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 this notice is preserved and that due credit is given 7 * to the University of California at Berkeley. The name of the University 8 * may not be used to endorse or promote products derived from this 9 * software without specific prior written permission. This software 10 * is provided ``as is'' without express or implied warranty. 11 * 12 * All recipients should regard themselves as participants in an ongoing 13 * research project and hence should feel obligated to report their 14 * experiences (good or bad) with these elementary function codes, using 15 * the sendbug(8) program, to the authors. 16 * 17 * @(#)trig.h 5.2 (Berkeley) 04/29/88 18 */ 19 20 #if defined(vax)||defined(tahoe) 21 #ifdef vax 22 #define _0x(A,B) 0x/**/A/**/B 23 #else /* vax */ 24 #define _0x(A,B) 0x/**/B/**/A 25 #endif /* vax */ 26 /*thresh = 2.6117239648121182150E-1 , Hex 2^ -1 * .85B8636B026EA0 */ 27 /*PIo4 = 7.8539816339744830676E-1 , Hex 2^ 0 * .C90FDAA22168C2 */ 28 /*PIo2 = 1.5707963267948966135E0 , Hex 2^ 1 * .C90FDAA22168C2 */ 29 /*PI3o4 = 2.3561944901923449203E0 , Hex 2^ 2 * .96CBE3F9990E92 */ 30 /*PI = 3.1415926535897932270E0 , Hex 2^ 2 * .C90FDAA22168C2 */ 31 /*PI2 = 6.2831853071795864540E0 ; Hex 2^ 3 * .C90FDAA22168C2 */ 32 static long threshx[] = { _0x(b863,3f85), _0x(6ea0,6b02)}; 33 static long PIo4x[] = { _0x(0fda,4049), _0x(68c2,a221)}; 34 static long PIo2x[] = { _0x(0fda,40c9), _0x(68c2,a221)}; 35 static long PI3o4x[] = { _0x(cbe3,4116), _0x(0e92,f999)}; 36 static long PIx[] = { _0x(0fda,4149), _0x(68c2,a221)}; 37 static long PI2x[] = { _0x(0fda,41c9), _0x(68c2,a221)}; 38 #define thresh (*(double*)threshx) 39 #define PIo4 (*(double*)PIo4x) 40 #define PIo2 (*(double*)PIo2x) 41 #define PI3o4 (*(double*)PI3o4x) 42 #define PI (*(double*)PIx) 43 #define PI2 (*(double*)PI2x) 44 #else /* defined(vax)||defined(tahoe) */ 45 static double 46 thresh = 2.6117239648121182150E-1 , /*Hex 2^ -2 * 1.0B70C6D604DD4 */ 47 PIo4 = 7.8539816339744827900E-1 , /*Hex 2^ -1 * 1.921FB54442D18 */ 48 PIo2 = 1.5707963267948965580E0 , /*Hex 2^ 0 * 1.921FB54442D18 */ 49 PI3o4 = 2.3561944901923448370E0 , /*Hex 2^ 1 * 1.2D97C7F3321D2 */ 50 PI = 3.1415926535897931160E0 , /*Hex 2^ 1 * 1.921FB54442D18 */ 51 PI2 = 6.2831853071795862320E0 ; /*Hex 2^ 2 * 1.921FB54442D18 */ 52 #ifdef national 53 static long fmaxx[] = { 0xffffffff, 0x7fefffff}; 54 #define fmax (*(double*)fmaxx) 55 #endif /* national */ 56 #endif /* defined(vax)||defined(tahoe) */ 57 static 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 #if defined(vax)||defined(tahoe) 111 /*S0 = -1.6666666666666646660E-1 , Hex 2^ -2 * -.AAAAAAAAAAAA71 */ 112 /*S1 = 8.3333333333297230413E-3 , Hex 2^ -6 * .8888888888477F */ 113 /*S2 = -1.9841269838362403710E-4 , Hex 2^-12 * -.D00D00CF8A1057 */ 114 /*S3 = 2.7557318019967078930E-6 , Hex 2^-18 * .B8EF1CA326BEDC */ 115 /*S4 = -2.5051841873876551398E-8 , Hex 2^-25 * -.D73195374CE1D3 */ 116 /*S5 = 1.6028995389845827653E-10 , Hex 2^-32 * .B03D9C6D26CCCC */ 117 /*S6 = -6.2723499671769283121E-13 ; Hex 2^-40 * -.B08D0B7561EA82 */ 118 static long S0x[] = { _0x(aaaa,bf2a), _0x(aa71,aaaa)}; 119 static long S1x[] = { _0x(8888,3d08), _0x(477f,8888)}; 120 static long S2x[] = { _0x(0d00,ba50), _0x(1057,cf8a)}; 121 static long S3x[] = { _0x(ef1c,3738), _0x(bedc,a326)}; 122 static long S4x[] = { _0x(3195,b3d7), _0x(e1d3,374c)}; 123 static long S5x[] = { _0x(3d9c,3030), _0x(cccc,6d26)}; 124 static long S6x[] = { _0x(8d0b,ac30), _0x(ea82,7561)}; 125 #define S0 (*(double*)S0x) 126 #define S1 (*(double*)S1x) 127 #define S2 (*(double*)S2x) 128 #define S3 (*(double*)S3x) 129 #define S4 (*(double*)S4x) 130 #define S5 (*(double*)S5x) 131 #define S6 (*(double*)S6x) 132 #else /* IEEE double */ 133 static double 134 S0 = -1.6666666666666463126E-1 , /*Hex 2^ -3 * -1.555555555550C */ 135 S1 = 8.3333333332992771264E-3 , /*Hex 2^ -7 * 1.111111110C461 */ 136 S2 = -1.9841269816180999116E-4 , /*Hex 2^-13 * -1.A01A019746345 */ 137 S3 = 2.7557309793219876880E-6 , /*Hex 2^-19 * 1.71DE3209CDCD9 */ 138 S4 = -2.5050225177523807003E-8 , /*Hex 2^-26 * -1.AE5C0E319A4EF */ 139 S5 = 1.5868926979889205164E-10 ; /*Hex 2^-33 * 1.5CF61DF672B13 */ 140 #endif 141 142 #if defined(vax)||defined(tahoe) 143 #define sin__S(z) (z*(S0+z*(S1+z*(S2+z*(S3+z*(S4+z*(S5+z*S6))))))) 144 #else /* defined(vax)||defined(tahoe) */ 145 #define sin__S(z) (z*(S0+z*(S1+z*(S2+z*(S3+z*(S4+z*S5)))))) 146 #endif /* defined(vax)||defined(tahoe) */ 147 148 /* cos__C(x*x) ... re-implemented as a macro 149 * DOUBLE PRECISION (VAX D FORMAT 56 BITS, IEEE DOUBLE 53 BITS) 150 * STATIC KERNEL FUNCTION OF SIN(X), COS(X), AND TAN(X) 151 * CODED IN C BY K.C. NG, 1/21/85; 152 * REVISED BY K.C. NG on 8/13/85. 153 * 154 * x*x 155 * RETURN cos(k*x) - 1 + ----- on [-PI/4,PI/4], where k = pi/PI, 156 * 2 157 * PI is the rounded value of pi in machine precision : 158 * 159 * Decimal: 160 * pi = 3.141592653589793 23846264338327 ..... 161 * 53 bits PI = 3.141592653589793 115997963 ..... , 162 * 56 bits PI = 3.141592653589793 227020265 ..... , 163 * 164 * Hexadecimal: 165 * pi = 3.243F6A8885A308D313198A2E.... 166 * 53 bits PI = 3.243F6A8885A30 = 2 * 1.921FB54442D18 167 * 56 bits PI = 3.243F6A8885A308 = 4 * .C90FDAA22168C2 168 * 169 * 170 * Method: 171 * 1. Let z=x*x. Create a polynomial approximation to 172 * cos(k*x)-1+z/2 = z*z*(C0 + C1*z^1 + ... + C5*z^5) 173 * then 174 * cos__C(z) = z*z*(C0 + C1*z^1 + ... + C5*z^5) 175 * 176 * The coefficient C's are obtained by a special Remez algorithm. 177 * 178 * Accuracy: 179 * In the absence of rounding error, the approximation has absolute error 180 * less than 2**(-64) for VAX D FORMAT, 2**(-58.3) for IEEE DOUBLE. 181 * 182 * 183 * Constants: 184 * The hexadecimal values are the intended ones for the following constants. 185 * The decimal values may be used, provided that the compiler will convert 186 * from decimal to binary accurately enough to produce the hexadecimal values 187 * shown. 188 * 189 */ 190 191 #if defined(vax)||defined(tahoe) 192 /*C0 = 4.1666666666666504759E-2 , Hex 2^ -4 * .AAAAAAAAAAA9F0 */ 193 /*C1 = -1.3888888888865302059E-3 , Hex 2^ -9 * -.B60B60B60A0CCA */ 194 /*C2 = 2.4801587285601038265E-5 , Hex 2^-15 * .D00D00CDCD098F */ 195 /*C3 = -2.7557313470902390219E-7 , Hex 2^-21 * -.93F27BB593E805 */ 196 /*C4 = 2.0875623401082232009E-9 , Hex 2^-28 * .8F74C8FA1E3FF0 */ 197 /*C5 = -1.1355178117642986178E-11 ; Hex 2^-36 * -.C7C32D0A5C5A63 */ 198 static long C0x[] = { _0x(aaaa,3e2a), _0x(a9f0,aaaa)}; 199 static long C1x[] = { _0x(0b60,bbb6), _0x(0cca,b60a)}; 200 static long C2x[] = { _0x(0d00,38d0), _0x(098f,cdcd)}; 201 static long C3x[] = { _0x(f27b,b593), _0x(e805,b593)}; 202 static long C4x[] = { _0x(74c8,320f), _0x(3ff0,fa1e)}; 203 static long C5x[] = { _0x(c32d,ae47), _0x(5a63,0a5c)}; 204 #define C0 (*(double*)C0x) 205 #define C1 (*(double*)C1x) 206 #define C2 (*(double*)C2x) 207 #define C3 (*(double*)C3x) 208 #define C4 (*(double*)C4x) 209 #define C5 (*(double*)C5x) 210 #else /* defined(vax)||defined(tahoe) */ 211 static double 212 C0 = 4.1666666666666504759E-2 , /*Hex 2^ -5 * 1.555555555553E */ 213 C1 = -1.3888888888865301516E-3 , /*Hex 2^-10 * -1.6C16C16C14199 */ 214 C2 = 2.4801587269650015769E-5 , /*Hex 2^-16 * 1.A01A01971CAEB */ 215 C3 = -2.7557304623183959811E-7 , /*Hex 2^-22 * -1.27E4F1314AD1A */ 216 C4 = 2.0873958177697780076E-9 , /*Hex 2^-29 * 1.1EE3B60DDDC8C */ 217 C5 = -1.1250289076471311557E-11 ; /*Hex 2^-37 * -1.8BD5986B2A52E */ 218 #endif /* defined(vax)||defined(tahoe) */ 219 220 #define cos__C(z) (z*z*(C0+z*(C1+z*(C2+z*(C3+z*(C4+z*C5)))))) 221 222 extern int finite(); 223 extern double copysign(),drem(); 224