int my_mpfr_lbeta(mpfr_t R, mpfr_t a, mpfr_t b, mpfr_rnd_t RND) { mpfr_prec_t p_a = mpfr_get_prec(a), p_b = mpfr_get_prec(b); if(p_a < p_b) p_a = p_b;// p_a := max(p_a, p_b) if(mpfr_get_prec(R) < p_a) mpfr_prec_round(R, p_a, RND);// so prec(R) = max( prec(a), prec(b) ) int ans; mpfr_t s; mpfr_init2(s, p_a); /* "FIXME": check each 'ans' below, and return when not ok ... */ ans = mpfr_add(s, a, b, RND); if(mpfr_integer_p(s) && mpfr_sgn(s) <= 0) { // (a + b) is integer <= 0 if(!mpfr_integer_p(a) && !mpfr_integer_p(b)) { // but a,b not integer ==> R = ln(finite / +-Inf) = ln(0) = -Inf : mpfr_set_inf (R, -1); mpfr_clear (s); return ans; }// else: sum is integer; at least one integer ==> both integer int sX = mpfr_sgn(a), sY = mpfr_sgn(b); if(sX * sY < 0) { // one negative, one positive integer // ==> special treatment here : if(sY < 0) // ==> sX > 0; swap the two mpfr_swap(a, b); /* now have --- a < 0 < b <= |a| integer ------------------ * ================ * --> see my_mpfr_beta() above */ unsigned long b_ = 0;// -Wall Rboolean b_fits_ulong = mpfr_fits_ulong_p(b, RND), small_b = b_fits_ulong && (b_ = mpfr_get_ui(b, RND)) < b_large; if(small_b) { //----------------- small b ------------------ // use GMP big integer arithmetic: mpz_t S; mpz_init(S); mpfr_get_z(S, s, RND); // S := s mpz_sub_ui (S, S, (unsigned long) 1); // S = s - 1 = (a+b-1) /* binomial coefficient choose(N, k) requires k a 'long int'; * here, b must fit into a long: */ mpz_bin_ui (S, S, b_); // S = choose(S, b) = choose(a+b-1, b) mpz_mul_ui (S, S, b_); // S = S*b = b * choose(a+b-1, b) // back to mpfr: R = log(|1 / S|) = - log(|S|) mpz_abs(S, S); mpfr_set_z(s, S, RND); // <mpfr> s := |S| mpfr_log(R, s, RND); // R := log(s) = log(|S|) mpfr_neg(R, R, RND); // R = -R = -log(|S|) mpz_clear(S); } else { // b is "large", use direct B(.,.) formula // a := (-1)^b -- not needed here, neither 'neg': want log( |.| ) // s' := 1-s = 1-a-b mpfr_ui_sub(s, 1, s, RND); // R := log(|B(1-a-b, b)|) = log(|B(s', b)|) my_mpfr_lbeta (R, s, b, RND); } mpfr_clear(s); return ans; } } ans = mpfr_lngamma(s, s, RND); // s = lngamma(a + b) ans = mpfr_lngamma(a, a, RND); ans = mpfr_lngamma(b, b, RND); ans = mpfr_add (b, b, a, RND); // b' = lngamma(a) + lngamma(b) ans = mpfr_sub (R, b, s, RND); mpfr_clear (s); return ans; }
static void special (void) { mpfr_t x, y; int inex; mpfr_init (x); mpfr_init (y); mpfr_set_nan (x); mpfr_lngamma (y, x, MPFR_RNDN); if (!mpfr_nan_p (y)) { printf ("Error for lngamma(NaN)\n"); exit (1); } mpfr_set_inf (x, -1); mpfr_lngamma (y, x, MPFR_RNDN); if (!mpfr_nan_p (y)) { printf ("Error for lngamma(-Inf)\n"); exit (1); } mpfr_set_inf (x, 1); mpfr_lngamma (y, x, MPFR_RNDN); if (!mpfr_inf_p (y) || mpfr_sgn (y) < 0) { printf ("Error for lngamma(+Inf)\n"); exit (1); } mpfr_set_ui (x, 0, MPFR_RNDN); mpfr_lngamma (y, x, MPFR_RNDN); if (!mpfr_inf_p (y) || mpfr_sgn (y) < 0) { printf ("Error for lngamma(+0)\n"); exit (1); } mpfr_set_ui (x, 0, MPFR_RNDN); mpfr_neg (x, x, MPFR_RNDN); mpfr_lngamma (y, x, MPFR_RNDN); if (!mpfr_nan_p (y)) { printf ("Error for lngamma(-0)\n"); exit (1); } mpfr_set_ui (x, 1, MPFR_RNDN); mpfr_lngamma (y, x, MPFR_RNDN); if (MPFR_IS_NAN (y) || mpfr_cmp_ui (y, 0) || MPFR_IS_NEG (y)) { printf ("Error for lngamma(1)\n"); exit (1); } mpfr_set_si (x, -1, MPFR_RNDN); mpfr_lngamma (y, x, MPFR_RNDN); if (!mpfr_nan_p (y)) { printf ("Error for lngamma(-1)\n"); exit (1); } mpfr_set_ui (x, 2, MPFR_RNDN); mpfr_lngamma (y, x, MPFR_RNDN); if (MPFR_IS_NAN (y) || mpfr_cmp_ui (y, 0) || MPFR_IS_NEG (y)) { printf ("Error for lngamma(2)\n"); exit (1); } mpfr_set_prec (x, 53); mpfr_set_prec (y, 53); #define CHECK_X1 "1.0762904832837976166" #define CHECK_Y1 "-0.039418362817587634939" mpfr_set_str (x, CHECK_X1, 10, MPFR_RNDN); mpfr_lngamma (y, x, MPFR_RNDN); mpfr_set_str (x, CHECK_Y1, 10, MPFR_RNDN); if (MPFR_IS_NAN (y) || mpfr_cmp (y, x)) { printf ("mpfr_lngamma("CHECK_X1") is wrong:\n" "expected "); mpfr_print_binary (x); putchar ('\n'); printf ("got "); mpfr_print_binary (y); putchar ('\n'); exit (1); } #define CHECK_X2 "9.23709516716202383435e-01" #define CHECK_Y2 "0.049010669407893718563" mpfr_set_str (x, CHECK_X2, 10, MPFR_RNDN); mpfr_lngamma (y, x, MPFR_RNDN); mpfr_set_str (x, CHECK_Y2, 10, MPFR_RNDN); if (MPFR_IS_NAN (y) || mpfr_cmp (y, x)) { printf ("mpfr_lngamma("CHECK_X2") is wrong:\n" "expected "); mpfr_print_binary (x); putchar ('\n'); printf ("got "); mpfr_print_binary (y); putchar ('\n'); exit (1); } mpfr_set_prec (x, 8); mpfr_set_prec (y, 175); mpfr_set_ui (x, 33, MPFR_RNDN); mpfr_lngamma (y, x, MPFR_RNDU); mpfr_set_prec (x, 175); mpfr_set_str_binary (x, "0.1010001100011101101011001101110010100001000001000001110011000001101100001111001001000101011011100100010101011110100111110101010100010011010010000101010111001100011000101111E7"); if (MPFR_IS_NAN (y) || mpfr_cmp (x, y)) { printf ("Error in mpfr_lngamma (1)\n"); exit (1); } mpfr_set_prec (x, 21); mpfr_set_prec (y, 8); mpfr_set_ui (y, 120, MPFR_RNDN); mpfr_lngamma (x, y, MPFR_RNDZ); mpfr_set_prec (y, 21); mpfr_set_str_binary (y, "0.111000101000001100101E9"); if (MPFR_IS_NAN (x) || mpfr_cmp (x, y)) { printf ("Error in mpfr_lngamma (120)\n"); printf ("Expected "); mpfr_print_binary (y); puts (""); printf ("Got "); mpfr_print_binary (x); puts (""); exit (1); } mpfr_set_prec (x, 3); mpfr_set_prec (y, 206); mpfr_set_str_binary (x, "0.110e10"); inex = mpfr_lngamma (y, x, MPFR_RNDN); mpfr_set_prec (x, 206); mpfr_set_str_binary (x, "0.10000111011000000011100010101001100110001110000111100011000100100110110010001011011110101001111011110110000001010100111011010000000011100110110101100111000111010011110010000100010111101010001101000110101001E13"); if (MPFR_IS_NAN (y) || mpfr_cmp (x, y)) { printf ("Error in mpfr_lngamma (768)\n"); exit (1); } if (inex >= 0) { printf ("Wrong flag for mpfr_lngamma (768)\n"); exit (1); } mpfr_set_prec (x, 4); mpfr_set_prec (y, 4); mpfr_set_str_binary (x, "0.1100E-66"); mpfr_lngamma (y, x, MPFR_RNDN); mpfr_set_str_binary (x, "0.1100E6"); if (MPFR_IS_NAN (y) || mpfr_cmp (x, y)) { printf ("Error for lngamma(0.1100E-66)\n"); exit (1); } mpfr_set_prec (x, 256); mpfr_set_prec (y, 32); mpfr_set_si_2exp (x, -1, 200, MPFR_RNDN); mpfr_add_ui (x, x, 1, MPFR_RNDN); mpfr_div_2ui (x, x, 1, MPFR_RNDN); mpfr_lngamma (y, x, MPFR_RNDN); mpfr_set_prec (x, 32); mpfr_set_str_binary (x, "-0.10001000111011111011000010100010E207"); if (MPFR_IS_NAN (y) || mpfr_cmp (x, y)) { printf ("Error for lngamma(-2^199+0.5)\n"); printf ("Got "); mpfr_dump (y); printf ("instead of "); mpfr_dump (x); exit (1); } mpfr_set_prec (x, 256); mpfr_set_prec (y, 32); mpfr_set_si_2exp (x, -1, 200, MPFR_RNDN); mpfr_sub_ui (x, x, 1, MPFR_RNDN); mpfr_div_2ui (x, x, 1, MPFR_RNDN); mpfr_lngamma (y, x, MPFR_RNDN); if (!mpfr_nan_p (y)) { printf ("Error for lngamma(-2^199-0.5)\n"); exit (1); } mpfr_clear (x); mpfr_clear (y); }
/* return in z a lower bound (for rnd = RNDD) or upper bound (for rnd = RNDU) of |zeta(s)|/2, using: log(|zeta(s)|/2) = (s-1)*log(2*Pi) + lngamma(1-s) + log(|sin(Pi*s/2)| * zeta(1-s)). Assumes s < 1/2 and s1 = 1-s exactly, thus s1 > 1/2. y and p are temporary variables. At input, p is Pi rounded down. The comments in the code are for rnd = RNDD. */ static void mpfr_reflection_overflow (mpfr_t z, mpfr_t s1, const mpfr_t s, mpfr_t y, mpfr_t p, mpfr_rnd_t rnd) { mpz_t sint; MPFR_ASSERTD (rnd == MPFR_RNDD || rnd == MPFR_RNDU); /* Since log is increasing, we want lower bounds on |sin(Pi*s/2)| and zeta(1-s). */ mpz_init (sint); mpfr_get_z (sint, s, MPFR_RNDD); /* sint = floor(s) */ /* We first compute a lower bound of |sin(Pi*s/2)|, which is a periodic function of period 2. Thus: if 2k < s < 2k+1, then |sin(Pi*s/2)| is increasing; if 2k-1 < s < 2k, then |sin(Pi*s/2)| is decreasing. These cases are distinguished by testing bit 0 of floor(s) as if represented in two's complement (or equivalently, as an unsigned integer mod 2): 0: sint = 0 mod 2, thus 2k < s < 2k+1 and |sin(Pi*s/2)| is increasing; 1: sint = 1 mod 2, thus 2k-1 < s < 2k and |sin(Pi*s/2)| is decreasing. Let's recall that the comments are for rnd = RNDD. */ if (mpz_tstbit (sint, 0) == 0) /* |sin(Pi*s/2)| is increasing: round down Pi*s to get a lower bound. */ { mpfr_mul (y, p, s, rnd); if (rnd == MPFR_RNDD) mpfr_nextabove (p); /* we will need p rounded above afterwards */ } else /* |sin(Pi*s/2)| is decreasing: round up Pi*s to get a lower bound. */ { if (rnd == MPFR_RNDD) mpfr_nextabove (p); mpfr_mul (y, p, s, MPFR_INVERT_RND(rnd)); } mpfr_div_2ui (y, y, 1, MPFR_RNDN); /* exact, rounding mode doesn't matter */ /* The rounding direction of sin depends on its sign. We have: if -4k-2 < s < -4k, then -2k-1 < s/2 < -2k, thus sin(Pi*s/2) < 0; if -4k < s < -4k+2, then -2k < s/2 < -2k+1, thus sin(Pi*s/2) > 0. These cases are distinguished by testing bit 1 of floor(s) as if represented in two's complement (or equivalently, as an unsigned integer mod 4): 0: sint = {0,1} mod 4, thus -2k < s/2 < -2k+1 and sin(Pi*s/2) > 0; 1: sint = {2,3} mod 4, thus -2k-1 < s/2 < -2k and sin(Pi*s/2) < 0. Let's recall that the comments are for rnd = RNDD. */ if (mpz_tstbit (sint, 1) == 0) /* -2k < s/2 < -2k+1; sin(Pi*s/2) > 0 */ { /* Round sin down to get a lower bound of |sin(Pi*s/2)|. */ mpfr_sin (y, y, rnd); } else /* -2k-1 < s/2 < -2k; sin(Pi*s/2) < 0 */ { /* Round sin up to get a lower bound of |sin(Pi*s/2)|. */ mpfr_sin (y, y, MPFR_INVERT_RND(rnd)); mpfr_abs (y, y, MPFR_RNDN); /* exact, rounding mode doesn't matter */ } mpz_clear (sint); /* now y <= |sin(Pi*s/2)| when rnd=RNDD, y >= |sin(Pi*s/2)| when rnd=RNDU */ mpfr_zeta_pos (z, s1, rnd); /* zeta(1-s) */ mpfr_mul (z, z, y, rnd); /* now z <= |sin(Pi*s/2)|*zeta(1-s) */ mpfr_log (z, z, rnd); /* now z <= log(|sin(Pi*s/2)|*zeta(1-s)) */ mpfr_lngamma (y, s1, rnd); mpfr_add (z, z, y, rnd); /* z <= lngamma(1-s) + log(|sin(Pi*s/2)|*zeta(1-s)) */ /* since s-1 < 0, we want to round log(2*pi) upwards */ mpfr_mul_2ui (y, p, 1, MPFR_INVERT_RND(rnd)); mpfr_log (y, y, MPFR_INVERT_RND(rnd)); mpfr_mul (y, y, s1, MPFR_INVERT_RND(rnd)); mpfr_sub (z, z, y, rnd); mpfr_exp (z, z, rnd); if (rnd == MPFR_RNDD) mpfr_nextbelow (p); /* restore original p */ }