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9b525f39 | 1 | /* |
38cc45a4 | 2 | * Copyright (c) 1987 Regents of the University of California. |
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3 | * All rights reserved. |
4 | * | |
5 | * Redistribution and use in source and binary forms are permitted | |
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6 | * provided that the above copyright notice and this paragraph are |
7 | * duplicated in all such forms and that any documentation, | |
8 | * advertising materials, and other materials related to such | |
9 | * distribution and use acknowledge that the software was developed | |
10 | * by the University of California, Berkeley. The name of the | |
11 | * University may not be used to endorse or promote products derived | |
12 | * from this software without specific prior written permission. | |
13 | * THIS SOFTWARE IS PROVIDED ``AS IS'' AND WITHOUT ANY EXPRESS OR | |
14 | * IMPLIED WARRANTIES, INCLUDING, WITHOUT LIMITATION, THE IMPLIED | |
15 | * WARRANTIES OF MERCHANTIBILITY AND FITNESS FOR A PARTICULAR PURPOSE. | |
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16 | * |
17 | * All recipients should regard themselves as participants in an ongoing | |
18 | * research project and hence should feel obligated to report their | |
19 | * experiences (good or bad) with these elementary function codes, using | |
20 | * the sendbug(8) program, to the authors. | |
21 | * | |
a399f6c8 | 22 | * @(#)trig.h 5.3 (Berkeley) %G% |
38cc45a4 | 23 | */ |
9b525f39 | 24 | |
38cc45a4 ZAL |
25 | #if defined(vax)||defined(tahoe) |
26 | #ifdef vax | |
27 | #define _0x(A,B) 0x/**/A/**/B | |
28 | #else /* vax */ | |
29 | #define _0x(A,B) 0x/**/B/**/A | |
30 | #endif /* vax */ | |
31 | /*thresh = 2.6117239648121182150E-1 , Hex 2^ -1 * .85B8636B026EA0 */ | |
32 | /*PIo4 = 7.8539816339744830676E-1 , Hex 2^ 0 * .C90FDAA22168C2 */ | |
33 | /*PIo2 = 1.5707963267948966135E0 , Hex 2^ 1 * .C90FDAA22168C2 */ | |
34 | /*PI3o4 = 2.3561944901923449203E0 , Hex 2^ 2 * .96CBE3F9990E92 */ | |
35 | /*PI = 3.1415926535897932270E0 , Hex 2^ 2 * .C90FDAA22168C2 */ | |
36 | /*PI2 = 6.2831853071795864540E0 ; Hex 2^ 3 * .C90FDAA22168C2 */ | |
37 | static long threshx[] = { _0x(b863,3f85), _0x(6ea0,6b02)}; | |
38 | static long PIo4x[] = { _0x(0fda,4049), _0x(68c2,a221)}; | |
39 | static long PIo2x[] = { _0x(0fda,40c9), _0x(68c2,a221)}; | |
40 | static long PI3o4x[] = { _0x(cbe3,4116), _0x(0e92,f999)}; | |
41 | static long PIx[] = { _0x(0fda,4149), _0x(68c2,a221)}; | |
42 | static long PI2x[] = { _0x(0fda,41c9), _0x(68c2,a221)}; | |
43 | #define thresh (*(double*)threshx) | |
44 | #define PIo4 (*(double*)PIo4x) | |
45 | #define PIo2 (*(double*)PIo2x) | |
46 | #define PI3o4 (*(double*)PI3o4x) | |
47 | #define PI (*(double*)PIx) | |
48 | #define PI2 (*(double*)PI2x) | |
49 | #else /* defined(vax)||defined(tahoe) */ | |
50 | static double | |
51 | thresh = 2.6117239648121182150E-1 , /*Hex 2^ -2 * 1.0B70C6D604DD4 */ | |
52 | PIo4 = 7.8539816339744827900E-1 , /*Hex 2^ -1 * 1.921FB54442D18 */ | |
53 | PIo2 = 1.5707963267948965580E0 , /*Hex 2^ 0 * 1.921FB54442D18 */ | |
54 | PI3o4 = 2.3561944901923448370E0 , /*Hex 2^ 1 * 1.2D97C7F3321D2 */ | |
55 | PI = 3.1415926535897931160E0 , /*Hex 2^ 1 * 1.921FB54442D18 */ | |
56 | PI2 = 6.2831853071795862320E0 ; /*Hex 2^ 2 * 1.921FB54442D18 */ | |
57 | #ifdef national | |
58 | static long fmaxx[] = { 0xffffffff, 0x7fefffff}; | |
59 | #define fmax (*(double*)fmaxx) | |
60 | #endif /* national */ | |
61 | #endif /* defined(vax)||defined(tahoe) */ | |
62 | static double | |
63 | zero = 0, | |
64 | one = 1, | |
65 | negone = -1, | |
66 | half = 1.0/2.0, | |
67 | small = 1E-10, /* 1+small**2 == 1; better values for small: | |
68 | * small = 1.5E-9 for VAX D | |
69 | * = 1.2E-8 for IEEE Double | |
70 | * = 2.8E-10 for IEEE Extended | |
71 | */ | |
72 | big = 1E20; /* big := 1/(small**2) */ | |
73 | ||
74 | /* sin__S(x*x) ... re-implemented as a macro | |
75 | * DOUBLE PRECISION (VAX D format 56 bits, IEEE DOUBLE 53 BITS) | |
76 | * STATIC KERNEL FUNCTION OF SIN(X), COS(X), AND TAN(X) | |
77 | * CODED IN C BY K.C. NG, 1/21/85; | |
78 | * REVISED BY K.C. NG on 8/13/85. | |
79 | * | |
80 | * sin(x*k) - x | |
81 | * RETURN --------------- on [-PI/4,PI/4] , where k=pi/PI, PI is the rounded | |
82 | * x | |
83 | * value of pi in machine precision: | |
84 | * | |
85 | * Decimal: | |
86 | * pi = 3.141592653589793 23846264338327 ..... | |
87 | * 53 bits PI = 3.141592653589793 115997963 ..... , | |
88 | * 56 bits PI = 3.141592653589793 227020265 ..... , | |
89 | * | |
90 | * Hexadecimal: | |
91 | * pi = 3.243F6A8885A308D313198A2E.... | |
92 | * 53 bits PI = 3.243F6A8885A30 = 2 * 1.921FB54442D18 | |
93 | * 56 bits PI = 3.243F6A8885A308 = 4 * .C90FDAA22168C2 | |
94 | * | |
95 | * Method: | |
96 | * 1. Let z=x*x. Create a polynomial approximation to | |
97 | * (sin(k*x)-x)/x = z*(S0 + S1*z^1 + ... + S5*z^5). | |
98 | * Then | |
99 | * sin__S(x*x) = z*(S0 + S1*z^1 + ... + S5*z^5) | |
100 | * | |
101 | * The coefficient S's are obtained by a special Remez algorithm. | |
102 | * | |
103 | * Accuracy: | |
104 | * In the absence of rounding error, the approximation has absolute error | |
105 | * less than 2**(-61.11) for VAX D FORMAT, 2**(-57.45) for IEEE DOUBLE. | |
106 | * | |
107 | * Constants: | |
108 | * The hexadecimal values are the intended ones for the following constants. | |
109 | * The decimal values may be used, provided that the compiler will convert | |
110 | * from decimal to binary accurately enough to produce the hexadecimal values | |
111 | * shown. | |
112 | * | |
113 | */ | |
114 | ||
115 | #if defined(vax)||defined(tahoe) | |
116 | /*S0 = -1.6666666666666646660E-1 , Hex 2^ -2 * -.AAAAAAAAAAAA71 */ | |
117 | /*S1 = 8.3333333333297230413E-3 , Hex 2^ -6 * .8888888888477F */ | |
118 | /*S2 = -1.9841269838362403710E-4 , Hex 2^-12 * -.D00D00CF8A1057 */ | |
119 | /*S3 = 2.7557318019967078930E-6 , Hex 2^-18 * .B8EF1CA326BEDC */ | |
120 | /*S4 = -2.5051841873876551398E-8 , Hex 2^-25 * -.D73195374CE1D3 */ | |
121 | /*S5 = 1.6028995389845827653E-10 , Hex 2^-32 * .B03D9C6D26CCCC */ | |
122 | /*S6 = -6.2723499671769283121E-13 ; Hex 2^-40 * -.B08D0B7561EA82 */ | |
123 | static long S0x[] = { _0x(aaaa,bf2a), _0x(aa71,aaaa)}; | |
124 | static long S1x[] = { _0x(8888,3d08), _0x(477f,8888)}; | |
125 | static long S2x[] = { _0x(0d00,ba50), _0x(1057,cf8a)}; | |
126 | static long S3x[] = { _0x(ef1c,3738), _0x(bedc,a326)}; | |
127 | static long S4x[] = { _0x(3195,b3d7), _0x(e1d3,374c)}; | |
128 | static long S5x[] = { _0x(3d9c,3030), _0x(cccc,6d26)}; | |
129 | static long S6x[] = { _0x(8d0b,ac30), _0x(ea82,7561)}; | |
130 | #define S0 (*(double*)S0x) | |
131 | #define S1 (*(double*)S1x) | |
132 | #define S2 (*(double*)S2x) | |
133 | #define S3 (*(double*)S3x) | |
134 | #define S4 (*(double*)S4x) | |
135 | #define S5 (*(double*)S5x) | |
136 | #define S6 (*(double*)S6x) | |
137 | #else /* IEEE double */ | |
138 | static double | |
139 | S0 = -1.6666666666666463126E-1 , /*Hex 2^ -3 * -1.555555555550C */ | |
140 | S1 = 8.3333333332992771264E-3 , /*Hex 2^ -7 * 1.111111110C461 */ | |
141 | S2 = -1.9841269816180999116E-4 , /*Hex 2^-13 * -1.A01A019746345 */ | |
142 | S3 = 2.7557309793219876880E-6 , /*Hex 2^-19 * 1.71DE3209CDCD9 */ | |
143 | S4 = -2.5050225177523807003E-8 , /*Hex 2^-26 * -1.AE5C0E319A4EF */ | |
144 | S5 = 1.5868926979889205164E-10 ; /*Hex 2^-33 * 1.5CF61DF672B13 */ | |
145 | #endif | |
146 | ||
147 | #if defined(vax)||defined(tahoe) | |
148 | #define sin__S(z) (z*(S0+z*(S1+z*(S2+z*(S3+z*(S4+z*(S5+z*S6))))))) | |
149 | #else /* defined(vax)||defined(tahoe) */ | |
150 | #define sin__S(z) (z*(S0+z*(S1+z*(S2+z*(S3+z*(S4+z*S5)))))) | |
151 | #endif /* defined(vax)||defined(tahoe) */ | |
152 | ||
153 | /* cos__C(x*x) ... re-implemented as a macro | |
154 | * DOUBLE PRECISION (VAX D FORMAT 56 BITS, IEEE DOUBLE 53 BITS) | |
155 | * STATIC KERNEL FUNCTION OF SIN(X), COS(X), AND TAN(X) | |
156 | * CODED IN C BY K.C. NG, 1/21/85; | |
157 | * REVISED BY K.C. NG on 8/13/85. | |
158 | * | |
159 | * x*x | |
160 | * RETURN cos(k*x) - 1 + ----- on [-PI/4,PI/4], where k = pi/PI, | |
161 | * 2 | |
162 | * PI is the rounded value of pi in machine precision : | |
163 | * | |
164 | * Decimal: | |
165 | * pi = 3.141592653589793 23846264338327 ..... | |
166 | * 53 bits PI = 3.141592653589793 115997963 ..... , | |
167 | * 56 bits PI = 3.141592653589793 227020265 ..... , | |
168 | * | |
169 | * Hexadecimal: | |
170 | * pi = 3.243F6A8885A308D313198A2E.... | |
171 | * 53 bits PI = 3.243F6A8885A30 = 2 * 1.921FB54442D18 | |
172 | * 56 bits PI = 3.243F6A8885A308 = 4 * .C90FDAA22168C2 | |
173 | * | |
174 | * | |
175 | * Method: | |
176 | * 1. Let z=x*x. Create a polynomial approximation to | |
177 | * cos(k*x)-1+z/2 = z*z*(C0 + C1*z^1 + ... + C5*z^5) | |
178 | * then | |
179 | * cos__C(z) = z*z*(C0 + C1*z^1 + ... + C5*z^5) | |
180 | * | |
181 | * The coefficient C's are obtained by a special Remez algorithm. | |
182 | * | |
183 | * Accuracy: | |
184 | * In the absence of rounding error, the approximation has absolute error | |
185 | * less than 2**(-64) for VAX D FORMAT, 2**(-58.3) for IEEE DOUBLE. | |
186 | * | |
187 | * | |
188 | * Constants: | |
189 | * The hexadecimal values are the intended ones for the following constants. | |
190 | * The decimal values may be used, provided that the compiler will convert | |
191 | * from decimal to binary accurately enough to produce the hexadecimal values | |
192 | * shown. | |
193 | * | |
194 | */ | |
195 | ||
196 | #if defined(vax)||defined(tahoe) | |
197 | /*C0 = 4.1666666666666504759E-2 , Hex 2^ -4 * .AAAAAAAAAAA9F0 */ | |
198 | /*C1 = -1.3888888888865302059E-3 , Hex 2^ -9 * -.B60B60B60A0CCA */ | |
199 | /*C2 = 2.4801587285601038265E-5 , Hex 2^-15 * .D00D00CDCD098F */ | |
200 | /*C3 = -2.7557313470902390219E-7 , Hex 2^-21 * -.93F27BB593E805 */ | |
201 | /*C4 = 2.0875623401082232009E-9 , Hex 2^-28 * .8F74C8FA1E3FF0 */ | |
202 | /*C5 = -1.1355178117642986178E-11 ; Hex 2^-36 * -.C7C32D0A5C5A63 */ | |
203 | static long C0x[] = { _0x(aaaa,3e2a), _0x(a9f0,aaaa)}; | |
204 | static long C1x[] = { _0x(0b60,bbb6), _0x(0cca,b60a)}; | |
205 | static long C2x[] = { _0x(0d00,38d0), _0x(098f,cdcd)}; | |
206 | static long C3x[] = { _0x(f27b,b593), _0x(e805,b593)}; | |
207 | static long C4x[] = { _0x(74c8,320f), _0x(3ff0,fa1e)}; | |
208 | static long C5x[] = { _0x(c32d,ae47), _0x(5a63,0a5c)}; | |
209 | #define C0 (*(double*)C0x) | |
210 | #define C1 (*(double*)C1x) | |
211 | #define C2 (*(double*)C2x) | |
212 | #define C3 (*(double*)C3x) | |
213 | #define C4 (*(double*)C4x) | |
214 | #define C5 (*(double*)C5x) | |
215 | #else /* defined(vax)||defined(tahoe) */ | |
216 | static double | |
217 | C0 = 4.1666666666666504759E-2 , /*Hex 2^ -5 * 1.555555555553E */ | |
218 | C1 = -1.3888888888865301516E-3 , /*Hex 2^-10 * -1.6C16C16C14199 */ | |
219 | C2 = 2.4801587269650015769E-5 , /*Hex 2^-16 * 1.A01A01971CAEB */ | |
220 | C3 = -2.7557304623183959811E-7 , /*Hex 2^-22 * -1.27E4F1314AD1A */ | |
221 | C4 = 2.0873958177697780076E-9 , /*Hex 2^-29 * 1.1EE3B60DDDC8C */ | |
222 | C5 = -1.1250289076471311557E-11 ; /*Hex 2^-37 * -1.8BD5986B2A52E */ | |
223 | #endif /* defined(vax)||defined(tahoe) */ | |
224 | ||
225 | #define cos__C(z) (z*z*(C0+z*(C1+z*(C2+z*(C3+z*(C4+z*C5)))))) | |
226 | ||
227 | extern int finite(); | |
228 | extern double copysign(),drem(); |