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 }