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/* mpfr_atanh -- Inverse Hyperbolic Tangente
Copyright 2001-2021 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"
/* Put in y an approximation of atanh(x) for x small.
We assume x <= 1/2, in which case:
x <= y ~ atanh(x) = x + x^3/3 + x^5/5 + x^7/7 + ... <= 2*x.
Return k such that the error is bounded by 2^k*ulp(y).
*/
static int
mpfr_atanh_small (mpfr_ptr y, mpfr_srcptr x)
{
mpfr_prec_t p = MPFR_PREC(y), err;
mpfr_t x2, t, u;
unsigned long i;
int k;
MPFR_ASSERTD(MPFR_GET_EXP (x) <= -1);
/* in the following, theta represents a value with |theta| <= 2^(1-p)
(might be a different value each time) */
mpfr_init2 (t, p);
mpfr_init2 (u, p);
mpfr_init2 (x2, p);
mpfr_set (t, x, MPFR_RNDF); /* t = x * (1 + theta) */
mpfr_set (y, t, MPFR_RNDF); /* exact */
mpfr_sqr (x2, x, MPFR_RNDF); /* x2 = x^2 * (1 + theta) */
for (i = 3; ; i += 2)
{
mpfr_mul (t, t, x2, MPFR_RNDF); /* t = x^i * (1 + theta)^i */
mpfr_div_ui (u, t, i, MPFR_RNDF); /* u = x^i/i * (1 + theta)^(i+1) */
if (MPFR_GET_EXP (u) <= MPFR_GET_EXP (y) - p) /* |u| < ulp(y) */
break;
mpfr_add (y, y, u, MPFR_RNDF); /* error <= ulp(y) */
}
/* We assume |(1 + theta)^(i+1)| <= 2.
The neglected part is at most |u| + |u|/4 + |u|/16 + ... <= 4/3*|u|,
which has to be multiplied by |(1 + theta)^(i+1)| <= 2, thus at most
3 ulp(y).
The rounding error on y is bounded by:
* for the (i-3)/2 add/sub, each error is bounded by ulp(y_i),
where y_i is the current value of y, which is bounded by ulp(y)
for y the final value (since it increases in absolute value),
this yields (i-3)/2*ulp(y)
* from Lemma 3.1 from [Higham02] (see algorithms.tex),
the relative error on u at step i is bounded by:
(i+1)*epsilon/(1-(i+1)*epsilon) where epsilon = 2^(1-p).
If (i+1)*epsilon <= 1/2, then the relative error on u at
step i is bounded by 2*(i+1)*epsilon, and since |u| <= 1/2^(i+1)
at step i, this gives an absolute error bound of;
2*epsilon*x*(4/2^4 + 6/2^6 + 8/2^8 + ...) = 2*2^(1-p)*x*(7/18) =
14/9*2^(-p)*x <= 2*ulp(x).
If (i+1)*epsilon <= 1/2, then the relative error on u at step i
is bounded by (i+1)*epsilon/(1-(i+1)*epsilon) <= 1, thus it follows
|(1 + theta)^(i+1)| <= 2.
Finally the total error is bounded by 3*ulp(y) + (i-3)/2*ulp(y) +2*ulp(x).
Since x <= 2*y, we have ulp(x) <= 2*ulp(y), thus the error is bounded by:
(i+7)/2*ulp(y).
*/
err = (i + 8) / 2; /* ceil((i+7)/2) */
k = __gmpfr_int_ceil_log2 (err);
MPFR_ASSERTN(k + 2 < p);
/* if k + 2 < p, since k = ceil(log2(err)), we have err <= 2^k <= 2^(p-3),
thus i+7 <= 2*err <= 2^(p-2), thus (i+7)*epsilon <= 1/2, which implies
our assumption (i+1)*epsilon <= 1/2. */
mpfr_clear (t);
mpfr_clear (u);
mpfr_clear (x2);
return k;
}
/* The computation of atanh is done by:
atanh = ln((1+x)/(1-x)) / 2
except when x is very small, in which case atanh = x + tiny error,
and when x is small, where we use directly the Taylor expansion.
*/
int
mpfr_atanh (mpfr_ptr y, mpfr_srcptr xt, mpfr_rnd_t rnd_mode)
{
int inexact;
mpfr_t x, t, te;
mpfr_prec_t Nx, Ny, Nt;
mpfr_exp_t err;
MPFR_ZIV_DECL (loop);
MPFR_SAVE_EXPO_DECL (expo);
MPFR_LOG_FUNC
(("x[%Pu]=%.*Rg rnd=%d", mpfr_get_prec (xt), mpfr_log_prec, xt, rnd_mode),
("y[%Pu]=%.*Rg inexact=%d",
mpfr_get_prec (y), mpfr_log_prec, y, inexact));
/* Special cases */
if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (xt)))
{
/* atanh(NaN) = NaN, and atanh(+/-Inf) = NaN since tanh gives a result
between -1 and 1 */
if (MPFR_IS_NAN (xt) || MPFR_IS_INF (xt))
{
MPFR_SET_NAN (y);
MPFR_RET_NAN;
}
else /* necessarily xt is 0 */
{
MPFR_ASSERTD (MPFR_IS_ZERO (xt));
MPFR_SET_ZERO (y); /* atanh(0) = 0 */
MPFR_SET_SAME_SIGN (y,xt);
MPFR_RET (0);
}
}
/* atanh (x) = NaN as soon as |x| > 1, and arctanh(+/-1) = +/-Inf */
if (MPFR_UNLIKELY (MPFR_GET_EXP (xt) > 0))
{
if (MPFR_GET_EXP (xt) == 1 && mpfr_powerof2_raw (xt))
{
MPFR_SET_INF (y);
MPFR_SET_SAME_SIGN (y, xt);
MPFR_SET_DIVBY0 ();
MPFR_RET (0);
}
MPFR_SET_NAN (y);
MPFR_RET_NAN;
}
/* atanh(x) = x + x^3/3 + ... so the error is < 2^(3*EXP(x)-1) */
MPFR_FAST_COMPUTE_IF_SMALL_INPUT (y, xt, -2 * MPFR_GET_EXP (xt), 1, 1,
rnd_mode, {});
MPFR_SAVE_EXPO_MARK (expo);
/* Compute initial precision */
Nx = MPFR_PREC (xt);
MPFR_TMP_INIT_ABS (x, xt);
Ny = MPFR_PREC (y);
Nt = MAX (Nx, Ny);
Nt = Nt + MPFR_INT_CEIL_LOG2 (Nt) + 4;
/* initialize of intermediary variable */
mpfr_init2 (t, Nt);
mpfr_init2 (te, Nt);
MPFR_ZIV_INIT (loop, Nt);
for (;;)
{
int k;
/* small case: assuming the AGM algorithm used by mpfr_log uses
log2(p) steps for a precision of p bits, we try the special
variant whenever EXP(x) <= -p/log2(p). */
k = 1 + __gmpfr_int_ceil_log2 (Ny); /* the +1 avoids a division by 0
when Ny=1 */
if (MPFR_GET_EXP (x) <= - 1 - (mpfr_exp_t) (Ny / k))
/* this implies EXP(x) <= -1 thus x < 1/2 */
{
err = Nt - mpfr_atanh_small (t, x);
goto round;
}
/* compute atanh */
mpfr_ui_sub (te, 1, x, MPFR_RNDU); /* (1-x) with x = |xt| */
mpfr_add_ui (t, x, 1, MPFR_RNDD); /* (1+x) */
mpfr_div (t, t, te, MPFR_RNDN); /* (1+x)/(1-x) */
mpfr_log (t, t, MPFR_RNDN); /* ln((1+x)/(1-x)) */
mpfr_div_2ui (t, t, 1, MPFR_RNDN); /* ln((1+x)/(1-x)) / 2 */
/* error estimate: see algorithms.tex */
/* FIXME: this does not correspond to the value in algorithms.tex!!! */
/* err = Nt - __gmpfr_ceil_log2(1+5*pow(2,1-MPFR_EXP(t))); */
err = Nt - (MAX (4 - MPFR_GET_EXP (t), 0) + 1);
round:
if (MPFR_LIKELY (MPFR_IS_ZERO (t)
|| MPFR_CAN_ROUND (t, err, Ny, rnd_mode)))
break;
/* reactualisation of the precision */
MPFR_ZIV_NEXT (loop, Nt);
mpfr_set_prec (t, Nt);
mpfr_set_prec (te, Nt);
}
MPFR_ZIV_FREE (loop);
inexact = mpfr_set4 (y, t, rnd_mode, MPFR_SIGN (xt));
mpfr_clear (t);
mpfr_clear (te);
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_check_range (y, inexact, rnd_mode);
}
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