/* mpfr_atanh -- Inverse Hyperbolic Tangente Copyright 2001-2023 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); }