void
arb_addmul(arb_t z, const arb_t x, const arb_t y, slong prec)
{
mag_t zr, xm, ym;
int inexact;
if (arb_is_exact(y))
{
arb_addmul_arf(z, x, arb_midref(y), prec);
}
else if (arb_is_exact(x))
{
arb_addmul_arf(z, y, arb_midref(x), prec);
}
else if (ARB_IS_LAGOM(x) && ARB_IS_LAGOM(y) && ARB_IS_LAGOM(z))
{
mag_fast_init_set_arf(xm, arb_midref(x));
mag_fast_init_set_arf(ym, arb_midref(y));
mag_fast_init_set(zr, arb_radref(z));
mag_fast_addmul(zr, xm, arb_radref(y));
mag_fast_addmul(zr, ym, arb_radref(x));
mag_fast_addmul(zr, arb_radref(x), arb_radref(y));
inexact = arf_addmul(arb_midref(z), arb_midref(x), arb_midref(y),
prec, ARF_RND_DOWN);
if (inexact)
arf_mag_fast_add_ulp(zr, zr, arb_midref(z), prec);
*arb_radref(z) = *zr;
}
else
{
mag_init_set_arf(xm, arb_midref(x));
mag_init_set_arf(ym, arb_midref(y));
mag_init_set(zr, arb_radref(z));
mag_addmul(zr, xm, arb_radref(y));
mag_addmul(zr, ym, arb_radref(x));
mag_addmul(zr, arb_radref(x), arb_radref(y));
inexact = arf_addmul(arb_midref(z), arb_midref(x), arb_midref(y),
prec, ARF_RND_DOWN);
if (inexact)
arf_mag_add_ulp(arb_radref(z), zr, arb_midref(z), prec);
else
mag_set(arb_radref(z), zr);
mag_clear(zr);
mag_clear(xm);
mag_clear(ym);
}
}
void
arb_addmul_arf(arb_t z, const arb_t x, const arf_t y, slong prec)
{
mag_t ym;
int inexact;
if (arb_is_exact(x))
{
inexact = arf_addmul(arb_midref(z), arb_midref(x), y, prec, ARB_RND);
if (inexact)
arf_mag_add_ulp(arb_radref(z), arb_radref(z), arb_midref(z), prec);
}
else if (ARB_IS_LAGOM(x) && ARF_IS_LAGOM(y) && ARB_IS_LAGOM(z))
{
mag_fast_init_set_arf(ym, y);
mag_fast_addmul(arb_radref(z), ym, arb_radref(x));
inexact = arf_addmul(arb_midref(z), arb_midref(x), y, prec, ARB_RND);
if (inexact)
arf_mag_fast_add_ulp(arb_radref(z), arb_radref(z), arb_midref(z), prec);
}
else
{
mag_init_set_arf(ym, y);
mag_addmul(arb_radref(z), ym, arb_radref(x));
inexact = arf_addmul(arb_midref(z), arb_midref(x), y, prec, ARB_RND);
if (inexact)
arf_mag_add_ulp(arb_radref(z), arb_radref(z), arb_midref(z), prec);
mag_clear(ym);
}
}
void
arb_pow(arb_t z, const arb_t x, const arb_t y, slong prec)
{
if (arb_is_zero(y))
{
arb_one(z);
return;
}
if (arb_is_zero(x))
{
if (arb_is_positive(y))
arb_zero(z);
else
arb_indeterminate(z);
return;
}
if (arb_is_exact(y) && !arf_is_special(arb_midref(x)))
{
const arf_struct * ymid = arb_midref(y);
/* small half-integer or integer */
if (arf_cmpabs_2exp_si(ymid, BINEXP_LIMIT) < 0 &&
arf_is_int_2exp_si(ymid, -1))
{
fmpz_t e;
fmpz_init(e);
if (arf_is_int(ymid))
{
arf_get_fmpz_fixed_si(e, ymid, 0);
arb_pow_fmpz_binexp(z, x, e, prec);
}
else
{
arf_get_fmpz_fixed_si(e, ymid, -1);
arb_sqrt(z, x, prec + fmpz_bits(e));
arb_pow_fmpz_binexp(z, z, e, prec);
}
fmpz_clear(e);
return;
}
else if (arf_is_int(ymid) && arf_sgn(arb_midref(x)) < 0)
{
/* use (-x)^n = (-1)^n * x^n to avoid NaNs
at least at high enough precision */
int odd = !arf_is_int_2exp_si(ymid, 1);
_arb_pow_exp(z, x, 1, y, prec);
if (odd)
arb_neg(z, z);
return;
}
}
_arb_pow_exp(z, x, 0, y, prec);
}
void
arb_sinc(arb_t z, const arb_t x, slong prec)
{
mag_t c, r;
mag_init(c);
mag_init(r);
mag_set_ui_2exp_si(c, 5, -1);
arb_get_mag_lower(r, x);
if (mag_cmp(c, r) < 0)
{
/* x is not near the origin */
_arb_sinc_direct(z, x, prec);
}
else if (mag_cmp_2exp_si(arb_radref(x), 1) < 0)
{
/* determine error magnitude using the derivative bound */
if (arb_is_exact(x))
{
mag_zero(c);
}
else
{
_arb_sinc_derivative_bound(r, x);
mag_mul(c, arb_radref(x), r);
}
/* evaluate sinc at the midpoint of x */
if (arf_is_zero(arb_midref(x)))
{
arb_one(z);
}
else
{
arb_get_mid_arb(z, x);
_arb_sinc_direct(z, z, prec);
}
/* add the error */
mag_add(arb_radref(z), arb_radref(z), c);
}
else
{
/* x has a large radius and includes points near the origin */
arf_zero(arb_midref(z));
mag_one(arb_radref(z));
}
mag_clear(c);
mag_clear(r);
}
void
arb_root_ui_algebraic(arb_t res, const arb_t x, ulong k, slong prec)
{
mag_t r, msubr, m1k, t;
if (arb_is_exact(x))
{
arb_root_arf(res, arb_midref(x), k, prec);
return;
}
if (!arb_is_nonnegative(x))
{
arb_indeterminate(res);
return;
}
mag_init(r);
mag_init(msubr);
mag_init(m1k);
mag_init(t);
/* x = [m-r, m+r] */
mag_set(r, arb_radref(x));
/* m - r */
arb_get_mag_lower(msubr, x);
/* m^(1/k) */
arb_root_arf(res, arb_midref(x), k, prec);
/* bound for m^(1/k) */
arb_get_mag(m1k, res);
/* C = min(1, log(1+r/(m-r))/k) */
mag_div(t, r, msubr);
mag_log1p(t, t);
mag_div_ui(t, t, k);
if (mag_cmp_2exp_si(t, 0) > 0)
mag_one(t);
/* C m^(1/k) */
mag_mul(t, m1k, t);
mag_add(arb_radref(res), arb_radref(res), t);
mag_clear(r);
mag_clear(msubr);
mag_clear(m1k);
mag_clear(t);
}
void
arb_div(arb_t z, const arb_t x, const arb_t y, long prec)
{
mag_t zr, xm, ym, yl, yw;
int inexact;
if (arb_is_exact(y))
{
arb_div_arf(z, x, arb_midref(y), prec);
}
else if (mag_is_inf(arb_radref(x)) || mag_is_inf(arb_radref(y)))
{
arf_div(arb_midref(z), arb_midref(x), arb_midref(y), prec, ARB_RND);
mag_inf(arb_radref(z));
}
else
{
mag_init_set_arf(xm, arb_midref(x));
mag_init_set_arf(ym, arb_midref(y));
mag_init(zr);
mag_init(yl);
mag_init(yw);
/* (|x|*yrad + |y|*xrad)/(y*(|y|-yrad)) */
mag_mul(zr, xm, arb_radref(y));
mag_addmul(zr, ym, arb_radref(x));
arb_get_mag_lower(yw, y);
arf_get_mag_lower(yl, arb_midref(y));
mag_mul_lower(yl, yl, yw);
mag_div(zr, zr, yl);
inexact = arf_div(arb_midref(z), arb_midref(x), arb_midref(y), prec, ARB_RND);
if (inexact)
arf_mag_add_ulp(arb_radref(z), zr, arb_midref(z), prec);
else
mag_swap(arb_radref(z), zr);
mag_clear(xm);
mag_clear(ym);
mag_clear(zr);
mag_clear(yl);
mag_clear(yw);
}
}
int
arb_contains_arf(const arb_t x, const arf_t y)
{
if (arf_is_nan(y))
{
return arf_is_nan(arb_midref(x));
}
else if (arf_is_nan(arb_midref(x)))
{
return 1;
}
else if (arb_is_exact(x))
{
return arf_equal(arb_midref(x), y);
}
else
{
arf_t t;
arf_struct tmp[3];
int result;
arf_init(t);
/* y >= xm - xr <=> 0 >= xm - xr - y */
arf_init_set_shallow(tmp + 0, arb_midref(x));
arf_init_neg_mag_shallow(tmp + 1, arb_radref(x));
arf_init_neg_shallow(tmp + 2, y);
arf_sum(t, tmp, 3, MAG_BITS, ARF_RND_DOWN);
result = (arf_sgn(t) <= 0);
if (result)
{
/* y <= xm + xr <=> 0 <= xm + xr - y */
arf_init_set_mag_shallow(tmp + 1, arb_radref(x));
arf_sum(t, tmp, 3, MAG_BITS, ARF_RND_DOWN);
result = (arf_sgn(t) >= 0);
}
arf_clear(t);
return result;
}
}
void
arb_div_arf(arb_t z, const arb_t x, const arf_t y, long prec)
{
mag_t zr, ym;
int inexact;
if (arf_is_zero(y))
{
arb_zero_pm_inf(z);
}
else if (arb_is_exact(x))
{
inexact = arf_div(arb_midref(z), arb_midref(x), y, prec, ARB_RND);
if (inexact)
arf_mag_set_ulp(arb_radref(z), arb_midref(z), prec);
else
mag_zero(arb_radref(z));
}
else if (mag_is_inf(arb_radref(x)))
{
arf_div(arb_midref(z), arb_midref(x), y, prec, ARB_RND);
mag_inf(arb_radref(z));
}
else
{
mag_init(ym);
mag_init(zr);
arf_get_mag_lower(ym, y);
mag_div(zr, arb_radref(x), ym);
inexact = arf_div(arb_midref(z), arb_midref(x), y, prec, ARB_RND);
if (inexact)
arf_mag_add_ulp(arb_radref(z), zr, arb_midref(z), prec);
else
mag_swap(arb_radref(z), zr);
mag_clear(ym);
mag_clear(zr);
}
}
void
arb_atan(arb_t z, const arb_t x, slong prec)
{
if (arb_is_exact(x))
{
arb_atan_arf(z, arb_midref(x), prec);
}
else
{
mag_t t, u;
mag_init(t);
mag_init(u);
arb_get_mag_lower(t, x);
if (mag_is_zero(t))
{
mag_set(t, arb_radref(x));
}
else
{
mag_mul_lower(t, t, t);
mag_one(u);
mag_add_lower(t, t, u);
mag_div(t, arb_radref(x), t);
}
if (mag_cmp_2exp_si(t, 0) > 0)
{
mag_const_pi(u);
mag_min(t, t, u);
}
arb_atan_arf(z, arb_midref(x), prec);
mag_add(arb_radref(z), arb_radref(z), t);
mag_clear(t);
mag_clear(u);
}
}
void
arb_log(arb_t y, const arb_t x, slong prec)
{
if (arb_is_exact(x))
{
arb_log_arf(y, arb_midref(x), prec);
}
else
{
/*
Let the input be [a-b, a+b]. We require a > b >= 0 (otherwise the
interval contains zero or a negative number and the logarithm is not
defined). The error is largest at a-b, and we have
log(a) - log(a-b) = log(1 + b/(a-b)).
*/
mag_t err;
mag_init(err);
arb_get_mag_lower_nonnegative(err, x);
if (mag_is_zero(err))
{
mag_inf(err);
}
else
{
mag_div(err, arb_radref(x), err);
mag_log1p(err, err);
}
arb_log_arf(y, arb_midref(x), prec);
mag_add(arb_radref(y), arb_radref(y), err);
mag_clear(err);
}
}
void
arb_mul_naive(arb_t z, const arb_t x, const arb_t y, slong prec)
{
arf_t zm_exact, zm_rounded, zr, t, u;
arf_init(zm_exact);
arf_init(zm_rounded);
arf_init(zr);
arf_init(t);
arf_init(u);
arf_mul(zm_exact, arb_midref(x), arb_midref(y), ARF_PREC_EXACT, ARF_RND_DOWN);
arf_set_round(zm_rounded, zm_exact, prec, ARB_RND);
/* rounding error */
if (arf_equal(zm_exact, zm_rounded))
{
arf_zero(zr);
}
else
{
fmpz_t e;
fmpz_init(e);
/* more accurate, but not what we are testing
arf_sub(zr, zm_exact, zm_rounded, MAG_BITS, ARF_RND_UP);
arf_abs(zr, zr); */
fmpz_sub_ui(e, ARF_EXPREF(zm_rounded), prec);
arf_one(zr);
arf_mul_2exp_fmpz(zr, zr, e);
fmpz_clear(e);
}
/* propagated error */
if (!arb_is_exact(x))
{
arf_set_mag(t, arb_radref(x));
arf_abs(u, arb_midref(y));
arf_addmul(zr, t, u, MAG_BITS, ARF_RND_UP);
}
if (!arb_is_exact(y))
{
arf_set_mag(t, arb_radref(y));
arf_abs(u, arb_midref(x));
arf_addmul(zr, t, u, MAG_BITS, ARF_RND_UP);
}
if (!arb_is_exact(x) && !arb_is_exact(y))
{
arf_set_mag(t, arb_radref(x));
arf_set_mag(u, arb_radref(y));
arf_addmul(zr, t, u, MAG_BITS, ARF_RND_UP);
}
arf_set(arb_midref(z), zm_rounded);
arf_get_mag(arb_radref(z), zr);
arf_clear(zm_exact);
arf_clear(zm_rounded);
arf_clear(zr);
arf_clear(t);
arf_clear(u);
}
void
_acb_poly_powsum_one_series_sieved(acb_ptr z, const acb_t s, slong n, slong len, slong prec)
{
slong * divisors;
slong powers_alloc;
slong i, j, k, ibound, kprev, power_of_two, horner_point;
int critical_line, integer;
acb_ptr powers;
acb_ptr t, u, x;
acb_ptr p1, p2;
arb_t logk, v, w;
critical_line = arb_is_exact(acb_realref(s)) &&
(arf_cmp_2exp_si(arb_midref(acb_realref(s)), -1) == 0);
integer = arb_is_zero(acb_imagref(s)) && arb_is_int(acb_realref(s));
divisors = flint_calloc(n / 2 + 1, sizeof(slong));
powers_alloc = (n / 6 + 1) * len;
powers = _acb_vec_init(powers_alloc);
ibound = n_sqrt(n);
for (i = 3; i <= ibound; i += 2)
if (DIVISOR(i) == 0)
for (j = i * i; j <= n; j += 2 * i)
DIVISOR(j) = i;
t = _acb_vec_init(len);
u = _acb_vec_init(len);
x = _acb_vec_init(len);
arb_init(logk);
arb_init(v);
arb_init(w);
power_of_two = 1;
while (power_of_two * 2 <= n)
power_of_two *= 2;
horner_point = n / power_of_two;
_acb_vec_zero(z, len);
kprev = 0;
COMPUTE_POWER(x, 2, kprev);
for (k = 1; k <= n; k += 2)
{
/* t = k^(-s) */
if (DIVISOR(k) == 0)
{
COMPUTE_POWER(t, k, kprev);
}
else
{
p1 = POWER(DIVISOR(k));
p2 = POWER(k / DIVISOR(k));
if (len == 1)
acb_mul(t, p1, p2, prec);
else
_acb_poly_mullow(t, p1, len, p2, len, len, prec);
}
if (k * 3 <= n)
_acb_vec_set(POWER(k), t, len);
_acb_vec_add(u, u, t, len, prec);
while (k == horner_point && power_of_two != 1)
{
_acb_poly_mullow(t, z, len, x, len, len, prec);
_acb_vec_add(z, t, u, len, prec);
power_of_two /= 2;
horner_point = n / power_of_two;
horner_point -= (horner_point % 2 == 0);
}
}
_acb_poly_mullow(t, z, len, x, len, len, prec);
_acb_vec_add(z, t, u, len, prec);
flint_free(divisors);
_acb_vec_clear(powers, powers_alloc);
_acb_vec_clear(t, len);
_acb_vec_clear(u, len);
_acb_vec_clear(x, len);
arb_clear(logk);
arb_clear(v);
arb_clear(w);
}
void
acb_dirichlet_zeta_rs_mid(acb_t res, const acb_t s, slong K, slong prec)
{
acb_t R1, R2, X, t;
slong wp;
if (arf_sgn(arb_midref(acb_imagref(s))) < 0)
{
acb_init(t);
acb_conj(t, s);
acb_dirichlet_zeta_rs(res, t, K, prec);
acb_conj(res, res);
acb_clear(t);
return;
}
acb_init(R1);
acb_init(R2);
acb_init(X);
acb_init(t);
/* rs_r increases the precision internally */
wp = prec;
acb_dirichlet_zeta_rs_r(R1, s, K, wp);
if (arb_is_exact(acb_realref(s)) &&
(arf_cmp_2exp_si(arb_midref(acb_realref(s)), -1) == 0))
{
acb_conj(R2, R1);
}
else
{
/* conj(R(conj(1-s))) */
arb_sub_ui(acb_realref(t), acb_realref(s), 1, 10 * wp);
arb_neg(acb_realref(t), acb_realref(t));
arb_set(acb_imagref(t), acb_imagref(s));
acb_dirichlet_zeta_rs_r(R2, t, K, wp);
acb_conj(R2, R2);
}
if (acb_is_finite(R1) && acb_is_finite(R2))
{
wp += 10 + arf_abs_bound_lt_2exp_si(arb_midref(acb_imagref(s)));
wp = FLINT_MAX(wp, 10);
/* X = pi^(s-1/2) gamma((1-s)/2) rgamma(s/2)
= (2 pi)^s rgamma(s) / (2 cos(pi s / 2)) */
acb_rgamma(X, s, wp);
acb_const_pi(t, wp);
acb_mul_2exp_si(t, t, 1);
acb_pow(t, t, s, wp);
acb_mul(X, X, t, wp);
acb_mul_2exp_si(t, s, -1);
acb_cos_pi(t, t, wp);
acb_mul_2exp_si(t, t, 1);
acb_div(X, X, t, wp);
acb_mul(R2, R2, X, wp);
}
/* R1 + X * R2 */
acb_add(res, R1, R2, prec);
acb_clear(R1);
acb_clear(R2);
acb_clear(X);
acb_clear(t);
}
int
arb_get_unique_fmpz(fmpz_t z, const arb_t x)
{
if (!arb_is_finite(x))
{
return 0;
}
else if (arb_is_exact(x))
{
/* x = b*2^e, e >= 0 */
if (arf_is_int(arb_midref(x)))
{
/* arf_get_fmpz aborts on overflow */
arf_get_fmpz(z, arb_midref(x), ARF_RND_DOWN);
return 1;
}
else
{
return 0;
}
}
/* if the radius is >= 1, there are at least two integers */
else if (mag_cmp_2exp_si(arb_radref(x), 0) >= 0)
{
return 0;
}
/* there are 0 or 1 integers if the radius is < 1 */
else
{
fmpz_t a, b, exp;
int res;
/* if the midpoint is exactly an integer, it is what we want */
if (arf_is_int(arb_midref(x)))
{
/* arf_get_fmpz aborts on overflow */
arf_get_fmpz(z, arb_midref(x), ARF_RND_DOWN);
return 1;
}
fmpz_init(a);
fmpz_init(b);
fmpz_init(exp);
/* if the radius is tiny, it can't be an integer */
arf_bot(a, arb_midref(x));
if (fmpz_cmp(a, MAG_EXPREF(arb_radref(x))) > 0)
{
res = 0;
}
else
{
arb_get_interval_fmpz_2exp(a, b, exp, x);
if (COEFF_IS_MPZ(*exp))
{
flint_printf("arb_get_unique_fmpz: input too large\n");
abort();
}
if (*exp >= 0)
{
res = fmpz_equal(a, b);
if (res)
{
fmpz_mul_2exp(a, a, *exp);
fmpz_mul_2exp(b, b, *exp);
}
}
else
{
fmpz_cdiv_q_2exp(a, a, -(*exp));
fmpz_fdiv_q_2exp(b, b, -(*exp));
res = fmpz_equal(a, b);
}
if (res)
fmpz_set(z, a);
}
fmpz_clear(a);
fmpz_clear(b);
fmpz_clear(exp);
return res;
}
}
