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Diffstat (limited to 'Build/source/libs/mpfr/mpfr-src/src/atan.c')
-rw-r--r-- | Build/source/libs/mpfr/mpfr-src/src/atan.c | 522 |
1 files changed, 0 insertions, 522 deletions
diff --git a/Build/source/libs/mpfr/mpfr-src/src/atan.c b/Build/source/libs/mpfr/mpfr-src/src/atan.c deleted file mode 100644 index 0f63416c95e..00000000000 --- a/Build/source/libs/mpfr/mpfr-src/src/atan.c +++ /dev/null @@ -1,522 +0,0 @@ -/* mpfr_atan -- arc-tangent of a floating-point number - -Copyright 2001-2020 Free Software Foundation, Inc. -Contributed by the AriC and Caramba projects, INRIA. - -This file is part of the GNU MPFR Library. - -The GNU MPFR Library is free software; you can redistribute it and/or modify -it under the terms of the GNU Lesser General Public License as published by -the Free Software Foundation; either version 3 of the License, or (at your -option) any later version. - -The GNU MPFR Library is distributed in the hope that it will be useful, but -WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY -or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public -License for more details. - -You should have received a copy of the GNU Lesser General Public License -along with the GNU MPFR Library; see the file COPYING.LESSER. If not, see -https://www.gnu.org/licenses/ or write to the Free Software Foundation, Inc., -51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA. */ - -#define MPFR_NEED_LONGLONG_H -#include "mpfr-impl.h" - -#if GMP_NUMB_BITS == 64 -/* for each pair (r,p), we store a 192-bit approximation of atan(x)/x for - x=p/2^r, with lowest limb first. - Sage code: - for p in range(1,2^ceil(r/2)): - x=p/2^r - l=floor(2^192*n(atan(x)/x, 300)).digits(2^64) - print ("{0x%x, 0x%x, 0x%x}, /"+"* (%d,%d) *"+"/") % (l[0],l[1],l[2],r,p) -*/ -static const mp_limb_t atan_table[][3] = { - {0x6e141587261cdf00, 0x6fe445ecbc3a8d03, 0xed63382b0dda7b45}, /* (1,1) */ - {0xaa7fa90388b3836b, 0x6dc79ef5f7a217e5, 0xfadbafc96406eb15}, /* (2,1) */ - {0x319c12cf59d4b2dc, 0xcb2792dc0e2e0d51, 0xffaaddb967ef4e36}, /* (4,1) */ - {0x8b3957d95d9ad922, 0xc897989f3e888ef7, 0xfeadd4d5617b6e32}, /* (4,2) */ - {0xc4e6abc8af62e439, 0x4eb9bf602625f0b4, 0xfd0fcdd343cac19b}, /* (4,3) */ - {0x7c18baeb9bc95789, 0xb12afb6b6d4f7e16, 0xffffaaaaddddb94b}, /* (8,1) */ - {0x6856a0171a2f001a, 0x62351fbbe60af47, 0xfffeaaadddd4b968}, /* (8,2) */ - {0x69164c094f49da06, 0xd517294f7373d07a, 0xfffd001032cb1179}, /* (8,3) */ - {0x20ef65c10deef460, 0xe78c564015f76048, 0xfffaaadddb94d5bb}, /* (8,4) */ - {0x3ce233aa002f0344, 0x9dd8ea342a65d4cc, 0xfff7ab27a1f32f95}, /* (8,5) */ - {0xa37f403c7279c5cb, 0x13ab53a1c8db8497, 0xfff40103192ce74d}, /* (8,6) */ - {0xe5a85657103c1aa8, 0xb8409e6c914191d3, 0xffefac8a9c40a26b}, /* (8,7) */ - {0x806d0294c0db8816, 0x779d776dda8c6213, 0xffeaaddd4bb12542}, /* (8,8) */ - {0x5545d1914ef21478, 0x3aea58d6660f5a12, 0xffe5051f0aebf73a}, /* (8,9) */ - {0x6e47a91d015f4133, 0xc085ab6b490b7f02, 0xffdeb2787d4adac1}, /* (8,10) */ - {0x4efc1f931f7ec9b3, 0xb7f43cd16195ef4b, 0xffd7b61702b09aad}, /* (8,11) */ - {0xd27d1dbf55fed60d, 0xd812c11d7d473e5e, 0xffd0102cb3c1bfbe}, /* (8,12) */ - {0xca629e927383fe97, 0x8c61aedf58e42206, 0xffc7c0f05db9d1b6}, /* (8,13) */ - {0x4eff0b53d4e905b7, 0x28ac1e800ca31e9d, 0xffbec89d7dddd7e9}, /* (8,14) */ - {0xb0a7931deec6fe60, 0xb46feea78588554b, 0xffb527743c8cdd8f} /* (8,15) */ - }; - -static void -set_table (mpfr_t y, const mp_limb_t x[3]) -{ - mpfr_prec_t p = MPFR_PREC(y); - mp_size_t n = MPFR_PREC2LIMBS(p); - mpfr_prec_t sh; - mp_limb_t *yp = MPFR_MANT(y); - - MPFR_UNSIGNED_MINUS_MODULO (sh, p); - MPFR_ASSERTD (n >= 1 && n <= 3); - mpn_copyi (yp, x + 3 - n, n); - yp[0] &= ~MPFR_LIMB_MASK(sh); - MPFR_SET_EXP(y, 0); -} -#endif - -/* If x = p/2^r, put in y an approximation to atan(x)/x using 2^m terms - for the series expansion, with an error of at most 1 ulp. - Assumes 0 < x < 1, thus 1 <= p < 2^r. - More precisely, p consists of the floor(r/2) bits of the binary expansion - of a number 0 < s < 1: - * the bit of weight 2^-1 is for r=1, thus p <= 1 - * the bit of weight 2^-2 is for r=2, thus p <= 1 - * the two bits of weight 2^-3 and 2^-4 are for r=4, thus p <= 3 - * more generally p < 2^(r/2). - - If X=x^2, we want 1 - X/3 + X^2/5 - ... + (-1)^k*X^k/(2k+1) + ... - - When we sum terms up to x^k/(2k+1), the denominator Q[0] is - 3*5*7*...*(2k+1) ~ (2k/e)^k. - - The tab[] array should have at least 3*(m+1) entries. -*/ -static void -mpfr_atan_aux (mpfr_ptr y, mpz_ptr p, unsigned long r, int m, mpz_t *tab) -{ - mpz_t *S, *Q, *ptoj; - mp_bitcnt_t n, h, j; /* unsigned type, which is >= unsigned long */ - mpfr_exp_t diff, expo; - int im, i, k, l, done; - mpfr_prec_t mult; - mpfr_prec_t accu[MPFR_PREC_BITS], log2_nb_terms[MPFR_PREC_BITS]; - mpfr_prec_t precy = MPFR_PREC(y); - - MPFR_ASSERTD (mpz_sgn (p) > 0); - MPFR_ASSERTD (m > 0); - MPFR_ASSERTD (m <= MPFR_PREC_BITS - 1); - -#if GMP_NUMB_BITS == 64 - /* tabulate values for small precision and small value of r (which are the - most expensive to compute) */ - if (precy <= 192) - { - unsigned long u; - - switch (r) - { - case 1: - /* p has 1 bit: necessarily p=1 */ - MPFR_ASSERTD(mpz_cmp_ui (p, 1) == 0); - set_table (y, atan_table[0]); - return; - case 2: - /* p has 1 bit: necessarily p=1 too */ - MPFR_ASSERTD(mpz_cmp_ui (p, 1) == 0); - set_table (y, atan_table[1]); - return; - case 4: - /* p has at most 2 bits: 1 <= p <= 3 */ - u = mpz_get_ui (p); - MPFR_ASSERTD(1 <= u && u <= 3); - set_table (y, atan_table[1 + u]); - return; - case 8: - /* p has at most 4 bits: 1 <= p <= 15 */ - u = mpz_get_ui (p); - MPFR_ASSERTD(1 <= u && u <= 15); - set_table (y, atan_table[4 + u]); - return; - } - } -#endif - - /* Set Tables */ - S = tab; /* S */ - ptoj = S + 1*(m+1); /* p^2^j Precomputed table */ - Q = S + 2*(m+1); /* Product of Odd integer table */ - - /* From p to p^2, and r to 2r */ - mpz_mul (p, p, p); - MPFR_ASSERTD (2 * r > r); - r = 2 * r; - - /* Normalize p */ - n = mpz_scan1 (p, 0); - if (n > 0) - { - mpz_tdiv_q_2exp (p, p, n); /* exact */ - MPFR_ASSERTD (r > n); - r -= n; - } - - /* Since |p/2^r| < 1, and p is a non-zero integer, necessarily r > 0. */ - MPFR_ASSERTD (mpz_sgn (p) > 0); - MPFR_ASSERTD (m > 0); - MPFR_ASSERTD (r > 0); - - /* check if p=1 (special case) */ - l = 0; - /* - We compute by binary splitting, with X = x^2 = p/2^r: - P(a,b) = p if a+1=b, P(a,c)*P(c,b) otherwise - Q(a,b) = (2a+1)*2^r if a+1=b [except Q(0,1)=1], Q(a,c)*Q(c,b) otherwise - S(a,b) = p*(2a+1) if a+1=b, Q(c,b)*S(a,c)+Q(a,c)*P(a,c)*S(c,b) otherwise - Then atan(x)/x ~ S(0,i)/Q(0,i) for i so that (p/2^r)^i/i is small enough. - The factor 2^(r*(b-a)) in Q(a,b) is implicit, thus we have to take it - into account when we compute with Q. - */ - accu[0] = 0; /* accu[k] = Mult[0] + ... + Mult[k], where Mult[j] is the - number of bits of the corresponding term S[j]/Q[j] */ - if (mpz_cmp_ui (p, 1) != 0) - { - /* p <> 1: precompute ptoj table */ - mpz_set (ptoj[0], p); - for (im = 1 ; im <= m ; im ++) - mpz_mul (ptoj[im], ptoj[im - 1], ptoj[im - 1]); - /* main loop */ - n = 1UL << m; - MPFR_ASSERTN (n != 0); /* no overflow */ - /* the i-th term being X^i/(2i+1) with X=p/2^r, we can stop when - p^i/2^(r*i) < 2^(-precy), i.e. r*i > precy + log2(p^i) */ - for (i = k = done = 0; (i < n) && (done == 0); i += 2, k ++) - { - /* initialize both S[k],Q[k] and S[k+1],Q[k+1] */ - mpz_set_ui (Q[k+1], 2 * i + 3); /* Q(i+1,i+2) */ - mpz_mul_ui (S[k+1], p, 2 * i + 1); /* S(i+1,i+2) */ - mpz_mul_2exp (S[k], Q[k+1], r); - mpz_sub (S[k], S[k], S[k+1]); /* S(i,i+2) */ - mpz_mul_ui (Q[k], Q[k+1], 2 * i + 1); /* Q(i,i+2) */ - log2_nb_terms[k] = 1; /* S[k]/Q[k] corresponds to 2 terms */ - for (j = (i + 2) >> 1, l = 1; (j & 1) == 0; l ++, j >>= 1, k --) - { - /* invariant: S[k-1]/Q[k-1] and S[k]/Q[k] correspond - to 2^l terms each. We combine them into S[k-1]/Q[k-1] */ - MPFR_ASSERTD (k > 0); - mpz_mul (S[k], S[k], Q[k-1]); - mpz_mul (S[k], S[k], ptoj[l]); - mpz_mul (S[k-1], S[k-1], Q[k]); - mpz_mul_2exp (S[k-1], S[k-1], r << l); - mpz_add (S[k-1], S[k-1], S[k]); - mpz_mul (Q[k-1], Q[k-1], Q[k]); - log2_nb_terms[k-1] = l + 1; - /* now S[k-1]/Q[k-1] corresponds to 2^(l+1) terms */ - MPFR_MPZ_SIZEINBASE2(mult, ptoj[l+1]); - mult = (r << (l + 1)) - mult - 1; - accu[k-1] = (k == 1) ? mult : accu[k-2] + mult; - if (accu[k-1] > precy) - done = 1; - } - } - } - else /* special case p=1: the i-th term being X^i/(2i+1) with X=1/2^r, - we can stop when r*i > precy i.e. i > precy/r */ - { - n = 1UL << m; - if (precy / r <= n) - n = (precy / r) + 1; - MPFR_ASSERTN (n != 0); /* no overflow */ - for (i = k = 0; i < n; i += 2, k ++) - { - mpz_set_ui (Q[k + 1], 2 * i + 3); - mpz_mul_2exp (S[k], Q[k+1], r); - mpz_sub_ui (S[k], S[k], 1 + 2 * i); - mpz_mul_ui (Q[k], Q[k + 1], 1 + 2 * i); - log2_nb_terms[k] = 1; /* S[k]/Q[k] corresponds to 2 terms */ - for (j = (i + 2) >> 1, l = 1; (j & 1) == 0; l++, j >>= 1, k --) - { - MPFR_ASSERTD (k > 0); - mpz_mul (S[k], S[k], Q[k-1]); - mpz_mul (S[k-1], S[k-1], Q[k]); - mpz_mul_2exp (S[k-1], S[k-1], r << l); - mpz_add (S[k-1], S[k-1], S[k]); - mpz_mul (Q[k-1], Q[k-1], Q[k]); - log2_nb_terms[k-1] = l + 1; - } - } - } - - /* we need to combine S[0]/Q[0]...S[k-1]/Q[k-1] */ - h = 0; /* number of terms accumulated in S[k]/Q[k] */ - while (k > 1) - { - k --; - /* combine S[k-1]/Q[k-1] and S[k]/Q[k] */ - mpz_mul (S[k], S[k], Q[k-1]); - if (mpz_cmp_ui (p, 1) != 0) - mpz_mul (S[k], S[k], ptoj[log2_nb_terms[k-1]]); - mpz_mul (S[k-1], S[k-1], Q[k]); - h += (mp_bitcnt_t) 1 << log2_nb_terms[k]; - mpz_mul_2exp (S[k-1], S[k-1], r * h); - mpz_add (S[k-1], S[k-1], S[k]); - mpz_mul (Q[k-1], Q[k-1], Q[k]); - } - - MPFR_MPZ_SIZEINBASE2 (diff, S[0]); - diff -= 2 * precy; - expo = diff; - if (diff >= 0) - mpz_tdiv_q_2exp (S[0], S[0], diff); - else - mpz_mul_2exp (S[0], S[0], -diff); - - MPFR_MPZ_SIZEINBASE2 (diff, Q[0]); - diff -= precy; - expo -= diff; - if (diff >= 0) - mpz_tdiv_q_2exp (Q[0], Q[0], diff); - else - mpz_mul_2exp (Q[0], Q[0], -diff); - - mpz_tdiv_q (S[0], S[0], Q[0]); - mpfr_set_z (y, S[0], MPFR_RNDD); - /* TODO: Check/prove that the following expression doesn't overflow. */ - expo = MPFR_GET_EXP (y) + expo - r * (i - 1); - MPFR_SET_EXP (y, expo); -} - -int -mpfr_atan (mpfr_ptr atan, mpfr_srcptr x, mpfr_rnd_t rnd_mode) -{ - mpfr_t xp, arctgt, sk, tmp, tmp2; - mpz_t ukz; - mpz_t tabz[3*(MPFR_PREC_BITS+1)]; - mpfr_exp_t exptol; - mpfr_prec_t prec, realprec, est_lost, lost; - unsigned long twopoweri, log2p, red; - int comparison, inexact; - int i, n0, oldn0; - MPFR_GROUP_DECL (group); - MPFR_SAVE_EXPO_DECL (expo); - MPFR_ZIV_DECL (loop); - - MPFR_LOG_FUNC - (("x[%Pu]=%.*Rg rnd=%d", mpfr_get_prec (x), mpfr_log_prec, x, rnd_mode), - ("atan[%Pu]=%.*Rg inexact=%d", - mpfr_get_prec (atan), mpfr_log_prec, atan, inexact)); - - /* Singular cases */ - if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x))) - { - if (MPFR_IS_NAN (x)) - { - MPFR_SET_NAN (atan); - MPFR_RET_NAN; - } - else if (MPFR_IS_INF (x)) - { - MPFR_SAVE_EXPO_MARK (expo); - if (MPFR_IS_POS (x)) /* arctan(+inf) = Pi/2 */ - inexact = mpfr_const_pi (atan, rnd_mode); - else /* arctan(-inf) = -Pi/2 */ - { - inexact = -mpfr_const_pi (atan, - MPFR_INVERT_RND (rnd_mode)); - MPFR_CHANGE_SIGN (atan); - } - mpfr_div_2ui (atan, atan, 1, rnd_mode); /* exact (no exceptions) */ - MPFR_SAVE_EXPO_FREE (expo); - return mpfr_check_range (atan, inexact, rnd_mode); - } - else /* x is necessarily 0 */ - { - MPFR_ASSERTD (MPFR_IS_ZERO (x)); - MPFR_SET_ZERO (atan); - MPFR_SET_SAME_SIGN (atan, x); - MPFR_RET (0); - } - } - - /* atan(x) = x - x^3/3 + x^5/5... - so the error is < 2^(3*EXP(x)-1) - so `EXP(x)-(3*EXP(x)-1)` = -2*EXP(x)+1 */ - MPFR_FAST_COMPUTE_IF_SMALL_INPUT (atan, x, -2 * MPFR_GET_EXP (x), 1, 0, - rnd_mode, {}); - - /* Set x_p=|x| */ - MPFR_TMP_INIT_ABS (xp, x); - - MPFR_SAVE_EXPO_MARK (expo); - - /* Other simple case arctan(-+1)=-+pi/4 */ - comparison = mpfr_cmp_ui (xp, 1); - if (MPFR_UNLIKELY (comparison == 0)) - { - int neg = MPFR_IS_NEG (x); - inexact = mpfr_const_pi (atan, MPFR_IS_POS (x) ? rnd_mode - : MPFR_INVERT_RND (rnd_mode)); - if (neg) - { - inexact = -inexact; - MPFR_CHANGE_SIGN (atan); - } - mpfr_div_2ui (atan, atan, 2, rnd_mode); /* exact (no exceptions) */ - MPFR_SAVE_EXPO_FREE (expo); - return mpfr_check_range (atan, inexact, rnd_mode); - } - - realprec = MPFR_PREC (atan) + MPFR_INT_CEIL_LOG2 (MPFR_PREC (atan)) + 4; - prec = realprec + GMP_NUMB_BITS; - - /* Initialisation */ - mpz_init2 (ukz, prec); /* ukz will need 'prec' bits below */ - MPFR_GROUP_INIT_4 (group, prec, sk, tmp, tmp2, arctgt); - oldn0 = 0; - - MPFR_ZIV_INIT (loop, prec); - for (;;) - { - /* First, if |x| < 1, we need to have more prec to be able to round (sup) - n0 = ceil(log(prec_requested + 2 + 1+ln(2.4)/ln(2))/log(2)) */ - mpfr_prec_t sup; - sup = MPFR_GET_EXP (xp) < 0 ? 2 - MPFR_GET_EXP (xp) : 1; /* sup >= 1 */ - - n0 = MPFR_INT_CEIL_LOG2 ((realprec + sup) + 3); - /* since realprec >= 4, n0 >= ceil(log2(8)) >= 3, thus 3*n0 > 2 */ - prec = (realprec + sup) + 1 + MPFR_INT_CEIL_LOG2 (3*n0-2); - - /* the number of lost bits due to argument reduction is - 9 - 2 * EXP(sk), which we estimate by 9 + 2*ceil(log2(p)) - since we manage that sk < 1/p */ - if (MPFR_PREC (atan) > 100) - { - log2p = MPFR_INT_CEIL_LOG2(prec) / 2 - 3; - est_lost = 9 + 2 * log2p; - prec += est_lost; - } - else - log2p = est_lost = 0; /* don't reduce the argument */ - - /* Initialisation */ - MPFR_GROUP_REPREC_4 (group, prec, sk, tmp, tmp2, arctgt); - MPFR_ASSERTD (n0 <= MPFR_PREC_BITS); - /* Note: the tabz[] entries are used to get a rational approximation - of atan(x) to precision 'prec', thus allocating them to 'prec' bits - should be good enough. */ - for (i = oldn0; i < 3 * (n0 + 1); i++) - mpz_init2 (tabz[i], prec); - oldn0 = 3 * (n0 + 1); - - /* The mpfr_ui_div below mustn't underflow. This is guaranteed by - MPFR_SAVE_EXPO_MARK, but let's check that for maintainability. */ - MPFR_ASSERTD (__gmpfr_emax <= 1 - __gmpfr_emin); - - if (comparison > 0) /* use atan(xp) = Pi/2 - atan(1/xp) */ - mpfr_ui_div (sk, 1, xp, MPFR_RNDN); - else - mpfr_set (sk, xp, MPFR_RNDN); - - /* now 0 < sk <= 1 */ - - /* Argument reduction: atan(x) = 2 atan((sqrt(1+x^2)-1)/x). - We want |sk| < k/sqrt(p) where p is the target precision. */ - lost = 0; - for (red = 0; MPFR_GET_EXP(sk) > - (mpfr_exp_t) log2p; red ++) - { - lost = 9 - 2 * MPFR_EXP(sk); - mpfr_sqr (tmp, sk, MPFR_RNDN); - mpfr_add_ui (tmp, tmp, 1, MPFR_RNDN); - mpfr_sqrt (tmp, tmp, MPFR_RNDN); - mpfr_sub_ui (tmp, tmp, 1, MPFR_RNDN); - if (red == 0 && comparison > 0) - /* use xp = 1/sk */ - mpfr_mul (sk, tmp, xp, MPFR_RNDN); - else - mpfr_div (sk, tmp, sk, MPFR_RNDN); - } - - /* We started from x0 = 1/|x| if |x| > 1, and |x| otherwise, thus - we had x0 = min(|x|, 1/|x|) <= 1, and applied 'red' times the - argument reduction x -> (sqrt(1+x^2)-1)/x, which keeps 0 < x <= 1 */ - - /* We first show that if the for-loop is executed at least once, then - sk < 1 after the loop. Indeed for 1/2 <= x <= 1, interval - arithmetic with precision 5 shows that (sqrt(1+x^2)-1)/x, - when evaluated with rounding to nearest, gives a value <= 0.875. - Now assume 2^(-k-1) <= x <= 2^(-k) for k >= 1. - Then o(x^2) <= 2^(-2k), o(1+x^2) <= 1+2^(-2k), - o(sqrt(1+x^2)) <= 1+2^(-2k-1), o(sqrt(1+x^2)-1) <= 2^(-2k-1), - and o((sqrt(1+x^2)-1)/x) <= 2^(-k) <= 1/2. - - Now if sk=1 before the loop, then EXP(sk)=1 and since log2p >= 0, - the loop is performed at least once, thus the case sk=1 cannot - happen below. - */ - - MPFR_ASSERTD(mpfr_cmp_ui (sk, 1) < 0); - - /* Assignation */ - MPFR_SET_ZERO (arctgt); - twopoweri = 1 << 0; - MPFR_ASSERTD (n0 >= 4); - for (i = 0 ; i < n0; i++) - { - if (MPFR_UNLIKELY (MPFR_IS_ZERO (sk))) - break; - /* Calculation of trunc(tmp) --> mpz */ - mpfr_mul_2ui (tmp, sk, twopoweri, MPFR_RNDN); - mpfr_trunc (tmp, tmp); - if (!MPFR_IS_ZERO (tmp)) - { - /* tmp = ukz*2^exptol */ - exptol = mpfr_get_z_2exp (ukz, tmp); - /* since the s_k are decreasing (see algorithms.tex), - and s_0 = min(|x|, 1/|x|) < 1, we have sk < 1, - thus exptol < 0 */ - MPFR_ASSERTD (exptol < 0); - mpz_tdiv_q_2exp (ukz, ukz, (unsigned long int) (-exptol)); - /* since tmp is a non-zero integer, and tmp = ukzold*2^exptol, - we now have ukz = tmp, thus ukz is non-zero */ - /* Calculation of arctan(Ak) */ - mpfr_set_z (tmp, ukz, MPFR_RNDN); - mpfr_div_2ui (tmp, tmp, twopoweri, MPFR_RNDN); - mpfr_atan_aux (tmp2, ukz, twopoweri, n0 - i, tabz); - mpfr_mul (tmp2, tmp2, tmp, MPFR_RNDN); - /* Addition */ - mpfr_add (arctgt, arctgt, tmp2, MPFR_RNDN); - /* Next iteration */ - mpfr_sub (tmp2, sk, tmp, MPFR_RNDN); - mpfr_mul (sk, sk, tmp, MPFR_RNDN); - mpfr_add_ui (sk, sk, 1, MPFR_RNDN); - mpfr_div (sk, tmp2, sk, MPFR_RNDN); - } - twopoweri <<= 1; - } - /* Add last step (Arctan(sk) ~= sk */ - mpfr_add (arctgt, arctgt, sk, MPFR_RNDN); - - /* argument reduction */ - mpfr_mul_2ui (arctgt, arctgt, red, MPFR_RNDN); - - if (comparison > 0) - { /* atan(x) = Pi/2-atan(1/x) for x > 0 */ - mpfr_const_pi (tmp, MPFR_RNDN); - mpfr_div_2ui (tmp, tmp, 1, MPFR_RNDN); - mpfr_sub (arctgt, tmp, arctgt, MPFR_RNDN); - } - MPFR_SET_POS (arctgt); - - if (MPFR_LIKELY (MPFR_CAN_ROUND (arctgt, realprec + est_lost - lost, - MPFR_PREC (atan), rnd_mode))) - break; - MPFR_ZIV_NEXT (loop, realprec); - } - MPFR_ZIV_FREE (loop); - - inexact = mpfr_set4 (atan, arctgt, rnd_mode, MPFR_SIGN (x)); - - for (i = 0 ; i < oldn0 ; i++) - mpz_clear (tabz[i]); - mpz_clear (ukz); - MPFR_GROUP_CLEAR (group); - - MPFR_SAVE_EXPO_FREE (expo); - return mpfr_check_range (atan, inexact, rnd_mode); -} |