void
acb_get_mag(mag_t u, const acb_t z)
{
if (arb_is_zero(acb_imagref(z)))
{
arb_get_mag(u, acb_realref(z));
}
else if (arb_is_zero(acb_realref(z)))
{
arb_get_mag(u, acb_imagref(z));
}
else
{
mag_t v;
mag_init(v);
arb_get_mag(u, acb_realref(z));
arb_get_mag(v, acb_imagref(z));
mag_mul(u, u, u);
mag_addmul(u, v, v);
mag_sqrt(u, u);
mag_clear(v);
}
}
static void
acb_rising_get_mag2_right(mag_t bound, const arb_t a, const arb_t b, ulong n)
{
mag_t t, u;
ulong k;
mag_init(t);
mag_init(u);
arb_get_mag(t, a);
arb_get_mag(u, b);
mag_mul(bound, t, t);
mag_addmul(bound, u, u);
mag_set(u, bound);
mag_mul_2exp_si(t, t, 1);
for (k = 1; k < n; k++)
{
mag_add_ui_2exp_si(u, u, 2 * k - 1, 0);
mag_add(u, u, t);
mag_mul(bound, bound, u);
}
mag_clear(t);
mag_clear(u);
}
void
arb_mat_bound_inf_norm(mag_t b, const arb_mat_t A)
{
slong i, j, r, c;
mag_t s, t;
r = arb_mat_nrows(A);
c = arb_mat_ncols(A);
mag_zero(b);
if (r == 0 || c == 0)
return;
mag_init(s);
mag_init(t);
for (i = 0; i < r; i++)
{
mag_zero(s);
for (j = 0; j < c; j++)
{
arb_get_mag(t, arb_mat_entry(A, i, j));
mag_add(s, s, t);
}
mag_max(b, b, s);
}
mag_clear(s);
mag_clear(t);
}
static void
arb_sqrt1pm1_tiny(arb_t r, const arb_t z, slong prec)
{
mag_t b, c;
arb_t t;
mag_init(b);
mag_init(c);
arb_init(t);
/* if |z| < 1, then |(sqrt(1+z)-1) - (z/2-z^2/8)| <= |z|^3/(1-|z|)/16 */
arb_get_mag(b, z);
mag_one(c);
mag_sub_lower(c, c, b);
mag_pow_ui(b, b, 3);
mag_div(b, b, c);
mag_mul_2exp_si(b, b, -4);
arb_mul(t, z, z, prec);
arb_mul_2exp_si(t, t, -2);
arb_sub(r, z, t, prec);
arb_mul_2exp_si(r, r, -1);
if (mag_is_finite(b))
arb_add_error_mag(r, b);
else
arb_indeterminate(r);
mag_clear(b);
mag_clear(c);
arb_clear(t);
}
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
_acb_poly_zeta_cpx_series(acb_ptr z, const acb_t s, const acb_t a, int deflate, slong d, slong prec)
{
ulong M, N;
slong i;
mag_t bound;
arb_ptr vb;
int is_real, const_is_real;
if (d < 1)
return;
if (!acb_is_finite(s) || !acb_is_finite(a))
{
_acb_vec_indeterminate(z, d);
return;
}
is_real = const_is_real = 0;
if (acb_is_real(s) && acb_is_real(a))
{
if (arb_is_positive(acb_realref(a)))
{
is_real = const_is_real = 1;
}
else if (arb_is_int(acb_realref(a)) &&
arb_is_int(acb_realref(s)) &&
arb_is_nonpositive(acb_realref(s)))
{
const_is_real = 1;
}
}
mag_init(bound);
vb = _arb_vec_init(d);
_acb_poly_zeta_em_choose_param(bound, &N, &M, s, a, FLINT_MIN(d, 2), prec, MAG_BITS);
_acb_poly_zeta_em_bound(vb, s, a, N, M, d, MAG_BITS);
_acb_poly_zeta_em_sum(z, s, a, deflate, N, M, d, prec);
for (i = 0; i < d; i++)
{
arb_get_mag(bound, vb + i);
arb_add_error_mag(acb_realref(z + i), bound);
if (!is_real && !(i == 0 && const_is_real))
arb_add_error_mag(acb_imagref(z + i), bound);
}
mag_clear(bound);
_arb_vec_clear(vb, d);
}
void
_arb_sinc_derivative_bound(mag_t d, const arb_t x)
{
/* |f'(x)| < min(arb_get_mag(x), 1) / 2 */
mag_t r, one;
mag_init(r);
mag_init(one);
arb_get_mag(r, x);
mag_one(one);
mag_min(d, r, one);
mag_mul_2exp_si(d, d, -1);
mag_clear(r);
mag_clear(one);
}
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);
}
/* check accuracy compared to reference algorithm */
void arb_exp_simple(arb_t res, const arb_t x, slong prec)
{
mag_t t, u;
mag_init_set(t, arb_radref(x));
mag_init(u);
arf_set(arb_midref(res), arb_midref(x));
mag_zero(arb_radref(res));
arb_exp(res, x, prec);
mag_expm1(t, t);
arb_get_mag(u, res);
mag_addmul(arb_radref(res), t, u);
mag_clear(t);
mag_clear(u);
}
void
arb_hypgeom_infsum(arb_t P, arb_t Q, hypgeom_t hyp, long target_prec, long prec)
{
mag_t err, z;
long n;
mag_init(err);
mag_init(z);
mag_set_fmpz(z, hyp->P->coeffs + hyp->P->length - 1);
mag_div_fmpz(z, z, hyp->Q->coeffs + hyp->Q->length - 1);
if (!hyp->have_precomputed)
{
hypgeom_precompute(hyp);
hyp->have_precomputed = 1;
}
n = hypgeom_bound(err, hyp->r, hyp->boundC, hyp->boundD,
hyp->boundK, hyp->MK, z, target_prec);
arb_hypgeom_sum(P, Q, hyp, n, prec);
if (arf_sgn(arb_midref(Q)) < 0)
{
arb_neg(P, P);
arb_neg(Q, Q);
}
/* We have p/q = s + err i.e. (p + q*err)/q = s */
{
mag_t u;
mag_init(u);
arb_get_mag(u, Q);
mag_mul(u, u, err);
mag_add(arb_radref(P), arb_radref(P), u);
mag_clear(u);
}
mag_clear(z);
mag_clear(err);
}
void
_arb_poly_mullow_block(arb_ptr z, arb_srcptr x, slong xlen,
arb_srcptr y, slong ylen, slong n, slong prec)
{
slong xmlen, xrlen, ymlen, yrlen, i;
fmpz *xz, *yz, *zz;
fmpz *xe, *ye;
slong *xblocks, *yblocks;
int squaring;
fmpz_t scale, t;
xlen = FLINT_MIN(xlen, n);
ylen = FLINT_MIN(ylen, n);
squaring = (x == y) && (xlen == ylen);
/* Strip trailing zeros */
xmlen = xrlen = xlen;
while (xmlen > 0 && arf_is_zero(arb_midref(x + xmlen - 1))) xmlen--;
while (xrlen > 0 && mag_is_zero(arb_radref(x + xrlen - 1))) xrlen--;
if (squaring)
{
ymlen = xmlen;
yrlen = xrlen;
}
else
{
ymlen = yrlen = ylen;
while (ymlen > 0 && arf_is_zero(arb_midref(y + ymlen - 1))) ymlen--;
while (yrlen > 0 && mag_is_zero(arb_radref(y + yrlen - 1))) yrlen--;
}
/* We don't know how to deal with infinities or NaNs */
if (!_arb_vec_is_finite(x, xlen) ||
(!squaring && !_arb_vec_is_finite(y, ylen)))
{
_arb_poly_mullow_classical(z, x, xlen, y, ylen, n, prec);
return;
}
xlen = FLINT_MAX(xmlen, xrlen);
ylen = FLINT_MAX(ymlen, yrlen);
/* Start with the zero polynomial */
_arb_vec_zero(z, n);
/* Nothing to do */
if (xlen == 0 || ylen == 0)
return;
n = FLINT_MIN(n, xlen + ylen - 1);
fmpz_init(scale);
fmpz_init(t);
xz = _fmpz_vec_init(xlen);
yz = _fmpz_vec_init(ylen);
zz = _fmpz_vec_init(n);
xe = _fmpz_vec_init(xlen);
ye = _fmpz_vec_init(ylen);
xblocks = flint_malloc(sizeof(slong) * (xlen + 1));
yblocks = flint_malloc(sizeof(slong) * (ylen + 1));
_arb_poly_get_scale(scale, x, xlen, y, ylen);
/* Error propagation */
/* (xm + xr)*(ym + yr) = (xm*ym) + (xr*ym + xm*yr + xr*yr)
= (xm*ym) + (xm*yr + xr*(ym + yr)) */
if (xrlen != 0 || yrlen != 0)
{
mag_ptr tmp;
double *xdbl, *ydbl;
tmp = _mag_vec_init(FLINT_MAX(xlen, ylen));
xdbl = flint_malloc(sizeof(double) * xlen);
ydbl = flint_malloc(sizeof(double) * ylen);
/* (xm + xr)^2 = (xm*ym) + (xr^2 + 2 xm xr)
= (xm*ym) + xr*(2 xm + xr) */
if (squaring)
{
_mag_vec_get_fmpz_2exp_blocks(xz, xdbl, xe, xblocks, scale, x, NULL, xrlen);
for (i = 0; i < xlen; i++)
{
arf_get_mag(tmp + i, arb_midref(x + i));
mag_mul_2exp_si(tmp + i, tmp + i, 1);
mag_add(tmp + i, tmp + i, arb_radref(x + i));
}
_mag_vec_get_fmpz_2exp_blocks(yz, ydbl, ye, yblocks, scale, NULL, tmp, xlen);
_arb_poly_addmullow_rad(z, zz, xz, xdbl, xe, xblocks, xrlen, yz, ydbl, ye, yblocks, xlen, n);
}
else if (yrlen == 0)
{
/* xr * |ym| */
_mag_vec_get_fmpz_2exp_blocks(xz, xdbl, xe, xblocks, scale, x, NULL, xrlen);
for (i = 0; i < ymlen; i++)
arf_get_mag(tmp + i, arb_midref(y + i));
_mag_vec_get_fmpz_2exp_blocks(yz, ydbl, ye, yblocks, scale, NULL, tmp, ymlen);
_arb_poly_addmullow_rad(z, zz, xz, xdbl, xe, xblocks, xrlen, yz, ydbl, ye, yblocks, ymlen, n);
}
else
{
/* |xm| * yr */
for (i = 0; i < xmlen; i++)
arf_get_mag(tmp + i, arb_midref(x + i));
_mag_vec_get_fmpz_2exp_blocks(xz, xdbl, xe, xblocks, scale, NULL, tmp, xmlen);
_mag_vec_get_fmpz_2exp_blocks(yz, ydbl, ye, yblocks, scale, y, NULL, yrlen);
_arb_poly_addmullow_rad(z, zz, xz, xdbl, xe, xblocks, xmlen, yz, ydbl, ye, yblocks, yrlen, n);
/* xr*(|ym| + yr) */
if (xrlen != 0)
{
_mag_vec_get_fmpz_2exp_blocks(xz, xdbl, xe, xblocks, scale, x, NULL, xrlen);
for (i = 0; i < ylen; i++)
arb_get_mag(tmp + i, y + i);
_mag_vec_get_fmpz_2exp_blocks(yz, ydbl, ye, yblocks, scale, NULL, tmp, ylen);
_arb_poly_addmullow_rad(z, zz, xz, xdbl, xe, xblocks, xrlen, yz, ydbl, ye, yblocks, ylen, n);
}
}
_mag_vec_clear(tmp, FLINT_MAX(xlen, ylen));
flint_free(xdbl);
flint_free(ydbl);
}
/* multiply midpoints */
if (xmlen != 0 && ymlen != 0)
{
_arb_vec_get_fmpz_2exp_blocks(xz, xe, xblocks, scale, x, xmlen, prec);
if (squaring)
{
_arb_poly_addmullow_block(z, zz, xz, xe, xblocks, xmlen, xz, xe, xblocks, xmlen, n, prec, 1);
}
else
{
_arb_vec_get_fmpz_2exp_blocks(yz, ye, yblocks, scale, y, ymlen, prec);
_arb_poly_addmullow_block(z, zz, xz, xe, xblocks, xmlen, yz, ye, yblocks, ymlen, n, prec, 0);
}
}
/* Unscale. */
if (!fmpz_is_zero(scale))
{
fmpz_zero(t);
for (i = 0; i < n; i++)
{
arb_mul_2exp_fmpz(z + i, z + i, t);
fmpz_add(t, t, scale);
}
}
_fmpz_vec_clear(xz, xlen);
_fmpz_vec_clear(yz, ylen);
_fmpz_vec_clear(zz, n);
_fmpz_vec_clear(xe, xlen);
_fmpz_vec_clear(ye, ylen);
flint_free(xblocks);
flint_free(yblocks);
fmpz_clear(scale);
fmpz_clear(t);
}
int main(int argc, char *argv[])
{
acb_t s, t, a, b;
mag_t tol;
slong prec, goal;
slong N;
ulong k;
int integral, ifrom, ito;
int i, twice, havegoal, havetol;
acb_calc_integrate_opt_t options;
ifrom = ito = -1;
for (i = 1; i < argc; i++)
{
if (!strcmp(argv[i], "-i"))
{
if (!strcmp(argv[i+1], "all"))
{
ifrom = 0;
ito = NUM_INTEGRALS - 1;
}
else
{
ifrom = ito = atol(argv[i+1]);
if (ito < 0 || ito >= NUM_INTEGRALS)
flint_abort();
}
}
}
if (ifrom == -1)
{
flint_printf("Compute integrals using acb_calc_integrate.\n");
flint_printf("Usage: integrals -i n [-prec p] [-tol eps] [-twice] [...]\n\n");
flint_printf("-i n - compute integral n (0 <= n <= %d), or \"-i all\"\n", NUM_INTEGRALS - 1);
flint_printf("-prec p - precision in bits (default p = 64)\n");
flint_printf("-goal p - approximate relative accuracy goal (default p)\n");
flint_printf("-tol eps - approximate absolute error goal (default 2^-p)\n");
flint_printf("-twice - run twice (to see overhead of computing nodes)\n");
flint_printf("-heap - use heap for subinterval queue\n");
flint_printf("-verbose - show information\n");
flint_printf("-verbose2 - show more information\n");
flint_printf("-deg n - use quadrature degree up to n\n");
flint_printf("-eval n - limit number of function evaluations to n\n");
flint_printf("-depth n - limit subinterval queue size to n\n\n");
flint_printf("Implemented integrals:\n");
for (integral = 0; integral < NUM_INTEGRALS; integral++)
flint_printf("I%d = %s\n", integral, descr[integral]);
flint_printf("\n");
return 1;
}
acb_calc_integrate_opt_init(options);
prec = 64;
twice = 0;
goal = 0;
havetol = havegoal = 0;
acb_init(a);
acb_init(b);
acb_init(s);
acb_init(t);
mag_init(tol);
for (i = 1; i < argc; i++)
{
if (!strcmp(argv[i], "-prec"))
{
prec = atol(argv[i+1]);
}
else if (!strcmp(argv[i], "-twice"))
{
twice = 1;
}
else if (!strcmp(argv[i], "-goal"))
{
goal = atol(argv[i+1]);
if (goal < 0)
{
flint_printf("expected goal >= 0\n");
return 1;
}
havegoal = 1;
}
else if (!strcmp(argv[i], "-tol"))
{
arb_t x;
arb_init(x);
arb_set_str(x, argv[i+1], 10);
arb_get_mag(tol, x);
arb_clear(x);
havetol = 1;
}
else if (!strcmp(argv[i], "-deg"))
{
options->deg_limit = atol(argv[i+1]);
}
else if (!strcmp(argv[i], "-eval"))
{
options->eval_limit = atol(argv[i+1]);
}
else if (!strcmp(argv[i], "-depth"))
{
options->depth_limit = atol(argv[i+1]);
}
else if (!strcmp(argv[i], "-verbose"))
{
options->verbose = 1;
}
else if (!strcmp(argv[i], "-verbose2"))
{
options->verbose = 2;
}
else if (!strcmp(argv[i], "-heap"))
{
options->use_heap = 1;
}
}
if (!havegoal)
goal = prec;
if (!havetol)
mag_set_ui_2exp_si(tol, 1, -prec);
for (integral = ifrom; integral <= ito; integral++)
{
flint_printf("I%d = %s ...\n", integral, descr[integral]);
for (i = 0; i < 1 + twice; i++)
{
TIMEIT_ONCE_START
switch (integral)
{
case 0:
acb_set_d(a, 0);
acb_set_d(b, 100);
acb_calc_integrate(s, f_sin, NULL, a, b, goal, tol, options, prec);
break;
case 1:
acb_set_d(a, 0);
acb_set_d(b, 1);
acb_calc_integrate(s, f_atanderiv, NULL, a, b, goal, tol, options, prec);
acb_mul_2exp_si(s, s, 2);
break;
case 2:
acb_set_d(a, 0);
acb_one(b);
acb_mul_2exp_si(b, b, goal);
acb_calc_integrate(s, f_atanderiv, NULL, a, b, goal, tol, options, prec);
arb_add_error_2exp_si(acb_realref(s), -goal);
acb_mul_2exp_si(s, s, 1);
break;
case 3:
acb_set_d(a, 0);
acb_set_d(b, 1);
acb_calc_integrate(s, f_circle, NULL, a, b, goal, tol, options, prec);
acb_mul_2exp_si(s, s, 2);
break;
case 4:
acb_set_d(a, 0);
acb_set_d(b, 8);
acb_calc_integrate(s, f_rump, NULL, a, b, goal, tol, options, prec);
break;
case 5:
acb_set_d(a, 1);
acb_set_d(b, 101);
acb_calc_integrate(s, f_floor, NULL, a, b, goal, tol, options, prec);
break;
case 6:
acb_set_d(a, 0);
acb_set_d(b, 1);
acb_calc_integrate(s, f_helfgott, NULL, a, b, goal, tol, options, prec);
break;
case 7:
acb_zero(s);
acb_set_d_d(a, -1.0, -1.0);
acb_set_d_d(b, 2.0, -1.0);
acb_calc_integrate(t, f_zeta, NULL, a, b, goal, tol, options, prec);
acb_add(s, s, t, prec);
acb_set_d_d(a, 2.0, -1.0);
acb_set_d_d(b, 2.0, 1.0);
acb_calc_integrate(t, f_zeta, NULL, a, b, goal, tol, options, prec);
acb_add(s, s, t, prec);
acb_set_d_d(a, 2.0, 1.0);
acb_set_d_d(b, -1.0, 1.0);
acb_calc_integrate(t, f_zeta, NULL, a, b, goal, tol, options, prec);
acb_add(s, s, t, prec);
acb_set_d_d(a, -1.0, 1.0);
acb_set_d_d(b, -1.0, -1.0);
acb_calc_integrate(t, f_zeta, NULL, a, b, goal, tol, options, prec);
acb_add(s, s, t, prec);
acb_const_pi(t, prec);
acb_div(s, s, t, prec);
acb_mul_2exp_si(s, s, -1);
acb_div_onei(s, s);
break;
case 8:
acb_set_d(a, 0);
acb_set_d(b, 1);
acb_calc_integrate(s, f_essing, NULL, a, b, goal, tol, options, prec);
break;
case 9:
acb_set_d(a, 0);
acb_set_d(b, 1);
acb_calc_integrate(s, f_essing2, NULL, a, b, goal, tol, options, prec);
break;
case 10:
acb_set_d(a, 0);
acb_set_d(b, 10000);
acb_calc_integrate(s, f_factorial1000, NULL, a, b, goal, tol, options, prec);
break;
case 11:
acb_set_d_d(a, 1.0, 0.0);
acb_set_d_d(b, 1.0, 1000.0);
acb_calc_integrate(s, f_gamma, NULL, a, b, goal, tol, options, prec);
break;
case 12:
acb_set_d(a, -10.0);
acb_set_d(b, 10.0);
acb_calc_integrate(s, f_sin_plus_small, NULL, a, b, goal, tol, options, prec);
break;
case 13:
acb_set_d(a, -1020.0);
acb_set_d(b, -1010.0);
acb_calc_integrate(s, f_exp, NULL, a, b, goal, tol, options, prec);
break;
case 14:
acb_set_d(a, 0);
acb_set_d(b, ceil(sqrt(goal * 0.693147181) + 1.0));
acb_calc_integrate(s, f_gaussian, NULL, a, b, goal, tol, options, prec);
acb_mul(b, b, b, prec);
acb_neg(b, b);
acb_exp(b, b, prec);
arb_add_error(acb_realref(s), acb_realref(b));
break;
case 15:
acb_set_d(a, 0.0);
acb_set_d(b, 1.0);
acb_calc_integrate(s, f_spike, NULL, a, b, goal, tol, options, prec);
break;
case 16:
acb_set_d(a, 0.0);
acb_set_d(b, 8.0);
acb_calc_integrate(s, f_monster, NULL, a, b, goal, tol, options, prec);
break;
case 17:
acb_set_d(a, 0);
acb_set_d(b, ceil(goal * 0.693147181 + 1.0));
acb_calc_integrate(s, f_sech, NULL, a, b, goal, tol, options, prec);
acb_neg(b, b);
acb_exp(b, b, prec);
acb_mul_2exp_si(b, b, 1);
arb_add_error(acb_realref(s), acb_realref(b));
break;
case 18:
acb_set_d(a, 0);
acb_set_d(b, ceil(goal * 0.693147181 / 3.0 + 2.0));
acb_calc_integrate(s, f_sech3, NULL, a, b, goal, tol, options, prec);
acb_neg(b, b);
acb_mul_ui(b, b, 3, prec);
acb_exp(b, b, prec);
acb_mul_2exp_si(b, b, 3);
acb_div_ui(b, b, 3, prec);
arb_add_error(acb_realref(s), acb_realref(b));
break;
case 19:
if (goal < 0)
abort();
/* error bound 2^-N (1+N) when truncated at 2^-N */
N = goal + FLINT_BIT_COUNT(goal);
acb_one(a);
acb_mul_2exp_si(a, a, -N);
acb_one(b);
acb_calc_integrate(s, f_log_div1p, NULL, a, b, goal, tol, options, prec);
acb_set_ui(b, N + 1);
acb_mul_2exp_si(b, b, -N);
arb_add_error(acb_realref(s), acb_realref(b));
break;
case 20:
if (goal < 0)
abort();
/* error bound (N+1) exp(-N) when truncated at N */
N = goal + FLINT_BIT_COUNT(goal);
acb_zero(a);
acb_set_ui(b, N);
acb_calc_integrate(s, f_log_div1p_transformed, NULL, a, b, goal, tol, options, prec);
acb_neg(b, b);
acb_exp(b, b, prec);
acb_mul_ui(b, b, N + 1, prec);
arb_add_error(acb_realref(s), acb_realref(b));
break;
case 21:
acb_zero(s);
N = 10;
acb_set_d_d(a, 0.5, -0.5);
acb_set_d_d(b, 0.5, 0.5);
acb_calc_integrate(t, f_elliptic_p_laurent_n, &N, a, b, goal, tol, options, prec);
acb_add(s, s, t, prec);
acb_set_d_d(a, 0.5, 0.5);
acb_set_d_d(b, -0.5, 0.5);
acb_calc_integrate(t, f_elliptic_p_laurent_n, &N, a, b, goal, tol, options, prec);
acb_add(s, s, t, prec);
acb_set_d_d(a, -0.5, 0.5);
acb_set_d_d(b, -0.5, -0.5);
acb_calc_integrate(t, f_elliptic_p_laurent_n, &N, a, b, goal, tol, options, prec);
acb_add(s, s, t, prec);
acb_set_d_d(a, -0.5, -0.5);
acb_set_d_d(b, 0.5, -0.5);
acb_calc_integrate(t, f_elliptic_p_laurent_n, &N, a, b, goal, tol, options, prec);
acb_add(s, s, t, prec);
acb_const_pi(t, prec);
acb_div(s, s, t, prec);
acb_mul_2exp_si(s, s, -1);
acb_div_onei(s, s);
break;
case 22:
acb_zero(s);
N = 1000;
acb_set_d_d(a, 100.0, 0.0);
acb_set_d_d(b, 100.0, N);
acb_calc_integrate(t, f_zeta_frac, NULL, a, b, goal, tol, options, prec);
acb_add(s, s, t, prec);
acb_set_d_d(a, 100, N);
acb_set_d_d(b, 0.5, N);
acb_calc_integrate(t, f_zeta_frac, NULL, a, b, goal, tol, options, prec);
acb_add(s, s, t, prec);
acb_div_onei(s, s);
arb_zero(acb_imagref(s));
acb_set_ui(t, N);
acb_dirichlet_hardy_theta(t, t, NULL, NULL, 1, prec);
acb_add(s, s, t, prec);
acb_const_pi(t, prec);
acb_div(s, s, t, prec);
acb_add_ui(s, s, 1, prec);
break;
case 23:
acb_set_d(a, 0.0);
acb_set_d(b, 1000.0);
acb_calc_integrate(s, f_lambertw, NULL, a, b, goal, tol, options, prec);
break;
case 24:
acb_set_d(a, 0.0);
acb_const_pi(b, prec);
acb_calc_integrate(s, f_max_sin_cos, NULL, a, b, goal, tol, options, prec);
break;
case 25:
acb_set_si(a, -1);
acb_set_si(b, 1);
acb_calc_integrate(s, f_erf_bent, NULL, a, b, goal, tol, options, prec);
break;
case 26:
acb_set_si(a, -10);
acb_set_si(b, 10);
acb_calc_integrate(s, f_airy_ai, NULL, a, b, goal, tol, options, prec);
break;
case 27:
acb_set_si(a, 0);
acb_set_si(b, 10);
acb_calc_integrate(s, f_horror, NULL, a, b, goal, tol, options, prec);
break;
case 28:
acb_set_d_d(a, -1, -1);
acb_set_d_d(b, -1, 1);
acb_calc_integrate(s, f_sqrt, NULL, a, b, goal, tol, options, prec);
break;
case 29:
acb_set_d(a, 0);
acb_set_d(b, ceil(sqrt(goal * 0.693147181) + 1.0));
acb_calc_integrate(s, f_gaussian_twist, NULL, a, b, goal, tol, options, prec);
acb_mul(b, b, b, prec);
acb_neg(b, b);
acb_exp(b, b, prec);
arb_add_error(acb_realref(s), acb_realref(b));
arb_add_error(acb_imagref(s), acb_realref(b));
break;
case 30:
acb_set_d(a, 0);
acb_set_d(b, ceil(goal * 0.693147181 + 1.0));
acb_calc_integrate(s, f_exp_airy, NULL, a, b, goal, tol, options, prec);
acb_neg(b, b);
acb_exp(b, b, prec);
acb_mul_2exp_si(b, b, 1);
arb_add_error(acb_realref(s), acb_realref(b));
break;
case 31:
acb_zero(a);
acb_const_pi(b, prec);
acb_calc_integrate(s, f_sin_cos_frac, NULL, a, b, goal, tol, options, prec);
break;
case 32:
acb_zero(a);
acb_set_ui(b, 3);
acb_calc_integrate(s, f_sin_near_essing, NULL, a, b, goal, tol, options, prec);
break;
case 33:
acb_zero(a);
acb_zero(b);
k = 3;
scaled_bessel_select_N(acb_realref(b), k, prec);
acb_calc_integrate(s, f_scaled_bessel, &k, a, b, goal, tol, options, prec);
scaled_bessel_tail_bound(acb_realref(a), k, acb_realref(b), prec);
arb_add_error(acb_realref(s), acb_realref(a));
break;
case 34:
acb_zero(a);
acb_zero(b);
k = 15;
scaled_bessel_select_N(acb_realref(b), k, prec);
acb_calc_integrate(s, f_scaled_bessel, &k, a, b, goal, tol, options, prec);
scaled_bessel_tail_bound(acb_realref(a), k, acb_realref(b), prec);
arb_add_error(acb_realref(s), acb_realref(a));
break;
case 35:
acb_set_d_d(a, -1, -1);
acb_set_d_d(b, -1, 1);
acb_calc_integrate(s, f_rsqrt, NULL, a, b, goal, tol, options, prec);
break;
default:
abort();
}
TIMEIT_ONCE_STOP
}
flint_printf("I%d = ", integral);
acb_printn(s, 3.333 * prec, 0);
flint_printf("\n\n");
}
acb_clear(a);
acb_clear(b);
acb_clear(s);
acb_clear(t);
mag_clear(tol);
flint_cleanup();
return 0;
}
/* derivatives: |8/sqrt(pi) sin(2z^2)|, |8/sqrt(pi) cos(2z^2)| <= 5 exp(4|xy|) */
void
acb_hypgeom_fresnel_erf_error(acb_t res1, acb_t res2, const acb_t z, slong prec)
{
mag_t re;
mag_t im;
acb_t zmid;
mag_init(re);
mag_init(im);
acb_init(zmid);
if (arf_cmpabs_ui(arb_midref(acb_realref(z)), 1000) < 0 &&
arf_cmpabs_ui(arb_midref(acb_imagref(z)), 1000) < 0)
{
arb_get_mag(re, acb_realref(z));
arb_get_mag(im, acb_imagref(z));
mag_mul(re, re, im);
mag_mul_2exp_si(re, re, 2);
mag_exp(re, re);
mag_mul_ui(re, re, 5);
}
else
{
arb_t t;
arb_init(t);
arb_mul(t, acb_realref(z), acb_imagref(z), prec);
arb_abs(t, t);
arb_mul_2exp_si(t, t, 2);
arb_exp(t, t, prec);
arb_get_mag(re, t);
mag_mul_ui(re, re, 5);
arb_clear(t);
}
mag_hypot(im, arb_radref(acb_realref(z)), arb_radref(acb_imagref(z)));
mag_mul(re, re, im);
if (arb_is_zero(acb_imagref(z)))
{
mag_set_ui(im, 8); /* For real x, |S(x)| < 4, |C(x)| < 4. */
mag_min(re, re, im);
mag_zero(im);
}
else if (arb_is_zero(acb_realref(z)))
{
mag_set_ui(im, 8);
mag_min(im, re, im);
mag_zero(re);
}
else
{
mag_set(im, re);
}
arf_set(arb_midref(acb_realref(zmid)), arb_midref(acb_realref(z)));
arf_set(arb_midref(acb_imagref(zmid)), arb_midref(acb_imagref(z)));
acb_hypgeom_fresnel_erf(res1, res2, zmid, prec);
if (res1 != NULL)
{
arb_add_error_mag(acb_realref(res1), re);
arb_add_error_mag(acb_imagref(res1), im);
}
if (res2 != NULL)
{
arb_add_error_mag(acb_realref(res2), re);
arb_add_error_mag(acb_imagref(res2), im);
}
mag_clear(re);
mag_clear(im);
acb_clear(zmid);
}
void
_arb_bell_sum_taylor(arb_t res, const fmpz_t n,
const fmpz_t a, const fmpz_t b, const fmpz_t mmag, long tol)
{
fmpz_t m, r, R, tmp;
mag_t B, C, D, bound;
arb_t t, u;
long wp, k, N;
if (_fmpz_sub_small(b, a) < 5)
{
arb_bell_sum_bsplit(res, n, a, b, mmag, tol);
return;
}
fmpz_init(m);
fmpz_init(r);
fmpz_init(R);
fmpz_init(tmp);
/* r = max(m - a, b - m) */
/* m = a + (b - a) / 2 */
fmpz_sub(r, b, a);
fmpz_cdiv_q_2exp(r, r, 1);
fmpz_add(m, a, r);
fmpz_mul_2exp(R, r, RADIUS_BITS);
mag_init(B);
mag_init(C);
mag_init(D);
mag_init(bound);
arb_init(t);
arb_init(u);
if (fmpz_cmp(R, m) >= 0)
{
mag_inf(C);
mag_inf(D);
}
else
{
/* C = exp(R * |F'(m)| + (1/2) R^2 * (n/(m-R)^2 + 1/(m-R))) */
/* C = exp(R * (|F'(m)| + (1/2) R * (n/(m-R) + 1)/(m-R))) */
/* D = (1/2) R * (n/(m-R) + 1)/(m-R) */
fmpz_sub(tmp, m, R);
mag_set_fmpz(D, n);
mag_div_fmpz(D, D, tmp);
mag_one(C);
mag_add(D, D, C);
mag_div_fmpz(D, D, tmp);
mag_mul_fmpz(D, D, R);
mag_mul_2exp_si(D, D, -1);
/* C = |F'(m)| */
wp = 20 + 1.05 * fmpz_bits(n);
arb_set_fmpz(t, n);
arb_div_fmpz(t, t, m, wp);
fmpz_add_ui(tmp, m, 1);
arb_set_fmpz(u, tmp);
arb_digamma(u, u, wp);
arb_sub(t, t, u, wp);
arb_get_mag(C, t);
/* C = exp(R * (C + D)) */
mag_add(C, C, D);
mag_mul_fmpz(C, C, R);
mag_exp(C, C);
}
if (mag_cmp_2exp_si(C, tol / 4 + 2) > 0)
{
_arb_bell_sum_taylor(res, n, a, m, mmag, tol);
_arb_bell_sum_taylor(t, n, m, b, mmag, tol);
arb_add(res, res, t, 2 * tol);
}
else
{
arb_ptr mx, ser1, ser2, ser3;
/* D = T(m) */
wp = 20 + 1.05 * fmpz_bits(n);
arb_set_fmpz(t, m);
arb_pow_fmpz(t, t, n, wp);
fmpz_add_ui(tmp, m, 1);
arb_gamma_fmpz(u, tmp, wp);
arb_div(t, t, u, wp);
arb_get_mag(D, t);
/* error bound: (b-a) * C * D * B^N / (1 - B), B = r/R */
/* ((b-a) * C * D * 2) * 2^(-N*RADIUS_BITS) */
/* ((b-a) * C * D * 2) */
mag_mul(bound, C, D);
mag_mul_2exp_si(bound, bound, 1);
fmpz_sub(tmp, b, a);
mag_mul_fmpz(bound, bound, tmp);
/* N = (tol + log2((b-a)*C*D*2) - mmag) / RADIUS_BITS */
if (mmag == NULL)
{
/* estimate D ~= 2^mmag */
fmpz_add_ui(tmp, MAG_EXPREF(C), tol);
fmpz_cdiv_q_ui(tmp, tmp, RADIUS_BITS);
}
else
{
fmpz_sub(tmp, MAG_EXPREF(bound), mmag);
fmpz_add_ui(tmp, tmp, tol);
fmpz_cdiv_q_ui(tmp, tmp, RADIUS_BITS);
}
if (fmpz_cmp_ui(tmp, 5 * tol / 4) > 0)
N = 5 * tol / 4;
else if (fmpz_cmp_ui(tmp, 2) < 0)
N = 2;
else
N = fmpz_get_ui(tmp);
/* multiply by 2^(-N*RADIUS_BITS) */
mag_mul_2exp_si(bound, bound, -N * RADIUS_BITS);
mx = _arb_vec_init(2);
ser1 = _arb_vec_init(N);
ser2 = _arb_vec_init(N);
ser3 = _arb_vec_init(N);
/* estimate (this should work for moderate n and tol) */
wp = 1.1 * tol + 1.05 * fmpz_bits(n) + 5;
/* increase precision until convergence */
while (1)
{
/* (m+x)^n / gamma(m+1+x) */
arb_set_fmpz(mx, m);
arb_one(mx + 1);
_arb_poly_log_series(ser1, mx, 2, N, wp);
for (k = 0; k < N; k++)
arb_mul_fmpz(ser1 + k, ser1 + k, n, wp);
arb_add_ui(mx, mx, 1, wp);
_arb_poly_lgamma_series(ser2, mx, 2, N, wp);
_arb_vec_sub(ser1, ser1, ser2, N, wp);
_arb_poly_exp_series(ser3, ser1, N, N, wp);
/* t = a - m, u = b - m */
arb_set_fmpz(t, a);
arb_sub_fmpz(t, t, m, wp);
arb_set_fmpz(u, b);
arb_sub_fmpz(u, u, m, wp);
arb_power_sum_vec(ser1, t, u, N, wp);
arb_zero(res);
for (k = 0; k < N; k++)
arb_addmul(res, ser3 + k, ser1 + k, wp);
if (mmag != NULL)
{
if (_fmpz_sub_small(MAG_EXPREF(arb_radref(res)), mmag) <= -tol)
break;
}
else
{
if (arb_rel_accuracy_bits(res) >= tol)
break;
}
wp = 2 * wp;
}
/* add the series truncation bound */
arb_add_error_mag(res, bound);
_arb_vec_clear(mx, 2);
_arb_vec_clear(ser1, N);
_arb_vec_clear(ser2, N);
_arb_vec_clear(ser3, N);
}
mag_clear(B);
mag_clear(C);
mag_clear(D);
mag_clear(bound);
arb_clear(t);
arb_clear(u);
fmpz_clear(m);
fmpz_clear(r);
fmpz_clear(R);
fmpz_clear(tmp);
}
int main()
{
slong iter;
flint_rand_t state;
flint_printf("approx_dot....");
fflush(stdout);
flint_randinit(state);
for (iter = 0; iter < 100000 * arb_test_multiplier(); iter++)
{
arb_ptr x, y;
arb_t s1, s2, z;
slong i, len, prec, xbits, ybits, ebits;
int initial, subtract, revx, revy;
if (n_randint(state, 100) == 0)
len = n_randint(state, 100);
else if (n_randint(state, 10) == 0)
len = n_randint(state, 10);
else
len = n_randint(state, 3);
if (n_randint(state, 10) != 0 || len > 10)
{
prec = 2 + n_randint(state, 500);
xbits = 2 + n_randint(state, 500);
ybits = 2 + n_randint(state, 500);
}
else
{
prec = 2 + n_randint(state, 4000);
xbits = 2 + n_randint(state, 4000);
ybits = 2 + n_randint(state, 4000);
}
if (n_randint(state, 100) == 0)
ebits = 2 + n_randint(state, 100);
else
ebits = 2 + n_randint(state, 10);
initial = n_randint(state, 2);
subtract = n_randint(state, 2);
revx = n_randint(state, 2);
revy = n_randint(state, 2);
x = _arb_vec_init(len);
y = _arb_vec_init(len);
arb_init(s1);
arb_init(s2);
arb_init(z);
switch (n_randint(state, 3))
{
case 0:
for (i = 0; i < len; i++)
{
arb_randtest(x + i, state, xbits, ebits);
arb_randtest(y + i, state, ybits, ebits);
}
break;
/* Test with cancellation */
case 1:
for (i = 0; i < len; i++)
{
if (i <= len / 2)
{
arb_randtest(x + i, state, xbits, ebits);
arb_randtest(y + i, state, ybits, ebits);
}
else
{
arb_neg(x + i, x + len - i - 1);
arb_set(y + i, y + len - i - 1);
}
}
break;
default:
for (i = 0; i < len; i++)
{
if (i <= len / 2)
{
arb_randtest(x + i, state, xbits, ebits);
arb_randtest(y + i, state, ybits, ebits);
}
else
{
arb_neg_round(x + i, x + len - i - 1, 2 + n_randint(state, 500));
arb_set_round(y + i, y + len - i - 1, 2 + n_randint(state, 500));
}
}
break;
}
arb_randtest(s1, state, 200, 100);
arb_randtest(s2, state, 200, 100);
arb_randtest(z, state, xbits, ebits);
arb_approx_dot(s1, initial ? z : NULL, subtract,
revx ? (x + len - 1) : x, revx ? -1 : 1,
revy ? (y + len - 1) : y, revy ? -1 : 1,
len, prec);
mag_zero(arb_radref(s1));
/* With the fast algorithm, we expect identical results when
reversing the vectors. */
if (ebits <= 12)
{
arb_approx_dot(s2, initial ? z : NULL, subtract,
!revx ? (x + len - 1) : x, !revx ? -1 : 1,
!revy ? (y + len - 1) : y, !revy ? -1 : 1,
len, prec);
mag_zero(arb_radref(s2));
if (!arb_equal(s1, s2))
{
flint_printf("FAIL (reversal)\n\n");
flint_printf("iter = %wd, len = %wd, prec = %wd, ebits = %wd\n\n", iter, len, prec, ebits);
if (initial)
{
flint_printf("z = ", i); arb_printn(z, 100, ARB_STR_MORE); flint_printf(" (%wd)\n\n", arb_bits(z));
}
for (i = 0; i < len; i++)
{
flint_printf("x[%wd] = ", i); arb_printn(x + i, 100, ARB_STR_MORE); flint_printf(" (%wd)\n", arb_bits(x + i));
flint_printf("y[%wd] = ", i); arb_printn(y + i, 100, ARB_STR_MORE); flint_printf(" (%wd)\n", arb_bits(y + i));
}
flint_printf("\n\n");
flint_printf("s1 = "); arb_printn(s1, 100, ARB_STR_MORE); flint_printf("\n\n");
flint_printf("s2 = "); arb_printn(s2, 100, ARB_STR_MORE); flint_printf("\n\n");
flint_abort();
}
}
/* Verify that radii are ignored */
for (i = 0; i < len; i++)
{
arb_get_mid_arb(x + i, x + i);
arb_get_mid_arb(y + i, y + i);
}
arb_get_mid_arb(z, z);
arb_approx_dot(s2, initial ? z : NULL, subtract,
revx ? (x + len - 1) : x, revx ? -1 : 1,
revy ? (y + len - 1) : y, revy ? -1 : 1,
len, prec);
mag_zero(arb_radref(s2));
if (!arb_equal(s1, s2))
{
flint_printf("FAIL (radii)\n\n");
flint_printf("iter = %wd, len = %wd, prec = %wd, ebits = %wd\n\n", iter, len, prec, ebits);
if (initial)
{
flint_printf("z = ", i); arb_printn(z, 100, ARB_STR_MORE); flint_printf(" (%wd)\n\n", arb_bits(z));
}
for (i = 0; i < len; i++)
{
flint_printf("x[%wd] = ", i); arb_printn(x + i, 100, ARB_STR_MORE); flint_printf(" (%wd)\n", arb_bits(x + i));
flint_printf("y[%wd] = ", i); arb_printn(y + i, 100, ARB_STR_MORE); flint_printf(" (%wd)\n", arb_bits(y + i));
}
flint_printf("\n\n");
flint_printf("s1 = "); arb_printn(s1, 100, ARB_STR_MORE); flint_printf("\n\n");
flint_printf("s2 = "); arb_printn(s2, 100, ARB_STR_MORE); flint_printf("\n\n");
flint_abort();
}
/* Compare with arb_dot */
arb_approx_dot(s2, initial ? z : NULL, subtract,
revx ? (x + len - 1) : x, revx ? -1 : 1,
revy ? (y + len - 1) : y, revy ? -1 : 1,
len, prec);
{
mag_t err, xx, yy;
mag_init(err);
mag_init(xx);
mag_init(yy);
if (initial)
arb_get_mag(err, z);
for (i = 0; i < len; i++)
{
arb_get_mag(xx, revx ? x + len - 1 - i : x + i);
arb_get_mag(yy, revx ? y + len - 1 - i : y + i);
mag_addmul(err, xx, yy);
}
mag_mul_2exp_si(err, err, -prec + 2);
arb_add_error_mag(s2, err);
if (!arb_contains(s2, s1))
{
flint_printf("FAIL (inclusion)\n\n");
flint_printf("iter = %wd, len = %wd, prec = %wd, ebits = %wd\n\n", iter, len, prec, ebits);
if (initial)
{
flint_printf("z = ", i); arb_printn(z, 100, ARB_STR_MORE); flint_printf(" (%wd)\n\n", arb_bits(z));
}
for (i = 0; i < len; i++)
{
flint_printf("x[%wd] = ", i); arb_printn(x + i, 100, ARB_STR_MORE); flint_printf(" (%wd)\n", arb_bits(x + i));
flint_printf("y[%wd] = ", i); arb_printn(y + i, 100, ARB_STR_MORE); flint_printf(" (%wd)\n", arb_bits(y + i));
}
flint_printf("\n\n");
flint_printf("s1 = "); arb_printn(s1, 100, ARB_STR_MORE); flint_printf("\n\n");
flint_printf("s2 = "); arb_printn(s2, 100, ARB_STR_MORE); flint_printf("\n\n");
flint_abort();
}
mag_clear(err);
mag_clear(xx);
mag_clear(yy);
}
arb_clear(s1);
arb_clear(s2);
arb_clear(z);
_arb_vec_clear(x, len);
_arb_vec_clear(y, len);
}
flint_randclear(state);
flint_cleanup();
flint_printf("PASS\n");
return EXIT_SUCCESS;
}
int main()
{
slong iter;
flint_rand_t state;
flint_printf("root_bound_fujiwara....");
fflush(stdout);
flint_randinit(state);
for (iter = 0; iter < 10000 * arb_test_multiplier(); iter++)
{
arb_poly_t a;
arb_ptr roots;
arb_t t;
mag_t mag1, mag2;
slong i, deg, prec;
prec = 10 + n_randint(state, 400);
deg = n_randint(state, 10);
arb_init(t);
arb_poly_init(a);
mag_init(mag1);
mag_init(mag2);
roots = _arb_vec_init(deg);
for (i = 0; i < deg; i++)
arb_randtest(roots + i, state, prec, 1 + n_randint(state, 20));
arb_poly_product_roots(a, roots, deg, prec);
arb_randtest(t, state, prec, 1 + n_randint(state, 20));
arb_poly_scalar_mul(a, a, t, prec);
arb_poly_root_bound_fujiwara(mag1, a);
for (i = 0; i < deg; i++)
{
arb_get_mag(mag2, roots + i);
/* arb_get_mag gives an upper bound which due to rounding
could be larger than mag1, so we pick a slightly
smaller number */
mag_mul_ui(mag2, mag2, 10000);
mag_div_ui(mag2, mag2, 10001);
if (mag_cmp(mag2, mag1) > 0)
{
flint_printf("FAIL\n");
flint_printf("a = "); arb_poly_printd(a, 15); flint_printf("\n\n");
flint_printf("root = "); arb_printd(roots + i, 15); flint_printf("\n\n");
flint_printf("mag1 = "); mag_printd(mag1, 10); flint_printf("\n\n");
flint_printf("mag2 = "); mag_printd(mag2, 10); flint_printf("\n\n");
abort();
}
}
_arb_vec_clear(roots, deg);
arb_clear(t);
arb_poly_clear(a);
mag_clear(mag1);
mag_clear(mag2);
}
flint_randclear(state);
flint_cleanup();
flint_printf("PASS\n");
return EXIT_SUCCESS;
}
void
acb_rising_ui_get_mag(mag_t bound, const acb_t s, ulong n)
{
if (n == 0)
{
mag_one(bound);
return;
}
if (n == 1)
{
acb_get_mag(bound, s);
return;
}
if (!acb_is_finite(s))
{
mag_inf(bound);
return;
}
if (arf_sgn(arb_midref(acb_realref(s))) >= 0)
{
acb_rising_get_mag2_right(bound, acb_realref(s), acb_imagref(s), n);
}
else
{
arb_t a;
long k;
mag_t bound2, t, u;
arb_init(a);
mag_init(bound2);
mag_init(t);
mag_init(u);
arb_get_mag(u, acb_imagref(s));
mag_mul(u, u, u);
mag_one(bound);
for (k = 0; k < n; k++)
{
arb_add_ui(a, acb_realref(s), k, MAG_BITS);
if (arf_sgn(arb_midref(a)) >= 0)
{
acb_rising_get_mag2_right(bound2, a, acb_imagref(s), n - k);
mag_mul(bound, bound, bound2);
break;
}
else
{
arb_get_mag(t, a);
mag_mul(t, t, t);
mag_add(t, t, u);
mag_mul(bound, bound, t);
}
}
arb_clear(a);
mag_clear(bound2);
mag_clear(t);
mag_clear(u);
}
mag_sqrt(bound, bound);
}