| /* mpfr_cos -- cosine of a floating-point number |
| |
| Copyright 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013 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) |
| { |
| MPFR_SAVE_EXPO_FREE (expo); |
| return mpfr_cos_fast (y, x, rnd_mode); |
| } |
| |
| 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); |
| } |
| |
| MPFR_SAVE_EXPO_FREE (expo); |
| return mpfr_check_range (y, inexact, rnd_mode); |
| } |