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Diffstat (limited to 'src/sin_cos.c')
| -rw-r--r-- | src/sin_cos.c | 662 |
1 files changed, 662 insertions, 0 deletions
diff --git a/src/sin_cos.c b/src/sin_cos.c new file mode 100644 index 0000000..8ee95e9 --- /dev/null +++ b/src/sin_cos.c @@ -0,0 +1,662 @@ +/* mpfr_sin_cos -- sine and cosine of a floating-point number + +Copyright 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" + +/* (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_SIGN (xr) > 0) + 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_RNDN, MPFR_RNDZ, + MPFR_PREC (z) + (rnd_mode == MPFR_RNDN))) + 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_RNDN, MPFR_RNDZ, + MPFR_PREC (y) + (rnd_mode == MPFR_RNDN))) + 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 = mpz_sizeinbase (R, 2); + + 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 = mpz_sizeinbase (S, 2); + unsigned long lc = mpz_sizeinbase (C, 2); + unsigned long l; + + 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. +*/ +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[GMP_NUMB_BITS], Q[GMP_NUMB_BITS], ptoj[GMP_NUMB_BITS], pp; + mpfr_prec_t log2_nb_terms[GMP_NUMB_BITS], mult[GMP_NUMB_BITS]; + mpfr_prec_t accu[GMP_NUMB_BITS], size_ptoj[GMP_NUMB_BITS]; + mpfr_prec_t prec_i_have, r0 = r; + unsigned long alloc, i, j, k; + mpfr_prec_t 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) */ + l = mpz_scan1 (p, 0); + mpz_fdiv_q_2exp (pp, p, l); /* p = pp * 2^l */ + mpz_mul (pp, pp, pp); + r = 2 * (r - l); /* 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 */ + size_ptoj[1] = mpz_sizeinbase (ptoj[1], 2); + + 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 * (...) */ + mult[0] = r - mpz_sizeinbase (pp, 2) + r0 - mpz_sizeinbase (p, 2); + /* 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 */ + { + alloc ++; + 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)) */ + size_ptoj[k+1] = mpz_sizeinbase (ptoj[k+1], 2); + } + /* 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). */ + log2_nb_terms[k] = 1; + mpz_set_ui (Q[k], (2 * i + 2) * (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) * (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] */ + mult[k] = mpz_sizeinbase (Q[k], 2) + 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 */ + { + 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 */ + prec_i_have = mpz_sizeinbase (Q[k], 2); + 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. */ + l = 0; /* number of accumulated terms in the right part T[k]/Q[k] */ + while (k > 0) + { + j = log2_nb_terms[k-1]; + mpz_mul (T[k], T[k], ptoj[j]); + mpz_mul (T[k-1], T[k-1], Q[k]); + l += 1 << log2_nb_terms[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]); + k--; + } + + l = r0 + r * (i - 1); /* implicit multiplier 2^r for Q0 */ + /* at this point T[0]/(2^l*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] */ + l += reduce (Q0, Q[0], prec); + l -= reduce (T[0], T[0], prec); + /* multiply by x = p/2^l */ + mpz_mul (S0, T[0], p); + l -= 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 (j = 0; j < alloc; j ++) + { + mpz_clear (T[j]); + mpz_clear (Q[j]); + mpz_clear (ptoj[j]); + } + + /* compute cos(X) from sin(X): sqrt(1-(S/Q)^2) = sqrt(Q^2-S^2)/Q + = sqrt(Q0^2*2^(2l)-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^l >= 2^(prec-1) */ + MPFR_ASSERTN(l + mpz_sizeinbase (Q0, 2) >= prec); + mpz_mul (C0, Q0, Q0); + mpz_mul_2exp (C0, C0, 2 * l); + mpz_submul (C0, S0, S0); + mpz_sqrt (C0, C0); + + return l; +} + +/* 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_2exp (x2, x2, sh, MPFR_RNDN); /* exact */ + mpfr_get_z (y, x2, MPFR_RNDZ); /* round towards 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_2exp (s, s, l, MPFR_RNDN); + + mpfr_set_z (c, C, MPFR_RNDN); + mpfr_div_z (c, c, Q, MPFR_RNDN); + mpfr_div_2exp (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_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_init2 (ts, w); + mpfr_init2 (tc, w); + + 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_init2 (x_red, MPFR_PREC(x)); + mpfr_neg (x_red, x, rnd); /* exact */ + err = sincos_aux (ts, tc, x_red, MPFR_RNDN); + mpfr_neg (ts, ts, MPFR_RNDN); + mpfr_clear (x_red); + } + 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_2exp (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_set_prec (ts, w); + mpfr_set_prec (tc, w); + } + MPFR_ZIV_FREE (loop); + + inexs = (s == NULL) ? 0 : mpfr_set (s, ts, rnd); + inexc = (c == NULL) ? 0 : mpfr_set (c, tc, rnd); + + mpfr_clear (ts); + mpfr_clear (tc); + return INEX(inexs,inexc); +} |