_INTERNALS) && defined(HAVE___GMPN_ROOTREM)
tsize = __gmpn_rootrem (rp, NULL, sp, rrsize, 2);
#else
tsize = mpn_sqrtrem (rp, NULL, sp, rrsize);
#endif
/* a return value of zero in mpn_sqrtrem indicates a perfect square */
sticky = sticky0 || tsize != 0;
/* truncate low bits of rp[0] */
sticky1 = rp[0] & ((sh < GMP_NUMB_BITS) ? MPFR_LIMB_MASK(sh)
: ~MPFR_LIMB_ZERO);
rp[0] -= sticky1;
sticky = sticky || sticky1;
expr = (MPFR_GET_EXP(u) + odd_exp) / 2; /* exact */
if (rnd_mode == MPFR_RNDZ || rnd_mode == MPFR_RNDD || sticky == MPFR_LIMB_ZERO)
{
inexact = (sticky == MPFR_LIMB_ZERO) ? 0 : -1;
goto truncate;
}
else if (rnd_mode == MPFR_RNDN)
{
/* if sh < GMP_NUMB_BITS, the round bit is bit (sh-1) of sticky1
and the sticky bit is formed by the low sh-1 bits from
sticky1, together with the sqrtrem remainder and sticky0. */
if (sh < GMP_NUMB_BITS)
{
if (sticky1 & (MPFR_LIMB_ONE << (sh - 1)))
{ /* round bit is set */
if (sticky1 == (MPFR_LIMB_ONE << (sh - 1)) && tsize == 0
&& sticky0 == 0)
goto even_rule;
else
goto add_one_ulp;
}
else /* round bit is zero */
goto truncate; /* with the default inexact=-1 */
}
else /* sh = GMP_NUMB_BITS: the round bit is the most significant bit
of rp[0], and the remaining GMP_NUMB_BITS-1 bits contribute to
the sticky bit */
{
if (sticky1 & MPFR_LIMB_HIGHBIT)
{ /* round bit is set */
if (sticky1 == MPFR_LIMB_HIGHBIT && tsize == 0 && sticky0 == 0)
goto even_rule;
else
goto add_one_ulp;
}
else /* round bit is zero */
goto truncate; /* with the default inexact=-1 */
}
}
else /* rnd_mode=GMP_RDNU, necessarily sticky <> 0, thus add 1 ulp */
goto add_one_ulp;
even_rule: /* has to set inexact */
if (sh < GMP_NUMB_BITS)
inexact = (rp[0] & (MPFR_LIMB_ONE << sh)) ? 1 : -1;
else
inexact = (rp[1] & MPFR_LIMB_ONE) ? 1 : -1;
if (inexact == -1)
goto truncate;
/* else go through add_one_ulp */
add_one_ulp:
inexact = 1; /* always here */
if (sh == GMP_NUMB_BITS)
{
rp ++;
rsize --;
sh = 0;
}
if (mpn_add_1 (rp0, rp, rsize, MPFR_LIMB_ONE << sh))
{
expr ++;
rp[rsize - 1] = MPFR_LIMB_HIGHBIT;
}
goto end;
truncate: /* inexact = 0 or -1 */
if (sh == GMP_NUMB_BITS)
MPN_COPY (rp0, rp + 1, rsize - 1);
end:
MPFR_ASSERTN (expr >= MPFR_EMIN_MIN && expr <= MPFR_EMAX_MAX);
MPFR_EXP (r) = expr;
MPFR_TMP_FREE(marker);
return mpfr_check_range (r, inexact, rnd_mode);
}
void
mpz_sqrtrem (mpz_ptr root, mpz_ptr rem, mpz_srcptr op)
{
mp_size_t op_size, root_size, rem_size;
mp_ptr root_ptr, op_ptr;
mp_ptr free_me = NULL;
mp_size_t free_me_size;
TMP_DECL;
TMP_MARK;
op_size = op->_mp_size;
if (op_size <= 0)
{
if (op_size < 0)
SQRT_OF_NEGATIVE;
SIZ(root) = 0;
SIZ(rem) = 0;
return;
}
if (rem->_mp_alloc < op_size)
_mpz_realloc (rem, op_size);
/* The size of the root is accurate after this simple calculation. */
root_size = (op_size + 1) / 2;
root_ptr = root->_mp_d;
op_ptr = op->_mp_d;
if (root->_mp_alloc < root_size)
{
if (root_ptr == op_ptr)
{
free_me = root_ptr;
free_me_size = root->_mp_alloc;
}
else
(*__gmp_free_func) (root_ptr, root->_mp_alloc * BYTES_PER_MP_LIMB);
root->_mp_alloc = root_size;
root_ptr = (mp_ptr) (*__gmp_allocate_func) (root_size * BYTES_PER_MP_LIMB);
root->_mp_d = root_ptr;
}
else
{
/* Make OP not overlap with ROOT. */
if (root_ptr == op_ptr)
{
/* ROOT and OP are identical. Allocate temporary space for OP. */
op_ptr = (mp_ptr) TMP_ALLOC (op_size * BYTES_PER_MP_LIMB);
/* Copy to the temporary space. Hack: Avoid temporary variable
by using ROOT_PTR. */
MPN_COPY (op_ptr, root_ptr, op_size);
}
}
rem_size = mpn_sqrtrem (root_ptr, rem->_mp_d, op_ptr, op_size);
root->_mp_size = root_size;
/* Write remainder size last, to enable us to define this function to
give only the square root remainder, if the user calls if with
ROOT == REM. */
rem->_mp_size = rem_size;
if (free_me != NULL)
(*__gmp_free_func) (free_me, free_me_size * BYTES_PER_MP_LIMB);
TMP_FREE;
}
static inline RawArray<mp_limb_t> sqrt_helper(RawArray<mp_limb_t> result, RawArray<const mp_limb_t> x) {
const auto s = result.slice(0,(1+x.size())/2);
mpn_sqrtrem(s.data(),0,x.data(),x.size());
return trim(s);
}