void
mag_sqrt(mag_t y, const mag_t x)
{
if (mag_is_special(x))
{
mag_set(y, x);
}
else
{
double t;
fmpz e;
t = MAG_MAN(x) * ldexp(1.0, -MAG_BITS);
e = MAG_EXP(x);
if (!COEFF_IS_MPZ(e))
{
if (e % 2 != 0)
{
e = (e - 1) >> 1;
t *= 2.0;
}
else
{
e >>= 1;
}
t = sqrt(t) * (1 + 1e-13);
mag_set_d_2exp_fmpz(y, t, &e);
}
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);
}
static void
_acb_mul_fast(acb_t z, const acb_t x, const acb_t y, slong prec)
{
int inexact;
mag_t am, bm, cm, dm, er, fr;
mag_fast_init_set_arf(am, arb_midref(a));
mag_fast_init_set_arf(bm, arb_midref(b));
mag_fast_init_set_arf(cm, arb_midref(c));
mag_fast_init_set_arf(dm, arb_midref(d));
mag_init(er);
mag_init(fr);
mag_fast_addmul(er, am, cr);
mag_fast_addmul(er, bm, dr);
mag_fast_addmul(er, cm, ar);
mag_fast_addmul(er, dm, br);
mag_fast_addmul(er, ar, cr);
mag_fast_addmul(er, br, dr);
mag_fast_addmul(fr, am, dr);
mag_fast_addmul(fr, bm, cr);
mag_fast_addmul(fr, cm, br);
mag_fast_addmul(fr, dm, ar);
mag_fast_addmul(fr, br, cr);
mag_fast_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_set(arb_radref(e), er);
if (inexact & 2)
arf_mag_add_ulp(arb_radref(f), fr, arb_midref(f), prec);
else
mag_set(arb_radref(f), fr);
}
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));
}
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_addmul(arb_t z, const arb_t x, const arb_t y, slong prec)
{
mag_t zr, xm, ym;
int inexact;
if (arb_is_exact(y))
{
arb_addmul_arf(z, x, arb_midref(y), prec);
}
else if (arb_is_exact(x))
{
arb_addmul_arf(z, y, arb_midref(x), prec);
}
else if (ARB_IS_LAGOM(x) && ARB_IS_LAGOM(y) && ARB_IS_LAGOM(z))
{
mag_fast_init_set_arf(xm, arb_midref(x));
mag_fast_init_set_arf(ym, arb_midref(y));
mag_fast_init_set(zr, arb_radref(z));
mag_fast_addmul(zr, xm, arb_radref(y));
mag_fast_addmul(zr, ym, arb_radref(x));
mag_fast_addmul(zr, arb_radref(x), arb_radref(y));
inexact = arf_addmul(arb_midref(z), arb_midref(x), arb_midref(y),
prec, ARF_RND_DOWN);
if (inexact)
arf_mag_fast_add_ulp(zr, zr, arb_midref(z), prec);
*arb_radref(z) = *zr;
}
else
{
mag_init_set_arf(xm, arb_midref(x));
mag_init_set_arf(ym, arb_midref(y));
mag_init_set(zr, arb_radref(z));
mag_addmul(zr, xm, arb_radref(y));
mag_addmul(zr, ym, arb_radref(x));
mag_addmul(zr, arb_radref(x), arb_radref(y));
inexact = arf_addmul(arb_midref(z), arb_midref(x), arb_midref(y),
prec, ARF_RND_DOWN);
if (inexact)
arf_mag_add_ulp(arb_radref(z), zr, arb_midref(z), prec);
else
mag_set(arb_radref(z), zr);
mag_clear(zr);
mag_clear(xm);
mag_clear(ym);
}
}
static void
_acb_sqr_fast(acb_t z, const acb_t x, slong prec)
{
int inexact;
mag_t am, bm, er, fr;
mag_fast_init_set_arf(am, arb_midref(a));
mag_fast_init_set_arf(bm, arb_midref(b));
mag_init(er);
mag_init(fr);
mag_fast_addmul(er, am, ar);
mag_fast_addmul(er, bm, br);
mag_fast_mul_2exp_si(er, er, 1);
mag_fast_addmul(er, ar, ar);
mag_fast_addmul(er, br, br);
mag_fast_addmul(fr, bm, ar);
mag_fast_addmul(fr, am, br);
mag_fast_addmul(fr, ar, br);
mag_fast_mul_2exp_si(fr, fr, 1);
inexact = arf_complex_sqr(arb_midref(e), arb_midref(f),
arb_midref(a), arb_midref(b), prec, ARB_RND);
if (inexact & 1)
arf_mag_add_ulp(arb_radref(e), er, arb_midref(e), prec);
else
mag_set(arb_radref(e), er);
if (inexact & 2)
arf_mag_add_ulp(arb_radref(f), fr, arb_midref(f), prec);
else
mag_set(arb_radref(f), fr);
}
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);
}
void
mag_root(mag_t y, const mag_t x, ulong n)
{
if (n == 0)
{
mag_inf(y);
}
else if (n == 1 || mag_is_special(x))
{
mag_set(y, x);
}
else if (n == 2)
{
mag_sqrt(y, x);
}
else if (n == 4)
{
mag_sqrt(y, x);
mag_sqrt(y, y);
}
else
{
fmpz_t e, f;
fmpz_init_set_ui(e, MAG_BITS);
fmpz_init(f);
/* We evaluate exp(log(1+2^(kn)x)/n) 2^-k where k is chosen
so that 2^(kn) x ~= 2^30. TODO: this rewriting is probably
unnecessary with the new exp/log functions. */
fmpz_sub(e, e, MAG_EXPREF(x));
fmpz_cdiv_q_ui(e, e, n);
fmpz_mul_ui(f, e, n);
mag_mul_2exp_fmpz(y, x, f);
mag_log1p(y, y);
mag_div_ui(y, y, n);
mag_exp(y, y);
fmpz_neg(e, e);
mag_mul_2exp_fmpz(y, y, e);
fmpz_clear(e);
fmpz_clear(f);
}
}
void
arb_atan(arb_t z, const arb_t x, slong prec)
{
if (arb_is_exact(x))
{
arb_atan_arf(z, arb_midref(x), prec);
}
else
{
mag_t t, u;
mag_init(t);
mag_init(u);
arb_get_mag_lower(t, x);
if (mag_is_zero(t))
{
mag_set(t, arb_radref(x));
}
else
{
mag_mul_lower(t, t, t);
mag_one(u);
mag_add_lower(t, t, u);
mag_div(t, arb_radref(x), t);
}
if (mag_cmp_2exp_si(t, 0) > 0)
{
mag_const_pi(u);
mag_min(t, t, u);
}
arb_atan_arf(z, arb_midref(x), prec);
mag_add(arb_radref(z), arb_radref(z), t);
mag_clear(t);
mag_clear(u);
}
}
void
mag_pow_ui_lower(mag_t z, const mag_t x, ulong e)
{
if (e <= 2)
{
if (e == 0)
mag_one(z);
else if (e == 1)
mag_set(z, x);
else
mag_mul_lower(z, x, x);
}
else if (mag_is_inf(x))
{
mag_inf(z);
}
else
{
mag_t y;
int i, bits;
mag_init_set(y, x);
bits = FLINT_BIT_COUNT(e);
for (i = bits - 2; i >= 0; i--)
{
mag_mul_lower(y, y, y);
if (e & (1UL << i))
mag_mul_lower(y, y, x);
}
mag_swap(z, y);
mag_clear(y);
}
}
/* 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);
}
/*
Given T(K), compute bound for T(n) z^n.
We need to multiply by
z^n * 1/rf(K+1,m)^r * (rf(K+1,m)/rf(K+1-A,m)) * (rf(K+1-B,m)/rf(K+1-2B,m))
where m = n - K. This is equal to
z^n *
(K+A)! (K-2B)! (K-B+m)!
----------------------- * ((K+m)! / K!)^(1-r)
(K-B)! (K-A+m)! (K-2B+m)!
*/
void
hypgeom_term_bound(mag_t Tn, const mag_t TK, slong K, slong A, slong B, int r, const mag_t z, slong n)
{
mag_t t, u, num;
slong m;
mag_init(t);
mag_init(u);
mag_init(num);
m = n - K;
if (m < 0)
{
flint_printf("hypgeom term bound\n");
abort();
}
/* TK * z^n */
mag_pow_ui(t, z, n);
mag_mul(num, TK, t);
/* numerator: (K+A)! (K-2B)! (K-B+m)! */
mag_fac_ui(t, K+A);
mag_mul(num, num, t);
mag_fac_ui(t, K-2*B);
mag_mul(num, num, t);
mag_fac_ui(t, K-B+m);
mag_mul(num, num, t);
/* denominator: (K-B)! (K-A+m)! (K-2B+m)! */
mag_rfac_ui(t, K-B);
mag_mul(num, num, t);
mag_rfac_ui(t, K-A+m);
mag_mul(num, num, t);
mag_rfac_ui(t, K-2*B+m);
mag_mul(num, num, t);
/* ((K+m)! / K!)^(1-r) */
if (r == 0)
{
mag_fac_ui(t, K+m);
mag_mul(num, num, t);
mag_rfac_ui(t, K);
mag_mul(num, num, t);
}
else if (r != 1)
{
mag_fac_ui(t, K);
mag_rfac_ui(u, K+m);
mag_mul(t, t, u);
mag_pow_ui(t, t, r-1);
mag_mul(num, num, t);
}
mag_set(Tn, num);
mag_clear(t);
mag_clear(u);
mag_clear(num);
}
slong
hypgeom_bound(mag_t error, int r,
slong A, slong B, slong K, const mag_t TK, const mag_t z, slong tol_2exp)
{
mag_t Tn, t, u, one, tol, num, den;
slong n, m;
mag_init(Tn);
mag_init(t);
mag_init(u);
mag_init(one);
mag_init(tol);
mag_init(num);
mag_init(den);
mag_one(one);
mag_set_ui_2exp_si(tol, UWORD(1), -tol_2exp);
/* approximate number of needed terms */
n = hypgeom_estimate_terms(z, r, tol_2exp);
/* required for 1 + O(1/k) part to be decreasing */
n = FLINT_MAX(n, K + 1);
/* required for z^k / (k!)^r to be decreasing */
m = hypgeom_root_bound(z, r);
n = FLINT_MAX(n, m);
/* We now have |R(k)| <= G(k) where G(k) is monotonically decreasing,
and can bound the tail using a geometric series as soon
as soon as G(k) < 1. */
/* bound T(n-1) */
hypgeom_term_bound(Tn, TK, K, A, B, r, z, n-1);
while (1)
{
/* bound R(n) */
mag_mul_ui(num, z, n);
mag_mul_ui(num, num, n - B);
mag_set_ui_lower(den, n - A);
mag_mul_ui_lower(den, den, n - 2*B);
if (r != 0)
{
mag_set_ui_lower(u, n);
mag_pow_ui_lower(u, u, r);
mag_mul_lower(den, den, u);
}
mag_div(t, num, den);
/* multiply bound for T(n-1) by bound for R(n) to bound T(n) */
mag_mul(Tn, Tn, t);
/* geometric series termination check */
/* u = max(1-t, 0), rounding down [lower bound] */
mag_sub_lower(u, one, t);
if (!mag_is_zero(u))
{
mag_div(u, Tn, u);
if (mag_cmp(u, tol) < 0)
{
mag_set(error, u);
break;
}
}
/* move on to next term */
n++;
}
mag_clear(Tn);
mag_clear(t);
mag_clear(u);
mag_clear(one);
mag_clear(tol);
mag_clear(num);
mag_clear(den);
return n;
}
void
arb_abs(arb_t y, const arb_t x)
{
arf_abs(arb_midref(y), arb_midref(x));
mag_set(arb_radref(y), arb_radref(x));
}
void
arb_set(arb_t x, const arb_t y)
{
arf_set(arb_midref(x), arb_midref(y));
mag_set(arb_radref(x), arb_radref(y));
}
/* computes the factors that are independent of n (all are upper bounds) */
void
acb_hypgeom_u_asymp_bound_factors(int * R, mag_t alpha,
mag_t nu, mag_t sigma, mag_t rho, mag_t zinv,
const acb_t a, const acb_t b, const acb_t z)
{
mag_t r, u, zre, zim, zlo, sigma_prime;
acb_t t;
mag_init(r);
mag_init(u);
mag_init(zre);
mag_init(zim);
mag_init(zlo);
mag_init(sigma_prime);
acb_init(t);
/* lower bounds for |re(z)|, |im(z)|, |z| */
arb_get_mag_lower(zre, acb_realref(z));
arb_get_mag_lower(zim, acb_imagref(z));
acb_get_mag_lower(zlo, z); /* todo: hypot */
/* upper bound for 1/|z| */
mag_one(u);
mag_div(zinv, u, zlo);
/* upper bound for r = |b - 2a| */
acb_mul_2exp_si(t, a, 1);
acb_sub(t, b, t, MAG_BITS);
acb_get_mag(r, t);
/* determine region */
*R = 0;
if (mag_cmp(zlo, r) >= 0)
{
int znonneg = arb_is_nonnegative(acb_realref(z));
if (znonneg && mag_cmp(zre, r) >= 0)
{
*R = 1;
}
else if (mag_cmp(zim, r) >= 0 || znonneg)
{
*R = 2;
}
else
{
mag_mul_2exp_si(u, r, 1);
if (mag_cmp(zlo, u) >= 0)
*R = 3;
}
}
if (R == 0)
{
mag_inf(alpha);
mag_inf(nu);
mag_inf(sigma);
mag_inf(rho);
}
else
{
/* sigma = |(b-2a)/z| */
mag_mul(sigma, r, zinv);
/* nu = (1/2 + 1/2 sqrt(1-4 sigma^2))^(-1/2) <= 1 + 2 sigma^2 */
if (mag_cmp_2exp_si(sigma, -1) <= 0)
{
mag_mul(nu, sigma, sigma);
mag_mul_2exp_si(nu, nu, 1);
mag_one(u);
mag_add(nu, nu, u);
}
else
{
mag_inf(nu);
}
/* modified sigma for alpha, beta, rho when in R3 */
if (*R == 3)
mag_mul(sigma_prime, sigma, nu);
else
mag_set(sigma_prime, sigma);
/* alpha = 1/(1-sigma') */
mag_one(alpha);
mag_sub_lower(alpha, alpha, sigma_prime);
mag_one(u);
mag_div(alpha, u, alpha);
/* rho = |2a^2-2ab+b|/2 + sigma'*(1+sigma'/4)/(1-sigma')^2 */
mag_mul_2exp_si(rho, sigma_prime, -2);
mag_one(u);
mag_add(rho, rho, u);
mag_mul(rho, rho, sigma_prime);
mag_mul(rho, rho, alpha);
mag_mul(rho, rho, alpha);
acb_sub(t, a, b, MAG_BITS);
acb_mul(t, t, a, MAG_BITS);
acb_mul_2exp_si(t, t, 1);
acb_add(t, t, b, MAG_BITS);
acb_get_mag(u, t);
mag_mul_2exp_si(u, u, -1);
mag_add(rho, rho, u);
}
mag_clear(r);
mag_clear(u);
mag_clear(zre);
mag_clear(zim);
mag_clear(zlo);
mag_clear(sigma_prime);
acb_clear(t);
}
void
mag_polylog_tail(mag_t u, const mag_t z, long sigma, ulong d, ulong N)
{
mag_t TN, UN, t;
if (N < 2)
{
mag_inf(u);
return;
}
mag_init(TN);
mag_init(UN);
mag_init(t);
if (mag_cmp_2exp_si(z, 0) >= 0)
{
mag_inf(u);
}
else
{
/* Bound T(N) */
mag_pow_ui(TN, z, N);
/* multiply by log(N)^d */
if (d > 0)
{
mag_log_ui(t, N);
mag_pow_ui(t, t, d);
mag_mul(TN, TN, t);
}
/* multiply by 1/k^s */
if (sigma > 0)
{
mag_set_ui_lower(t, N);
mag_pow_ui_lower(t, t, sigma);
mag_div(TN, TN, t);
}
else if (sigma < 0)
{
mag_set_ui(t, N);
mag_pow_ui(t, t, -sigma);
mag_mul(TN, TN, t);
}
/* Bound U(N) */
mag_set(UN, z);
/* multiply by (1 + 1/N)**S */
if (sigma < 0)
{
mag_binpow_uiui(t, N, -sigma);
mag_mul(UN, UN, t);
}
/* multiply by (1 + 1/(N log(N)))^d */
if (d > 0)
{
ulong nl;
/* rounds down */
nl = mag_d_log_lower_bound(N) * N * (1 - 1e-13);
mag_binpow_uiui(t, nl, d);
mag_mul(UN, UN, t);
}
/* T(N) / (1 - U(N)) */
if (mag_cmp_2exp_si(UN, 0) >= 0)
{
mag_inf(u);
}
else
{
mag_one(t);
mag_sub_lower(t, t, UN);
mag_div(u, TN, t);
}
}
mag_clear(TN);
mag_clear(UN);
mag_clear(t);
}
void
acb_hypgeom_2f1_continuation(acb_t res, acb_t res1,
const acb_t a, const acb_t b, const acb_t c, const acb_t y,
const acb_t z, const acb_t f0, const acb_t f1, long prec)
{
mag_t A, nu, N, w, err, err1, R, T, goal;
acb_t x;
long j, k;
mag_init(A);
mag_init(nu);
mag_init(N);
mag_init(err);
mag_init(err1);
mag_init(w);
mag_init(R);
mag_init(T);
mag_init(goal);
acb_init(x);
bound(A, nu, N, a, b, c, y, f0, f1);
acb_sub(x, z, y, prec);
/* |T(k)| <= A * binomial(N+k, k) * nu^k * |x|^k */
acb_get_mag(w, x);
mag_mul(w, w, nu); /* w = nu |x| */
mag_mul_2exp_si(goal, A, -prec-2);
/* bound for T(0) */
mag_set(T, A);
mag_inf(R);
for (k = 1; k < 100 * prec; k++)
{
/* T(k) = T(k) * R(k), R(k) = (N+k)/k * w = (1 + N/k) w */
mag_div_ui(R, N, k);
mag_add_ui(R, R, 1);
mag_mul(R, R, w);
/* T(k) */
mag_mul(T, T, R);
if (mag_cmp(T, goal) <= 0 && mag_cmp_2exp_si(R, 0) < 0)
break;
}
/* T(k) [1 + R + R^2 + R^3 + ...] */
mag_geom_series(err, R, 0);
mag_mul(err, T, err);
/* Now compute T, R for the derivative */
/* Coefficients are A * (k+1) * binomial(N+k+1, k+1) */
mag_add_ui(T, N, 1);
mag_mul(T, T, A);
mag_inf(R);
for (j = 1; j <= k; j++)
{
mag_add_ui(R, N, k + 1);
mag_div_ui(R, R, k);
mag_mul(R, R, w);
mag_mul(T, T, R);
}
mag_geom_series(err1, R, 0);
mag_mul(err1, T, err1);
if (mag_is_inf(err))
{
acb_indeterminate(res);
acb_indeterminate(res1);
}
else
{
evaluate_sum(res, res1, a, b, c, y, x, f0, f1, k, prec);
acb_add_error_mag(res, err);
acb_add_error_mag(res1, err1);
}
mag_clear(A);
mag_clear(nu);
mag_clear(N);
mag_clear(err);
mag_clear(err1);
mag_clear(w);
mag_clear(R);
mag_clear(T);
mag_clear(goal);
acb_clear(x);
}
void
mag_log1p(mag_t z, const mag_t x)
{
if (mag_is_special(x))
{
if (mag_is_zero(x))
mag_zero(z);
else
mag_inf(z);
}
else
{
fmpz exp = MAG_EXP(x);
if (!COEFF_IS_MPZ(exp))
{
/* Quick bound by x */
if (exp < -10)
{
mag_set(z, x);
return;
}
else if (exp < 1000)
{
double t;
t = ldexp(MAG_MAN(x), exp - MAG_BITS);
t = (1.0 + t) * (1 + 1e-14);
t = mag_d_log_upper_bound(t);
mag_set_d(z, t);
return;
}
}
else if (fmpz_sgn(MAG_EXPREF(x)) < 0)
{
/* Quick bound by x */
mag_set(z, x);
return;
}
/* Now we must have x >= 2^1000 */
/* Use log(2^(exp-1) * (2*v)) = exp*log(2) + log(2*v) */
{
double t;
fmpz_t b;
mag_t u;
mag_init(u);
fmpz_init(b);
/* incrementing the mantissa gives an upper bound for x+1 */
t = ldexp(MAG_MAN(x) + 1, 1 - MAG_BITS);
t = mag_d_log_upper_bound(t);
mag_set_d(u, t);
/* log(2) < 744261118/2^30 */
_fmpz_add_fast(b, MAG_EXPREF(x), -1);
fmpz_mul_ui(b, b, 744261118);
mag_set_fmpz(z, b);
_fmpz_add_fast(MAG_EXPREF(z), MAG_EXPREF(z), -30);
mag_add(z, z, u);
mag_clear(u);
fmpz_clear(b);
}
}
}
int main()
{
slong iter;
flint_rand_t state;
flint_printf("rel_accuracy_bits....");
fflush(stdout);
flint_randinit(state);
/* test aliasing of c and a */
for (iter = 0; iter < 10000 * arb_test_multiplier(); iter++)
{
arb_t x;
acb_t z;
slong a1, a2;
arb_init(x);
acb_init(z);
arb_randtest_special(x, state, 1 + n_randint(state, 200), 1 + n_randint(state, 200));
acb_set_arb(z, x);
a1 = arb_rel_accuracy_bits(x);
a2 = acb_rel_accuracy_bits(z);
if (a1 != a2)
{
flint_printf("FAIL: acb != arb\n\n");
flint_printf("x = "); arb_print(x); flint_printf("\n\n");
flint_printf("z = "); acb_print(z); flint_printf("\n\n");
flint_printf("a1 = %wd, a2 = %wd\n\n", a1, a2);
abort();
}
acb_randtest_special(z, state, 1 + n_randint(state, 200), 1 + n_randint(state, 200));
a1 = acb_rel_accuracy_bits(z);
if (n_randint(state, 2))
arf_swap(arb_midref(acb_realref(z)), arb_midref(acb_imagref(z)));
if (n_randint(state, 2))
mag_swap(arb_radref(acb_realref(z)), arb_radref(acb_imagref(z)));
a2 = acb_rel_accuracy_bits(z);
if (a1 != a2)
{
flint_printf("FAIL: swapping\n\n");
flint_printf("z = "); acb_print(z); flint_printf("\n\n");
flint_printf("a1 = %wd, a2 = %wd\n\n", a1, a2);
abort();
}
acb_randtest_special(z, state, 1 + n_randint(state, 200), 1 + n_randint(state, 200));
if (arf_cmpabs(arb_midref(acb_realref(z)), arb_midref(acb_imagref(z))) >= 0)
arf_set(arb_midref(x), arb_midref(acb_realref(z)));
else
arf_set(arb_midref(x), arb_midref(acb_imagref(z)));
if (mag_cmp(arb_radref(acb_realref(z)), arb_radref(acb_imagref(z))) >= 0)
mag_set(arb_radref(x), arb_radref(acb_realref(z)));
else
mag_set(arb_radref(x), arb_radref(acb_imagref(z)));
a1 = acb_rel_accuracy_bits(z);
a2 = arb_rel_accuracy_bits(x);
if (a1 != a2)
{
flint_printf("FAIL: acb != arb (2)\n\n");
flint_printf("x = "); arb_print(x); flint_printf("\n\n");
flint_printf("z = "); acb_print(z); flint_printf("\n\n");
flint_printf("a1 = %wd, a2 = %wd\n\n", a1, a2);
abort();
}
arb_clear(x);
acb_clear(z);
}
flint_randclear(state);
flint_cleanup();
flint_printf("PASS\n");
return EXIT_SUCCESS;
}
void
_arb_sin_cos_generic(arb_t s, arb_t c, const arf_t x, const mag_t xrad, slong prec)
{
int want_sin, want_cos;
slong maglim;
want_sin = (s != NULL);
want_cos = (c != NULL);
if (arf_is_zero(x) && mag_is_zero(xrad))
{
if (want_sin) arb_zero(s);
if (want_cos) arb_one(c);
return;
}
if (!arf_is_finite(x) || !mag_is_finite(xrad))
{
if (arf_is_nan(x))
{
if (want_sin) arb_indeterminate(s);
if (want_cos) arb_indeterminate(c);
}
else
{
if (want_sin) arb_zero_pm_one(s);
if (want_cos) arb_zero_pm_one(c);
}
return;
}
maglim = FLINT_MAX(65536, 4 * prec);
if (mag_cmp_2exp_si(xrad, -16) > 0 || arf_cmpabs_2exp_si(x, maglim) > 0)
{
_arb_sin_cos_wide(s, c, x, xrad, prec);
return;
}
if (arf_cmpabs_2exp_si(x, -(prec/2) - 2) <= 0)
{
mag_t t, u, v;
mag_init(t);
mag_init(u);
mag_init(v);
arf_get_mag(t, x);
mag_add(t, t, xrad);
mag_mul(u, t, t);
/* |sin(z)-z| <= z^3/6 */
if (want_sin)
{
arf_set(arb_midref(s), x);
mag_set(arb_radref(s), xrad);
arb_set_round(s, s, prec);
mag_mul(v, u, t);
mag_div_ui(v, v, 6);
arb_add_error_mag(s, v);
}
/* |cos(z)-1| <= z^2/2 */
if (want_cos)
{
arf_one(arb_midref(c));
mag_mul_2exp_si(arb_radref(c), u, -1);
}
mag_clear(t);
mag_clear(u);
mag_clear(v);
return;
}
if (mag_is_zero(xrad))
{
arb_sin_cos_arf_generic(s, c, x, prec);
}
else
{
mag_t t;
slong exp, radexp;
mag_init_set(t, xrad);
exp = arf_abs_bound_lt_2exp_si(x);
radexp = MAG_EXP(xrad);
if (radexp < MAG_MIN_LAGOM_EXP || radexp > MAG_MAX_LAGOM_EXP)
radexp = MAG_MIN_LAGOM_EXP;
if (want_cos && exp < -2)
prec = FLINT_MIN(prec, 20 - FLINT_MAX(exp, radexp) - radexp);
else
prec = FLINT_MIN(prec, 20 - radexp);
arb_sin_cos_arf_generic(s, c, x, prec);
/* todo: could use quadratic bound */
if (want_sin) mag_add(arb_radref(s), arb_radref(s), t);
if (want_cos) mag_add(arb_radref(c), arb_radref(c), t);
mag_clear(t);
}
}