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_set_interval_arf(arb_t x, const arf_t a, const arf_t b, slong prec)
{
arf_t t;
int inexact;
if (arf_is_inf(a) && arf_equal(a, b))
{
/* [-inf, -inf] or [+inf, +inf] */
arf_set(arb_midref(x), a);
mag_zero(arb_radref(x));
return;
}
arf_init(t);
arf_sub(t, b, a, MAG_BITS, ARF_RND_UP);
if (arf_sgn(t) < 0)
{
flint_printf("exception: arb_set_interval_arf: endpoints not ordered\n");
abort();
}
arf_get_mag(arb_radref(x), t);
inexact = arf_add(arb_midref(x), a, b, prec, ARB_RND);
if (inexact)
arf_mag_add_ulp(arb_radref(x), arb_radref(x), arb_midref(x), prec);
arb_mul_2exp_si(x, x, -1);
arf_clear(t);
}
void arb_square(arb_t out, const arb_t in, slong prec) {
mag_mul(arb_radref(out), arb_radref(in), arb_radref(in));
int inexact = arf_mul(arb_midref(out), arb_midref(in), arb_midref(in), ARB_RND, prec);
if (inexact)
arf_mag_add_ulp(arb_radref(out), arb_radref(out), arb_midref(out), prec);
}
static __inline__ void
arb_nonnegative_part(arb_t z, const arb_t x, long prec)
{
if (arb_contains_negative(x))
{
arf_t t;
arf_init(t);
arf_set_mag(t, arb_radref(x));
arf_add(arb_midref(z), arb_midref(x), t, MAG_BITS, ARF_RND_CEIL);
if (arf_sgn(arb_midref(z)) <= 0)
{
mag_zero(arb_radref(z));
}
else
{
arf_mul_2exp_si(arb_midref(z), arb_midref(z), -1);
arf_get_mag(arb_radref(z), arb_midref(z));
/* XXX: needed since arf_get_mag is inexact */
arf_set_mag(arb_midref(z), arb_radref(z));
}
arf_clear(t);
}
else
{
arb_set(z, x);
}
}
void
arb_max(arb_t z, const arb_t x, const arb_t y, slong prec)
{
arf_t left, right, t, xr, yr;
if (arf_is_nan(arb_midref(x)) || arf_is_nan(arb_midref(y)))
{
arb_indeterminate(z);
return;
}
arf_init(left);
arf_init(right);
arf_init(t);
arf_init_set_mag_shallow(xr, arb_radref(x));
arf_init_set_mag_shallow(yr, arb_radref(y));
arf_sub(left, arb_midref(x), xr, prec, ARF_RND_FLOOR);
arf_sub(t, arb_midref(y), yr, prec, ARF_RND_FLOOR);
arf_max(left, left, t);
arf_add(right, arb_midref(x), xr, prec, ARF_RND_CEIL);
arf_add(t, arb_midref(y), yr, prec, ARF_RND_CEIL);
arf_max(right, right, t);
arb_set_interval_arf(z, left, right, prec);
arf_clear(left);
arf_clear(right);
arf_clear(t);
}
static void
acb_set_dddd(acb_t z, double a, double ar, double b, double br)
{
arf_set_d(arb_midref(acb_realref(z)), a);
mag_set_d(arb_radref(acb_realref(z)), ar);
arf_set_d(arb_midref(acb_imagref(z)), b);
mag_set_d(arb_radref(acb_imagref(z)), br);
}
static void
arb_root_arf(arb_t z, const arf_t x, ulong k, slong prec)
{
int inexact = arf_root(arb_midref(z), x, k, prec, ARB_RND);
if (inexact)
arf_mag_set_ulp(arb_radref(z), arb_midref(z), prec);
else
mag_zero(arb_radref(z));
}
void
arb_sub_fmpz(arb_t z, const arb_t x, const fmpz_t y, slong prec)
{
int inexact;
inexact = arf_sub_fmpz(arb_midref(z), arb_midref(x), y, prec, ARB_RND);
if (inexact)
arf_mag_add_ulp(arb_radref(z), arb_radref(x), arb_midref(z), prec);
else
mag_set(arb_radref(z), arb_radref(x));
}
static void
_arb_arf_div_rounded_den_add_err(arb_t res, const arf_t x, const arf_t y, int yinexact, slong prec)
{
int inexact = arf_div(arb_midref(res), x, y, prec, ARB_RND);
if (yinexact && !arf_is_special(arb_midref(res)))
arf_mag_add_ulp(arb_radref(res), arb_radref(res), arb_midref(res), prec - 1);
else if (inexact)
arf_mag_add_ulp(arb_radref(res), arb_radref(res), arb_midref(res), prec);
}
void
acb_hypgeom_erf_propagated_error(mag_t re, mag_t im, const acb_t z)
{
mag_t x, y;
mag_init(x);
mag_init(y);
/* |exp(-(x+y)^2)| = exp(y^2-x^2) */
arb_get_mag(y, acb_imagref(z));
mag_mul(y, y, y);
arb_get_mag_lower(x, acb_realref(z));
mag_mul_lower(x, x, x);
if (mag_cmp(y, x) >= 0)
{
mag_sub(re, y, x);
mag_exp(re, re);
}
else
{
mag_sub_lower(re, x, y);
mag_expinv(re, re);
}
/* Radius. */
mag_hypot(x, arb_radref(acb_realref(z)), arb_radref(acb_imagref(z)));
mag_mul(re, re, x);
/* 2/sqrt(pi) < 289/256 */
mag_mul_ui(re, re, 289);
mag_mul_2exp_si(re, re, -8);
if (arb_is_zero(acb_imagref(z)))
{
/* todo: could bound magnitude even for complex numbers */
mag_set_ui(y, 2);
mag_min(re, re, y);
mag_zero(im);
}
else if (arb_is_zero(acb_realref(z)))
{
mag_swap(im, re);
mag_zero(re);
}
else
{
mag_set(im, re);
}
mag_clear(x);
mag_clear(y);
}
void
arb_set_round_fmpz(arb_t y, const fmpz_t x, slong prec)
{
int inexact;
inexact = arf_set_round_fmpz(arb_midref(y), x, prec, ARB_RND);
if (inexact)
arf_mag_set_ulp(arb_radref(y), arb_midref(y), prec);
else
mag_zero(arb_radref(y));
}
int main()
{
flint_printf("spherical_y....");
fflush(stdout);
{
slong i, n, m;
acb_t z, w, x, y;
acb_init(z);
acb_init(w);
acb_init(x);
acb_init(y);
i = 0;
arb_set_str(acb_realref(x), "0.2", 64);
arb_set_str(acb_imagref(x), "0.3", 64);
arb_set_str(acb_realref(y), "0.3", 64);
arb_set_str(acb_imagref(y), "0.4", 64);
for (n = -4; n <= 4; n++)
{
for (m = -4; m <= 4; m++)
{
acb_hypgeom_spherical_y(z, n, m, x, y, 64);
acb_set_d_d(w, testdata[2 * i], testdata[2 * i + 1]);
mag_set_d(arb_radref(acb_realref(w)), 1e-13);
mag_set_d(arb_radref(acb_imagref(w)), 1e-13);
if (!acb_overlaps(z, w))
{
flint_printf("FAIL: value\n\n");
flint_printf("n = %wd, m = %wd\n", n, m);
flint_printf("z = "); acb_printd(z, 20); flint_printf("\n\n");
flint_printf("w = "); acb_printd(w, 20); flint_printf("\n\n");
abort();
}
i++;
}
}
acb_clear(z);
acb_clear(w);
acb_clear(x);
acb_clear(y);
}
flint_cleanup();
flint_printf("PASS\n");
return EXIT_SUCCESS;
}
void
arb_sub(arb_t z, const arb_t x, const arb_t y, slong prec)
{
int inexact;
inexact = arf_sub(arb_midref(z), arb_midref(x), arb_midref(y), prec, ARB_RND);
mag_add(arb_radref(z), arb_radref(x), arb_radref(y));
if (inexact)
arf_mag_add_ulp(arb_radref(z), arb_radref(z), arb_midref(z), prec);
}
void
arb_fmpz_div_fmpz(arb_t z, const fmpz_t x, const fmpz_t y, long prec)
{
int inexact;
inexact = arf_fmpz_div_fmpz(arb_midref(z), x, y, prec, ARB_RND);
if (inexact)
arf_mag_set_ulp(arb_radref(z), arb_midref(z), prec);
else
mag_zero(arb_radref(z));
}
void
arb_sin_cos_pi(arb_t s, arb_t c, const arb_t x, long prec)
{
arb_t t;
arb_t u;
fmpz_t v;
if (arf_cmpabs_2exp_si(arb_midref(x), FLINT_MAX(65536, (4*prec))) > 0)
{
arf_zero(arb_midref(s));
mag_one(arb_radref(s));
arf_zero(arb_midref(c));
mag_one(arb_radref(c));
return;
}
arb_init(t);
arb_init(u);
fmpz_init(v);
arb_mul_2exp_si(t, x, 1);
arf_get_fmpz(v, arb_midref(t), ARF_RND_NEAR);
arb_sub_fmpz(t, t, v, prec);
arb_const_pi(u, prec);
arb_mul(t, t, u, prec);
arb_mul_2exp_si(t, t, -1);
switch (fmpz_fdiv_ui(v, 4))
{
case 0:
arb_sin_cos(s, c, t, prec);
break;
case 1:
arb_sin_cos(c, s, t, prec);
arb_neg(c, c);
break;
case 2:
arb_sin_cos(s, c, t, prec);
arb_neg(s, s);
arb_neg(c, c);
break;
default:
arb_sin_cos(c, s, t, prec);
arb_neg(s, s);
break;
}
fmpz_clear(v);
arb_clear(t);
arb_clear(u);
}
int check_accuracy(acb_ptr vec, slong len, slong prec)
{
slong i;
for (i = 0; i < len; i++)
{
if (mag_cmp_2exp_si(arb_radref(acb_realref(vec + i)), -prec) >= 0
|| mag_cmp_2exp_si(arb_radref(acb_imagref(vec + i)), -prec) >= 0)
return 0;
}
return 1;
}
/* note: the tmp variable t should have zero radius */
static void
acb_approx_div(acb_t z, const acb_t x, const acb_t y, acb_t t, slong prec)
{
arf_set(arb_midref(acb_realref(t)), arb_midref(acb_realref(y)));
arf_set(arb_midref(acb_imagref(t)), arb_midref(acb_imagref(y)));
acb_inv(t, t, prec);
mag_zero(arb_radref(acb_realref(t)));
mag_zero(arb_radref(acb_imagref(t)));
acb_approx_mul(z, x, t, prec);
}
static void
_acb_mul_slow(acb_t z, const acb_t x, const acb_t y, slong prec)
{
int inexact;
mag_t am, bm, cm, dm, er, fr;
mag_init_set_arf(am, arb_midref(a));
mag_init_set_arf(bm, arb_midref(b));
mag_init_set_arf(cm, arb_midref(c));
mag_init_set_arf(dm, arb_midref(d));
mag_init(er);
mag_init(fr);
mag_addmul(er, am, cr);
mag_addmul(er, bm, dr);
mag_addmul(er, cm, ar);
mag_addmul(er, dm, br);
mag_addmul(er, ar, cr);
mag_addmul(er, br, dr);
mag_addmul(fr, am, dr);
mag_addmul(fr, bm, cr);
mag_addmul(fr, cm, br);
mag_addmul(fr, dm, ar);
mag_addmul(fr, br, cr);
mag_addmul(fr, ar, dr);
inexact = arf_complex_mul(arb_midref(e), arb_midref(f),
arb_midref(a), arb_midref(b),
arb_midref(c), arb_midref(d), prec, ARB_RND);
if (inexact & 1)
arf_mag_add_ulp(arb_radref(e), er, arb_midref(e), prec);
else
mag_swap(arb_radref(e), er);
if (inexact & 2)
arf_mag_add_ulp(arb_radref(f), fr, arb_midref(f), prec);
else
mag_swap(arb_radref(f), fr);
mag_clear(am);
mag_clear(bm);
mag_clear(cm);
mag_clear(dm);
mag_clear(er);
mag_clear(fr);
}
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);
}
/* assumes no aliasing */
slong
acb_lambertw_initial(acb_t res, const acb_t z, const acb_t ez1, const fmpz_t k, slong prec)
{
/* Handle z very close to 0 on the principal branch. */
if (fmpz_is_zero(k) &&
(arf_cmpabs_2exp_si(arb_midref(acb_realref(z)), -20) <= 0 &&
arf_cmpabs_2exp_si(arb_midref(acb_imagref(z)), -20) <= 0))
{
acb_set(res, z);
acb_submul(res, res, res, prec);
return 40; /* could be tightened... */
}
/* For moderate input not close to the branch point, compute a double
approximation as the initial value. */
if (fmpz_is_zero(k) &&
arf_cmpabs_2exp_si(arb_midref(acb_realref(z)), 400) < 0 &&
arf_cmpabs_2exp_si(arb_midref(acb_imagref(z)), 400) < 0 &&
(arf_cmp_d(arb_midref(acb_realref(z)), -0.37) < 0 ||
arf_cmp_d(arb_midref(acb_realref(z)), -0.36) > 0 ||
arf_cmpabs_d(arb_midref(acb_imagref(z)), 0.01) > 0))
{
acb_lambertw_principal_d(res, z);
return 48;
}
/* Check if we are close to the branch point at -1/e. */
if ((fmpz_is_zero(k) || (fmpz_is_one(k) && arb_is_negative(acb_imagref(z)))
|| (fmpz_equal_si(k, -1) && arb_is_nonnegative(acb_imagref(z))))
&& ((arf_cmpabs_2exp_si(arb_midref(acb_realref(ez1)), -2) <= 0 &&
arf_cmpabs_2exp_si(arb_midref(acb_imagref(ez1)), -2) <= 0)))
{
acb_t t;
acb_init(t);
acb_mul_2exp_si(t, ez1, 1);
mag_zero(arb_radref(acb_realref(t)));
mag_zero(arb_radref(acb_imagref(t)));
acb_mul_ui(t, t, 3, prec);
acb_sqrt(t, t, prec);
if (!fmpz_is_zero(k))
acb_neg(t, t);
acb_lambertw_branchpoint_series(res, t, 0, prec);
acb_clear(t);
return 1; /* todo: estimate */
}
acb_lambertw_initial_asymp(res, z, k, prec);
return 1; /* todo: estimate */
}
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);
}
int
_arb_poly_newton_step(arb_t xnew, arb_srcptr poly, long len,
const arb_t x,
const arb_t convergence_interval,
const arf_t convergence_factor, long prec)
{
arf_t err;
arb_t t, u, v;
int result;
arf_init(err);
arb_init(t);
arb_init(u);
arb_init(v);
arf_set_mag(err, arb_radref(x));
arf_mul(err, err, err, MAG_BITS, ARF_RND_UP);
arf_mul(err, err, convergence_factor, MAG_BITS, ARF_RND_UP);
arf_set(arb_midref(t), arb_midref(x));
mag_zero(arb_radref(t));
_arb_poly_evaluate2(u, v, poly, len, t, prec);
arb_div(u, u, v, prec);
arb_sub(u, t, u, prec);
arb_add_error_arf(u, err);
if (arb_contains(convergence_interval, u) &&
(mag_cmp(arb_radref(u), arb_radref(x)) < 0))
{
arb_swap(xnew, u);
result = 1;
}
else
{
arb_set(xnew, x);
result = 0;
}
arb_clear(t);
arb_clear(u);
arb_clear(v);
arf_clear(err);
return result;
}
void
arb_fprintd(FILE * file, const arb_t x, slong digits)
{
arf_fprintd(file, arb_midref(x), FLINT_MAX(digits, 1));
flint_fprintf(file, " +/- ");
mag_fprintd(file, arb_radref(x), 5);
}
void
arb_fprint(FILE * file, const arb_t x)
{
arf_fprint(file, arb_midref(x));
flint_fprintf(file, " +/- ");
mag_fprint(file, arb_radref(x));
}
void
acb_lambertw_cleared_cut_fix_small(acb_t res, const acb_t z,
const acb_t ez1, const fmpz_t k, int flags, slong prec)
{
acb_t zz, zmid, zmide1;
arf_t eps;
acb_init(zz);
acb_init(zmid);
acb_init(zmide1);
arf_init(eps);
arf_mul_2exp_si(eps, arb_midref(acb_realref(z)), -prec);
acb_set(zz, z);
if (arf_sgn(arb_midref(acb_realref(zz))) < 0 &&
(!fmpz_is_zero(k) || arf_sgn(arb_midref(acb_realref(ez1))) < 0) &&
arf_cmpabs(arb_midref(acb_imagref(zz)), eps) < 0)
{
/* now the value must be in [0,2eps] */
arf_get_mag(arb_radref(acb_imagref(zz)), eps);
arf_set_mag(arb_midref(acb_imagref(zz)), arb_radref(acb_imagref(zz)));
if (arf_sgn(arb_midref(acb_imagref(z))) >= 0)
{
acb_lambertw_cleared_cut(res, zz, k, flags, prec);
}
else
{
fmpz_t kk;
fmpz_init(kk);
fmpz_neg(kk, k);
acb_lambertw_cleared_cut(res, zz, kk, flags, prec);
acb_conj(res, res);
fmpz_clear(kk);
}
}
else
{
acb_lambertw_cleared_cut(res, zz, k, flags, prec);
}
acb_clear(zz);
acb_clear(zmid);
acb_clear(zmide1);
arf_clear(eps);
}
static void
arb_supremum(arf_t res, const arb_t x)
{
if (arf_is_nan(arb_midref(x)))
{
arf_nan(res);
}
else if (mag_is_inf(arb_radref(x)))
{
arf_pos_inf(res);
}
else
{
arf_set_mag(res, arb_radref(x));
arf_add(res, res, arb_midref(x), ARF_PREC_EXACT, ARF_RND_CEIL);
}
}
static void
arb_infimum(arf_t res, const arb_t x)
{
if (arf_is_nan(arb_midref(x)))
{
arf_nan(res);
}
else if (mag_is_inf(arb_radref(x)))
{
arf_neg_inf(res);
}
else
{
arf_set_mag(res, arb_radref(x));
arf_sub(res, arb_midref(x), res, ARF_PREC_EXACT, ARF_RND_FLOOR);
}
}
void
arb_get_lbound_arf(arf_t u, const arb_t x, long prec)
{
arf_t t;
arf_init_set_mag_shallow(t, arb_radref(x));
arf_sub(u, arb_midref(x), t, prec, ARF_RND_FLOOR);
}
void
arb_zeta_ui_borwein_bsplit(arb_t x, ulong s, slong prec)
{
zeta_bsplit_t sum;
mag_t err;
slong wp, n;
/* zeta(0) = -1/2 */
if (s == 0)
{
arb_set_si(x, -1);
arb_mul_2exp_si(x, x, -1);
return;
}
if (s == 1)
{
flint_printf("zeta_ui_borwein_bsplit: zeta(1)");
abort();
}
n = prec / ERROR_B + 2;
wp = prec + 30;
zeta_bsplit_init(sum);
zeta_bsplit(sum, 0, n + 1, n, s, 0, wp);
/* A/Q3 - B/Q3 / (C/Q1) = (A*C - B*Q1) / (Q3*C) */
arb_mul(sum->A, sum->A, sum->C, wp);
arb_mul(sum->B, sum->B, sum->Q1, wp);
arb_sub(sum->A, sum->A, sum->B, wp);
arb_mul(sum->Q3, sum->Q3, sum->C, wp);
arb_div(sum->C, sum->A, sum->Q3, wp);
mag_init(err);
mag_borwein_error(err, n);
mag_add(arb_radref(sum->C), arb_radref(sum->C), err);
mag_clear(err);
/* convert from eta(s) to zeta(s) */
arb_div_2expm1_ui(x, sum->C, s - 1, wp);
arb_mul_2exp_si(x, x, s - 1);
zeta_bsplit_clear(sum);
}
void
arb_bernoulli_fmpz(arb_t res, const fmpz_t n, slong prec)
{
if (fmpz_cmp_ui(n, UWORD_MAX) <= 0)
{
if (fmpz_sgn(n) >= 0)
arb_bernoulli_ui(res, fmpz_get_ui(n), prec);
else
arb_zero(res);
}
else if (fmpz_is_odd(n))
{
arb_zero(res);
}
else
{
arb_t t;
slong wp;
arb_init(t);
wp = prec + 2 * fmpz_bits(n);
/* zeta(n) ~= 1 */
arf_one(arb_midref(res));
mag_one(arb_radref(res));
mag_mul_2exp_si(arb_radref(res), arb_radref(res), WORD_MIN);
/* |B_n| = 2 * n! / (2*pi)^n * zeta(n) */
arb_gamma_fmpz(t, n, wp);
arb_mul_fmpz(t, t, n, wp);
arb_mul(res, res, t, wp);
arb_const_pi(t, wp);
arb_mul_2exp_si(t, t, 1);
arb_pow_fmpz(t, t, n, wp);
arb_div(res, res, t, prec);
arb_mul_2exp_si(res, res, 1);
if (fmpz_fdiv_ui(n, 4) == 0)
arb_neg(res, res);
arb_clear(t);
}
}
