1 /* 2 * CDDL HEADER START 3 * 4 * The contents of this file are subject to the terms of the 5 * Common Development and Distribution License (the "License"). 6 * You may not use this file except in compliance with the License. 7 * 8 * You can obtain a copy of the license at usr/src/OPENSOLARIS.LICENSE 9 * or http://www.opensolaris.org/os/licensing. 10 * See the License for the specific language governing permissions 11 * and limitations under the License. 12 * 13 * When distributing Covered Code, include this CDDL HEADER in each 14 * file and include the License file at usr/src/OPENSOLARIS.LICENSE. 15 * If applicable, add the following below this CDDL HEADER, with the 16 * fields enclosed by brackets "[]" replaced with your own identifying 17 * information: Portions Copyright [yyyy] [name of copyright owner] 18 * 19 * CDDL HEADER END 20 */ 21 /* 22 * Copyright 2011 Nexenta Systems, Inc. All rights reserved. 23 */ 24 /* 25 * Copyright 2005 Sun Microsystems, Inc. All rights reserved. 26 * Use is subject to license terms. 27 */ 28 29 #pragma weak sincos = __sincos 30 31 /* INDENT OFF */ 32 /* 33 * sincos(x,s,c) 34 * Accurate Table look-up algorithm by K.C. Ng, 2000. 35 * 36 * 1. Reduce x to x>0 by cos(-x)=cos(x), sin(-x)=-sin(x). 37 * 2. For 0<= x < 8, let i = (64*x chopped)-10. Let d = x - a[i], where 38 * a[i] is a double that is close to (i+10.5)/64 (and hence |d|< 10.5/64) 39 * and such that sin(a[i]) and cos(a[i]) is close to a double (with error 40 * less than 2**-8 ulp). Then 41 * 42 * cos(x) = cos(a[i]+d) = cos(a[i])cos(d) - sin(a[i])*sin(d) 43 * = TBL_cos_a[i]*(1+QQ1*d^2+QQ2*d^4) - 44 * TBL_sin_a[i]*(d+PP1*d^3+PP2*d^5) 45 * = TBL_cos_a[i] + (TBL_cos_a[i]*d^2*(QQ1+QQ2*d^2) - 46 * TBL_sin_a[i]*(d+PP1*d^3+PP2*d^5)) 47 * 48 * sin(x) = sin(a[i]+d) = sin(a[i])cos(d) + cos(a[i])*sin(d) 49 * = TBL_sin_a[i]*(1+QQ1*d^2+QQ2*d^4) + 50 * TBL_cos_a[i]*(d+PP1*d^3+PP2*d^5) 51 * = TBL_sin_a[i] + (TBL_sin_a[i]*d^2*(QQ1+QQ2*d^2) + 52 * TBL_cos_a[i]*(d+PP1*d^3+PP2*d^5)) 53 * 54 * Note: for x close to n*pi/2, special treatment is need for either 55 * sin or cos: 56 * i in [81, 100] ( pi/2 +-10.5/64 => tiny cos(x) = sin(pi/2-x) 57 * i in [181,200] ( pi +-10.5/64 => tiny sin(x) = sin(pi-x) 58 * i in [282,301] ( 3pi/2+-10.5/64 => tiny cos(x) = sin(x-3pi/2) 59 * i in [382,401] ( 2pi +-10.5/64 => tiny sin(x) = sin(x-2pi) 60 * i in [483,502] ( 5pi/2+-10.5/64 => tiny cos(x) = sin(5pi/2-x) 61 * 62 * 3. For x >= 8.0, use kernel function __rem_pio2 to perform argument 63 * reduction and call __k_sincos_ to compute sin and cos. 64 * 65 * kernel function: 66 * __rem_pio2 ... argument reduction routine 67 * __k_sincos_ ... sine and cosine function on [-pi/4,pi/4] 68 * 69 * Method. 70 * Let S and C denote the sin and cos respectively on [-PI/4, +PI/4]. 71 * 1. Assume the argument x is reduced to y1+y2 = x-k*pi/2 in 72 * [-pi/2 , +pi/2], and let n = k mod 4. 73 * 2. Let S=S(y1+y2), C=C(y1+y2). Depending on n, we have 74 * 75 * n sin(x) cos(x) tan(x) 76 * ---------------------------------------------------------- 77 * 0 S C S/C 78 * 1 C -S -C/S 79 * 2 -S -C S/C 80 * 3 -C S -C/S 81 * ---------------------------------------------------------- 82 * 83 * Special cases: 84 * Let trig be any of sin, cos, or tan. 85 * trig(+-INF) is NaN, with signals; 86 * trig(NaN) is that NaN; 87 * 88 * Accuracy: 89 * TRIG(x) returns trig(x) nearly rounded (less than 1 ulp) 90 */ 91 92 #include "libm.h" 93 94 static const double sc[] = { 95 /* ONE = */ 1.0, 96 /* NONE = */ -1.0, 97 /* 98 * |sin(x) - (x+pp1*x^3+pp2*x^5)| <= 2^-58.79 for |x| < 0.008 99 */ 100 /* PP1 = */ -0.166666666666316558867252052378889521480627858683055567, 101 /* PP2 = */ .008333315652997472323564894248466758248475374977974017927, 102 /* 103 * |(sin(x) - (x+p1*x^3+...+p4*x^9)| 104 * |------------------------------ | <= 2^-57.63 for |x| < 0.1953125 105 * | x | 106 */ 107 /* P1 = */ -1.666666666666629669805215138920301589656e-0001, 108 /* P2 = */ 8.333333332390951295683993455280336376663e-0003, 109 /* P3 = */ -1.984126237997976692791551778230098403960e-0004, 110 /* P4 = */ 2.753403624854277237649987622848330351110e-0006, 111 /* 112 * |cos(x) - (1+qq1*x^2+qq2*x^4)| <= 2^-55.99 for |x| <= 0.008 (0x3f80624d) 113 */ 114 /* QQ1 = */ -0.4999999999975492381842911981948418542742729, 115 /* QQ2 = */ 0.041666542904352059294545209158357640398771740, 116 /* Q1 = */ -0.5, 117 /* Q2 = */ 4.166666666500350703680945520860748617445e-0002, 118 /* Q3 = */ -1.388888596436972210694266290577848696006e-0003, 119 /* Q4 = */ 2.478563078858589473679519517892953492192e-0005, 120 /* PIO2_H = */ 1.570796326794896557999, 121 /* PIO2_L = */ 6.123233995736765886130e-17, 122 /* PIO2_L0 = */ 6.123233995727922165564e-17, 123 /* PIO2_L1 = */ 8.843720566135701120255e-29, 124 /* PI_H = */ 3.1415926535897931159979634685, 125 /* PI_L = */ 1.22464679914735317722606593227425e-16, 126 /* PI_L0 = */ 1.22464679914558443311283879205095e-16, 127 /* PI_L1 = */ 1.768744113227140223300005233735517376e-28, 128 /* PI3O2_H = */ 4.712388980384689673997, 129 /* PI3O2_L = */ 1.836970198721029765839e-16, 130 /* PI3O2_L0 = */ 1.836970198720396133587e-16, 131 /* PI3O2_L1 = */ 6.336322524749201142226e-29, 132 /* PI2_H = */ 6.2831853071795862319959269370, 133 /* PI2_L = */ 2.44929359829470635445213186454850e-16, 134 /* PI2_L0 = */ 2.44929359829116886622567758410190e-16, 135 /* PI2_L1 = */ 3.537488226454280446600010467471034752e-28, 136 /* PI5O2_H = */ 7.853981633974482789995, 137 /* PI5O2_L = */ 3.061616997868382943065e-16, 138 /* PI5O2_L0 = */ 3.061616997861941598865e-16, 139 /* PI5O2_L1 = */ 6.441344200433640781982e-28, 140 }; 141 /* INDENT ON */ 142 143 #define ONE sc[0] 144 #define PP1 sc[2] 145 #define PP2 sc[3] 146 #define P1 sc[4] 147 #define P2 sc[5] 148 #define P3 sc[6] 149 #define P4 sc[7] 150 #define QQ1 sc[8] 151 #define QQ2 sc[9] 152 #define Q1 sc[10] 153 #define Q2 sc[11] 154 #define Q3 sc[12] 155 #define Q4 sc[13] 156 #define PIO2_H sc[14] 157 #define PIO2_L sc[15] 158 #define PIO2_L0 sc[16] 159 #define PIO2_L1 sc[17] 160 #define PI_H sc[18] 161 #define PI_L sc[19] 162 #define PI_L0 sc[20] 163 #define PI_L1 sc[21] 164 #define PI3O2_H sc[22] 165 #define PI3O2_L sc[23] 166 #define PI3O2_L0 sc[24] 167 #define PI3O2_L1 sc[25] 168 #define PI2_H sc[26] 169 #define PI2_L sc[27] 170 #define PI2_L0 sc[28] 171 #define PI2_L1 sc[29] 172 #define PI5O2_H sc[30] 173 #define PI5O2_L sc[31] 174 #define PI5O2_L0 sc[32] 175 #define PI5O2_L1 sc[33] 176 #define PoS(x, z) ((x * z) * (PP1 + z * PP2)) 177 #define PoL(x, z) ((x * z) * ((P1 + z * P2) + (z * z) * (P3 + z * P4))) 178 179 extern const double _TBL_sincos[], _TBL_sincosx[]; 180 181 void 182 sincos(double x, double *s, double *c) { 183 double z, y[2], w, t, v, p, q; 184 int i, j, n, hx, ix, lx; 185 186 hx = ((int *)&x)[HIWORD]; 187 lx = ((int *)&x)[LOWORD]; 188 ix = hx & ~0x80000000; 189 190 if (ix <= 0x3fc50000) { /* |x| < 10.5/64 = 0.164062500 */ 191 if (ix < 0x3e400000) { /* |x| < 2**-27 */ 192 if ((int)x == 0) 193 *c = ONE; 194 *s = x; 195 } else { 196 z = x * x; 197 if (ix < 0x3f800000) { /* |x| < 0.008 */ 198 q = z * (QQ1 + z * QQ2); 199 p = PoS(x, z); 200 } else { 201 q = z * ((Q1 + z * Q2) + (z * z) * 202 (Q3 + z * Q4)); 203 p = PoL(x, z); 204 } 205 *c = ONE + q; 206 *s = x + p; 207 } 208 return; 209 } 210 211 n = ix >> 20; 212 i = (((ix >> 12) & 0xff) | 0x100) >> (0x401 - n); 213 j = i - 10; 214 if (n < 0x402) { /* |x| < 8 */ 215 x = fabs(x); 216 v = x - _TBL_sincosx[j]; 217 t = v * v; 218 w = _TBL_sincos[(j<<1)]; 219 z = _TBL_sincos[(j<<1)+1]; 220 p = v + PoS(v, t); 221 q = t * (QQ1 + t * QQ2); 222 if ((((j - 81) ^ (j - 101)) | 223 ((j - 282) ^ (j - 302)) | 224 ((j - 483) ^ (j - 503)) | 225 ((j - 181) ^ (j - 201)) | 226 ((j - 382) ^ (j - 402))) < 0) { 227 if (j <= 101) { 228 /* near pi/2, cos(x) = sin(pi/2-x) */ 229 t = w * q + z * p; 230 *s = (hx >= 0)? w + t : -w - t; 231 p = PIO2_H - x; 232 i = ix - 0x3ff921fb; 233 x = p + PIO2_L; 234 if ((i | ((lx - 0x54442D00) & 235 0xffffff00)) == 0) { 236 /* very close to pi/2 */ 237 x = p + PIO2_L0; 238 *c = x + PIO2_L1; 239 } else { 240 z = x * x; 241 if (((ix - 0x3ff92000) >> 12) == 0) { 242 /* |pi/2-x|<2**-8 */ 243 w = PIO2_L + PoS(x, z); 244 } else { 245 w = PIO2_L + PoL(x, z); 246 } 247 *c = p + w; 248 } 249 } else if (j <= 201) { 250 /* near pi, sin(x) = sin(pi-x) */ 251 *c = z - (w * p - z * q); 252 p = PI_H - x; 253 i = ix - 0x400921fb; 254 x = p + PI_L; 255 if ((i | ((lx - 0x54442D00) & 256 0xffffff00)) == 0) { 257 /* very close to pi */ 258 x = p + PI_L0; 259 *s = (hx >= 0)? x + PI_L1 : 260 -(x + PI_L1); 261 } else { 262 z = x * x; 263 if (((ix - 0x40092000) >> 11) == 0) { 264 /* |pi-x|<2**-8 */ 265 w = PI_L + PoS(x, z); 266 } else { 267 w = PI_L + PoL(x, z); 268 } 269 *s = (hx >= 0)? p + w : -p - w; 270 } 271 } else if (j <= 302) { 272 /* near 3/2pi, cos(x)=sin(x-3/2pi) */ 273 t = w * q + z * p; 274 *s = (hx >= 0)? w + t : -w - t; 275 p = x - PI3O2_H; 276 i = ix - 0x4012D97C; 277 x = p - PI3O2_L; 278 if ((i | ((lx - 0x7f332100) & 279 0xffffff00)) == 0) { 280 /* very close to 3/2pi */ 281 x = p - PI3O2_L0; 282 *c = x - PI3O2_L1; 283 } else { 284 z = x * x; 285 if (((ix - 0x4012D800) >> 9) == 0) { 286 /* |3/2pi-x|<2**-8 */ 287 w = PoS(x, z) - PI3O2_L; 288 } else { 289 w = PoL(x, z) - PI3O2_L; 290 } 291 *c = p + w; 292 } 293 } else if (j <= 402) { 294 /* near 2pi, sin(x)=sin(x-2pi) */ 295 *c = z - (w * p - z * q); 296 p = x - PI2_H; 297 i = ix - 0x401921fb; 298 x = p - PI2_L; 299 if ((i | ((lx - 0x54442D00) & 300 0xffffff00)) == 0) { 301 /* very close to 2pi */ 302 x = p - PI2_L0; 303 *s = (hx >= 0)? x - PI2_L1 : 304 -(x - PI2_L1); 305 } else { 306 z = x * x; 307 if (((ix - 0x40192000) >> 10) == 0) { 308 /* |x-2pi|<2**-8 */ 309 w = PoS(x, z) - PI2_L; 310 } else { 311 w = PoL(x, z) - PI2_L; 312 } 313 *s = (hx >= 0)? p + w : -p - w; 314 } 315 } else { 316 /* near 5pi/2, cos(x) = sin(5pi/2-x) */ 317 t = w * q + z * p; 318 *s = (hx >= 0)? w + t : -w - t; 319 p = PI5O2_H - x; 320 i = ix - 0x401F6A7A; 321 x = p + PI5O2_L; 322 if ((i | ((lx - 0x29553800) & 323 0xffffff00)) == 0) { 324 /* very close to pi/2 */ 325 x = p + PI5O2_L0; 326 *c = x + PI5O2_L1; 327 } else { 328 z = x * x; 329 if (((ix - 0x401F6A7A) >> 7) == 0) { 330 /* |5pi/2-x|<2**-8 */ 331 w = PI5O2_L + PoS(x, z); 332 } else { 333 w = PI5O2_L + PoL(x, z); 334 } 335 *c = p + w; 336 } 337 } 338 } else { 339 *c = z - (w * p - z * q); 340 t = w * q + z * p; 341 *s = (hx >= 0)? w + t : -w - t; 342 } 343 return; 344 } 345 346 if (ix >= 0x7ff00000) { 347 *s = *c = x / x; 348 return; 349 } 350 351 /* argument reduction needed */ 352 n = __rem_pio2(x, y); 353 switch (n & 3) { 354 case 0: 355 *s = __k_sincos(y[0], y[1], c); 356 break; 357 case 1: 358 *c = -__k_sincos(y[0], y[1], s); 359 break; 360 case 2: 361 *s = -__k_sincos(y[0], y[1], c); 362 *c = -*c; 363 break; 364 default: 365 *c = __k_sincos(y[0], y[1], s); 366 *s = -*s; 367 } 368 } 369