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 /*
23 * Copyright 2011 Nexenta Systems, Inc. All rights reserved.
24 */
25
26 /*
27 * Copyright 2005 Sun Microsystems, Inc. All rights reserved.
28 * Use is subject to license terms.
29 */
30
31 #pragma weak __sincos = sincos
32
33 /*
34 * sincos(x,s,c)
35 * Accurate Table look-up algorithm by K.C. Ng, 2000.
36 *
37 * 1. Reduce x to x>0 by cos(-x)=cos(x), sin(-x)=-sin(x).
38 * 2. For 0<= x < 8, let i = (64*x chopped)-10. Let d = x - a[i], where
39 * a[i] is a double that is close to (i+10.5)/64 (and hence |d|< 10.5/64)
40 * and such that sin(a[i]) and cos(a[i]) is close to a double (with error
41 * less than 2**-8 ulp). Then
42 *
43 * cos(x) = cos(a[i]+d) = cos(a[i])cos(d) - sin(a[i])*sin(d)
44 * = TBL_cos_a[i]*(1+QQ1*d^2+QQ2*d^4) -
45 * TBL_sin_a[i]*(d+PP1*d^3+PP2*d^5)
46 * = TBL_cos_a[i] + (TBL_cos_a[i]*d^2*(QQ1+QQ2*d^2) -
47 * TBL_sin_a[i]*(d+PP1*d^3+PP2*d^5))
48 *
49 * sin(x) = sin(a[i]+d) = sin(a[i])cos(d) + cos(a[i])*sin(d)
50 * = TBL_sin_a[i]*(1+QQ1*d^2+QQ2*d^4) +
51 * TBL_cos_a[i]*(d+PP1*d^3+PP2*d^5)
52 * = TBL_sin_a[i] + (TBL_sin_a[i]*d^2*(QQ1+QQ2*d^2) +
53 * TBL_cos_a[i]*(d+PP1*d^3+PP2*d^5))
54 *
55 * Note: for x close to n*pi/2, special treatment is need for either
56 * sin or cos:
57 * i in [81, 100] ( pi/2 +-10.5/64 => tiny cos(x) = sin(pi/2-x)
58 * i in [181,200] ( pi +-10.5/64 => tiny sin(x) = sin(pi-x)
59 * i in [282,301] ( 3pi/2+-10.5/64 => tiny cos(x) = sin(x-3pi/2)
60 * i in [382,401] ( 2pi +-10.5/64 => tiny sin(x) = sin(x-2pi)
61 * i in [483,502] ( 5pi/2+-10.5/64 => tiny cos(x) = sin(5pi/2-x)
62 *
63 * 3. For x >= 8.0, use kernel function __rem_pio2 to perform argument
64 * reduction and call __k_sincos_ to compute sin and cos.
65 *
66 * kernel function:
67 * __rem_pio2 ... argument reduction routine
68 * __k_sincos_ ... sine and cosine function on [-pi/4,pi/4]
69 *
70 * Method.
71 * Let S and C denote the sin and cos respectively on [-PI/4, +PI/4].
72 * 1. Assume the argument x is reduced to y1+y2 = x-k*pi/2 in
73 * [-pi/2 , +pi/2], and let n = k mod 4.
74 * 2. Let S=S(y1+y2), C=C(y1+y2). Depending on n, we have
75 *
76 * n sin(x) cos(x) tan(x)
77 * ----------------------------------------------------------
78 * 0 S C S/C
79 * 1 C -S -C/S
80 * 2 -S -C S/C
81 * 3 -C S -C/S
82 * ----------------------------------------------------------
83 *
84 * Special cases:
85 * Let trig be any of sin, cos, or tan.
86 * trig(+-INF) is NaN, with signals;
87 * trig(NaN) is that NaN;
88 *
89 * Accuracy:
90 * TRIG(x) returns trig(x) nearly rounded (less than 1 ulp)
91 */
92
93 #include "libm.h"
94
95 /* BEGIN CSTYLED */
96 static const double sc[] = {
97 /* ONE = */
98 1.0,
99 /* NONE = */ -1.0,
100
101 /*
102 * |sin(x) - (x+pp1*x^3+pp2*x^5)| <= 2^-58.79 for |x| < 0.008
103 */
104 /* PP1 = */-0.166666666666316558867252052378889521480627858683055567,
105 /* PP2 = */.008333315652997472323564894248466758248475374977974017927,
106
107 /*
108 * |(sin(x) - (x+p1*x^3+...+p4*x^9)|
109 * |------------------------------ | <= 2^-57.63 for |x| < 0.1953125
110 * | x |
111 */
112 /* P1 = */ -1.666666666666629669805215138920301589656e-0001,
113 /* P2 = */ 8.333333332390951295683993455280336376663e-0003,
114 /* P3 = */ -1.984126237997976692791551778230098403960e-0004,
115 /* P4 = */ 2.753403624854277237649987622848330351110e-0006,
116
117 /*
118 * |cos(x) - (1+qq1*x^2+qq2*x^4)| <= 2^-55.99 for |x| <= 0.008 (0x3f80624d)
119 */
120 /* QQ1 = */-0.4999999999975492381842911981948418542742729,
121 /* QQ2 = */0.041666542904352059294545209158357640398771740,
122 /* Q1 = */ -0.5,
123 /* Q2 = */ 4.166666666500350703680945520860748617445e-0002,
124 /* Q3 = */ -1.388888596436972210694266290577848696006e-0003,
125 /* Q4 = */ 2.478563078858589473679519517892953492192e-0005,
126 /* PIO2_H = */ 1.570796326794896557999,
127 /* PIO2_L = */ 6.123233995736765886130e-17,
128 /* PIO2_L0 = */ 6.123233995727922165564e-17,
129 /* PIO2_L1 = */ 8.843720566135701120255e-29,
130 /* PI_H = */ 3.1415926535897931159979634685,
131 /* PI_L = */ 1.22464679914735317722606593227425e-16,
132 /* PI_L0 = */ 1.22464679914558443311283879205095e-16,
133 /* PI_L1 = */ 1.768744113227140223300005233735517376e-28,
134 /* PI3O2_H = */ 4.712388980384689673997,
135 /* PI3O2_L = */ 1.836970198721029765839e-16,
136 /* PI3O2_L0 = */ 1.836970198720396133587e-16,
137 /* PI3O2_L1 = */ 6.336322524749201142226e-29,
138 /* PI2_H = */ 6.2831853071795862319959269370,
139 /* PI2_L = */ 2.44929359829470635445213186454850e-16,
140 /* PI2_L0 = */ 2.44929359829116886622567758410190e-16,
141 /* PI2_L1 = */ 3.537488226454280446600010467471034752e-28,
142 /* PI5O2_H = */ 7.853981633974482789995,
143 /* PI5O2_L = */ 3.061616997868382943065e-16,
144 /* PI5O2_L0 = */ 3.061616997861941598865e-16,
145 /* PI5O2_L1 = */ 6.441344200433640781982e-28,
146 };
147 /* END CSTYLED */
148
149 #define ONE sc[0]
150 #define PP1 sc[2]
151 #define PP2 sc[3]
152 #define P1 sc[4]
153 #define P2 sc[5]
154 #define P3 sc[6]
155 #define P4 sc[7]
156 #define QQ1 sc[8]
157 #define QQ2 sc[9]
158 #define Q1 sc[10]
159 #define Q2 sc[11]
160 #define Q3 sc[12]
161 #define Q4 sc[13]
162 #define PIO2_H sc[14]
163 #define PIO2_L sc[15]
164 #define PIO2_L0 sc[16]
165 #define PIO2_L1 sc[17]
166 #define PI_H sc[18]
167 #define PI_L sc[19]
168 #define PI_L0 sc[20]
169 #define PI_L1 sc[21]
170 #define PI3O2_H sc[22]
171 #define PI3O2_L sc[23]
172 #define PI3O2_L0 sc[24]
173 #define PI3O2_L1 sc[25]
174 #define PI2_H sc[26]
175 #define PI2_L sc[27]
176 #define PI2_L0 sc[28]
177 #define PI2_L1 sc[29]
178 #define PI5O2_H sc[30]
179 #define PI5O2_L sc[31]
180 #define PI5O2_L0 sc[32]
181 #define PI5O2_L1 sc[33]
182 #define PoS(x, z) ((x * z) * (PP1 + z * PP2))
183 #define PoL(x, z) ((x * z) * ((P1 + z * P2) + (z * z) * \
184 (P3 + z * P4)))
185
186 extern const double _TBL_sincos[], _TBL_sincosx[];
187
188 void
189 sincos(double x, double *s, double *c)
190 {
191 double z, y[2], w, t, v, p, q;
192 int i, j, n, hx, ix, lx;
193
194 hx = ((int *)&x)[HIWORD];
195 lx = ((int *)&x)[LOWORD];
196 ix = hx & ~0x80000000;
197
198 if (ix <= 0x3fc50000) { /* |x| < 10.5/64 = 0.164062500 */
199 if (ix < 0x3e400000) { /* |x| < 2**-27 */
200 if ((int)x == 0)
201 *c = ONE;
202
203 *s = x;
204 } else {
205 z = x * x;
206
207 if (ix < 0x3f800000) { /* |x| < 0.008 */
208 q = z * (QQ1 + z * QQ2);
209 p = PoS(x, z);
210 } else {
211 q = z * ((Q1 + z * Q2) + (z * z) * (Q3 + z *
212 Q4));
213 p = PoL(x, z);
214 }
215
216 *c = ONE + q;
217 *s = x + p;
218 }
219
220 return;
221 }
222
223 n = ix >> 20;
224 i = (((ix >> 12) & 0xff) | 0x100) >> (0x401 - n);
225 j = i - 10;
226
227 if (n < 0x402) { /* |x| < 8 */
228 x = fabs(x);
229 v = x - _TBL_sincosx[j];
230 t = v * v;
231 w = _TBL_sincos[(j << 1)];
232 z = _TBL_sincos[(j << 1) + 1];
233 p = v + PoS(v, t);
234 q = t * (QQ1 + t * QQ2);
235
236 if ((((j - 81) ^ (j - 101)) | ((j - 282) ^ (j - 302)) | ((j -
237 483) ^ (j - 503)) | ((j - 181) ^ (j - 201)) | ((j - 382) ^
238 (j - 402))) < 0) {
239 if (j <= 101) {
240 /* near pi/2, cos(x) = sin(pi/2-x) */
241 t = w * q + z * p;
242 *s = (hx >= 0) ? w + t : -w - t;
243 p = PIO2_H - x;
244 i = ix - 0x3ff921fb;
245 x = p + PIO2_L;
246
247 if ((i | ((lx - 0x54442D00) & 0xffffff00)) ==
248 0) {
249 /* very close to pi/2 */
250 x = p + PIO2_L0;
251 *c = x + PIO2_L1;
252 } else {
253 z = x * x;
254
255 if (((ix - 0x3ff92000) >> 12) == 0) {
256 /* |pi/2-x|<2**-8 */
257 w = PIO2_L + PoS(x, z);
258 } else {
259 w = PIO2_L + PoL(x, z);
260 }
261
262 *c = p + w;
263 }
264 } else if (j <= 201) {
265 /* near pi, sin(x) = sin(pi-x) */
266 *c = z - (w * p - z * q);
267 p = PI_H - x;
268 i = ix - 0x400921fb;
269 x = p + PI_L;
270
271 if ((i | ((lx - 0x54442D00) & 0xffffff00)) ==
272 0) {
273 /* very close to pi */
274 x = p + PI_L0;
275 *s = (hx >= 0) ? x + PI_L1 : -(x +
276 PI_L1);
277 } else {
278 z = x * x;
279
280 if (((ix - 0x40092000) >> 11) == 0) {
281 /* |pi-x|<2**-8 */
282 w = PI_L + PoS(x, z);
283 } else {
284 w = PI_L + PoL(x, z);
285 }
286
287 *s = (hx >= 0) ? p + w : -p - w;
288 }
289 } else if (j <= 302) {
290 /* near 3/2pi, cos(x)=sin(x-3/2pi) */
291 t = w * q + z * p;
292 *s = (hx >= 0) ? w + t : -w - t;
293 p = x - PI3O2_H;
294 i = ix - 0x4012D97C;
295 x = p - PI3O2_L;
296
297 if ((i | ((lx - 0x7f332100) & 0xffffff00)) ==
298 0) {
299 /* very close to 3/2pi */
300 x = p - PI3O2_L0;
301 *c = x - PI3O2_L1;
302 } else {
303 z = x * x;
304
305 if (((ix - 0x4012D800) >> 9) == 0) {
306 /* |3/2pi-x|<2**-8 */
307 w = PoS(x, z) - PI3O2_L;
308 } else {
309 w = PoL(x, z) - PI3O2_L;
310 }
311
312 *c = p + w;
313 }
314 } else if (j <= 402) {
315 /* near 2pi, sin(x)=sin(x-2pi) */
316 *c = z - (w * p - z * q);
317 p = x - PI2_H;
318 i = ix - 0x401921fb;
319 x = p - PI2_L;
320
321 if ((i | ((lx - 0x54442D00) & 0xffffff00)) ==
322 0) {
323 /* very close to 2pi */
324 x = p - PI2_L0;
325 *s = (hx >= 0) ? x - PI2_L1 : -(x -
326 PI2_L1);
327 } else {
328 z = x * x;
329
330 if (((ix - 0x40192000) >> 10) == 0) {
331 /* |x-2pi|<2**-8 */
332 w = PoS(x, z) - PI2_L;
333 } else {
334 w = PoL(x, z) - PI2_L;
335 }
336
337 *s = (hx >= 0) ? p + w : -p - w;
338 }
339 } else {
340 /* near 5pi/2, cos(x) = sin(5pi/2-x) */
341 t = w * q + z * p;
342 *s = (hx >= 0) ? w + t : -w - t;
343 p = PI5O2_H - x;
344 i = ix - 0x401F6A7A;
345 x = p + PI5O2_L;
346
347 if ((i | ((lx - 0x29553800) & 0xffffff00)) ==
348 0) {
349 /* very close to pi/2 */
350 x = p + PI5O2_L0;
351 *c = x + PI5O2_L1;
352 } else {
353 z = x * x;
354
355 if (((ix - 0x401F6A7A) >> 7) == 0) {
356 /* |5pi/2-x|<2**-8 */
357 w = PI5O2_L + PoS(x, z);
358 } else {
359 w = PI5O2_L + PoL(x, z);
360 }
361
362 *c = p + w;
363 }
364 }
365 } else {
366 *c = z - (w * p - z * q);
367 t = w * q + z * p;
368 *s = (hx >= 0) ? w + t : -w - t;
369 }
370
371 return;
372 }
373
374 if (ix >= 0x7ff00000) {
375 *s = *c = x / x;
376 return;
377 }
378
379 /* argument reduction needed */
380 n = __rem_pio2(x, y);
381
382 switch (n & 3) {
383 case 0:
384 *s = __k_sincos(y[0], y[1], c);
385 break;
386 case 1:
387 *c = -__k_sincos(y[0], y[1], s);
388 break;
389 case 2:
390 *s = -__k_sincos(y[0], y[1], c);
391 *c = -*c;
392 break;
393 default:
394 *c = __k_sincos(y[0], y[1], s);
395 *s = -*s;
396 }
397 }