// Refine the random square
mpfr_prec_t Refine(gmp_randstate_t r, mpfr_prec_t prec, long num = 1)
const {
if (num <= 0) return prec;
// Use _vx as scratch
prec += num * chunk_;
mpfr_div_2ui(_eps, _eps, num * chunk_, MPFR_RNDN);
mpz_urandomb(_ui, r, num * chunk_);
mpfr_set_prec(_up, prec);
mpfr_set_z_2exp(_up, _ui, -prec, MPFR_RNDN);
mpfr_set_prec(_vx, prec);
mpfr_add(_vx, _u, _up, MPFR_RNDN);
mpfr_swap(_u, _vx); // u = vx
mpfr_add(_up, _u, _eps, MPFR_RNDN);
mpz_urandomb(_vi, r, num * chunk_);
mpfr_set_prec(_vp, prec);
mpfr_set_z_2exp(_vp, _vi, -prec, MPFR_RNDN);
mpfr_set_prec(_vx, prec);
mpfr_add(_vx, _v, _vp, MPFR_RNDN);
mpfr_swap(_v, _vx); // v = vx
mpfr_add(_vp, _v, _eps, MPFR_RNDN);
return prec;
}
static void
check0 (void)
{
mpz_t y;
mpfr_t x;
int inexact, r;
mpfr_exp_t e;
/* Check for +0 */
mpfr_init (x);
mpz_init (y);
mpz_set_si (y, 0);
for (r = 0; r < MPFR_RND_MAX; r++)
{
e = randexp ();
inexact = mpfr_set_z_2exp (x, y, e, (mpfr_rnd_t) r);
if (!MPFR_IS_ZERO(x) || !MPFR_IS_POS(x) || inexact)
{
printf ("mpfr_set_z_2exp(x,0,e) failed for e=");
if (e < LONG_MIN)
printf ("(<LONG_MIN)");
else if (e > LONG_MAX)
printf ("(>LONG_MAX)");
else
printf ("%ld", (long) e);
printf (", rnd=%s\n", mpfr_print_rnd_mode ((mpfr_rnd_t) r));
exit (1);
}
}
mpfr_clear(x);
mpz_clear(y);
}
int
fmpr_get_mpfr(mpfr_t x, const fmpr_t y, mpfr_rnd_t rnd)
{
int r;
if (fmpr_is_special(y))
{
if (fmpr_is_zero(y)) mpfr_set_zero(x, 0);
else if (fmpr_is_pos_inf(y)) mpfr_set_inf(x, 1);
else if (fmpr_is_neg_inf(y)) mpfr_set_inf(x, -1);
else mpfr_set_nan(x);
r = 0;
}
else if (COEFF_IS_MPZ(*fmpr_expref(y)))
{
flint_printf("exception: exponent too large to convert to mpfr");
abort();
}
else
{
if (!COEFF_IS_MPZ(*fmpr_manref(y)))
#if defined(__MINGW64__)
r = mpfr_set_sj_2exp(x, *fmpr_manref(y), *fmpr_expref(y), rnd);
#else
r = mpfr_set_si_2exp(x, *fmpr_manref(y), *fmpr_expref(y), rnd);
#endif
else
r = mpfr_set_z_2exp(x, COEFF_TO_PTR(*fmpr_manref(y)), *fmpr_expref(y), rnd);
if (!mpfr_regular_p(x))
{
flint_printf("exception: exponent too large to convert to mpfr");
abort();
}
}
static void
check (long i, mpfr_rnd_t rnd)
{
mpfr_t f;
mpz_t z;
mpfr_exp_t e;
int inex;
/* using CHAR_BIT * sizeof(long) bits of precision ensures that
mpfr_set_z_2exp is exact below */
mpfr_init2 (f, CHAR_BIT * sizeof(long));
mpz_init (z);
mpz_set_ui (z, i);
/* the following loop ensures that no overflow occurs */
do
e = randexp ();
while (e > mpfr_get_emax () - CHAR_BIT * sizeof(long));
inex = mpfr_set_z_2exp (f, z, e, rnd);
if (inex != 0)
{
printf ("Error in mpfr_set_z_2exp for i=%ld, e=%ld,"
" wrong ternary value\n", i, (long) e);
printf ("expected 0, got %d\n", inex);
exit (1);
}
mpfr_div_2si (f, f, e, rnd);
if (mpfr_get_si (f, MPFR_RNDZ) != i)
{
printf ("Error in mpfr_set_z_2exp for i=%ld e=", i);
if (e < LONG_MIN)
printf ("(<LONG_MIN)");
else if (e > LONG_MAX)
printf ("(>LONG_MAX)");
else
printf ("%ld", (long) e);
printf (" rnd_mode=%d\n", rnd);
printf ("expected %ld\n", i);
printf ("got "); mpfr_dump (f);
exit (1);
}
mpfr_clear (f);
mpz_clear (z);
}
/* set f to the integer z */
int
mpfr_set_z (mpfr_ptr f, mpz_srcptr z, mpfr_rnd_t rnd_mode)
{
return mpfr_set_z_2exp (f, z, 0, rnd_mode);
}
/**
* Sample from the normal distribution with mean 0 and variance 1.
*
* @param[out] val the sample from the normal distribution
* @param[in,out] r a GMP random generator.
* @param[in] round the rounding direction.
* @return the MPFR ternary result (±1 if val is larger/smaller than
* the exact sample).
**********************************************************************/
int operator()(mpfr_t val, gmp_randstate_t r, mpfr_rnd_t round) const {
const double
s = 0.449871, // Constants from Leva
t = -0.386595,
a = 0.19600 ,
b = 0.25472 ,
r1 = 0.27597 ,
r2 = 0.27846 ,
u1 = 0.606530, // sqrt(1/e) rounded down and up
u2 = 0.606531,
scale = std::pow(2.0, -chunk_); // for turning randoms into doubles
while (true) {
mpz_urandomb(_vi, r, chunk_);
if (mpz_cmp_ui(_vi, m) >= 0) continue; // Very early reject
double vf = (mpz_get_ui(_vi) + 0.5) * scale;
mpz_urandomb(_ui, r, chunk_);
double uf = (mpz_get_ui(_ui) + 0.5) * scale;
double
x = uf - s,
y = vf - t,
Q = x*x + y * (a*y - b*x);
if (Q >= r2) continue; // Early reject
mpfr_set_ui_2exp(_eps, 1u, -chunk_, MPFR_RNDN);
mpfr_prec_t prec = chunk_;
mpfr_set_prec(_u, prec);
mpfr_set_prec(_v, prec);
// (u,v) = sw corner of range
mpfr_set_z_2exp(_u, _ui, -prec, MPFR_RNDN);
mpfr_set_z_2exp(_v, _vi, -prec, MPFR_RNDN);
mpfr_set_prec(_up, prec);
mpfr_set_prec(_vp, prec);
// (up,vp) = ne corner of range
mpfr_add(_up, _u, _eps, MPFR_RNDN);
mpfr_add(_vp, _v, _eps, MPFR_RNDN);
// Estimate how many extra bits will be needed to achieve the desired
// precision.
mpfr_prec_t prec_guard = 3 + chunk_ -
(std::max)(mpz_sizeinbase(_ui, 2), mpz_sizeinbase(_vi, 2));
if (Q > r1) {
int reject;
while (true) {
// Rejection curve v^2 + 4 * u^2 * log(u) < 0 has a peak at u =
// exp(-1/2) = 0.60653066. So treat uf in (0.606530, 0.606531) =
// (u1, u2) specially
// Try for rejection first
if (uf <= u1)
reject = Reject(_u, _vp, prec, MPFR_RNDU);
else if (uf >= u2)
reject = Reject(_up, _vp, prec, MPFR_RNDU);
else { // u in (u1, u2)
mpfr_set_prec(_vx, prec);
mpfr_add(_vx, _vp, _eps, MPFR_RNDN);
reject = Reject(_u, _vx, prec, MPFR_RNDU); // Could use _up too
}
if (reject < 0) break; // tried to reject but failed, so accept
// Try for acceptance
if (uf <= u1)
reject = Reject(_up, _v, prec, MPFR_RNDD);
else if (uf >= u2)
reject = Reject(_u, _v, prec, MPFR_RNDD);
else { // u in (u2, u2)
mpfr_sub(_vx, _v, _eps, MPFR_RNDN);
reject = Reject(_u, _vx, prec, MPFR_RNDD); // Could use _up too
}
if (reject > 0) break; // tried to accept but failed, so reject
prec = Refine(r, prec); // still can't decide, to refine
}
if (reject > 0) continue; // reject, back to outer loop
}
// Now evaluate v/u to the necessary precision
mpfr_prec_t prec0 = mpfr_get_prec (val);
// while (prec < prec0 + prec_guard) prec = Refine(r, prec);
if (prec < prec0 + prec_guard)
prec = Refine(r, prec,
(prec0 + prec_guard - prec + chunk_ - 1) / chunk_);
mpfr_set_prec(_x1, prec0);
mpfr_set_prec(_x2, prec0);
int flag;
while (true) {
int
f1 = mpfr_div(_x1, _v, _up, round), // min slope
f2 = mpfr_div(_x2, _vp, _u, round); // max slope
if (f1 == f2 && mpfr_equal_p(_x1, _x2)) {
flag = f1;
break;
}
prec = Refine(r, prec);
}
mpz_urandomb(_ui, r, 1);
if (mpz_tstbit(_ui, 0)) {
flag = -flag;
mpfr_neg(val, _x1, MPFR_RNDN);
} else
mpfr_set(val, _x1, MPFR_RNDN);
// std::cerr << uf << " " << vf << " " << Q << "\n";
return flag;
}
}
static int
mpfr_rem1 (mpfr_ptr rem, long *quo, mpfr_rnd_t rnd_q,
mpfr_srcptr x, mpfr_srcptr y, mpfr_rnd_t rnd)
{
mpfr_exp_t ex, ey;
int compare, inex, q_is_odd, sign, signx = MPFR_SIGN (x);
mpz_t mx, my, r;
int tiny = 0;
MPFR_ASSERTD (rnd_q == MPFR_RNDN || rnd_q == MPFR_RNDZ);
if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x) || MPFR_IS_SINGULAR (y)))
{
if (MPFR_IS_NAN (x) || MPFR_IS_NAN (y) || MPFR_IS_INF (x)
|| MPFR_IS_ZERO (y))
{
/* for remquo, quo is undefined */
MPFR_SET_NAN (rem);
MPFR_RET_NAN;
}
else /* either y is Inf and x is 0 or non-special,
or x is 0 and y is non-special,
in both cases the quotient is zero. */
{
if (quo)
*quo = 0;
return mpfr_set (rem, x, rnd);
}
}
/* now neither x nor y is NaN, Inf or zero */
mpz_init (mx);
mpz_init (my);
mpz_init (r);
ex = mpfr_get_z_2exp (mx, x); /* x = mx*2^ex */
ey = mpfr_get_z_2exp (my, y); /* y = my*2^ey */
/* to get rid of sign problems, we compute it separately:
quo(-x,-y) = quo(x,y), rem(-x,-y) = -rem(x,y)
quo(-x,y) = -quo(x,y), rem(-x,y) = -rem(x,y)
thus quo = sign(x/y)*quo(|x|,|y|), rem = sign(x)*rem(|x|,|y|) */
sign = (signx == MPFR_SIGN (y)) ? 1 : -1;
mpz_abs (mx, mx);
mpz_abs (my, my);
q_is_odd = 0;
/* divide my by 2^k if possible to make operations mod my easier */
{
unsigned long k = mpz_scan1 (my, 0);
ey += k;
mpz_fdiv_q_2exp (my, my, k);
}
if (ex <= ey)
{
/* q = x/y = mx/(my*2^(ey-ex)) */
/* First detect cases where q=0, to avoid creating a huge number
my*2^(ey-ex): if sx = mpz_sizeinbase (mx, 2) and sy =
mpz_sizeinbase (my, 2), we have x < 2^(ex + sx) and
y >= 2^(ey + sy - 1), thus if ex + sx <= ey + sy - 1
the quotient is 0 */
if (ex + (mpfr_exp_t) mpz_sizeinbase (mx, 2) <
ey + (mpfr_exp_t) mpz_sizeinbase (my, 2))
{
tiny = 1;
mpz_set (r, mx);
mpz_set_ui (mx, 0);
}
else
{
mpz_mul_2exp (my, my, ey - ex); /* divide mx by my*2^(ey-ex) */
/* since mx > 0 and my > 0, we can use mpz_tdiv_qr in all cases */
mpz_tdiv_qr (mx, r, mx, my);
/* 0 <= |r| <= |my|, r has the same sign as mx */
}
if (rnd_q == MPFR_RNDN)
q_is_odd = mpz_tstbit (mx, 0);
if (quo) /* mx is the quotient */
{
mpz_tdiv_r_2exp (mx, mx, WANTED_BITS);
*quo = mpz_get_si (mx);
}
}
else /* ex > ey */
{
if (quo) /* remquo case */
/* for remquo, to get the low WANTED_BITS more bits of the quotient,
we first compute R = X mod Y*2^WANTED_BITS, where X and Y are
defined below. Then the low WANTED_BITS of the quotient are
floor(R/Y). */
mpz_mul_2exp (my, my, WANTED_BITS); /* 2^WANTED_BITS*Y */
else if (rnd_q == MPFR_RNDN) /* remainder case */
/* Let X = mx*2^(ex-ey) and Y = my. Then both X and Y are integers.
Assume X = R mod Y, then x = X*2^ey = R*2^ey mod (Y*2^ey=y).
To be able to perform the rounding, we need the least significant
bit of the quotient, i.e., one more bit in the remainder,
which is obtained by dividing by 2Y. */
mpz_mul_2exp (my, my, 1); /* 2Y */
mpz_set_ui (r, 2);
mpz_powm_ui (r, r, ex - ey, my); /* 2^(ex-ey) mod my */
mpz_mul (r, r, mx);
mpz_mod (r, r, my);
if (quo) /* now 0 <= r < 2^WANTED_BITS*Y */
{
mpz_fdiv_q_2exp (my, my, WANTED_BITS); /* back to Y */
mpz_tdiv_qr (mx, r, r, my);
/* oldr = mx*my + newr */
*quo = mpz_get_si (mx);
q_is_odd = *quo & 1;
}
else if (rnd_q == MPFR_RNDN) /* now 0 <= r < 2Y in the remainder case */
{
mpz_fdiv_q_2exp (my, my, 1); /* back to Y */
/* least significant bit of q */
q_is_odd = mpz_cmpabs (r, my) >= 0;
if (q_is_odd)
mpz_sub (r, r, my);
}
/* now 0 <= |r| < |my|, and if needed,
q_is_odd is the least significant bit of q */
}
if (mpz_cmp_ui (r, 0) == 0)
{
inex = mpfr_set_ui (rem, 0, MPFR_RNDN);
/* take into account sign of x */
if (signx < 0)
mpfr_neg (rem, rem, MPFR_RNDN);
}
else
{
if (rnd_q == MPFR_RNDN)
{
/* FIXME: the comparison 2*r < my could be done more efficiently
at the mpn level */
mpz_mul_2exp (r, r, 1);
/* if tiny=1, we should compare r with my*2^(ey-ex) */
if (tiny)
{
if (ex + (mpfr_exp_t) mpz_sizeinbase (r, 2) <
ey + (mpfr_exp_t) mpz_sizeinbase (my, 2))
compare = 0; /* r*2^ex < my*2^ey */
else
{
mpz_mul_2exp (my, my, ey - ex);
compare = mpz_cmpabs (r, my);
}
}
else
compare = mpz_cmpabs (r, my);
mpz_fdiv_q_2exp (r, r, 1);
compare = ((compare > 0) ||
((rnd_q == MPFR_RNDN) && (compare == 0) && q_is_odd));
/* if compare != 0, we need to subtract my to r, and add 1 to quo */
if (compare)
{
mpz_sub (r, r, my);
if (quo && (rnd_q == MPFR_RNDN))
*quo += 1;
}
}
/* take into account sign of x */
if (signx < 0)
mpz_neg (r, r);
inex = mpfr_set_z_2exp (rem, r, ex > ey ? ey : ex, rnd);
}
if (quo)
*quo *= sign;
mpz_clear (mx);
mpz_clear (my);
mpz_clear (r);
return inex;
}
