/* mpfr_cos -- cosine of a floating-point number Copyright 2001-2015 Free Software Foundation, Inc. Contributed by the AriC and Caramel 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 http://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" static int mpfr_cos_fast (mpfr_ptr y, mpfr_srcptr x, mpfr_rnd_t rnd_mode) { int inex; inex = mpfr_sincos_fast (NULL, y, x, rnd_mode); inex = inex >> 2; /* 0: exact, 1: rounded up, 2: rounded down */ return (inex == 2) ? -1 : inex; } /* f <- 1 - r/2! + r^2/4! + ... + (-1)^l r^l/(2l)! + ... Assumes |r| < 1/2, and f, r have the same precision. Returns e such that the error on f is bounded by 2^e ulps. */ static int mpfr_cos2_aux (mpfr_ptr f, mpfr_srcptr r) { mpz_t x, t, s; mpfr_exp_t ex, l, m; mpfr_prec_t p, q; unsigned long i, maxi, imax; MPFR_ASSERTD(mpfr_get_exp (r) <= -1); /* compute minimal i such that i*(i+1) does not fit in an unsigned long, assuming that there are no padding bits. */ maxi = 1UL << (CHAR_BIT * sizeof(unsigned long) / 2); if (maxi * (maxi / 2) == 0) /* test checked at compile time */ { /* can occur only when there are padding bits. */ /* maxi * (maxi-1) is representable iff maxi * (maxi / 2) != 0 */ do maxi /= 2; while (maxi * (maxi / 2) == 0); } mpz_init (x); mpz_init (s); mpz_init (t); ex = mpfr_get_z_2exp (x, r); /* r = x*2^ex */ /* remove trailing zeroes */ l = mpz_scan1 (x, 0); ex += l; mpz_fdiv_q_2exp (x, x, l); /* since |r| < 1, r = x*2^ex, and x is an integer, necessarily ex < 0 */ p = mpfr_get_prec (f); /* same than r */ /* bound for number of iterations */ imax = p / (-mpfr_get_exp (r)); imax += (imax == 0); q = 2 * MPFR_INT_CEIL_LOG2(imax) + 4; /* bound for (3l)^2 */ mpz_set_ui (s, 1); /* initialize sum with 1 */ mpz_mul_2exp (s, s, p + q); /* scale all values by 2^(p+q) */ mpz_set (t, s); /* invariant: t is previous term */ for (i = 1; (m = mpz_sizeinbase (t, 2)) >= q; i += 2) { /* adjust precision of x to that of t */ l = mpz_sizeinbase (x, 2); if (l > m) { l -= m; mpz_fdiv_q_2exp (x, x, l); ex += l; } /* multiply t by r */ mpz_mul (t, t, x); mpz_fdiv_q_2exp (t, t, -ex); /* divide t by i*(i+1) */ if (i < maxi) mpz_fdiv_q_ui (t, t, i * (i + 1)); else { mpz_fdiv_q_ui (t, t, i); mpz_fdiv_q_ui (t, t, i + 1); } /* if m is the (current) number of bits of t, we can consider that all operations on t so far had precision >= m, so we can prove by induction that the relative error on t is of the form (1+u)^(3l)-1, where |u| <= 2^(-m), and l=(i+1)/2 is the # of loops. Since |(1+x^2)^(1/x) - 1| <= 4x/3 for |x| <= 1/2, for |u| <= 1/(3l)^2, the absolute error is bounded by 4/3*(3l)*2^(-m)*t <= 4*l since |t| < 2^m. Therefore the error on s is bounded by 2*l*(l+1). */ /* add or subtract to s */ if (i % 4 == 1) mpz_sub (s, s, t); else mpz_add (s, s, t); } mpfr_set_z (f, s, MPFR_RNDN); mpfr_div_2ui (f, f, p + q, MPFR_RNDN); mpz_clear (x); mpz_clear (s); mpz_clear (t); l = (i - 1) / 2; /* number of iterations */ return 2 * MPFR_INT_CEIL_LOG2 (l + 1) + 1; /* bound is 2l(l+1) */ } int mpfr_cos (mpfr_ptr y, mpfr_srcptr x, mpfr_rnd_t rnd_mode) { mpfr_prec_t K0, K, precy, m, k, l; int inexact, reduce = 0; mpfr_t r, s, xr, c; mpfr_exp_t exps, cancel = 0, expx; MPFR_ZIV_DECL (loop); MPFR_SAVE_EXPO_DECL (expo); MPFR_GROUP_DECL (group); MPFR_LOG_FUNC ( ("x[%Pu]=%*.Rg rnd=%d", mpfr_get_prec (x), mpfr_log_prec, x, rnd_mode), ("y[%Pu]=%*.Rg inexact=%d", mpfr_get_prec (y), mpfr_log_prec, y, inexact)); if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x))) { if (MPFR_IS_NAN (x) || MPFR_IS_INF (x)) { MPFR_SET_NAN (y); MPFR_RET_NAN; } else { MPFR_ASSERTD (MPFR_IS_ZERO (x)); return mpfr_set_ui (y, 1, rnd_mode); } } MPFR_SAVE_EXPO_MARK (expo); /* cos(x) = 1-x^2/2 + ..., so error < 2^(2*EXP(x)-1) */ expx = MPFR_GET_EXP (x); MPFR_SMALL_INPUT_AFTER_SAVE_EXPO (y, __gmpfr_one, -2 * expx, 1, 0, rnd_mode, expo, {}); /* Compute initial precision */ precy = MPFR_PREC (y); if (precy >= MPFR_SINCOS_THRESHOLD) { inexact = mpfr_cos_fast (y, x, rnd_mode); goto end; } K0 = __gmpfr_isqrt (precy / 3); m = precy + 2 * MPFR_INT_CEIL_LOG2 (precy) + 2 * K0; if (expx >= 3) { reduce = 1; /* As expx + m - 1 will silently be converted into mpfr_prec_t in the mpfr_init2 call, the assert below may be useful to avoid undefined behavior. */ MPFR_ASSERTN (expx + m - 1 <= MPFR_PREC_MAX); mpfr_init2 (c, expx + m - 1); mpfr_init2 (xr, m); } MPFR_GROUP_INIT_2 (group, m, r, s); MPFR_ZIV_INIT (loop, m); for (;;) { /* If |x| >= 4, first reduce x cmod (2*Pi) into xr, using mpfr_remainder: let e = EXP(x) >= 3, and m the target precision: (1) c <- 2*Pi [precision e+m-1, nearest] (2) xr <- remainder (x, c) [precision m, nearest] We have |c - 2*Pi| <= 1/2ulp(c) = 2^(3-e-m) |xr - x - k c| <= 1/2ulp(xr) <= 2^(1-m) |k| <= |x|/(2*Pi) <= 2^(e-2) Thus |xr - x - 2kPi| <= |k| |c - 2Pi| + 2^(1-m) <= 2^(2-m). It follows |cos(xr) - cos(x)| <= 2^(2-m). */ if (reduce) { mpfr_const_pi (c, MPFR_RNDN); mpfr_mul_2ui (c, c, 1, MPFR_RNDN); /* 2Pi */ mpfr_remainder (xr, x, c, MPFR_RNDN); if (MPFR_IS_ZERO(xr)) goto ziv_next; /* now |xr| <= 4, thus r <= 16 below */ mpfr_mul (r, xr, xr, MPFR_RNDU); /* err <= 1 ulp */ } else mpfr_mul (r, x, x, MPFR_RNDU); /* err <= 1 ulp */ /* now |x| < 4 (or xr if reduce = 1), thus |r| <= 16 */ /* we need |r| < 1/2 for mpfr_cos2_aux, i.e., EXP(r) - 2K <= -1 */ K = K0 + 1 + MAX(0, MPFR_GET_EXP(r)) / 2; /* since K0 >= 0, if EXP(r) < 0, then K >= 1, thus EXP(r) - 2K <= -3; otherwise if EXP(r) >= 0, then K >= 1/2 + EXP(r)/2, thus EXP(r) - 2K <= -1 */ MPFR_SET_EXP (r, MPFR_GET_EXP (r) - 2 * K); /* Can't overflow! */ /* s <- 1 - r/2! + ... + (-1)^l r^l/(2l)! */ l = mpfr_cos2_aux (s, r); /* l is the error bound in ulps on s */ MPFR_SET_ONE (r); for (k = 0; k < K; k++) { mpfr_sqr (s, s, MPFR_RNDU); /* err <= 2*olderr */ MPFR_SET_EXP (s, MPFR_GET_EXP (s) + 1); /* Can't overflow */ mpfr_sub (s, s, r, MPFR_RNDN); /* err <= 4*olderr */ if (MPFR_IS_ZERO(s)) goto ziv_next; MPFR_ASSERTD (MPFR_GET_EXP (s) <= 1); } /* The absolute error on s is bounded by (2l+1/3)*2^(2K-m) 2l+1/3 <= 2l+1. If |x| >= 4, we need to add 2^(2-m) for the argument reduction by 2Pi: if K = 0, this amounts to add 4 to 2l+1/3, i.e., to add 2 to l; if K >= 1, this amounts to add 1 to 2*l+1/3. */ l = 2 * l + 1; if (reduce) l += (K == 0) ? 4 : 1; k = MPFR_INT_CEIL_LOG2 (l) + 2*K; /* now the error is bounded by 2^(k-m) = 2^(EXP(s)-err) */ exps = MPFR_GET_EXP (s); if (MPFR_LIKELY (MPFR_CAN_ROUND (s, exps + m - k, precy, rnd_mode))) break; if (MPFR_UNLIKELY (exps == 1)) /* s = 1 or -1, and except x=0 which was already checked above, cos(x) cannot be 1 or -1, so we can round if the error is less than 2^(-precy) for directed rounding, or 2^(-precy-1) for rounding to nearest. */ { if (m > k && (m - k >= precy + (rnd_mode == MPFR_RNDN))) { /* If round to nearest or away, result is s = 1 or -1, otherwise it is round(nexttoward (s, 0)). However in order to have the inexact flag correctly set below, we set |s| to 1 - 2^(-m) in all cases. */ mpfr_nexttozero (s); break; } } if (exps < cancel) { m += cancel - exps; cancel = exps; } ziv_next: MPFR_ZIV_NEXT (loop, m); MPFR_GROUP_REPREC_2 (group, m, r, s); if (reduce) { mpfr_set_prec (xr, m); mpfr_set_prec (c, expx + m - 1); } } MPFR_ZIV_FREE (loop); inexact = mpfr_set (y, s, rnd_mode); MPFR_GROUP_CLEAR (group); if (reduce) { mpfr_clear (xr); mpfr_clear (c); } end: MPFR_SAVE_EXPO_FREE (expo); return mpfr_check_range (y, inexact, rnd_mode); }