--- /dev/null
+/* mpfr_cos -- cosine of a floating-point number
+
+Copyright 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009 Free Software Foundation, Inc.
+Contributed by the Arenaire and Cacao 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 2.1 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.LIB. If not, 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"
+
+/* 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;
+ mp_exp_t ex, l, m;
+ mp_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_exp (x, r); /* r = x*2^ex */
+
+ /* remove trailing zeroes */
+ l = mpz_scan1 (x, 0);
+ ex += l;
+ mpz_div_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_div_2exp (x, x, l);
+ ex += l;
+ }
+ /* multiply t by r */
+ mpz_mul (t, t, x);
+ mpz_div_2exp (t, t, -ex);
+ /* divide t by i*(i+1) */
+ if (i < maxi)
+ mpz_div_ui (t, t, i * (i + 1));
+ else
+ {
+ mpz_div_ui (t, t, i);
+ mpz_div_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, GMP_RNDN);
+ mpfr_div_2ui (f, f, p + q, GMP_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, mp_rnd_t rnd_mode)
+{
+ mp_prec_t K0, K, precy, m, k, l;
+ int inexact, reduce = 0;
+ mpfr_t r, s, xr, c;
+ mp_exp_t exps, cancel = 0, expx;
+ MPFR_ZIV_DECL (loop);
+ MPFR_SAVE_EXPO_DECL (expo);
+ MPFR_GROUP_DECL (group);
+
+ MPFR_LOG_FUNC (("x[%#R]=%R rnd=%d", x, x, rnd_mode),
+ ("y[%#R]=%R inexact=%d", y, 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);
+ K0 = __gmpfr_isqrt (precy / 3);
+ m = precy + 2 * MPFR_INT_CEIL_LOG2 (precy) + 2 * K0;
+
+ if (expx >= 3)
+ {
+ reduce = 1;
+ mpfr_init2 (xr, m);
+ mpfr_init2 (c, expx + m - 1);
+ }
+
+ 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, GMP_RNDN);
+ mpfr_mul_2ui (c, c, 1, GMP_RNDN); /* 2Pi */
+ mpfr_remainder (xr, x, c, GMP_RNDN);
+ /* now |xr| <= 4, thus r <= 16 below */
+ mpfr_mul (r, xr, xr, GMP_RNDU); /* err <= 1 ulp */
+ }
+ else
+ mpfr_mul (r, x, x, GMP_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_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, GMP_RNDU); /* err <= 2*olderr */
+ MPFR_SET_EXP (s, MPFR_GET_EXP (s) + 1); /* Can't overflow */
+ mpfr_sub (s, s, r, GMP_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 == GMP_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);
+}
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