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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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