projects / unix-history / blame
| Commit | Line | Data |
|---|---|---|
| 38cc45a4 ZAL |
1 | /* |
| 2 | * Copyright (c) 1987 Regents of the University of California. | |
| 3 | * | |
| 4 | * Use and reproduction of this software are granted in accordance with | |
| 5 | * the terms and conditions specified in the Berkeley Software License | |
| 6 | * Agreement (in particular, this entails acknowledgement of the programs' | |
| 7 | * source, and inclusion of this notice) with the additional understanding | |
| 8 | * that all recipients should regard themselves as participants in an | |
| 9 | * ongoing research project and hence should feel obligated to report | |
| 10 | * their experiences (good or bad) with these elementary function codes, | |
| 11 | * using "sendbug 4bsd-bugs@BERKELEY", to the authors. | |
| 12 | */ | |
| 2893dfdc | 13 | /* @(#)trig.h 5.1 5.1 (ucb.elefunt) %G% */ |
| 38cc45a4 ZAL |
14 | #if defined(vax)||defined(tahoe) |
| 15 | #ifdef vax | |
| 16 | #define _0x(A,B) 0x/**/A/**/B | |
| 17 | #else /* vax */ | |
| 18 | #define _0x(A,B) 0x/**/B/**/A | |
| 19 | #endif /* vax */ | |
| 20 | /*thresh = 2.6117239648121182150E-1 , Hex 2^ -1 * .85B8636B026EA0 */ | |
| 21 | /*PIo4 = 7.8539816339744830676E-1 , Hex 2^ 0 * .C90FDAA22168C2 */ | |
| 22 | /*PIo2 = 1.5707963267948966135E0 , Hex 2^ 1 * .C90FDAA22168C2 */ | |
| 23 | /*PI3o4 = 2.3561944901923449203E0 , Hex 2^ 2 * .96CBE3F9990E92 */ | |
| 24 | /*PI = 3.1415926535897932270E0 , Hex 2^ 2 * .C90FDAA22168C2 */ | |
| 25 | /*PI2 = 6.2831853071795864540E0 ; Hex 2^ 3 * .C90FDAA22168C2 */ | |
| 26 | static long threshx[] = { _0x(b863,3f85), _0x(6ea0,6b02)}; | |
| 27 | static long PIo4x[] = { _0x(0fda,4049), _0x(68c2,a221)}; | |
| 28 | static long PIo2x[] = { _0x(0fda,40c9), _0x(68c2,a221)}; | |
| 29 | static long PI3o4x[] = { _0x(cbe3,4116), _0x(0e92,f999)}; | |
| 30 | static long PIx[] = { _0x(0fda,4149), _0x(68c2,a221)}; | |
| 31 | static long PI2x[] = { _0x(0fda,41c9), _0x(68c2,a221)}; | |
| 32 | #define thresh (*(double*)threshx) | |
| 33 | #define PIo4 (*(double*)PIo4x) | |
| 34 | #define PIo2 (*(double*)PIo2x) | |
| 35 | #define PI3o4 (*(double*)PI3o4x) | |
| 36 | #define PI (*(double*)PIx) | |
| 37 | #define PI2 (*(double*)PI2x) | |
| 38 | #else /* defined(vax)||defined(tahoe) */ | |
| 39 | static double | |
| 40 | thresh = 2.6117239648121182150E-1 , /*Hex 2^ -2 * 1.0B70C6D604DD4 */ | |
| 41 | PIo4 = 7.8539816339744827900E-1 , /*Hex 2^ -1 * 1.921FB54442D18 */ | |
| 42 | PIo2 = 1.5707963267948965580E0 , /*Hex 2^ 0 * 1.921FB54442D18 */ | |
| 43 | PI3o4 = 2.3561944901923448370E0 , /*Hex 2^ 1 * 1.2D97C7F3321D2 */ | |
| 44 | PI = 3.1415926535897931160E0 , /*Hex 2^ 1 * 1.921FB54442D18 */ | |
| 45 | PI2 = 6.2831853071795862320E0 ; /*Hex 2^ 2 * 1.921FB54442D18 */ | |
| 46 | #ifdef national | |
| 47 | static long fmaxx[] = { 0xffffffff, 0x7fefffff}; | |
| 48 | #define fmax (*(double*)fmaxx) | |
| 49 | #endif /* national */ | |
| 50 | #endif /* defined(vax)||defined(tahoe) */ | |
| 51 | static double | |
| 52 | zero = 0, | |
| 53 | one = 1, | |
| 54 | negone = -1, | |
| 55 | half = 1.0/2.0, | |
| 56 | small = 1E-10, /* 1+small**2 == 1; better values for small: | |
| 57 | * small = 1.5E-9 for VAX D | |
| 58 | * = 1.2E-8 for IEEE Double | |
| 59 | * = 2.8E-10 for IEEE Extended | |
| 60 | */ | |
| 61 | big = 1E20; /* big := 1/(small**2) */ | |
| 62 | ||
| 63 | /* sin__S(x*x) ... re-implemented as a macro | |
| 64 | * DOUBLE PRECISION (VAX D format 56 bits, IEEE DOUBLE 53 BITS) | |
| 65 | * STATIC KERNEL FUNCTION OF SIN(X), COS(X), AND TAN(X) | |
| 66 | * CODED IN C BY K.C. NG, 1/21/85; | |
| 67 | * REVISED BY K.C. NG on 8/13/85. | |
| 68 | * | |
| 69 | * sin(x*k) - x | |
| 70 | * RETURN --------------- on [-PI/4,PI/4] , where k=pi/PI, PI is the rounded | |
| 71 | * x | |
| 72 | * value of pi in machine precision: | |
| 73 | * | |
| 74 | * Decimal: | |
| 75 | * pi = 3.141592653589793 23846264338327 ..... | |
| 76 | * 53 bits PI = 3.141592653589793 115997963 ..... , | |
| 77 | * 56 bits PI = 3.141592653589793 227020265 ..... , | |
| 78 | * | |
| 79 | * Hexadecimal: | |
| 80 | * pi = 3.243F6A8885A308D313198A2E.... | |
| 81 | * 53 bits PI = 3.243F6A8885A30 = 2 * 1.921FB54442D18 | |
| 82 | * 56 bits PI = 3.243F6A8885A308 = 4 * .C90FDAA22168C2 | |
| 83 | * | |
| 84 | * Method: | |
| 85 | * 1. Let z=x*x. Create a polynomial approximation to | |
| 86 | * (sin(k*x)-x)/x = z*(S0 + S1*z^1 + ... + S5*z^5). | |
| 87 | * Then | |
| 88 | * sin__S(x*x) = z*(S0 + S1*z^1 + ... + S5*z^5) | |
| 89 | * | |
| 90 | * The coefficient S's are obtained by a special Remez algorithm. | |
| 91 | * | |
| 92 | * Accuracy: | |
| 93 | * In the absence of rounding error, the approximation has absolute error | |
| 94 | * less than 2**(-61.11) for VAX D FORMAT, 2**(-57.45) for IEEE DOUBLE. | |
| 95 | * | |
| 96 | * Constants: | |
| 97 | * The hexadecimal values are the intended ones for the following constants. | |
| 98 | * The decimal values may be used, provided that the compiler will convert | |
| 99 | * from decimal to binary accurately enough to produce the hexadecimal values | |
| 100 | * shown. | |
| 101 | * | |
| 102 | */ | |
| 103 | ||
| 104 | #if defined(vax)||defined(tahoe) | |
| 105 | /*S0 = -1.6666666666666646660E-1 , Hex 2^ -2 * -.AAAAAAAAAAAA71 */ | |
| 106 | /*S1 = 8.3333333333297230413E-3 , Hex 2^ -6 * .8888888888477F */ | |
| 107 | /*S2 = -1.9841269838362403710E-4 , Hex 2^-12 * -.D00D00CF8A1057 */ | |
| 108 | /*S3 = 2.7557318019967078930E-6 , Hex 2^-18 * .B8EF1CA326BEDC */ | |
| 109 | /*S4 = -2.5051841873876551398E-8 , Hex 2^-25 * -.D73195374CE1D3 */ | |
| 110 | /*S5 = 1.6028995389845827653E-10 , Hex 2^-32 * .B03D9C6D26CCCC */ | |
| 111 | /*S6 = -6.2723499671769283121E-13 ; Hex 2^-40 * -.B08D0B7561EA82 */ | |
| 112 | static long S0x[] = { _0x(aaaa,bf2a), _0x(aa71,aaaa)}; | |
| 113 | static long S1x[] = { _0x(8888,3d08), _0x(477f,8888)}; | |
| 114 | static long S2x[] = { _0x(0d00,ba50), _0x(1057,cf8a)}; | |
| 115 | static long S3x[] = { _0x(ef1c,3738), _0x(bedc,a326)}; | |
| 116 | static long S4x[] = { _0x(3195,b3d7), _0x(e1d3,374c)}; | |
| 117 | static long S5x[] = { _0x(3d9c,3030), _0x(cccc,6d26)}; | |
| 118 | static long S6x[] = { _0x(8d0b,ac30), _0x(ea82,7561)}; | |
| 119 | #define S0 (*(double*)S0x) | |
| 120 | #define S1 (*(double*)S1x) | |
| 121 | #define S2 (*(double*)S2x) | |
| 122 | #define S3 (*(double*)S3x) | |
| 123 | #define S4 (*(double*)S4x) | |
| 124 | #define S5 (*(double*)S5x) | |
| 125 | #define S6 (*(double*)S6x) | |
| 126 | #else /* IEEE double */ | |
| 127 | static double | |
| 128 | S0 = -1.6666666666666463126E-1 , /*Hex 2^ -3 * -1.555555555550C */ | |
| 129 | S1 = 8.3333333332992771264E-3 , /*Hex 2^ -7 * 1.111111110C461 */ | |
| 130 | S2 = -1.9841269816180999116E-4 , /*Hex 2^-13 * -1.A01A019746345 */ | |
| 131 | S3 = 2.7557309793219876880E-6 , /*Hex 2^-19 * 1.71DE3209CDCD9 */ | |
| 132 | S4 = -2.5050225177523807003E-8 , /*Hex 2^-26 * -1.AE5C0E319A4EF */ | |
| 133 | S5 = 1.5868926979889205164E-10 ; /*Hex 2^-33 * 1.5CF61DF672B13 */ | |
| 134 | #endif | |
| 135 | ||
| 136 | #if defined(vax)||defined(tahoe) | |
| 137 | #define sin__S(z) (z*(S0+z*(S1+z*(S2+z*(S3+z*(S4+z*(S5+z*S6))))))) | |
| 138 | #else /* defined(vax)||defined(tahoe) */ | |
| 139 | #define sin__S(z) (z*(S0+z*(S1+z*(S2+z*(S3+z*(S4+z*S5)))))) | |
| 140 | #endif /* defined(vax)||defined(tahoe) */ | |
| 141 | ||
| 142 | /* cos__C(x*x) ... re-implemented as a macro | |
| 143 | * DOUBLE PRECISION (VAX D FORMAT 56 BITS, IEEE DOUBLE 53 BITS) | |
| 144 | * STATIC KERNEL FUNCTION OF SIN(X), COS(X), AND TAN(X) | |
| 145 | * CODED IN C BY K.C. NG, 1/21/85; | |
| 146 | * REVISED BY K.C. NG on 8/13/85. | |
| 147 | * | |
| 148 | * x*x | |
| 149 | * RETURN cos(k*x) - 1 + ----- on [-PI/4,PI/4], where k = pi/PI, | |
| 150 | * 2 | |
| 151 | * PI is the rounded value of pi in machine precision : | |
| 152 | * | |
| 153 | * Decimal: | |
| 154 | * pi = 3.141592653589793 23846264338327 ..... | |
| 155 | * 53 bits PI = 3.141592653589793 115997963 ..... , | |
| 156 | * 56 bits PI = 3.141592653589793 227020265 ..... , | |
| 157 | * | |
| 158 | * Hexadecimal: | |
| 159 | * pi = 3.243F6A8885A308D313198A2E.... | |
| 160 | * 53 bits PI = 3.243F6A8885A30 = 2 * 1.921FB54442D18 | |
| 161 | * 56 bits PI = 3.243F6A8885A308 = 4 * .C90FDAA22168C2 | |
| 162 | * | |
| 163 | * | |
| 164 | * Method: | |
| 165 | * 1. Let z=x*x. Create a polynomial approximation to | |
| 166 | * cos(k*x)-1+z/2 = z*z*(C0 + C1*z^1 + ... + C5*z^5) | |
| 167 | * then | |
| 168 | * cos__C(z) = z*z*(C0 + C1*z^1 + ... + C5*z^5) | |
| 169 | * | |
| 170 | * The coefficient C's are obtained by a special Remez algorithm. | |
| 171 | * | |
| 172 | * Accuracy: | |
| 173 | * In the absence of rounding error, the approximation has absolute error | |
| 174 | * less than 2**(-64) for VAX D FORMAT, 2**(-58.3) for IEEE DOUBLE. | |
| 175 | * | |
| 176 | * | |
| 177 | * Constants: | |
| 178 | * The hexadecimal values are the intended ones for the following constants. | |
| 179 | * The decimal values may be used, provided that the compiler will convert | |
| 180 | * from decimal to binary accurately enough to produce the hexadecimal values | |
| 181 | * shown. | |
| 182 | * | |
| 183 | */ | |
| 184 | ||
| 185 | #if defined(vax)||defined(tahoe) | |
| 186 | /*C0 = 4.1666666666666504759E-2 , Hex 2^ -4 * .AAAAAAAAAAA9F0 */ | |
| 187 | /*C1 = -1.3888888888865302059E-3 , Hex 2^ -9 * -.B60B60B60A0CCA */ | |
| 188 | /*C2 = 2.4801587285601038265E-5 , Hex 2^-15 * .D00D00CDCD098F */ | |
| 189 | /*C3 = -2.7557313470902390219E-7 , Hex 2^-21 * -.93F27BB593E805 */ | |
| 190 | /*C4 = 2.0875623401082232009E-9 , Hex 2^-28 * .8F74C8FA1E3FF0 */ | |
| 191 | /*C5 = -1.1355178117642986178E-11 ; Hex 2^-36 * -.C7C32D0A5C5A63 */ | |
| 192 | static long C0x[] = { _0x(aaaa,3e2a), _0x(a9f0,aaaa)}; | |
| 193 | static long C1x[] = { _0x(0b60,bbb6), _0x(0cca,b60a)}; | |
| 194 | static long C2x[] = { _0x(0d00,38d0), _0x(098f,cdcd)}; | |
| 195 | static long C3x[] = { _0x(f27b,b593), _0x(e805,b593)}; | |
| 196 | static long C4x[] = { _0x(74c8,320f), _0x(3ff0,fa1e)}; | |
| 197 | static long C5x[] = { _0x(c32d,ae47), _0x(5a63,0a5c)}; | |
| 198 | #define C0 (*(double*)C0x) | |
| 199 | #define C1 (*(double*)C1x) | |
| 200 | #define C2 (*(double*)C2x) | |
| 201 | #define C3 (*(double*)C3x) | |
| 202 | #define C4 (*(double*)C4x) | |
| 203 | #define C5 (*(double*)C5x) | |
| 204 | #else /* defined(vax)||defined(tahoe) */ | |
| 205 | static double | |
| 206 | C0 = 4.1666666666666504759E-2 , /*Hex 2^ -5 * 1.555555555553E */ | |
| 207 | C1 = -1.3888888888865301516E-3 , /*Hex 2^-10 * -1.6C16C16C14199 */ | |
| 208 | C2 = 2.4801587269650015769E-5 , /*Hex 2^-16 * 1.A01A01971CAEB */ | |
| 209 | C3 = -2.7557304623183959811E-7 , /*Hex 2^-22 * -1.27E4F1314AD1A */ | |
| 210 | C4 = 2.0873958177697780076E-9 , /*Hex 2^-29 * 1.1EE3B60DDDC8C */ | |
| 211 | C5 = -1.1250289076471311557E-11 ; /*Hex 2^-37 * -1.8BD5986B2A52E */ | |
| 212 | #endif /* defined(vax)||defined(tahoe) */ | |
| 213 | ||
| 214 | #define cos__C(z) (z*z*(C0+z*(C1+z*(C2+z*(C3+z*(C4+z*C5)))))) | |
| 215 | ||
| 216 | extern int finite(); | |
| 217 | extern double copysign(),drem(); |