void
fmprb_sqrt(fmprb_t z, const fmprb_t x, slong prec)
{
slong r;
if (fmprb_contains_negative(x))
{
fmpr_nan(fmprb_midref(z));
fmpr_pos_inf(fmprb_radref(z));
return;
}
if (fmprb_is_exact(x))
{
r = fmpr_sqrt(fmprb_midref(z), fmprb_midref(x), prec, FMPR_RND_DOWN);
fmpr_set_error_result(fmprb_radref(z), fmprb_midref(z), r);
}
else
{
fmpr_t err;
fmpr_init(err);
fmpr_sub(err, fmprb_midref(x), fmprb_radref(x), FMPRB_RAD_PREC, FMPR_RND_DOWN);
fmpr_rsqrt(err, err, FMPRB_RAD_PREC, FMPR_RND_UP);
fmpr_mul(err, fmprb_radref(x), err, FMPRB_RAD_PREC, FMPR_RND_UP);
fmpr_mul_2exp_si(err, err, -1);
r = fmpr_sqrt(fmprb_midref(z), fmprb_midref(x), prec, FMPR_RND_DOWN);
fmpr_add_error_result(fmprb_radref(z), err, fmprb_midref(z), r,
FMPRB_RAD_PREC, FMPR_RND_UP);
fmpr_clear(err);
}
fmprb_adjust(z);
}
void
gamma_taylor_bound_remainder(fmpr_t err, const fmpr_t z, long n)
{
fmpr_t t, u;
gamma_taylor_bound_extend_cache(n + 1);
fmpr_init(t);
fmpr_init(u);
/* denominator: 1 - r(n+1) * z, rounded down */
fmpr_mul(t, gamma_taylor_bound_ratio_cache + n + 1,
z, FMPRB_RAD_PREC, FMPR_RND_UP);
fmpr_one(u);
fmpr_sub(u, u, t, FMPRB_RAD_PREC, FMPR_RND_DOWN);
if (fmpr_sgn(u) <= 0)
{
fmpr_pos_inf(err);
}
else
{
fmpr_pow_sloppy_ui(t, z, n, FMPRB_RAD_PREC, FMPR_RND_UP);
fmpr_mul_2exp_si(t, t, gamma_taylor_bound_mag_cache[n + 1]);
fmpr_div(err, t, u, FMPRB_RAD_PREC, FMPR_RND_UP);
}
fmpr_clear(t);
fmpr_clear(u);
}
int main()
{
slong iter;
flint_rand_t state;
flint_printf("set_d....");
fflush(stdout);
flint_randinit(state);
for (iter = 0; iter < 100000 * arb_test_multiplier(); iter++)
{
fmpr_t a, b, c;
mag_t m;
double x;
fmpr_init(a);
fmpr_init(b);
fmpr_init(c);
mag_init(m);
x = d_randtest2(state);
x = ldexp(x, 100 - n_randint(state, 200));
if (n_randint(state, 100) == 0)
x = 0.0;
fmpr_set_d(a, x);
mag_set_d(m, x);
mag_get_fmpr(b, m);
fmpr_set(c, a);
fmpr_mul_ui(c, c, 1025, MAG_BITS, FMPR_RND_UP);
fmpr_mul_2exp_si(c, c, -10);
MAG_CHECK_BITS(m)
if (!(fmpr_cmpabs(a, b) <= 0 && fmpr_cmpabs(b, c) <= 0))
{
flint_printf("FAIL\n\n");
flint_printf("a = "); fmpr_print(a); flint_printf("\n\n");
flint_printf("b = "); fmpr_print(b); flint_printf("\n\n");
flint_printf("c = "); fmpr_print(c); flint_printf("\n\n");
abort();
}
fmpr_clear(a);
fmpr_clear(b);
fmpr_clear(c);
mag_clear(m);
}
flint_randclear(state);
flint_cleanup();
flint_printf("PASS\n");
return EXIT_SUCCESS;
}
int main()
{
slong iter;
flint_rand_t state;
flint_printf("mul_2exp_si....");
fflush(stdout);
flint_randinit(state);
for (iter = 0; iter < 100000 * arb_test_multiplier(); iter++)
{
fmpr_t x, y, z;
mag_t xb, yb;
slong e;
fmpr_init(x);
fmpr_init(y);
fmpr_init(z);
mag_init(xb);
mag_init(yb);
mag_randtest_special(xb, state, 100);
e = z_randtest(state);
mag_get_fmpr(x, xb);
mag_mul_2exp_si(yb, xb, e);
fmpr_mul_2exp_si(y, x, e);
mag_get_fmpr(z, yb);
MAG_CHECK_BITS(yb)
if (!fmpr_equal(z, y))
{
flint_printf("FAIL\n\n");
flint_printf("x = "); fmpr_printd(x, 15); flint_printf("\n\n");
flint_printf("y = "); fmpr_printd(y, 15); flint_printf("\n\n");
flint_printf("z = "); fmpr_printd(z, 15); flint_printf("\n\n");
abort();
}
fmpr_clear(x);
fmpr_clear(y);
fmpr_clear(z);
mag_clear(xb);
mag_clear(yb);
}
flint_randclear(state);
flint_cleanup();
flint_printf("PASS\n");
return EXIT_SUCCESS;
}
/* Absolute value of rising factorial (could speed up once complex gamma is available). */
void
fmpcb_rfac_abs_ubound2(fmpr_t bound, const fmpcb_t s, ulong n, long prec)
{
fmpr_t term, t;
ulong k;
/* M(k) = (a+k)^2 + b^2
M(0) = a^2 + b^2
M(k+1) = M(k) + 2*a + (2*k+1)
*/
fmpr_init(t);
fmpr_init(term);
fmpr_one(bound);
/* M(0) = a^2 + b^2 */
fmprb_get_abs_ubound_fmpr(t, fmpcb_realref(s), prec);
fmpr_mul(term, t, t, prec, FMPR_RND_UP);
fmprb_get_abs_ubound_fmpr(t, fmpcb_imagref(s), prec);
fmpr_mul(t, t, t, prec, FMPR_RND_UP);
fmpr_add(term, term, t, prec, FMPR_RND_UP);
/* we add t = 2*a to each term. note that this can be signed;
we always want the most positive value */
fmpr_add(t, fmprb_midref(fmpcb_realref(s)),
fmprb_radref(fmpcb_realref(s)), prec, FMPR_RND_CEIL);
fmpr_mul_2exp_si(t, t, 1);
for (k = 0; k < n; k++)
{
fmpr_mul(bound, bound, term, prec, FMPR_RND_UP);
fmpr_add_ui(term, term, 2 * k + 1, prec, FMPR_RND_UP);
fmpr_add(term, term, t, prec, FMPR_RND_UP);
}
fmpr_sqrt(bound, bound, prec, FMPR_RND_UP);
fmpr_clear(t);
fmpr_clear(term);
}
static __inline__ void
fmprb_nonnegative_part(fmprb_t z, const fmprb_t x, slong prec)
{
if (fmprb_contains_negative(x))
{
fmpr_add(fmprb_midref(z), fmprb_midref(x), fmprb_radref(x), prec, FMPR_RND_CEIL);
if (fmpr_sgn(fmprb_midref(z)) <= 0)
{
fmpr_zero(fmprb_radref(z));
}
else
{
fmpr_mul_2exp_si(fmprb_midref(z), fmprb_midref(z), -1);
fmpr_set(fmprb_midref(z), fmprb_radref(z));
}
}
else
{
fmprb_set(z, x);
}
}
void
fmprb_sqrtpos(fmprb_t z, const fmprb_t x, slong prec)
{
if (!fmprb_is_finite(x))
{
if (fmpr_is_zero(fmprb_radref(x)) && fmpr_is_pos_inf(fmprb_midref(x)))
fmprb_pos_inf(z);
else
fmprb_zero_pm_inf(z);
}
else if (fmprb_contains_nonpositive(x))
{
fmpr_t t;
fmpr_init(t);
fmpr_add(t, fmprb_midref(x), fmprb_radref(x),
FMPRB_RAD_PREC, FMPR_RND_CEIL);
if (fmpr_sgn(t) <= 0)
{
fmprb_zero(z);
}
else
{
fmpr_sqrt(t, t, FMPRB_RAD_PREC, FMPR_RND_CEIL);
fmpr_mul_2exp_si(t, t, -1);
fmpr_set(fmprb_midref(z), t);
fmpr_set(fmprb_radref(z), t);
}
fmpr_clear(t);
}
else
{
fmprb_sqrt(z, x, prec);
}
fmprb_nonnegative_part(z, z, prec);
}
void _fmprb_poly_mullow_ztrunc(fmprb_ptr C,
fmprb_srcptr A, long lenA,
fmprb_srcptr B, long lenB, long n, long prec)
{
fmpz * Acoeffs, * Bcoeffs, * Ccoeffs;
fmpz_t Aexp, Bexp, Cexp;
fmpr_t Aerr, Berr, Anorm, Bnorm, err;
long i;
int squaring;
lenA = FLINT_MIN(lenA, n);
lenB = FLINT_MIN(lenB, n);
squaring = (A == B) && (lenA == lenB);
/* TODO: make the code below work correctly with out this workaround */
if (_fmprb_vec_rad_has_inf_nan(A, lenA) ||
(!squaring && _fmprb_vec_rad_has_inf_nan(B, lenB)))
{
_fmprb_vec_indeterminate(C, n);
return;
}
fmpz_init(Aexp);
fmpz_init(Bexp);
fmpz_init(Cexp);
Acoeffs = _fmpz_vec_init(lenA);
Bcoeffs = _fmpz_vec_init(lenB);
Ccoeffs = _fmpz_vec_init(n);
fmpr_init(Aerr);
fmpr_init(Berr);
fmpr_init(Anorm);
fmpr_init(Bnorm);
fmpr_init(err);
_fmprb_poly_get_fmpz_poly_2exp(Aerr, Aexp, Acoeffs, A, lenA, prec);
if (squaring)
{
_fmpz_poly_sqrlow(Ccoeffs, Acoeffs, lenA, n);
fmpz_add(Cexp, Aexp, Aexp);
/* cross-multiply error bounds: (A+r)(B+s) = A^2 + 2Ar + r^2 */
_fmpr_fmpz_vec_max_norm(Anorm, Acoeffs, lenA, FMPRB_RAD_PREC);
fmpr_mul_2exp_fmpz(Anorm, Anorm, Aexp);
fmpr_mul(err, Anorm, Aerr, FMPRB_RAD_PREC, FMPR_RND_UP);
fmpr_mul_2exp_si(err, err, 1);
fmpr_addmul(err, Aerr, Aerr, FMPRB_RAD_PREC, FMPR_RND_UP);
}
else
{
_fmprb_poly_get_fmpz_poly_2exp(Berr, Bexp, Bcoeffs, B, lenB, prec);
/* main multiplication */
if (lenA >= lenB)
_fmpz_poly_mullow(Ccoeffs, Acoeffs, lenA, Bcoeffs, lenB, n);
else
_fmpz_poly_mullow(Ccoeffs, Bcoeffs, lenB, Acoeffs, lenA, n);
fmpz_add(Cexp, Aexp, Bexp);
/* cross-multiply error bounds: (A+r)(B+s) = AB + As + Br + rs */
_fmpr_fmpz_vec_max_norm(Anorm, Acoeffs, lenA, FMPRB_RAD_PREC);
fmpr_mul_2exp_fmpz(Anorm, Anorm, Aexp);
_fmpr_fmpz_vec_max_norm(Bnorm, Bcoeffs, lenB, FMPRB_RAD_PREC);
fmpr_mul_2exp_fmpz(Bnorm, Bnorm, Bexp);
fmpr_mul(err, Aerr, Berr, FMPRB_RAD_PREC, FMPR_RND_UP);
fmpr_addmul(err, Anorm, Berr, FMPRB_RAD_PREC, FMPR_RND_UP);
fmpr_addmul(err, Bnorm, Aerr, FMPRB_RAD_PREC, FMPR_RND_UP);
}
for (i = 0; i < n; i++)
{
fmprb_set_round_fmpz_2exp(C + i, Ccoeffs + i, Cexp, prec);
/* there are at most (i+1) error terms for coefficient i */
/* TODO: make this tight */
fmpr_addmul_ui(fmprb_radref(C + i), err, i + 1,
FMPRB_RAD_PREC, FMPR_RND_UP);
}
fmpr_clear(Aerr);
fmpr_clear(Berr);
fmpr_clear(Anorm);
fmpr_clear(Bnorm);
fmpr_clear(err);
_fmpz_vec_clear(Acoeffs, lenA);
_fmpz_vec_clear(Bcoeffs, lenB);
_fmpz_vec_clear(Ccoeffs, n);
fmpz_clear(Aexp);
fmpz_clear(Bexp);
fmpz_clear(Cexp);
}
