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1.1 root 1: /*
2: * Copyright (c) 1987 Regents of the University of California.
3: * All rights reserved.
4: *
5: * Redistribution and use in source and binary forms are permitted
6: * provided that: (1) source distributions retain this entire copyright
7: * notice and comment, and (2) distributions including binaries display
8: * the following acknowledgement: ``This product includes software
9: * developed by the University of California, Berkeley and its contributors''
10: * in the documentation or other materials provided with the distribution
11: * and in all advertising materials mentioning features or use of this
12: * software. Neither the name of the University nor the names of its
13: * contributors may be used to endorse or promote products derived
14: * from this software without specific prior written permission.
15: * THIS SOFTWARE IS PROVIDED ``AS IS'' AND WITHOUT ANY EXPRESS OR
16: * IMPLIED WARRANTIES, INCLUDING, WITHOUT LIMITATION, THE IMPLIED
17: * WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE.
18: *
19: * All recipients should regard themselves as participants in an ongoing
20: * research project and hence should feel obligated to report their
21: * experiences (good or bad) with these elementary function codes, using
22: * the sendbug(8) program, to the authors.
23: *
24: * @(#)trig.h 5.5 (Berkeley) 6/1/90
25: */
26:
27: #include "mathimpl.h"
28:
29: vc(thresh, 2.6117239648121182150E-1 ,b863,3f85,6ea0,6b02, -1, .85B8636B026EA0)
30: vc(PIo4, 7.8539816339744830676E-1 ,0fda,4049,68c2,a221, 0, .C90FDAA22168C2)
31: vc(PIo2, 1.5707963267948966135E0 ,0fda,40c9,68c2,a221, 1, .C90FDAA22168C2)
32: vc(PI3o4, 2.3561944901923449203E0 ,cbe3,4116,0e92,f999, 2, .96CBE3F9990E92)
33: vc(PI, 3.1415926535897932270E0 ,0fda,4149,68c2,a221, 2, .C90FDAA22168C2)
34: vc(PI2, 6.2831853071795864540E0 ,0fda,41c9,68c2,a221, 3, .C90FDAA22168C2)
35:
36: ic(thresh, 2.6117239648121182150E-1 , -2, 1.0B70C6D604DD4)
37: ic(PIo4, 7.8539816339744827900E-1 , -1, 1.921FB54442D18)
38: ic(PIo2, 1.5707963267948965580E0 , 0, 1.921FB54442D18)
39: ic(PI3o4, 2.3561944901923448370E0 , 1, 1.2D97C7F3321D2)
40: ic(PI, 3.1415926535897931160E0 , 1, 1.921FB54442D18)
41: ic(PI2, 6.2831853071795862320E0 , 2, 1.921FB54442D18)
42:
43: #ifdef vccast
44: #define thresh vccast(thresh)
45: #define PIo4 vccast(PIo4)
46: #define PIo2 vccast(PIo2)
47: #define PI3o4 vccast(PI3o4)
48: #define PI vccast(PI)
49: #define PI2 vccast(PI2)
50: #endif
51:
52: #ifdef national
53: static long fmaxx[] = { 0xffffffff, 0x7fefffff};
54: #define fmax (*(double*)fmaxx)
55: #endif /* national */
56:
57: static const double
58: zero = 0,
59: one = 1,
60: negone = -1,
61: half = 1.0/2.0,
62: small = 1E-10, /* 1+small**2 == 1; better values for small:
63: * small = 1.5E-9 for VAX D
64: * = 1.2E-8 for IEEE Double
65: * = 2.8E-10 for IEEE Extended
66: */
67: big = 1E20; /* big := 1/(small**2) */
68:
69: /* sin__S(x*x) ... re-implemented as a macro
70: * DOUBLE PRECISION (VAX D format 56 bits, IEEE DOUBLE 53 BITS)
71: * STATIC KERNEL FUNCTION OF SIN(X), COS(X), AND TAN(X)
72: * CODED IN C BY K.C. NG, 1/21/85;
73: * REVISED BY K.C. NG on 8/13/85.
74: *
75: * sin(x*k) - x
76: * RETURN --------------- on [-PI/4,PI/4] , where k=pi/PI, PI is the rounded
77: * x
78: * value of pi in machine precision:
79: *
80: * Decimal:
81: * pi = 3.141592653589793 23846264338327 .....
82: * 53 bits PI = 3.141592653589793 115997963 ..... ,
83: * 56 bits PI = 3.141592653589793 227020265 ..... ,
84: *
85: * Hexadecimal:
86: * pi = 3.243F6A8885A308D313198A2E....
87: * 53 bits PI = 3.243F6A8885A30 = 2 * 1.921FB54442D18
88: * 56 bits PI = 3.243F6A8885A308 = 4 * .C90FDAA22168C2
89: *
90: * Method:
91: * 1. Let z=x*x. Create a polynomial approximation to
92: * (sin(k*x)-x)/x = z*(S0 + S1*z^1 + ... + S5*z^5).
93: * Then
94: * sin__S(x*x) = z*(S0 + S1*z^1 + ... + S5*z^5)
95: *
96: * The coefficient S's are obtained by a special Remez algorithm.
97: *
98: * Accuracy:
99: * In the absence of rounding error, the approximation has absolute error
100: * less than 2**(-61.11) for VAX D FORMAT, 2**(-57.45) for IEEE DOUBLE.
101: *
102: * Constants:
103: * The hexadecimal values are the intended ones for the following constants.
104: * The decimal values may be used, provided that the compiler will convert
105: * from decimal to binary accurately enough to produce the hexadecimal values
106: * shown.
107: *
108: */
109:
110: vc(S0, -1.6666666666666646660E-1 ,aaaa,bf2a,aa71,aaaa, -2, -.AAAAAAAAAAAA71)
111: vc(S1, 8.3333333333297230413E-3 ,8888,3d08,477f,8888, -6, .8888888888477F)
112: vc(S2, -1.9841269838362403710E-4 ,0d00,ba50,1057,cf8a, -12, -.D00D00CF8A1057)
113: vc(S3, 2.7557318019967078930E-6 ,ef1c,3738,bedc,a326, -18, .B8EF1CA326BEDC)
114: vc(S4, -2.5051841873876551398E-8 ,3195,b3d7,e1d3,374c, -25, -.D73195374CE1D3)
115: vc(S5, 1.6028995389845827653E-10 ,3d9c,3030,cccc,6d26, -32, .B03D9C6D26CCCC)
116: vc(S6, -6.2723499671769283121E-13 ,8d0b,ac30,ea82,7561, -40, -.B08D0B7561EA82)
117:
118: ic(S0, -1.6666666666666463126E-1 , -3, -1.555555555550C)
119: ic(S1, 8.3333333332992771264E-3 , -7, 1.111111110C461)
120: ic(S2, -1.9841269816180999116E-4 , -13, -1.A01A019746345)
121: ic(S3, 2.7557309793219876880E-6 , -19, 1.71DE3209CDCD9)
122: ic(S4, -2.5050225177523807003E-8 , -26, -1.AE5C0E319A4EF)
123: ic(S5, 1.5868926979889205164E-10 , -33, 1.5CF61DF672B13)
124:
125: #ifdef vccast
126: #define S0 vccast(S0)
127: #define S1 vccast(S1)
128: #define S2 vccast(S2)
129: #define S3 vccast(S3)
130: #define S4 vccast(S4)
131: #define S5 vccast(S5)
132: #define S6 vccast(S6)
133: #endif
134:
135: #if defined(vax)||defined(tahoe)
136: # define sin__S(z) (z*(S0+z*(S1+z*(S2+z*(S3+z*(S4+z*(S5+z*S6)))))))
137: #else /* defined(vax)||defined(tahoe) */
138: # define sin__S(z) (z*(S0+z*(S1+z*(S2+z*(S3+z*(S4+z*S5))))))
139: #endif /* defined(vax)||defined(tahoe) */
140:
141: /* cos__C(x*x) ... re-implemented as a macro
142: * DOUBLE PRECISION (VAX D FORMAT 56 BITS, IEEE DOUBLE 53 BITS)
143: * STATIC KERNEL FUNCTION OF SIN(X), COS(X), AND TAN(X)
144: * CODED IN C BY K.C. NG, 1/21/85;
145: * REVISED BY K.C. NG on 8/13/85.
146: *
147: * x*x
148: * RETURN cos(k*x) - 1 + ----- on [-PI/4,PI/4], where k = pi/PI,
149: * 2
150: * PI is the rounded value of pi in machine precision :
151: *
152: * Decimal:
153: * pi = 3.141592653589793 23846264338327 .....
154: * 53 bits PI = 3.141592653589793 115997963 ..... ,
155: * 56 bits PI = 3.141592653589793 227020265 ..... ,
156: *
157: * Hexadecimal:
158: * pi = 3.243F6A8885A308D313198A2E....
159: * 53 bits PI = 3.243F6A8885A30 = 2 * 1.921FB54442D18
160: * 56 bits PI = 3.243F6A8885A308 = 4 * .C90FDAA22168C2
161: *
162: *
163: * Method:
164: * 1. Let z=x*x. Create a polynomial approximation to
165: * cos(k*x)-1+z/2 = z*z*(C0 + C1*z^1 + ... + C5*z^5)
166: * then
167: * cos__C(z) = z*z*(C0 + C1*z^1 + ... + C5*z^5)
168: *
169: * The coefficient C's are obtained by a special Remez algorithm.
170: *
171: * Accuracy:
172: * In the absence of rounding error, the approximation has absolute error
173: * less than 2**(-64) for VAX D FORMAT, 2**(-58.3) for IEEE DOUBLE.
174: *
175: *
176: * Constants:
177: * The hexadecimal values are the intended ones for the following constants.
178: * The decimal values may be used, provided that the compiler will convert
179: * from decimal to binary accurately enough to produce the hexadecimal values
180: * shown.
181: */
182:
183: vc(C0, 4.1666666666666504759E-2 ,aaaa,3e2a,a9f0,aaaa, -4, .AAAAAAAAAAA9F0)
184: vc(C1, -1.3888888888865302059E-3 ,0b60,bbb6,0cca,b60a, -9, -.B60B60B60A0CCA)
185: vc(C2, 2.4801587285601038265E-5 ,0d00,38d0,098f,cdcd, -15, .D00D00CDCD098F)
186: vc(C3, -2.7557313470902390219E-7 ,f27b,b593,e805,b593, -21, -.93F27BB593E805)
187: vc(C4, 2.0875623401082232009E-9 ,74c8,320f,3ff0,fa1e, -28, .8F74C8FA1E3FF0)
188: vc(C5, -1.1355178117642986178E-11 ,c32d,ae47,5a63,0a5c, -36, -.C7C32D0A5C5A63)
189:
190: ic(C0, 4.1666666666666504759E-2 , -5, 1.555555555553E)
191: ic(C1, -1.3888888888865301516E-3 , -10, -1.6C16C16C14199)
192: ic(C2, 2.4801587269650015769E-5 , -16, 1.A01A01971CAEB)
193: ic(C3, -2.7557304623183959811E-7 , -22, -1.27E4F1314AD1A)
194: ic(C4, 2.0873958177697780076E-9 , -29, 1.1EE3B60DDDC8C)
195: ic(C5, -1.1250289076471311557E-11 , -37, -1.8BD5986B2A52E)
196:
197: #ifdef vccast
198: #define C0 vccast(C0)
199: #define C1 vccast(C1)
200: #define C2 vccast(C2)
201: #define C3 vccast(C3)
202: #define C4 vccast(C4)
203: #define C5 vccast(C5)
204: #endif
205:
206: #define cos__C(z) (z*z*(C0+z*(C1+z*(C2+z*(C3+z*(C4+z*C5))))))
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