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/* 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);
}
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