```   1 /*
3  *
4  * The contents of this file are subject to the terms of the
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
15  * If applicable, add the following below this CDDL HEADER, with the
16  * fields enclosed by brackets "[]" replaced with your own identifying
18  *
20  */
21 /*
23  */
24 /*
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 }
```