diff options
Diffstat (limited to 'Build/source/libs/mpfr/mpfr-src/src/sin_cos.c')
-rw-r--r-- | Build/source/libs/mpfr/mpfr-src/src/sin_cos.c | 671 |
1 files changed, 0 insertions, 671 deletions
diff --git a/Build/source/libs/mpfr/mpfr-src/src/sin_cos.c b/Build/source/libs/mpfr/mpfr-src/src/sin_cos.c deleted file mode 100644 index daf3040ee9b..00000000000 --- a/Build/source/libs/mpfr/mpfr-src/src/sin_cos.c +++ /dev/null @@ -1,671 +0,0 @@ -/* mpfr_sin_cos -- sine and cosine of a floating-point number - -Copyright 2002-2020 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" - -/* (y, z) <- (sin(x), cos(x)), return value is 0 iff both results are exact - ie, iff x = 0 */ -int -mpfr_sin_cos (mpfr_ptr y, mpfr_ptr z, mpfr_srcptr x, mpfr_rnd_t rnd_mode) -{ - mpfr_prec_t prec, m; - int neg, reduce; - mpfr_t c, xr; - mpfr_srcptr xx; - mpfr_exp_t err, expx; - int inexy, inexz; - MPFR_ZIV_DECL (loop); - MPFR_SAVE_EXPO_DECL (expo); - - MPFR_ASSERTN (y != z); - - if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x))) - { - if (MPFR_IS_NAN(x) || MPFR_IS_INF(x)) - { - MPFR_SET_NAN (y); - MPFR_SET_NAN (z); - MPFR_RET_NAN; - } - else /* x is zero */ - { - MPFR_ASSERTD (MPFR_IS_ZERO (x)); - MPFR_SET_ZERO (y); - MPFR_SET_SAME_SIGN (y, x); - /* y = 0, thus exact, but z is inexact in case of underflow - or overflow */ - inexy = 0; /* y is exact */ - inexz = mpfr_set_ui (z, 1, rnd_mode); - return INEX(inexy,inexz); - } - } - - MPFR_LOG_FUNC - (("x[%Pu]=%.*Rg rnd=%d", mpfr_get_prec (x), mpfr_log_prec, x, rnd_mode), - ("sin[%Pu]=%.*Rg cos[%Pu]=%.*Rg", mpfr_get_prec(y), mpfr_log_prec, y, - mpfr_get_prec (z), mpfr_log_prec, z)); - - MPFR_SAVE_EXPO_MARK (expo); - - prec = MAX (MPFR_PREC (y), MPFR_PREC (z)); - m = prec + MPFR_INT_CEIL_LOG2 (prec) + 13; - expx = MPFR_GET_EXP (x); - - /* When x is close to 0, say 2^(-k), then there is a cancellation of about - 2k bits in 1-cos(x)^2. FIXME: in that case, it would be more efficient - to compute sin(x) directly. VL: This is partly done by using - MPFR_FAST_COMPUTE_IF_SMALL_INPUT from the mpfr_sin and mpfr_cos - functions. Moreover, any overflow on m is avoided. */ - if (expx < 0) - { - /* Warning: in case y = x, and the first call to - MPFR_FAST_COMPUTE_IF_SMALL_INPUT succeeds but the second fails, - we will have clobbered the original value of x. - The workaround is to first compute z = cos(x) in that case, since - y and z are different. */ - if (y != x) - /* y and x differ, thus we can safely try to compute y first */ - { - MPFR_FAST_COMPUTE_IF_SMALL_INPUT ( - y, x, -2 * expx, 2, 0, rnd_mode, - { inexy = _inexact; - goto small_input; }); - if (0) - { - small_input: - /* we can go here only if we can round sin(x) */ - MPFR_FAST_COMPUTE_IF_SMALL_INPUT ( - z, __gmpfr_one, -2 * expx, 1, 0, rnd_mode, - { inexz = _inexact; - MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, __gmpfr_flags); - goto end; }); - } - - /* if we go here, one of the two MPFR_FAST_COMPUTE_IF_SMALL_INPUT - calls failed */ - } - else /* y and x are the same variable: try to compute z first, which - necessarily differs */ - { - MPFR_FAST_COMPUTE_IF_SMALL_INPUT ( - z, __gmpfr_one, -2 * expx, 1, 0, rnd_mode, - { inexz = _inexact; - goto small_input2; }); - if (0) - { - small_input2: - /* we can go here only if we can round cos(x) */ - MPFR_FAST_COMPUTE_IF_SMALL_INPUT ( - y, x, -2 * expx, 2, 0, rnd_mode, - { inexy = _inexact; - MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, __gmpfr_flags); - goto end; }); - } - } - m += 2 * (-expx); - } - - if (prec >= MPFR_SINCOS_THRESHOLD) - { - MPFR_SAVE_EXPO_FREE (expo); - return mpfr_sincos_fast (y, z, x, rnd_mode); - } - - mpfr_init (c); - mpfr_init (xr); - - MPFR_ZIV_INIT (loop, m); - for (;;) - { - /* the following is copied from sin.c */ - if (expx >= 2) /* reduce the argument */ - { - reduce = 1; - mpfr_set_prec (c, expx + m - 1); - mpfr_set_prec (xr, m); - mpfr_const_pi (c, MPFR_RNDN); - mpfr_mul_2ui (c, c, 1, MPFR_RNDN); - mpfr_remainder (xr, x, c, MPFR_RNDN); - mpfr_div_2ui (c, c, 1, MPFR_RNDN); - if (MPFR_IS_POS (xr)) - mpfr_sub (c, c, xr, MPFR_RNDZ); - else - mpfr_add (c, c, xr, MPFR_RNDZ); - if (MPFR_IS_ZERO(xr) - || MPFR_EXP(xr) < (mpfr_exp_t) 3 - (mpfr_exp_t) m - || MPFR_EXP(c) < (mpfr_exp_t) 3 - (mpfr_exp_t) m) - goto next_step; - xx = xr; - } - else /* the input argument is already reduced */ - { - reduce = 0; - xx = x; - } - - neg = MPFR_IS_NEG (xx); /* gives sign of sin(x) */ - mpfr_set_prec (c, m); - mpfr_cos (c, xx, MPFR_RNDZ); - /* If no argument reduction was performed, the error is at most ulp(c), - otherwise it is at most ulp(c) + 2^(2-m). Since |c| < 1, we have - ulp(c) <= 2^(-m), thus the error is bounded by 2^(3-m) in that later - case. */ - if (reduce == 0) - err = m; - else - err = MPFR_GET_EXP (c) + (mpfr_exp_t) (m - 3); - if (!MPFR_CAN_ROUND (c, err, MPFR_PREC (z), rnd_mode)) - goto next_step; - - /* we can't set z now, because in case z = x, and the MPFR_CAN_ROUND() - call below fails, we will have clobbered the input */ - mpfr_set_prec (xr, MPFR_PREC(c)); - mpfr_swap (xr, c); /* save the approximation of the cosine in xr */ - mpfr_sqr (c, xr, MPFR_RNDU); /* the absolute error is bounded by - 2^(5-m) if reduce=1, and by 2^(2-m) - otherwise */ - mpfr_ui_sub (c, 1, c, MPFR_RNDN); /* error bounded by 2^(6-m) if reduce - is 1, and 2^(3-m) otherwise */ - mpfr_sqrt (c, c, MPFR_RNDN); /* the absolute error is bounded by - 2^(6-m-Exp(c)) if reduce=1, and - 2^(3-m-Exp(c)) otherwise */ - err = 3 + 3 * reduce - MPFR_GET_EXP (c); - if (neg) - MPFR_CHANGE_SIGN (c); - - /* the absolute error on c is at most 2^(err-m), which we must put - in the form 2^(EXP(c)-err). */ - err = MPFR_GET_EXP (c) + (mpfr_exp_t) m - err; - if (MPFR_CAN_ROUND (c, err, MPFR_PREC (y), rnd_mode)) - break; - /* check for huge cancellation */ - if (err < (mpfr_exp_t) MPFR_PREC (y)) - m += MPFR_PREC (y) - err; - /* Check if near 1 */ - if (MPFR_GET_EXP (c) == 1 - && MPFR_MANT (c)[MPFR_LIMB_SIZE (c)-1] == MPFR_LIMB_HIGHBIT) - m += m; - - next_step: - MPFR_ZIV_NEXT (loop, m); - mpfr_set_prec (c, m); - } - MPFR_ZIV_FREE (loop); - - inexy = mpfr_set (y, c, rnd_mode); - inexz = mpfr_set (z, xr, rnd_mode); - - mpfr_clear (c); - mpfr_clear (xr); - - end: - MPFR_SAVE_EXPO_FREE (expo); - /* FIXME: add a test for bug before revision 7355 */ - inexy = mpfr_check_range (y, inexy, rnd_mode); - inexz = mpfr_check_range (z, inexz, rnd_mode); - MPFR_RET (INEX(inexy,inexz)); -} - -/*************** asymptotically fast implementation below ********************/ - -/* truncate Q from R to at most prec bits. - Return the number of truncated bits. - */ -static mpfr_prec_t -reduce (mpz_t Q, mpz_srcptr R, mpfr_prec_t prec) -{ - mpfr_prec_t l; - - MPFR_MPZ_SIZEINBASE2(l, R); - l = (l > prec) ? l - prec : 0; - mpz_fdiv_q_2exp (Q, R, l); - return l; -} - -/* truncate S and C so that the smaller has prec bits. - Return the number of truncated bits. - */ -static unsigned long -reduce2 (mpz_t S, mpz_t C, mpfr_prec_t prec) -{ - unsigned long ls; - unsigned long lc; - unsigned long l; - - MPFR_MPZ_SIZEINBASE2(ls, S); - MPFR_MPZ_SIZEINBASE2(lc, C); - - l = (ls < lc) ? ls : lc; /* smaller length */ - l = (l > prec) ? l - prec : 0; - mpz_fdiv_q_2exp (S, S, l); - mpz_fdiv_q_2exp (C, C, l); - return l; -} - -/* return in S0/Q0 a rational approximation of sin(X) with absolute error - bounded by 9*2^(-prec), where 0 <= X=p/2^r <= 1/2, - and in C0/Q0 a rational approximation of cos(X), with relative error - bounded by 9*2^(-prec) (and also absolute error, since - |cos(X)| <= 1). - We have sin(X)/X = sum((-1)^i*(p/2^r)^i/(2i+1)!, i=0..infinity). - We use the following binary splitting formula: - P(a,b) = (-p)^(b-a) - Q(a,b) = (2a)*(2a+1)*2^r if a+1=b [except Q(0,1)=1], Q(a,c)*Q(c,b) otherwise - T(a,b) = 1 if a+1=b, Q(c,b)*T(a,c)+P(a,c)*T(c,b) otherwise. - - Since we use P(a,b) for b-a=2^k only, we compute only p^(2^k). - We do not store the factor 2^r in Q(). - - Then sin(X)/X ~ T(0,i)/Q(0,i) for i so that (p/2^r)^i/i! is small enough. - - Return l such that Q0 has to be multiplied by 2^l. - - Assumes prec >= 10. -*/ - -#define KMAX 64 -static unsigned long -sin_bs_aux (mpz_t Q0, mpz_t S0, mpz_t C0, mpz_srcptr p, mpfr_prec_t r, - mpfr_prec_t prec) -{ - mpz_t T[KMAX], Q[KMAX], ptoj[KMAX], pp; - mpfr_prec_t log2_nb_terms[KMAX], mult[KMAX]; - mpfr_prec_t accu[KMAX], size_ptoj[KMAX]; - mpfr_prec_t prec_i_have, h, r0 = r, pp_s, p_s; - unsigned long i, j, m; - int alloc, k, l; - - if (MPFR_UNLIKELY(mpz_cmp_ui (p, 0) == 0)) /* sin(x)/x -> 1 */ - { - mpz_set_ui (Q0, 1); - mpz_set_ui (S0, 1); - mpz_set_ui (C0, 1); - return 0; - } - - /* check that X=p/2^r <= 1/2 */ - MPFR_ASSERTN(mpz_sizeinbase (p, 2) - (mpfr_exp_t) r <= -1); - - mpz_init (pp); - - /* normalize p (non-zero here) */ - h = mpz_scan1 (p, 0); - mpz_fdiv_q_2exp (pp, p, h); /* p = pp * 2^h */ - mpz_mul (pp, pp, pp); - r = 2 * (r - h); /* x^2 = (p/2^r0)^2 = pp / 2^r */ - - /* now p is odd */ - alloc = 2; - mpz_init_set_ui (T[0], 6); - mpz_init_set_ui (Q[0], 6); - mpz_init_set (ptoj[0], pp); /* ptoj[i] = pp^(2^i) */ - mpz_init (T[1]); - mpz_init (Q[1]); - mpz_init (ptoj[1]); - mpz_mul (ptoj[1], pp, pp); /* ptoj[1] = pp^2 */ - MPFR_MPZ_SIZEINBASE2(size_ptoj[1], ptoj[1]); - - mpz_mul_2exp (T[0], T[0], r); - mpz_sub (T[0], T[0], pp); /* 6*2^r - pp = 6*2^r*(1 - x^2/6) */ - log2_nb_terms[0] = 1; - - /* already take into account the factor x=p/2^r in sin(x) = x * (...) */ - MPFR_MPZ_SIZEINBASE2(pp_s, pp); - MPFR_MPZ_SIZEINBASE2(p_s, p); - mult[0] = r - pp_s + r0 - p_s; - /* we have x^3 < 1/2^mult[0] */ - - for (i = 2, k = 0, prec_i_have = mult[0]; prec_i_have < prec; i += 2) - { - /* i is even here */ - /* invariant: Q[0]*Q[1]*...*Q[k] equals (2i-1)!, - we have already summed terms of index < i - in S[0]/Q[0], ..., S[k]/Q[k] */ - k ++; - if (k + 1 >= alloc) /* necessarily k + 1 = alloc */ - { - MPFR_ASSERTD (k + 1 == alloc); - alloc ++; - MPFR_ASSERTN (k + 1 < KMAX); - mpz_init (T[k+1]); - mpz_init (Q[k+1]); - mpz_init (ptoj[k+1]); - mpz_mul (ptoj[k+1], ptoj[k], ptoj[k]); /* pp^(2^(k+1)) */ - MPFR_MPZ_SIZEINBASE2(size_ptoj[k+1], ptoj[k+1]); - } - /* for i even, we have Q[k] = (2*i)*(2*i+1), T[k] = 1, - then Q[k+1] = (2*i+2)*(2*i+3), T[k+1] = 1, - which reduces to T[k] = (2*i+2)*(2*i+3)*2^r-pp, - Q[k] = (2*i)*(2*i+1)*(2*i+2)*(2*i+3). */ - MPFR_ASSERTN (k < KMAX); - log2_nb_terms[k] = 1; - mpz_set_ui (Q[k], 2 * i + 2); - mpz_mul_ui (Q[k], Q[k], 2 * i + 3); - mpz_mul_2exp (T[k], Q[k], r); - mpz_sub (T[k], T[k], pp); - mpz_mul_ui (Q[k], Q[k], 2 * i); - mpz_mul_ui (Q[k], Q[k], 2 * i + 1); - /* the next term of the series is divided by Q[k] and multiplied - by pp^2/2^(2r), thus the mult. factor < 1/2^mult[k] */ - MPFR_MPZ_SIZEINBASE2(mult[k], Q[k]); - mult[k] += 2 * r - size_ptoj[1] - 1; - /* the absolute contribution of the next term is 1/2^accu[k] */ - accu[k] = (k == 0) ? mult[k] : mult[k] + accu[k-1]; - prec_i_have = accu[k]; /* the current term is < 1/2^accu[k] */ - j = (i + 2) / 2; - l = 1; - while ((j & 1) == 0) /* combine and reduce */ - { - MPFR_ASSERTN (k >= 1); - mpz_mul (T[k], T[k], ptoj[l]); - mpz_mul (T[k-1], T[k-1], Q[k]); - mpz_mul_2exp (T[k-1], T[k-1], r << l); - mpz_add (T[k-1], T[k-1], T[k]); - mpz_mul (Q[k-1], Q[k-1], Q[k]); - log2_nb_terms[k-1] ++; /* number of terms in S[k-1] - is a power of 2 by construction */ - MPFR_MPZ_SIZEINBASE2(prec_i_have, Q[k]); - mult[k-1] += prec_i_have + (r << l) - size_ptoj[l] - 1; - accu[k-1] = (k == 1) ? mult[k-1] : mult[k-1] + accu[k-2]; - prec_i_have = accu[k-1]; - l ++; - j >>= 1; - k --; - } - } - - /* accumulate all products in T[0] and Q[0]. Warning: contrary to above, - here we do not have log2_nb_terms[k-1] = log2_nb_terms[k]+1. */ - h = 0; /* number of accumulated terms in the right part T[k]/Q[k] */ - while (k > 0) - { - mpz_mul (T[k], T[k], ptoj[log2_nb_terms[k-1]]); - mpz_mul (T[k-1], T[k-1], Q[k]); - h += (mpfr_prec_t) 1 << log2_nb_terms[k]; - mpz_mul_2exp (T[k-1], T[k-1], r * h); - mpz_add (T[k-1], T[k-1], T[k]); - mpz_mul (Q[k-1], Q[k-1], Q[k]); - k--; - } - - m = r0 + r * (i - 1); /* implicit multiplier 2^r for Q0 */ - /* at this point T[0]/(2^m*Q[0]) is an approximation of sin(x) where the 1st - neglected term has contribution < 1/2^prec, thus since the series has - alternate signs, the error is < 1/2^prec */ - - /* we truncate Q0 to prec bits: the relative error is at most 2^(1-prec), - which means that Q0 = Q[0] * (1+theta) with |theta| <= 2^(1-prec) - [up to a power of two] */ - m += reduce (Q0, Q[0], prec); - m -= reduce (T[0], T[0], prec); - /* multiply by x = p/2^m */ - mpz_mul (S0, T[0], p); - m -= reduce (S0, S0, prec); /* S0 = T[0] * (1 + theta)^2 up to power of 2 */ - /* sin(X) ~ S0/Q0*(1 + theta)^3 + err with |theta| <= 2^(1-prec) and - |err| <= 2^(-prec), thus since |S0/Q0| <= 1: - |sin(X) - S0/Q0| <= 4*|theta*S0/Q0| + |err| <= 9*2^(-prec) */ - - mpz_clear (pp); - for (k = 0; k < alloc; k ++) - { - mpz_clear (T[k]); - mpz_clear (Q[k]); - mpz_clear (ptoj[k]); - } - - /* compute cos(X) from sin(X): sqrt(1-(S/Q)^2) = sqrt(Q^2-S^2)/Q - = sqrt(Q0^2*2^(2m)-S0^2)/Q0. - Write S/Q = sin(X) + eps with |eps| <= 9*2^(-prec), - then sqrt(Q^2-S^2) = sqrt(Q^2-Q^2*(sin(X)+eps)^2) - = sqrt(Q^2*cos(X)^2-Q^2*(2*sin(X)*eps+eps^2)) - = sqrt(Q^2*cos(X)^2-Q^2*eps1) with |eps1|<=9*2^(-prec) - [using X<=1/2 and eps<=9*2^(-prec) and prec>=10] - - Since we truncate the square root, we get: - sqrt(Q^2*cos(X)^2-Q^2*eps1)+eps2 with |eps2|<1 - = Q*sqrt(cos(X)^2-eps1)+eps2 - = Q*cos(X)*(1+eps3)+eps2 with |eps3| <= 6*2^(-prec) - = Q*cos(X)*(1+eps3+eps2/(Q*cos(X))) - = Q*cos(X)*(1+eps4) with |eps4| <= 9*2^(-prec) - since |Q| >= 2^(prec-1) */ - /* we assume that Q0*2^m >= 2^(prec-1) */ - MPFR_ASSERTN(m + mpz_sizeinbase (Q0, 2) >= prec); - mpz_mul (C0, Q0, Q0); - mpz_mul_2exp (C0, C0, 2 * m); - mpz_submul (C0, S0, S0); - mpz_sqrt (C0, C0); - - return m; -} - -/* Put in s and c approximations of sin(x) and cos(x) respectively. - Assumes 0 < x < Pi/4 and PREC(s) = PREC(c) >= 10. - Return err such that the relative error is bounded by 2^err ulps. -*/ -static int -sincos_aux (mpfr_t s, mpfr_t c, mpfr_srcptr x, mpfr_rnd_t rnd_mode) -{ - mpfr_prec_t prec_s, sh; - mpz_t Q, S, C, Q2, S2, C2, y; - mpfr_t x2; - unsigned long l, l2, j, err; - - MPFR_ASSERTD(MPFR_PREC(s) == MPFR_PREC(c)); - - prec_s = MPFR_PREC(s); - - mpfr_init2 (x2, MPFR_PREC(x)); - mpz_init (Q); - mpz_init (S); - mpz_init (C); - mpz_init (Q2); - mpz_init (S2); - mpz_init (C2); - mpz_init (y); - - mpfr_set (x2, x, MPFR_RNDN); /* exact */ - mpz_set_ui (Q, 1); - l = 0; - mpz_set_ui (S, 0); /* sin(0) = S/(2^l*Q), exact */ - mpz_set_ui (C, 1); /* cos(0) = C/(2^l*Q), exact */ - - /* Invariant: x = X + x2/2^(sh-1), where the part X was already treated, - S/(2^l*Q) ~ sin(X), C/(2^l*Q) ~ cos(X), and x2/2^(sh-1) < Pi/4. - 'sh-1' is the number of already shifted bits in x2. - */ - - for (sh = 1, j = 0; mpfr_cmp_ui (x2, 0) != 0 && sh <= prec_s; sh <<= 1, j++) - { - if (sh > prec_s / 2) /* sin(x) = x + O(x^3), cos(x) = 1 + O(x^2) */ - { - l2 = -mpfr_get_z_2exp (S2, x2); /* S2/2^l2 = x2 */ - l2 += sh - 1; - mpz_set_ui (Q2, 1); - mpz_set_ui (C2, 1); - mpz_mul_2exp (C2, C2, l2); - mpfr_set_ui (x2, 0, MPFR_RNDN); - } - else - { - /* y <- trunc(x2 * 2^sh) = trunc(x * 2^(2*sh-1)) */ - mpfr_mul_2ui (x2, x2, sh, MPFR_RNDN); /* exact */ - mpfr_get_z (y, x2, MPFR_RNDZ); /* round toward zero: now - 0 <= x2 < 2^sh, thus - 0 <= x2/2^(sh-1) < 2^(1-sh) */ - if (mpz_cmp_ui (y, 0) == 0) - continue; - mpfr_sub_z (x2, x2, y, MPFR_RNDN); /* should be exact */ - l2 = sin_bs_aux (Q2, S2, C2, y, 2 * sh - 1, prec_s); - /* we now have |S2/Q2/2^l2 - sin(X)| <= 9*2^(prec_s) - and |C2/Q2/2^l2 - cos(X)| <= 6*2^(prec_s), with X=y/2^(2sh-1) */ - } - if (sh == 1) /* S=0, C=1 */ - { - l = l2; - mpz_swap (Q, Q2); - mpz_swap (S, S2); - mpz_swap (C, C2); - } - else - { - /* s <- s*c2+c*s2, c <- c*c2-s*s2, using Karatsuba: - a = s+c, b = s2+c2, t = a*b, d = s*s2, e = c*c2, - s <- t - d - e, c <- e - d */ - mpz_add (y, S, C); /* a */ - mpz_mul (C, C, C2); /* e */ - mpz_add (C2, C2, S2); /* b */ - mpz_mul (S2, S, S2); /* d */ - mpz_mul (y, y, C2); /* a*b */ - mpz_sub (S, y, S2); /* t - d */ - mpz_sub (S, S, C); /* t - d - e */ - mpz_sub (C, C, S2); /* e - d */ - mpz_mul (Q, Q, Q2); - /* after j loops, the error is <= (11j-2)*2^(prec_s) */ - l += l2; - /* reduce Q to prec_s bits */ - l += reduce (Q, Q, prec_s); - /* reduce S,C to prec_s bits, error <= 11*j*2^(prec_s) */ - l -= reduce2 (S, C, prec_s); - } - } - - j = 11 * j; - for (err = 0; j > 1; j = (j + 1) / 2, err ++); - - mpfr_set_z (s, S, MPFR_RNDN); - mpfr_div_z (s, s, Q, MPFR_RNDN); - mpfr_div_2ui (s, s, l, MPFR_RNDN); - - mpfr_set_z (c, C, MPFR_RNDN); - mpfr_div_z (c, c, Q, MPFR_RNDN); - mpfr_div_2ui (c, c, l, MPFR_RNDN); - - mpz_clear (Q); - mpz_clear (S); - mpz_clear (C); - mpz_clear (Q2); - mpz_clear (S2); - mpz_clear (C2); - mpz_clear (y); - mpfr_clear (x2); - return err; -} - -/* Assumes x is neither NaN, +/-Inf, nor +/- 0. - One of s and c might be NULL, in which case the corresponding value is - not computed. - Assumes s differs from c. - */ -int -mpfr_sincos_fast (mpfr_t s, mpfr_t c, mpfr_srcptr x, mpfr_rnd_t rnd) -{ - int inexs, inexc; - mpfr_t x_red, ts, tc; - mpfr_prec_t w; - mpfr_exp_t err, errs, errc; - MPFR_GROUP_DECL (group); - MPFR_ZIV_DECL (loop); - - MPFR_ASSERTN(s != c); - if (s == NULL) - w = MPFR_PREC(c); - else if (c == NULL) - w = MPFR_PREC(s); - else - w = MPFR_PREC(s) >= MPFR_PREC(c) ? MPFR_PREC(s) : MPFR_PREC(c); - w += MPFR_INT_CEIL_LOG2(w) + 9; /* ensures w >= 10 (needed by sincos_aux) */ - - MPFR_GROUP_INIT_2(group, w, ts, tc); - - MPFR_ZIV_INIT (loop, w); - for (;;) - { - /* if 0 < x <= Pi/4, we can call sincos_aux directly */ - if (MPFR_IS_POS(x) && mpfr_cmp_ui_2exp (x, 1686629713, -31) <= 0) - { - err = sincos_aux (ts, tc, x, MPFR_RNDN); - } - /* if -Pi/4 <= x < 0, use sin(-x)=-sin(x) */ - else if (MPFR_IS_NEG(x) && mpfr_cmp_si_2exp (x, -1686629713, -31) >= 0) - { - MPFR_ALIAS(x_red, x, MPFR_SIGN_POS, MPFR_GET_EXP(x)); - err = sincos_aux (ts, tc, x_red, MPFR_RNDN); - MPFR_CHANGE_SIGN(ts); - } - else /* argument reduction is needed */ - { - long q; - mpfr_t pi; - int neg = 0; - - mpfr_init2 (x_red, w); - mpfr_init2 (pi, (MPFR_EXP(x) > 0) ? w + MPFR_EXP(x) : w); - mpfr_const_pi (pi, MPFR_RNDN); - mpfr_div_2ui (pi, pi, 1, MPFR_RNDN); /* Pi/2 */ - mpfr_remquo (x_red, &q, x, pi, MPFR_RNDN); - /* x = q * (Pi/2 + eps1) + x_red + eps2, - where |eps1| <= 1/2*ulp(Pi/2) = 2^(-w-MAX(0,EXP(x))), - and eps2 <= 1/2*ulp(x_red) <= 1/2*ulp(Pi/2) = 2^(-w) - Since |q| <= x/(Pi/2) <= |x|, we have - q*|eps1| <= 2^(-w), thus - |x - q * Pi/2 - x_red| <= 2^(1-w) */ - /* now -Pi/4 <= x_red <= Pi/4: if x_red < 0, consider -x_red */ - if (MPFR_IS_NEG(x_red)) - { - mpfr_neg (x_red, x_red, MPFR_RNDN); - neg = 1; - } - err = sincos_aux (ts, tc, x_red, MPFR_RNDN); - err ++; /* to take into account the argument reduction */ - if (neg) /* sin(-x) = -sin(x), cos(-x) = cos(x) */ - mpfr_neg (ts, ts, MPFR_RNDN); - if (q & 2) /* sin(x+Pi) = -sin(x), cos(x+Pi) = -cos(x) */ - { - mpfr_neg (ts, ts, MPFR_RNDN); - mpfr_neg (tc, tc, MPFR_RNDN); - } - if (q & 1) /* sin(x+Pi/2) = cos(x), cos(x+Pi/2) = -sin(x) */ - { - mpfr_neg (ts, ts, MPFR_RNDN); - mpfr_swap (ts, tc); - } - mpfr_clear (x_red); - mpfr_clear (pi); - } - /* adjust errors with respect to absolute values */ - errs = err - MPFR_EXP(ts); - errc = err - MPFR_EXP(tc); - if ((s == NULL || MPFR_CAN_ROUND (ts, w - errs, MPFR_PREC(s), rnd)) && - (c == NULL || MPFR_CAN_ROUND (tc, w - errc, MPFR_PREC(c), rnd))) - break; - MPFR_ZIV_NEXT (loop, w); - MPFR_GROUP_REPREC_2(group, w, ts, tc); - } - MPFR_ZIV_FREE (loop); - - inexs = (s == NULL) ? 0 : mpfr_set (s, ts, rnd); - inexc = (c == NULL) ? 0 : mpfr_set (c, tc, rnd); - - MPFR_GROUP_CLEAR (group); - return INEX(inexs,inexc); -} |