/* mpfr_tanu -- tanu(x) = tan(2*pi*x/u) mpfr_tanpi -- tanpi(x) = tan(pi*x) Copyright 2020-2023 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 the correctly rounded value of tan(2*pi*x/u) */ int mpfr_tanu (mpfr_ptr y, mpfr_srcptr x, unsigned long u, mpfr_rnd_t rnd_mode) { mpfr_srcptr xp; mpfr_prec_t precy, prec; mpfr_exp_t expx, expt, err; mpfr_t t, xr; int inexact = 0, nloops = 0, underflow = 0; MPFR_ZIV_DECL (loop); MPFR_SAVE_EXPO_DECL (expo); MPFR_LOG_FUNC ( ("x[%Pd]=%.*Rg u=%lu rnd=%d", mpfr_get_prec (x), mpfr_log_prec, x, u, rnd_mode), ("y[%Pd]=%.*Rg inexact=%d", mpfr_get_prec (y), mpfr_log_prec, y, inexact)); if (u == 0 || MPFR_UNLIKELY (MPFR_IS_SINGULAR (x))) { /* for u=0, return NaN */ if (u == 0 || MPFR_IS_NAN (x) || MPFR_IS_INF (x)) { MPFR_SET_NAN (y); MPFR_RET_NAN; } else /* x is zero */ { MPFR_ASSERTD (MPFR_IS_ZERO (x)); MPFR_SET_ZERO (y); MPFR_SET_SAME_SIGN (y, x); MPFR_RET (0); } } MPFR_SAVE_EXPO_MARK (expo); /* Range reduction. We do not need to reduce the argument if it is already reduced (|x| < u). Note that the case |x| = u is better in the "else" branch as it will give xr = 0. */ if (mpfr_cmpabs_ui (x, u) < 0) { xp = x; } else { mpfr_exp_t p = MPFR_GET_PREC (x) - MPFR_GET_EXP (x); int inex; /* Let's compute xr = x mod u, with signbit(xr) = signbit(x), which may be important when x is a multiple of u, in which case xr = 0 (but this property is actually not needed in the code below). The precision of xr is chosen to ensure that x mod u is exactly representable in xr, e.g., the maximum size of u + the length of the fractional part of x. Note that since |x| >= u in this branch, the additional memory amount will not be more than the one of x. Note that due to the rules on the special values, we needed to consider a period of u instead of u/2. */ mpfr_init2 (xr, sizeof (unsigned long) * CHAR_BIT + (p < 0 ? 0 : p)); MPFR_DBGRES (inex = mpfr_fmod_ui (xr, x, u, MPFR_RNDN)); /* exact */ MPFR_ASSERTD (inex == 0); if (MPFR_IS_ZERO (xr)) { mpfr_clear (xr); MPFR_SAVE_EXPO_FREE (expo); MPFR_SET_ZERO (y); MPFR_SET_SAME_SIGN (y, x); MPFR_RET (0); } xp = xr; } /* now |xp/u| < 1 */ precy = MPFR_GET_PREC (y); expx = MPFR_GET_EXP (xp); /* For x large, since argument reduction is expensive, we want to avoid any failure in Ziv's strategy, thus we take into account expx too. */ prec = precy + MAX(expx,MPFR_INT_CEIL_LOG2(precy)) + 8; MPFR_ASSERTD(prec >= 2); mpfr_init2 (t, prec); MPFR_ZIV_INIT (loop, prec); for (;;) { int inex; nloops ++; /* In the error analysis below, xp stands for x. We first compute an approximation t of 2*pi*x/u, then call tan(t). If t = 2*pi*x/u + s, then |tan(t) - tan(2*pi*x/u)| = |s| * (1 + tan(v)^2) where v is in the interval [t, t+s]. If we ensure that |t| >= |2*pi*x/u|, since tan() is increasing, we can bound tan(v)^2 by tan(t)^2. */ mpfr_set_prec (t, prec); mpfr_const_pi (t, MPFR_RNDU); /* t = pi * (1 + theta1) where |theta1| <= 2^(1-prec) */ mpfr_mul_2ui (t, t, 1, MPFR_RNDN); /* t = 2*pi * (1 + theta1) */ mpfr_mul (t, t, xp, MPFR_RNDA); /* t = 2*pi*x * (1 + theta2)^2 where |theta2| <= 2^(1-prec) */ inex = mpfr_div_ui (t, t, u, MPFR_RNDN); /* t = 2*pi*x/u * (1 + theta3)^3 where |theta3| <= 2^(1-prec) */ /* if t is zero here, it means the division by u underflows, then tan(t) also underflows, since |tan(x)| <= |x|. */ if (MPFR_UNLIKELY (MPFR_IS_ZERO (t))) { inexact = mpfr_underflow (y, rnd_mode, MPFR_SIGN(t)); MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, MPFR_FLAGS_INEXACT | MPFR_FLAGS_UNDERFLOW); underflow = 1; goto end; } /* emulate mpfr_div_ui (t, t, u, MPFR_RNDA) above, so that t is rounded away from zero */ if (MPFR_SIGN(t) > 0 && inex < 0) mpfr_nextabove (t); else if (MPFR_SIGN(t) < 0 && inex > 0) mpfr_nextbelow (t); expt = MPFR_GET_EXP (t); /* since prec >= 3, |(1 + theta3)^3 - 1| <= 4*theta3 <= 2^(3-prec) thus |s| = |t - 2*pi*x/u| <= |t| * 2^(3-prec) */ mpfr_tan (t, t, MPFR_RNDA); { /* compute an upper bound for 1+tan(t)^2 */ mpfr_t z; mpfr_init2 (z, 64); mpfr_sqr (z, t, MPFR_RNDU); mpfr_add_ui (z, z, 1, MPFR_RNDU); expt += MPFR_GET_EXP (z); /* now |t - tan(2*pi*x/u)| <= ulp(t) + 2^(expt + 3 - prec) */ mpfr_clear (z); } /* t cannot be zero here, since we excluded t=0 before, which is the only exact case where tan(t)=0, and we round away from zero */ err = expt + 3 - prec; expt = MPFR_GET_EXP (t); /* new exponent of t */ /* the total error is bounded by 2^err + ulp(t) = 2^err + 2^(expt-prec) thus if err <= expt-prec, it is bounded by 2^(expt-prec+1), otherwise it is bounded by 2^(err+1). */ err = (err <= expt - prec) ? expt - prec + 1 : err + 1; /* normalize err for mpfr_can_round */ err = expt - err; if (MPFR_CAN_ROUND (t, err, precy, rnd_mode)) break; /* Check exact cases only after the first level of Ziv' strategy, to avoid slowing down the average case. Exact cases are when 2*pi*x/u is a multiple of pi/4, i.e., x/u a multiple of 1/8: (a) x/u = {0,1/2} mod 1: return +0 or -0 (b) x/u = {1/4,3/4} mod 1: return +Inf or -Inf (c) x/u = {1/8,3/8,5/8,7/8} mod 1: return 1 or -1 */ if (nloops == 1) { inexact = mpfr_div_ui (t, xp, u, MPFR_RNDA); mpfr_mul_2ui (t, t, 3, MPFR_RNDA); if (inexact == 0 && mpfr_integer_p (t)) { mpz_t z; unsigned long mod8; mpz_init (z); inexact = mpfr_get_z (z, t, MPFR_RNDZ); MPFR_ASSERTN(inexact == 0); mod8 = mpz_fdiv_ui (z, 8); mpz_clear (z); if (mod8 == 0 || mod8 == 4) /* case (a) */ mpfr_set_zero (y, ((mod8 == 0) ? +1 : -1) * MPFR_SIGN (x)); else if (mod8 == 2 || mod8 == 6) /* case (b) */ { mpfr_set_inf (y, (mod8 == 2) ? +1 : -1); MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, MPFR_FLAGS_DIVBY0); } else /* case (c) */ { if (mod8 == 1 || mod8 == 5) mpfr_set_ui (y, 1, rnd_mode); else mpfr_set_si (y, -1, rnd_mode); } goto end; } } MPFR_ZIV_NEXT (loop, prec); } MPFR_ZIV_FREE (loop); inexact = mpfr_set (y, t, rnd_mode); end: mpfr_clear (t); if (xp != x) { MPFR_ASSERTD (xp == xr); mpfr_clear (xr); } MPFR_SAVE_EXPO_FREE (expo); return underflow ? inexact : mpfr_check_range (y, inexact, rnd_mode); } int mpfr_tanpi (mpfr_ptr y, mpfr_srcptr x, mpfr_rnd_t rnd_mode) { return mpfr_tanu (y, x, 2, rnd_mode); }