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); }