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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 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.
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: *
22: * @(#)trig.h 5.3 (Berkeley) 6/30/88
23: */
24:
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();
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