summaryrefslogtreecommitdiff
path: root/Build/source/libs/mpfr/mpfr-src/src/atan.c
diff options
context:
space:
mode:
Diffstat (limited to 'Build/source/libs/mpfr/mpfr-src/src/atan.c')
-rw-r--r--Build/source/libs/mpfr/mpfr-src/src/atan.c522
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);
-}