1/* mpfr_sin_cos -- sine and cosine of a floating-point number 2 3Copyright 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013 Free Software Foundation, Inc. 4Contributed by the AriC and Caramel projects, INRIA. 5 6This file is part of the GNU MPFR Library. 7 8The GNU MPFR Library is free software; you can redistribute it and/or modify 9it under the terms of the GNU Lesser General Public License as published by 10the Free Software Foundation; either version 3 of the License, or (at your 11option) any later version. 12 13The GNU MPFR Library is distributed in the hope that it will be useful, but 14WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY 15or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public 16License for more details. 17 18You should have received a copy of the GNU Lesser General Public License 19along with the GNU MPFR Library; see the file COPYING.LESSER. If not, see 20http://www.gnu.org/licenses/ or write to the Free Software Foundation, Inc., 2151 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA. */ 22 23#define MPFR_NEED_LONGLONG_H 24#include "mpfr-impl.h" 25 26/* (y, z) <- (sin(x), cos(x)), return value is 0 iff both results are exact 27 ie, iff x = 0 */ 28int 29mpfr_sin_cos (mpfr_ptr y, mpfr_ptr z, mpfr_srcptr x, mpfr_rnd_t rnd_mode) 30{ 31 mpfr_prec_t prec, m; 32 int neg, reduce; 33 mpfr_t c, xr; 34 mpfr_srcptr xx; 35 mpfr_exp_t err, expx; 36 int inexy, inexz; 37 MPFR_ZIV_DECL (loop); 38 MPFR_SAVE_EXPO_DECL (expo); 39 40 MPFR_ASSERTN (y != z); 41 42 if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x))) 43 { 44 if (MPFR_IS_NAN(x) || MPFR_IS_INF(x)) 45 { 46 MPFR_SET_NAN (y); 47 MPFR_SET_NAN (z); 48 MPFR_RET_NAN; 49 } 50 else /* x is zero */ 51 { 52 MPFR_ASSERTD (MPFR_IS_ZERO (x)); 53 MPFR_SET_ZERO (y); 54 MPFR_SET_SAME_SIGN (y, x); 55 /* y = 0, thus exact, but z is inexact in case of underflow 56 or overflow */ 57 inexy = 0; /* y is exact */ 58 inexz = mpfr_set_ui (z, 1, rnd_mode); 59 return INEX(inexy,inexz); 60 } 61 } 62 63 MPFR_LOG_FUNC 64 (("x[%Pu]=%.*Rg rnd=%d", mpfr_get_prec (x), mpfr_log_prec, x, rnd_mode), 65 ("sin[%Pu]=%.*Rg cos[%Pu]=%.*Rg", mpfr_get_prec(y), mpfr_log_prec, y, 66 mpfr_get_prec (z), mpfr_log_prec, z)); 67 68 MPFR_SAVE_EXPO_MARK (expo); 69 70 prec = MAX (MPFR_PREC (y), MPFR_PREC (z)); 71 m = prec + MPFR_INT_CEIL_LOG2 (prec) + 13; 72 expx = MPFR_GET_EXP (x); 73 74 /* When x is close to 0, say 2^(-k), then there is a cancellation of about 75 2k bits in 1-cos(x)^2. FIXME: in that case, it would be more efficient 76 to compute sin(x) directly. VL: This is partly done by using 77 MPFR_FAST_COMPUTE_IF_SMALL_INPUT from the mpfr_sin and mpfr_cos 78 functions. Moreover, any overflow on m is avoided. */ 79 if (expx < 0) 80 { 81 /* Warning: in case y = x, and the first call to 82 MPFR_FAST_COMPUTE_IF_SMALL_INPUT succeeds but the second fails, 83 we will have clobbered the original value of x. 84 The workaround is to first compute z = cos(x) in that case, since 85 y and z are different. */ 86 if (y != x) 87 /* y and x differ, thus we can safely try to compute y first */ 88 { 89 MPFR_FAST_COMPUTE_IF_SMALL_INPUT ( 90 y, x, -2 * expx, 2, 0, rnd_mode, 91 { inexy = _inexact; 92 goto small_input; }); 93 if (0) 94 { 95 small_input: 96 /* we can go here only if we can round sin(x) */ 97 MPFR_FAST_COMPUTE_IF_SMALL_INPUT ( 98 z, __gmpfr_one, -2 * expx, 1, 0, rnd_mode, 99 { inexz = _inexact; 100 MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, __gmpfr_flags); 101 goto end; }); 102 } 103 104 /* if we go here, one of the two MPFR_FAST_COMPUTE_IF_SMALL_INPUT 105 calls failed */ 106 } 107 else /* y and x are the same variable: try to compute z first, which 108 necessarily differs */ 109 { 110 MPFR_FAST_COMPUTE_IF_SMALL_INPUT ( 111 z, __gmpfr_one, -2 * expx, 1, 0, rnd_mode, 112 { inexz = _inexact; 113 goto small_input2; }); 114 if (0) 115 { 116 small_input2: 117 /* we can go here only if we can round cos(x) */ 118 MPFR_FAST_COMPUTE_IF_SMALL_INPUT ( 119 y, x, -2 * expx, 2, 0, rnd_mode, 120 { inexy = _inexact; 121 MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, __gmpfr_flags); 122 goto end; }); 123 } 124 } 125 m += 2 * (-expx); 126 } 127 128 if (prec >= MPFR_SINCOS_THRESHOLD) 129 { 130 MPFR_SAVE_EXPO_FREE (expo); 131 return mpfr_sincos_fast (y, z, x, rnd_mode); 132 } 133 134 mpfr_init (c); 135 mpfr_init (xr); 136 137 MPFR_ZIV_INIT (loop, m); 138 for (;;) 139 { 140 /* the following is copied from sin.c */ 141 if (expx >= 2) /* reduce the argument */ 142 { 143 reduce = 1; 144 mpfr_set_prec (c, expx + m - 1); 145 mpfr_set_prec (xr, m); 146 mpfr_const_pi (c, MPFR_RNDN); 147 mpfr_mul_2ui (c, c, 1, MPFR_RNDN); 148 mpfr_remainder (xr, x, c, MPFR_RNDN); 149 mpfr_div_2ui (c, c, 1, MPFR_RNDN); 150 if (MPFR_SIGN (xr) > 0) 151 mpfr_sub (c, c, xr, MPFR_RNDZ); 152 else 153 mpfr_add (c, c, xr, MPFR_RNDZ); 154 if (MPFR_IS_ZERO(xr) 155 || MPFR_EXP(xr) < (mpfr_exp_t) 3 - (mpfr_exp_t) m 156 || MPFR_EXP(c) < (mpfr_exp_t) 3 - (mpfr_exp_t) m) 157 goto next_step; 158 xx = xr; 159 } 160 else /* the input argument is already reduced */ 161 { 162 reduce = 0; 163 xx = x; 164 } 165 166 neg = MPFR_IS_NEG (xx); /* gives sign of sin(x) */ 167 mpfr_set_prec (c, m); 168 mpfr_cos (c, xx, MPFR_RNDZ); 169 /* If no argument reduction was performed, the error is at most ulp(c), 170 otherwise it is at most ulp(c) + 2^(2-m). Since |c| < 1, we have 171 ulp(c) <= 2^(-m), thus the error is bounded by 2^(3-m) in that later 172 case. */ 173 if (reduce == 0) 174 err = m; 175 else 176 err = MPFR_GET_EXP (c) + (mpfr_exp_t) (m - 3); 177 if (!mpfr_can_round (c, err, MPFR_RNDN, MPFR_RNDZ, 178 MPFR_PREC (z) + (rnd_mode == MPFR_RNDN))) 179 goto next_step; 180 181 /* we can't set z now, because in case z = x, and the mpfr_can_round() 182 call below fails, we will have clobbered the input */ 183 mpfr_set_prec (xr, MPFR_PREC(c)); 184 mpfr_swap (xr, c); /* save the approximation of the cosine in xr */ 185 mpfr_sqr (c, xr, MPFR_RNDU); /* the absolute error is bounded by 186 2^(5-m) if reduce=1, and by 2^(2-m) 187 otherwise */ 188 mpfr_ui_sub (c, 1, c, MPFR_RNDN); /* error bounded by 2^(6-m) if reduce 189 is 1, and 2^(3-m) otherwise */ 190 mpfr_sqrt (c, c, MPFR_RNDN); /* the absolute error is bounded by 191 2^(6-m-Exp(c)) if reduce=1, and 192 2^(3-m-Exp(c)) otherwise */ 193 err = 3 + 3 * reduce - MPFR_GET_EXP (c); 194 if (neg) 195 MPFR_CHANGE_SIGN (c); 196 197 /* the absolute error on c is at most 2^(err-m), which we must put 198 in the form 2^(EXP(c)-err). */ 199 err = MPFR_GET_EXP (c) + (mpfr_exp_t) m - err; 200 if (mpfr_can_round (c, err, MPFR_RNDN, MPFR_RNDZ, 201 MPFR_PREC (y) + (rnd_mode == MPFR_RNDN))) 202 break; 203 /* check for huge cancellation */ 204 if (err < (mpfr_exp_t) MPFR_PREC (y)) 205 m += MPFR_PREC (y) - err; 206 /* Check if near 1 */ 207 if (MPFR_GET_EXP (c) == 1 208 && MPFR_MANT (c)[MPFR_LIMB_SIZE (c)-1] == MPFR_LIMB_HIGHBIT) 209 m += m; 210 211 next_step: 212 MPFR_ZIV_NEXT (loop, m); 213 mpfr_set_prec (c, m); 214 } 215 MPFR_ZIV_FREE (loop); 216 217 inexy = mpfr_set (y, c, rnd_mode); 218 inexz = mpfr_set (z, xr, rnd_mode); 219 220 mpfr_clear (c); 221 mpfr_clear (xr); 222 223 end: 224 MPFR_SAVE_EXPO_FREE (expo); 225 /* FIXME: add a test for bug before revision 7355 */ 226 inexy = mpfr_check_range (y, inexy, rnd_mode); 227 inexz = mpfr_check_range (z, inexz, rnd_mode); 228 MPFR_RET (INEX(inexy,inexz)); 229} 230 231/*************** asymptotically fast implementation below ********************/ 232 233/* truncate Q from R to at most prec bits. 234 Return the number of truncated bits. 235 */ 236static mpfr_prec_t 237reduce (mpz_t Q, mpz_srcptr R, mpfr_prec_t prec) 238{ 239 mpfr_prec_t l = mpz_sizeinbase (R, 2); 240 241 l = (l > prec) ? l - prec : 0; 242 mpz_fdiv_q_2exp (Q, R, l); 243 return l; 244} 245 246/* truncate S and C so that the smaller has prec bits. 247 Return the number of truncated bits. 248 */ 249static unsigned long 250reduce2 (mpz_t S, mpz_t C, mpfr_prec_t prec) 251{ 252 unsigned long ls = mpz_sizeinbase (S, 2); 253 unsigned long lc = mpz_sizeinbase (C, 2); 254 unsigned long l; 255 256 l = (ls < lc) ? ls : lc; /* smaller length */ 257 l = (l > prec) ? l - prec : 0; 258 mpz_fdiv_q_2exp (S, S, l); 259 mpz_fdiv_q_2exp (C, C, l); 260 return l; 261} 262 263/* return in S0/Q0 a rational approximation of sin(X) with absolute error 264 bounded by 9*2^(-prec), where 0 <= X=p/2^r <= 1/2, 265 and in C0/Q0 a rational approximation of cos(X), with relative error 266 bounded by 9*2^(-prec) (and also absolute error, since 267 |cos(X)| <= 1). 268 We have sin(X)/X = sum((-1)^i*(p/2^r)^i/(2i+1)!, i=0..infinity). 269 We use the following binary splitting formula: 270 P(a,b) = (-p)^(b-a) 271 Q(a,b) = (2a)*(2a+1)*2^r if a+1=b [except Q(0,1)=1], Q(a,c)*Q(c,b) otherwise 272 T(a,b) = 1 if a+1=b, Q(c,b)*T(a,c)+P(a,c)*T(c,b) otherwise. 273 274 Since we use P(a,b) for b-a=2^k only, we compute only p^(2^k). 275 We do not store the factor 2^r in Q(). 276 277 Then sin(X)/X ~ T(0,i)/Q(0,i) for i so that (p/2^r)^i/i! is small enough. 278 279 Return l such that Q0 has to be multiplied by 2^l. 280 281 Assumes prec >= 10. 282*/ 283static unsigned long 284sin_bs_aux (mpz_t Q0, mpz_t S0, mpz_t C0, mpz_srcptr p, mpfr_prec_t r, 285 mpfr_prec_t prec) 286{ 287 mpz_t T[GMP_NUMB_BITS], Q[GMP_NUMB_BITS], ptoj[GMP_NUMB_BITS], pp; 288 mpfr_prec_t log2_nb_terms[GMP_NUMB_BITS], mult[GMP_NUMB_BITS]; 289 mpfr_prec_t accu[GMP_NUMB_BITS], size_ptoj[GMP_NUMB_BITS]; 290 mpfr_prec_t prec_i_have, r0 = r; 291 unsigned long alloc, i, j, k; 292 mpfr_prec_t l; 293 294 if (MPFR_UNLIKELY(mpz_cmp_ui (p, 0) == 0)) /* sin(x)/x -> 1 */ 295 { 296 mpz_set_ui (Q0, 1); 297 mpz_set_ui (S0, 1); 298 mpz_set_ui (C0, 1); 299 return 0; 300 } 301 302 /* check that X=p/2^r <= 1/2 */ 303 MPFR_ASSERTN(mpz_sizeinbase (p, 2) - (mpfr_exp_t) r <= -1); 304 305 mpz_init (pp); 306 307 /* normalize p (non-zero here) */ 308 l = mpz_scan1 (p, 0); 309 mpz_fdiv_q_2exp (pp, p, l); /* p = pp * 2^l */ 310 mpz_mul (pp, pp, pp); 311 r = 2 * (r - l); /* x^2 = (p/2^r0)^2 = pp / 2^r */ 312 313 /* now p is odd */ 314 alloc = 2; 315 mpz_init_set_ui (T[0], 6); 316 mpz_init_set_ui (Q[0], 6); 317 mpz_init_set (ptoj[0], pp); /* ptoj[i] = pp^(2^i) */ 318 mpz_init (T[1]); 319 mpz_init (Q[1]); 320 mpz_init (ptoj[1]); 321 mpz_mul (ptoj[1], pp, pp); /* ptoj[1] = pp^2 */ 322 size_ptoj[1] = mpz_sizeinbase (ptoj[1], 2); 323 324 mpz_mul_2exp (T[0], T[0], r); 325 mpz_sub (T[0], T[0], pp); /* 6*2^r - pp = 6*2^r*(1 - x^2/6) */ 326 log2_nb_terms[0] = 1; 327 328 /* already take into account the factor x=p/2^r in sin(x) = x * (...) */ 329 mult[0] = r - mpz_sizeinbase (pp, 2) + r0 - mpz_sizeinbase (p, 2); 330 /* we have x^3 < 1/2^mult[0] */ 331 332 for (i = 2, k = 0, prec_i_have = mult[0]; prec_i_have < prec; i += 2) 333 { 334 /* i is even here */ 335 /* invariant: Q[0]*Q[1]*...*Q[k] equals (2i-1)!, 336 we have already summed terms of index < i 337 in S[0]/Q[0], ..., S[k]/Q[k] */ 338 k ++; 339 if (k + 1 >= alloc) /* necessarily k + 1 = alloc */ 340 { 341 alloc ++; 342 mpz_init (T[k+1]); 343 mpz_init (Q[k+1]); 344 mpz_init (ptoj[k+1]); 345 mpz_mul (ptoj[k+1], ptoj[k], ptoj[k]); /* pp^(2^(k+1)) */ 346 size_ptoj[k+1] = mpz_sizeinbase (ptoj[k+1], 2); 347 } 348 /* for i even, we have Q[k] = (2*i)*(2*i+1), T[k] = 1, 349 then Q[k+1] = (2*i+2)*(2*i+3), T[k+1] = 1, 350 which reduces to T[k] = (2*i+2)*(2*i+3)*2^r-pp, 351 Q[k] = (2*i)*(2*i+1)*(2*i+2)*(2*i+3). */ 352 log2_nb_terms[k] = 1; 353 mpz_set_ui (Q[k], (2 * i + 2) * (2 * i + 3)); 354 mpz_mul_2exp (T[k], Q[k], r); 355 mpz_sub (T[k], T[k], pp); 356 mpz_mul_ui (Q[k], Q[k], (2 * i) * (2 * i + 1)); 357 /* the next term of the series is divided by Q[k] and multiplied 358 by pp^2/2^(2r), thus the mult. factor < 1/2^mult[k] */ 359 mult[k] = mpz_sizeinbase (Q[k], 2) + 2 * r - size_ptoj[1] - 1; 360 /* the absolute contribution of the next term is 1/2^accu[k] */ 361 accu[k] = (k == 0) ? mult[k] : mult[k] + accu[k-1]; 362 prec_i_have = accu[k]; /* the current term is < 1/2^accu[k] */ 363 j = (i + 2) / 2; 364 l = 1; 365 while ((j & 1) == 0) /* combine and reduce */ 366 { 367 mpz_mul (T[k], T[k], ptoj[l]); 368 mpz_mul (T[k-1], T[k-1], Q[k]); 369 mpz_mul_2exp (T[k-1], T[k-1], r << l); 370 mpz_add (T[k-1], T[k-1], T[k]); 371 mpz_mul (Q[k-1], Q[k-1], Q[k]); 372 log2_nb_terms[k-1] ++; /* number of terms in S[k-1] 373 is a power of 2 by construction */ 374 prec_i_have = mpz_sizeinbase (Q[k], 2); 375 mult[k-1] += prec_i_have + (r << l) - size_ptoj[l] - 1; 376 accu[k-1] = (k == 1) ? mult[k-1] : mult[k-1] + accu[k-2]; 377 prec_i_have = accu[k-1]; 378 l ++; 379 j >>= 1; 380 k --; 381 } 382 } 383 384 /* accumulate all products in T[0] and Q[0]. Warning: contrary to above, 385 here we do not have log2_nb_terms[k-1] = log2_nb_terms[k]+1. */ 386 l = 0; /* number of accumulated terms in the right part T[k]/Q[k] */ 387 while (k > 0) 388 { 389 j = log2_nb_terms[k-1]; 390 mpz_mul (T[k], T[k], ptoj[j]); 391 mpz_mul (T[k-1], T[k-1], Q[k]); 392 l += 1 << log2_nb_terms[k]; 393 mpz_mul_2exp (T[k-1], T[k-1], r * l); 394 mpz_add (T[k-1], T[k-1], T[k]); 395 mpz_mul (Q[k-1], Q[k-1], Q[k]); 396 k--; 397 } 398 399 l = r0 + r * (i - 1); /* implicit multiplier 2^r for Q0 */ 400 /* at this point T[0]/(2^l*Q[0]) is an approximation of sin(x) where the 1st 401 neglected term has contribution < 1/2^prec, thus since the series has 402 alternate signs, the error is < 1/2^prec */ 403 404 /* we truncate Q0 to prec bits: the relative error is at most 2^(1-prec), 405 which means that Q0 = Q[0] * (1+theta) with |theta| <= 2^(1-prec) 406 [up to a power of two] */ 407 l += reduce (Q0, Q[0], prec); 408 l -= reduce (T[0], T[0], prec); 409 /* multiply by x = p/2^l */ 410 mpz_mul (S0, T[0], p); 411 l -= reduce (S0, S0, prec); /* S0 = T[0] * (1 + theta)^2 up to power of 2 */ 412 /* sin(X) ~ S0/Q0*(1 + theta)^3 + err with |theta| <= 2^(1-prec) and 413 |err| <= 2^(-prec), thus since |S0/Q0| <= 1: 414 |sin(X) - S0/Q0| <= 4*|theta*S0/Q0| + |err| <= 9*2^(-prec) */ 415 416 mpz_clear (pp); 417 for (j = 0; j < alloc; j ++) 418 { 419 mpz_clear (T[j]); 420 mpz_clear (Q[j]); 421 mpz_clear (ptoj[j]); 422 } 423 424 /* compute cos(X) from sin(X): sqrt(1-(S/Q)^2) = sqrt(Q^2-S^2)/Q 425 = sqrt(Q0^2*2^(2l)-S0^2)/Q0. 426 Write S/Q = sin(X) + eps with |eps| <= 9*2^(-prec), 427 then sqrt(Q^2-S^2) = sqrt(Q^2-Q^2*(sin(X)+eps)^2) 428 = sqrt(Q^2*cos(X)^2-Q^2*(2*sin(X)*eps+eps^2)) 429 = sqrt(Q^2*cos(X)^2-Q^2*eps1) with |eps1|<=9*2^(-prec) 430 [using X<=1/2 and eps<=9*2^(-prec) and prec>=10] 431 432 Since we truncate the square root, we get: 433 sqrt(Q^2*cos(X)^2-Q^2*eps1)+eps2 with |eps2|<1 434 = Q*sqrt(cos(X)^2-eps1)+eps2 435 = Q*cos(X)*(1+eps3)+eps2 with |eps3| <= 6*2^(-prec) 436 = Q*cos(X)*(1+eps3+eps2/(Q*cos(X))) 437 = Q*cos(X)*(1+eps4) with |eps4| <= 9*2^(-prec) 438 since |Q| >= 2^(prec-1) */ 439 /* we assume that Q0*2^l >= 2^(prec-1) */ 440 MPFR_ASSERTN(l + mpz_sizeinbase (Q0, 2) >= prec); 441 mpz_mul (C0, Q0, Q0); 442 mpz_mul_2exp (C0, C0, 2 * l); 443 mpz_submul (C0, S0, S0); 444 mpz_sqrt (C0, C0); 445 446 return l; 447} 448 449/* Put in s and c approximations of sin(x) and cos(x) respectively. 450 Assumes 0 < x < Pi/4 and PREC(s) = PREC(c) >= 10. 451 Return err such that the relative error is bounded by 2^err ulps. 452*/ 453static int 454sincos_aux (mpfr_t s, mpfr_t c, mpfr_srcptr x, mpfr_rnd_t rnd_mode) 455{ 456 mpfr_prec_t prec_s, sh; 457 mpz_t Q, S, C, Q2, S2, C2, y; 458 mpfr_t x2; 459 unsigned long l, l2, j, err; 460 461 MPFR_ASSERTD(MPFR_PREC(s) == MPFR_PREC(c)); 462 463 prec_s = MPFR_PREC(s); 464 465 mpfr_init2 (x2, MPFR_PREC(x)); 466 mpz_init (Q); 467 mpz_init (S); 468 mpz_init (C); 469 mpz_init (Q2); 470 mpz_init (S2); 471 mpz_init (C2); 472 mpz_init (y); 473 474 mpfr_set (x2, x, MPFR_RNDN); /* exact */ 475 mpz_set_ui (Q, 1); 476 l = 0; 477 mpz_set_ui (S, 0); /* sin(0) = S/(2^l*Q), exact */ 478 mpz_set_ui (C, 1); /* cos(0) = C/(2^l*Q), exact */ 479 480 /* Invariant: x = X + x2/2^(sh-1), where the part X was already treated, 481 S/(2^l*Q) ~ sin(X), C/(2^l*Q) ~ cos(X), and x2/2^(sh-1) < Pi/4. 482 'sh-1' is the number of already shifted bits in x2. 483 */ 484 485 for (sh = 1, j = 0; mpfr_cmp_ui (x2, 0) != 0 && sh <= prec_s; sh <<= 1, j++) 486 { 487 if (sh > prec_s / 2) /* sin(x) = x + O(x^3), cos(x) = 1 + O(x^2) */ 488 { 489 l2 = -mpfr_get_z_2exp (S2, x2); /* S2/2^l2 = x2 */ 490 l2 += sh - 1; 491 mpz_set_ui (Q2, 1); 492 mpz_set_ui (C2, 1); 493 mpz_mul_2exp (C2, C2, l2); 494 mpfr_set_ui (x2, 0, MPFR_RNDN); 495 } 496 else 497 { 498 /* y <- trunc(x2 * 2^sh) = trunc(x * 2^(2*sh-1)) */ 499 mpfr_mul_2exp (x2, x2, sh, MPFR_RNDN); /* exact */ 500 mpfr_get_z (y, x2, MPFR_RNDZ); /* round towards zero: now 501 0 <= x2 < 2^sh, thus 502 0 <= x2/2^(sh-1) < 2^(1-sh) */ 503 if (mpz_cmp_ui (y, 0) == 0) 504 continue; 505 mpfr_sub_z (x2, x2, y, MPFR_RNDN); /* should be exact */ 506 l2 = sin_bs_aux (Q2, S2, C2, y, 2 * sh - 1, prec_s); 507 /* we now have |S2/Q2/2^l2 - sin(X)| <= 9*2^(prec_s) 508 and |C2/Q2/2^l2 - cos(X)| <= 6*2^(prec_s), with X=y/2^(2sh-1) */ 509 } 510 if (sh == 1) /* S=0, C=1 */ 511 { 512 l = l2; 513 mpz_swap (Q, Q2); 514 mpz_swap (S, S2); 515 mpz_swap (C, C2); 516 } 517 else 518 { 519 /* s <- s*c2+c*s2, c <- c*c2-s*s2, using Karatsuba: 520 a = s+c, b = s2+c2, t = a*b, d = s*s2, e = c*c2, 521 s <- t - d - e, c <- e - d */ 522 mpz_add (y, S, C); /* a */ 523 mpz_mul (C, C, C2); /* e */ 524 mpz_add (C2, C2, S2); /* b */ 525 mpz_mul (S2, S, S2); /* d */ 526 mpz_mul (y, y, C2); /* a*b */ 527 mpz_sub (S, y, S2); /* t - d */ 528 mpz_sub (S, S, C); /* t - d - e */ 529 mpz_sub (C, C, S2); /* e - d */ 530 mpz_mul (Q, Q, Q2); 531 /* after j loops, the error is <= (11j-2)*2^(prec_s) */ 532 l += l2; 533 /* reduce Q to prec_s bits */ 534 l += reduce (Q, Q, prec_s); 535 /* reduce S,C to prec_s bits, error <= 11*j*2^(prec_s) */ 536 l -= reduce2 (S, C, prec_s); 537 } 538 } 539 540 j = 11 * j; 541 for (err = 0; j > 1; j = (j + 1) / 2, err ++); 542 543 mpfr_set_z (s, S, MPFR_RNDN); 544 mpfr_div_z (s, s, Q, MPFR_RNDN); 545 mpfr_div_2exp (s, s, l, MPFR_RNDN); 546 547 mpfr_set_z (c, C, MPFR_RNDN); 548 mpfr_div_z (c, c, Q, MPFR_RNDN); 549 mpfr_div_2exp (c, c, l, MPFR_RNDN); 550 551 mpz_clear (Q); 552 mpz_clear (S); 553 mpz_clear (C); 554 mpz_clear (Q2); 555 mpz_clear (S2); 556 mpz_clear (C2); 557 mpz_clear (y); 558 mpfr_clear (x2); 559 return err; 560} 561 562/* Assumes x is neither NaN, +/-Inf, nor +/- 0. 563 One of s and c might be NULL, in which case the corresponding value is 564 not computed. 565 Assumes s differs from c. 566 */ 567int 568mpfr_sincos_fast (mpfr_t s, mpfr_t c, mpfr_srcptr x, mpfr_rnd_t rnd) 569{ 570 int inexs, inexc; 571 mpfr_t x_red, ts, tc; 572 mpfr_prec_t w; 573 mpfr_exp_t err, errs, errc; 574 MPFR_ZIV_DECL (loop); 575 576 MPFR_ASSERTN(s != c); 577 if (s == NULL) 578 w = MPFR_PREC(c); 579 else if (c == NULL) 580 w = MPFR_PREC(s); 581 else 582 w = MPFR_PREC(s) >= MPFR_PREC(c) ? MPFR_PREC(s) : MPFR_PREC(c); 583 w += MPFR_INT_CEIL_LOG2(w) + 9; /* ensures w >= 10 (needed by sincos_aux) */ 584 mpfr_init2 (ts, w); 585 mpfr_init2 (tc, w); 586 587 MPFR_ZIV_INIT (loop, w); 588 for (;;) 589 { 590 /* if 0 < x <= Pi/4, we can call sincos_aux directly */ 591 if (MPFR_IS_POS(x) && mpfr_cmp_ui_2exp (x, 1686629713, -31) <= 0) 592 { 593 err = sincos_aux (ts, tc, x, MPFR_RNDN); 594 } 595 /* if -Pi/4 <= x < 0, use sin(-x)=-sin(x) */ 596 else if (MPFR_IS_NEG(x) && mpfr_cmp_si_2exp (x, -1686629713, -31) >= 0) 597 { 598 mpfr_init2 (x_red, MPFR_PREC(x)); 599 mpfr_neg (x_red, x, rnd); /* exact */ 600 err = sincos_aux (ts, tc, x_red, MPFR_RNDN); 601 mpfr_neg (ts, ts, MPFR_RNDN); 602 mpfr_clear (x_red); 603 } 604 else /* argument reduction is needed */ 605 { 606 long q; 607 mpfr_t pi; 608 int neg = 0; 609 610 mpfr_init2 (x_red, w); 611 mpfr_init2 (pi, (MPFR_EXP(x) > 0) ? w + MPFR_EXP(x) : w); 612 mpfr_const_pi (pi, MPFR_RNDN); 613 mpfr_div_2exp (pi, pi, 1, MPFR_RNDN); /* Pi/2 */ 614 mpfr_remquo (x_red, &q, x, pi, MPFR_RNDN); 615 /* x = q * (Pi/2 + eps1) + x_red + eps2, 616 where |eps1| <= 1/2*ulp(Pi/2) = 2^(-w-MAX(0,EXP(x))), 617 and eps2 <= 1/2*ulp(x_red) <= 1/2*ulp(Pi/2) = 2^(-w) 618 Since |q| <= x/(Pi/2) <= |x|, we have 619 q*|eps1| <= 2^(-w), thus 620 |x - q * Pi/2 - x_red| <= 2^(1-w) */ 621 /* now -Pi/4 <= x_red <= Pi/4: if x_red < 0, consider -x_red */ 622 if (MPFR_IS_NEG(x_red)) 623 { 624 mpfr_neg (x_red, x_red, MPFR_RNDN); 625 neg = 1; 626 } 627 err = sincos_aux (ts, tc, x_red, MPFR_RNDN); 628 err ++; /* to take into account the argument reduction */ 629 if (neg) /* sin(-x) = -sin(x), cos(-x) = cos(x) */ 630 mpfr_neg (ts, ts, MPFR_RNDN); 631 if (q & 2) /* sin(x+Pi) = -sin(x), cos(x+Pi) = -cos(x) */ 632 { 633 mpfr_neg (ts, ts, MPFR_RNDN); 634 mpfr_neg (tc, tc, MPFR_RNDN); 635 } 636 if (q & 1) /* sin(x+Pi/2) = cos(x), cos(x+Pi/2) = -sin(x) */ 637 { 638 mpfr_neg (ts, ts, MPFR_RNDN); 639 mpfr_swap (ts, tc); 640 } 641 mpfr_clear (x_red); 642 mpfr_clear (pi); 643 } 644 /* adjust errors with respect to absolute values */ 645 errs = err - MPFR_EXP(ts); 646 errc = err - MPFR_EXP(tc); 647 if ((s == NULL || MPFR_CAN_ROUND (ts, w - errs, MPFR_PREC(s), rnd)) && 648 (c == NULL || MPFR_CAN_ROUND (tc, w - errc, MPFR_PREC(c), rnd))) 649 break; 650 MPFR_ZIV_NEXT (loop, w); 651 mpfr_set_prec (ts, w); 652 mpfr_set_prec (tc, w); 653 } 654 MPFR_ZIV_FREE (loop); 655 656 inexs = (s == NULL) ? 0 : mpfr_set (s, ts, rnd); 657 inexc = (c == NULL) ? 0 : mpfr_set (c, tc, rnd); 658 659 mpfr_clear (ts); 660 mpfr_clear (tc); 661 return INEX(inexs,inexc); 662} 663