/* mpfr_cosu -- cosu(x) = cos(2*pi*x/u) mpfr_cospi -- cospi(x) = cos(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 cos(2*pi*x/u) */ int mpfr_cosu (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, log2u, erra, errb; 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: cos(0) = 1 */ { MPFR_ASSERTD (MPFR_IS_ZERO (x)); return mpfr_set_ui (y, 1, rnd_mode); } } 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), though this doesn't matter. 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. */ 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); return mpfr_set_ui (y, 1, rnd_mode); } xp = xr; } #define CLEAR_XR \ do \ if (xp != x) \ { \ MPFR_ASSERTD (xp == xr); \ mpfr_clear (xr); \ } \ while (0) /* now |xp/u| < 1 */ /* for x small, we have |cos(2*pi*x/u)-1| < 1/2*(2*pi*x/u)^2 < 2^5*(x/u)^2 */ expx = MPFR_GET_EXP (xp); log2u = u == 1 ? 0 : MPFR_INT_CEIL_LOG2 (u) - 1; /* u >= 2^log2u thus 1/u <= 2^(-log2u) */ erra = -2 * expx; errb = 5 - 2 * log2u; /* The 3rd argument (err1) of MPFR_SMALL_INPUT_AFTER_SAVE_EXPO should be erra - errb, but it may overflow. The negative overflow is avoided by the test erra > errb: if erra - errb <= 0, the macro is no-op. Saturate to MPFR_EXP_MAX in case of positive overflow, as the error test in MPFR_SMALL_INPUT_AFTER_SAVE_EXPO will always be true for any value >= MPFR_PREC_MAX + 1, and this includes MPFR_EXP_MAX (from the definition of MPFR_PREC_MAX and mpfr_exp_t >= mpfr_prec_t). */ if (erra > errb) { mpfr_exp_t err1 = errb >= 0 || erra < MPFR_EXP_MAX + errb ? erra - errb : MPFR_EXP_MAX; MPFR_SMALL_INPUT_AFTER_SAVE_EXPO (y, __gmpfr_one, err1, 0, 0, rnd_mode, expo, CLEAR_XR); } precy = MPFR_GET_PREC (y); /* 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 (;;) { nloops ++; /* In the error analysis below, xp stands for x. We first compute an approximation t of 2*pi*x/u, then call cos(t). If t = 2*pi*x/u + s, then |cos(t) - cos(2*pi*x/u)| <= |s|. */ mpfr_set_prec (t, prec); mpfr_const_pi (t, MPFR_RNDN); /* t = pi * (1 + theta1) where |theta1| <= 2^-prec */ mpfr_mul_2ui (t, t, 1, MPFR_RNDN); /* t = 2*pi * (1 + theta1) */ mpfr_mul (t, t, xp, MPFR_RNDN); /* t = 2*pi*x * (1 + theta2)^2 where |theta2| <= 2^-prec */ mpfr_div_ui (t, t, u, MPFR_RNDN); /* t = 2*pi*x/u * (1 + theta3)^3 where |theta3| <= 2^-prec */ /* if t is zero here, it means the division by u underflowd */ if (MPFR_UNLIKELY (MPFR_IS_ZERO (t))) { mpfr_set_ui (y, 1, MPFR_RNDZ); if (MPFR_IS_LIKE_RNDZ(rnd_mode,0)) { inexact = -1; mpfr_nextbelow (y); } else inexact = 1; goto end; } /* since prec >= 2, |(1 + theta3)^3 - 1| <= 4*theta3 <= 2^(2-prec) */ expt = MPFR_GET_EXP (t); /* we have |s| <= 2^(expt + 2 - prec) */ mpfr_cos (t, t, MPFR_RNDN); err = expt + 2 - prec; expt = MPFR_GET_EXP (t); /* new exponent of t */ /* the total error is at most 2^err + ulp(t)/2 = 2^err + 2^(expt-prec-1) thus if err <= expt-prec-1, it is bounded by 2^(expt-prec), otherwise it is bounded by 2^(err+1). */ err = (err <= expt - prec - 1) ? expt - prec : 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: (a) 2*pi*x/u is a multiple of pi/2, i.e., x/u is a multiple of 1/4 (b) 2*pi*x/u is {pi/3,2pi/3,4pi/3,5pi/3} mod 2pi */ if (nloops == 1) { /* detect case (a) */ inexact = mpfr_div_ui (t, xp, u, MPFR_RNDZ); mpfr_mul_2ui (t, t, 2, MPFR_RNDZ); if (inexact == 0 && mpfr_integer_p (t)) { if (mpfr_odd_p (t)) /* t is odd: we have kpi+pi/2, thus cosu = 0, for the sign, we always return +0, following IEEE 754-2019: cosPi(n + 1/2) is +0 for any integer n when n + 1/2 is representable. */ mpfr_set_zero (y, +1); else /* t is even: case kpi */ { mpfr_div_2ui (t, t, 1, MPFR_RNDZ); if (!mpfr_odd_p (t)) /* case 2kpi: cosu = 1 */ mpfr_set_ui (y, 1, MPFR_RNDZ); else mpfr_set_si (y, -1, MPFR_RNDZ); } goto end; } /* detect case (b): this can only occur if u is divisible by 3 */ if ((u % 3) == 0) { inexact = mpfr_div_ui (t, xp, u / 3, MPFR_RNDZ); /* t should be in {1/2,2/2,4/2,5/2} */ mpfr_mul_2ui (t, t, 1, MPFR_RNDZ); /* t should be {1,2,4,5} mod 6: t = 1 mod 6: case pi/3: return 1/2 t = 2 mod 6: case 2pi/3: return -1/2 t = 4 mod 6: case 4pi/3: return -1/2 t = 5 mod 6: case 5pi/3: return 1/2 */ if (inexact == 0 && mpfr_integer_p (t)) { mpz_t z; unsigned long mod6; mpz_init (z); inexact = mpfr_get_z (z, t, MPFR_RNDZ); MPFR_ASSERTN(inexact == 0); mod6 = mpz_fdiv_ui (z, 6); mpz_clear (z); if (mod6 == 1 || mod6 == 5) { mpfr_set_ui_2exp (y, 1, -1, MPFR_RNDZ); goto end; } else /* we cannot have mod6 = 0 or 3 since those case belong to (a) */ { MPFR_ASSERTD(mod6 == 2 || mod6 == 4); mpfr_set_si_2exp (y, -1, -1, MPFR_RNDZ); goto end; } } } } MPFR_ZIV_NEXT (loop, prec); } MPFR_ZIV_FREE (loop); inexact = mpfr_set (y, t, rnd_mode); end: mpfr_clear (t); CLEAR_XR; MPFR_SAVE_EXPO_FREE (expo); return underflow ? inexact : mpfr_check_range (y, inexact, rnd_mode); } int mpfr_cospi (mpfr_ptr y, mpfr_srcptr x, mpfr_rnd_t rnd_mode) { return mpfr_cosu (y, x, 2, rnd_mode); }