int main()
{
slong i, bound;
double a, b;
fmpq_t q;
fmpr_t t;
flint_printf("bound_2exp_si....");
fflush(stdout);
fmpq_init(q);
fmpr_init(t);
for (i = 0; i < 1000; i++)
{
arith_bernoulli_number(q, i);
bound = bernoulli_bound_2exp_si(i);
fmpr_set_round_fmpz(t, fmpq_numref(q), 32, FMPR_RND_UP);
fmpr_div_fmpz(t, t, fmpq_denref(q), 32, FMPR_RND_UP);
if (fmpr_cmpabs_2exp_si(t, bound) > 0)
{
flint_printf("FAIL: %wd\n", i);
fmpr_print(t); flint_printf("\n\n");
flint_printf("%wd\n", bound); flint_printf("\n\n");
abort();
}
}
fmpq_clear(q);
fmpr_clear(t);
for (i = 100; i < 4000000; i += 1)
{
i += (i & 1);
a = bernoulli_bound_2exp_si(i);
b = log2bern_approx(i);
if (a < b || a > 1.01 * b)
{
flint_printf("FAIL: %wd\n", i);
flint_printf("%wd: %f %f %f\n", i, a, b, (float) a / b);
abort();
}
}
flint_cleanup();
flint_printf("PASS\n");
return EXIT_SUCCESS;
}
void
gamma_stirling_eval_fmprb(fmprb_t s, const fmprb_t z, long nterms, int digamma, long prec)
{
fmprb_t b, t, logz, zinv, zinv2;
fmpr_t err;
long k, term_prec;
double z_mag, term_mag;
fmprb_init(b);
fmprb_init(t);
fmprb_init(logz);
fmprb_init(zinv);
fmprb_init(zinv2);
fmprb_log(logz, z, prec);
fmprb_ui_div(zinv, 1UL, z, prec);
nterms = FLINT_MAX(nterms, 1);
fmprb_zero(s);
if (nterms > 1)
{
fmprb_mul(zinv2, zinv, zinv, prec);
z_mag = fmpr_get_d(fmprb_midref(logz), FMPR_RND_UP) * 1.44269504088896;
for (k = nterms - 1; k >= 1; k--)
{
term_mag = bernoulli_bound_2exp_si(2 * k);
term_mag -= (2 * k - 1) * z_mag;
term_prec = prec + term_mag;
term_prec = FLINT_MIN(term_prec, prec);
term_prec = FLINT_MAX(term_prec, 10);
if (prec > 2000)
{
fmprb_set_round(t, zinv2, term_prec);
fmprb_mul(s, s, t, term_prec);
}
else
fmprb_mul(s, s, zinv2, term_prec);
gamma_stirling_coeff(b, k, digamma, term_prec);
fmprb_add(s, s, b, term_prec);
}
if (digamma)
fmprb_mul(s, s, zinv2, prec);
else
fmprb_mul(s, s, zinv, prec);
}
/* remainder bound */
fmpr_init(err);
gamma_stirling_bound_fmprb(err, z, digamma ? 1 : 0, 1, nterms);
fmprb_add_error_fmpr(s, err);
fmpr_clear(err);
if (digamma)
{
fmprb_neg(s, s);
fmprb_mul_2exp_si(zinv, zinv, -1);
fmprb_sub(s, s, zinv, prec);
fmprb_add(s, s, logz, prec);
}
else
{
/* (z-0.5)*log(z) - z + log(2*pi)/2 */
fmprb_one(t);
fmprb_mul_2exp_si(t, t, -1);
fmprb_sub(t, z, t, prec);
fmprb_mul(t, logz, t, prec);
fmprb_add(s, s, t, prec);
fmprb_sub(s, s, z, prec);
fmprb_const_log_sqrt2pi(t, prec);
fmprb_add(s, s, t, prec);
}
fmprb_clear(t);
fmprb_clear(b);
fmprb_clear(zinv);
fmprb_clear(zinv2);
fmprb_clear(logz);
}
void
acb_gamma_stirling_eval(acb_t s, const acb_t z, long nterms, int digamma, long prec)
{
acb_t t, logz, zinv, zinv2;
arb_t b;
mag_t err;
long k, term_prec;
double z_mag, term_mag;
acb_init(t);
acb_init(logz);
acb_init(zinv);
acb_init(zinv2);
arb_init(b);
acb_log(logz, z, prec);
acb_inv(zinv, z, prec);
nterms = FLINT_MAX(nterms, 1);
acb_zero(s);
if (nterms > 1)
{
acb_mul(zinv2, zinv, zinv, prec);
z_mag = arf_get_d(arb_midref(acb_realref(logz)), ARF_RND_UP) * 1.44269504088896;
for (k = nterms - 1; k >= 1; k--)
{
term_mag = bernoulli_bound_2exp_si(2 * k);
term_mag -= (2 * k - 1) * z_mag;
term_prec = prec + term_mag;
term_prec = FLINT_MIN(term_prec, prec);
term_prec = FLINT_MAX(term_prec, 10);
arb_gamma_stirling_coeff(b, k, digamma, term_prec);
if (prec > 2000)
{
acb_set_round(t, zinv2, term_prec);
acb_mul(s, s, t, term_prec);
}
else
acb_mul(s, s, zinv2, term_prec);
arb_add(acb_realref(s), acb_realref(s), b, term_prec);
}
if (digamma)
acb_mul(s, s, zinv2, prec);
else
acb_mul(s, s, zinv, prec);
}
/* remainder bound */
mag_init(err);
acb_gamma_stirling_bound(err, z, digamma ? 1 : 0, 1, nterms);
mag_add(arb_radref(acb_realref(s)), arb_radref(acb_realref(s)), err);
mag_add(arb_radref(acb_imagref(s)), arb_radref(acb_imagref(s)), err);
mag_clear(err);
if (digamma)
{
acb_neg(s, s);
acb_mul_2exp_si(zinv, zinv, -1);
acb_sub(s, s, zinv, prec);
acb_add(s, s, logz, prec);
}
else
{
/* (z-0.5)*log(z) - z + log(2*pi)/2 */
arb_one(b);
arb_mul_2exp_si(b, b, -1);
arb_set(acb_imagref(t), acb_imagref(z));
arb_sub(acb_realref(t), acb_realref(z), b, prec);
acb_mul(t, logz, t, prec);
acb_add(s, s, t, prec);
acb_sub(s, s, z, prec);
arb_const_log_sqrt2pi(b, prec);
arb_add(acb_realref(s), acb_realref(s), b, prec);
}
acb_clear(t);
acb_clear(logz);
acb_clear(zinv);
acb_clear(zinv2);
arb_clear(b);
}