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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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