1 /*
2 * IBM Accurate Mathematical Library
3 * written by International Business Machines Corp.
4 * Copyright (C) 2001-2022 Free Software Foundation, Inc.
5 *
6 * This program is free software; you can redistribute it and/or modify
7 * it under the terms of the GNU Lesser General Public License as published by
8 * the Free Software Foundation; either version 2.1 of the License, or
9 * (at your option) any later version.
10 *
11 * This program is distributed in the hope that it will be useful,
12 * but WITHOUT ANY WARRANTY; without even the implied warranty of
13 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
14 * GNU Lesser General Public License for more details.
15 *
16 * You should have received a copy of the GNU Lesser General Public License
17 * along with this program; if not, see <https://www.gnu.org/licenses/>.
18 */
19 /****************************************************************************/
20 /* */
21 /* MODULE_NAME:usncs.c */
22 /* */
23 /* FUNCTIONS: usin */
24 /* ucos */
25 /* FILES NEEDED: dla.h endian.h mpa.h mydefs.h usncs.h */
26 /* branred.c sincos.tbl */
27 /* */
28 /* An ultimate sin and cos routine. Given an IEEE double machine number x */
29 /* it computes sin(x) or cos(x) with ~0.55 ULP. */
30 /* Assumption: Machine arithmetic operations are performed in */
31 /* round to nearest mode of IEEE 754 standard. */
32 /* */
33 /****************************************************************************/
34
35
36 #include <errno.h>
37 #include <float.h>
38 #include "endian.h"
39 #include "mydefs.h"
40 #include "usncs.h"
41 #include <math.h>
42 #include <math_private.h>
43 #include <fenv_private.h>
44 #include <math-underflow.h>
45 #include <libm-alias-double.h>
46 #include <fenv.h>
47
48 /* Helper macros to compute sin of the input values. */
49 #define POLYNOMIAL2(xx) ((((s5 * (xx) + s4) * (xx) + s3) * (xx) + s2) * (xx))
50
51 #define POLYNOMIAL(xx) (POLYNOMIAL2 (xx) + s1)
52
53 /* The computed polynomial is a variation of the Taylor series expansion for
54 sin(x):
55
56 x - x^3/3! + x^5/5! - x^7/7! + x^9/9! - dx*x^2/2 + dx
57
58 The constants s1, s2, s3, etc. are pre-computed values of 1/3!, 1/5! and so
59 on. The result is returned to LHS. */
60 #define TAYLOR_SIN(xx, x, dx) \
61 ({ \
62 double t = ((POLYNOMIAL (xx) * (x) - 0.5 * (dx)) * (xx) + (dx)); \
63 double res = (x) + t; \
64 res; \
65 })
66
67 #define SINCOS_TABLE_LOOKUP(u, sn, ssn, cs, ccs) \
68 ({ \
69 int4 k = u.i[LOW_HALF] << 2; \
70 sn = __sincostab.x[k]; \
71 ssn = __sincostab.x[k + 1]; \
72 cs = __sincostab.x[k + 2]; \
73 ccs = __sincostab.x[k + 3]; \
74 })
75
76 #ifndef SECTION
77 # define SECTION
78 #endif
79
80 extern const union
81 {
82 int4 i[880];
83 double x[440];
84 } __sincostab attribute_hidden;
85
86 static const double
87 sn3 = -1.66666666666664880952546298448555E-01,
88 sn5 = 8.33333214285722277379541354343671E-03,
89 cs2 = 4.99999999999999999999950396842453E-01,
90 cs4 = -4.16666666666664434524222570944589E-02,
91 cs6 = 1.38888874007937613028114285595617E-03;
92
93 int __branred (double x, double *a, double *aa);
94
95 /* Given a number partitioned into X and DX, this function computes the cosine
96 of the number by combining the sin and cos of X (as computed by a variation
97 of the Taylor series) with the values looked up from the sin/cos table to
98 get the result. */
99 static __always_inline double
do_cos(double x,double dx)100 do_cos (double x, double dx)
101 {
102 mynumber u;
103
104 if (x < 0)
105 dx = -dx;
106
107 u.x = big + fabs (x);
108 x = fabs (x) - (u.x - big) + dx;
109
110 double xx, s, sn, ssn, c, cs, ccs, cor;
111 xx = x * x;
112 s = x + x * xx * (sn3 + xx * sn5);
113 c = xx * (cs2 + xx * (cs4 + xx * cs6));
114 SINCOS_TABLE_LOOKUP (u, sn, ssn, cs, ccs);
115 cor = (ccs - s * ssn - cs * c) - sn * s;
116 return cs + cor;
117 }
118
119 /* Given a number partitioned into X and DX, this function computes the sine of
120 the number by combining the sin and cos of X (as computed by a variation of
121 the Taylor series) with the values looked up from the sin/cos table to get
122 the result. */
123 static __always_inline double
do_sin(double x,double dx)124 do_sin (double x, double dx)
125 {
126 double xold = x;
127 /* Max ULP is 0.501 if |x| < 0.126, otherwise ULP is 0.518. */
128 if (fabs (x) < 0.126)
129 return TAYLOR_SIN (x * x, x, dx);
130
131 mynumber u;
132
133 if (x <= 0)
134 dx = -dx;
135 u.x = big + fabs (x);
136 x = fabs (x) - (u.x - big);
137
138 double xx, s, sn, ssn, c, cs, ccs, cor;
139 xx = x * x;
140 s = x + (dx + x * xx * (sn3 + xx * sn5));
141 c = x * dx + xx * (cs2 + xx * (cs4 + xx * cs6));
142 SINCOS_TABLE_LOOKUP (u, sn, ssn, cs, ccs);
143 cor = (ssn + s * ccs - sn * c) + cs * s;
144 return copysign (sn + cor, xold);
145 }
146
147 /* Reduce range of x to within PI/2 with abs (x) < 105414350. The high part
148 is written to *a, the low part to *da. Range reduction is accurate to 136
149 bits so that when x is large and *a very close to zero, all 53 bits of *a
150 are correct. */
151 static __always_inline int4
reduce_sincos(double x,double * a,double * da)152 reduce_sincos (double x, double *a, double *da)
153 {
154 mynumber v;
155
156 double t = (x * hpinv + toint);
157 double xn = t - toint;
158 v.x = t;
159 double y = (x - xn * mp1) - xn * mp2;
160 int4 n = v.i[LOW_HALF] & 3;
161
162 double b, db, t1, t2;
163 t1 = xn * pp3;
164 t2 = y - t1;
165 db = (y - t2) - t1;
166
167 t1 = xn * pp4;
168 b = t2 - t1;
169 db += (t2 - b) - t1;
170
171 *a = b;
172 *da = db;
173 return n;
174 }
175
176 /* Compute sin or cos (A + DA) for the given quadrant N. */
177 static __always_inline double
do_sincos(double a,double da,int4 n)178 do_sincos (double a, double da, int4 n)
179 {
180 double retval;
181
182 if (n & 1)
183 /* Max ULP is 0.513. */
184 retval = do_cos (a, da);
185 else
186 /* Max ULP is 0.501 if xx < 0.01588, otherwise ULP is 0.518. */
187 retval = do_sin (a, da);
188
189 return (n & 2) ? -retval : retval;
190 }
191
192
193 /*******************************************************************/
194 /* An ultimate sin routine. Given an IEEE double machine number x */
195 /* it computes the rounded value of sin(x). */
196 /*******************************************************************/
197 #ifndef IN_SINCOS
198 double
199 SECTION
__sin(double x)200 __sin (double x)
201 {
202 double t, a, da;
203 mynumber u;
204 int4 k, m, n;
205 double retval = 0;
206
207 SET_RESTORE_ROUND_53BIT (FE_TONEAREST);
208
209 u.x = x;
210 m = u.i[HIGH_HALF];
211 k = 0x7fffffff & m; /* no sign */
212 if (k < 0x3e500000) /* if x->0 =>sin(x)=x */
213 {
214 math_check_force_underflow (x);
215 retval = x;
216 }
217 /*--------------------------- 2^-26<|x|< 0.855469---------------------- */
218 else if (k < 0x3feb6000)
219 {
220 /* Max ULP is 0.548. */
221 retval = do_sin (x, 0);
222 } /* else if (k < 0x3feb6000) */
223
224 /*----------------------- 0.855469 <|x|<2.426265 ----------------------*/
225 else if (k < 0x400368fd)
226 {
227 t = hp0 - fabs (x);
228 /* Max ULP is 0.51. */
229 retval = copysign (do_cos (t, hp1), x);
230 } /* else if (k < 0x400368fd) */
231
232 /*-------------------------- 2.426265<|x|< 105414350 ----------------------*/
233 else if (k < 0x419921FB)
234 {
235 n = reduce_sincos (x, &a, &da);
236 retval = do_sincos (a, da, n);
237 } /* else if (k < 0x419921FB ) */
238
239 /* --------------------105414350 <|x| <2^1024------------------------------*/
240 else if (k < 0x7ff00000)
241 {
242 n = __branred (x, &a, &da);
243 retval = do_sincos (a, da, n);
244 }
245 /*--------------------- |x| > 2^1024 ----------------------------------*/
246 else
247 {
248 if (k == 0x7ff00000 && u.i[LOW_HALF] == 0)
249 __set_errno (EDOM);
250 retval = x / x;
251 }
252
253 return retval;
254 }
255
256
257 /*******************************************************************/
258 /* An ultimate cos routine. Given an IEEE double machine number x */
259 /* it computes the rounded value of cos(x). */
260 /*******************************************************************/
261
262 double
263 SECTION
__cos(double x)264 __cos (double x)
265 {
266 double y, a, da;
267 mynumber u;
268 int4 k, m, n;
269
270 double retval = 0;
271
272 SET_RESTORE_ROUND_53BIT (FE_TONEAREST);
273
274 u.x = x;
275 m = u.i[HIGH_HALF];
276 k = 0x7fffffff & m;
277
278 /* |x|<2^-27 => cos(x)=1 */
279 if (k < 0x3e400000)
280 retval = 1.0;
281
282 else if (k < 0x3feb6000)
283 { /* 2^-27 < |x| < 0.855469 */
284 /* Max ULP is 0.51. */
285 retval = do_cos (x, 0);
286 } /* else if (k < 0x3feb6000) */
287
288 else if (k < 0x400368fd)
289 { /* 0.855469 <|x|<2.426265 */ ;
290 y = hp0 - fabs (x);
291 a = y + hp1;
292 da = (y - a) + hp1;
293 /* Max ULP is 0.501 if xx < 0.01588 or 0.518 otherwise.
294 Range reduction uses 106 bits here which is sufficient. */
295 retval = do_sin (a, da);
296 } /* else if (k < 0x400368fd) */
297
298 else if (k < 0x419921FB)
299 { /* 2.426265<|x|< 105414350 */
300 n = reduce_sincos (x, &a, &da);
301 retval = do_sincos (a, da, n + 1);
302 } /* else if (k < 0x419921FB ) */
303
304 /* 105414350 <|x| <2^1024 */
305 else if (k < 0x7ff00000)
306 {
307 n = __branred (x, &a, &da);
308 retval = do_sincos (a, da, n + 1);
309 }
310
311 else
312 {
313 if (k == 0x7ff00000 && u.i[LOW_HALF] == 0)
314 __set_errno (EDOM);
315 retval = x / x; /* |x| > 2^1024 */
316 }
317
318 return retval;
319 }
320
321 #ifndef __cos
322 libm_alias_double (__cos, cos)
323 #endif
324 #ifndef __sin
325 libm_alias_double (__sin, sin)
326 #endif
327
328 #endif
329