1*31931Szliu /* 2*31931Szliu * Copyright (c) 1987 Regents of the University of California. 3*31931Szliu * 4*31931Szliu * Use and reproduction of this software are granted in accordance with 5*31931Szliu * the terms and conditions specified in the Berkeley Software License 6*31931Szliu * Agreement (in particular, this entails acknowledgement of the programs' 7*31931Szliu * source, and inclusion of this notice) with the additional understanding 8*31931Szliu * that all recipients should regard themselves as participants in an 9*31931Szliu * ongoing research project and hence should feel obligated to report 10*31931Szliu * their experiences (good or bad) with these elementary function codes, 11*31931Szliu * using "sendbug 4bsd-bugs@BERKELEY", to the authors. 12*31931Szliu */ 13*31931Szliu /* @(#)trig.h 1.1 1.1 (ucb.elefunt) 07/24/87 */ 14*31931Szliu #if defined(vax)||defined(tahoe) 15*31931Szliu #ifdef vax 16*31931Szliu #define _0x(A,B) 0x/**/A/**/B 17*31931Szliu #else /* vax */ 18*31931Szliu #define _0x(A,B) 0x/**/B/**/A 19*31931Szliu #endif /* vax */ 20*31931Szliu /*thresh = 2.6117239648121182150E-1 , Hex 2^ -1 * .85B8636B026EA0 */ 21*31931Szliu /*PIo4 = 7.8539816339744830676E-1 , Hex 2^ 0 * .C90FDAA22168C2 */ 22*31931Szliu /*PIo2 = 1.5707963267948966135E0 , Hex 2^ 1 * .C90FDAA22168C2 */ 23*31931Szliu /*PI3o4 = 2.3561944901923449203E0 , Hex 2^ 2 * .96CBE3F9990E92 */ 24*31931Szliu /*PI = 3.1415926535897932270E0 , Hex 2^ 2 * .C90FDAA22168C2 */ 25*31931Szliu /*PI2 = 6.2831853071795864540E0 ; Hex 2^ 3 * .C90FDAA22168C2 */ 26*31931Szliu static long threshx[] = { _0x(b863,3f85), _0x(6ea0,6b02)}; 27*31931Szliu static long PIo4x[] = { _0x(0fda,4049), _0x(68c2,a221)}; 28*31931Szliu static long PIo2x[] = { _0x(0fda,40c9), _0x(68c2,a221)}; 29*31931Szliu static long PI3o4x[] = { _0x(cbe3,4116), _0x(0e92,f999)}; 30*31931Szliu static long PIx[] = { _0x(0fda,4149), _0x(68c2,a221)}; 31*31931Szliu static long PI2x[] = { _0x(0fda,41c9), _0x(68c2,a221)}; 32*31931Szliu #define thresh (*(double*)threshx) 33*31931Szliu #define PIo4 (*(double*)PIo4x) 34*31931Szliu #define PIo2 (*(double*)PIo2x) 35*31931Szliu #define PI3o4 (*(double*)PI3o4x) 36*31931Szliu #define PI (*(double*)PIx) 37*31931Szliu #define PI2 (*(double*)PI2x) 38*31931Szliu #else /* defined(vax)||defined(tahoe) */ 39*31931Szliu static double 40*31931Szliu thresh = 2.6117239648121182150E-1 , /*Hex 2^ -2 * 1.0B70C6D604DD4 */ 41*31931Szliu PIo4 = 7.8539816339744827900E-1 , /*Hex 2^ -1 * 1.921FB54442D18 */ 42*31931Szliu PIo2 = 1.5707963267948965580E0 , /*Hex 2^ 0 * 1.921FB54442D18 */ 43*31931Szliu PI3o4 = 2.3561944901923448370E0 , /*Hex 2^ 1 * 1.2D97C7F3321D2 */ 44*31931Szliu PI = 3.1415926535897931160E0 , /*Hex 2^ 1 * 1.921FB54442D18 */ 45*31931Szliu PI2 = 6.2831853071795862320E0 ; /*Hex 2^ 2 * 1.921FB54442D18 */ 46*31931Szliu #ifdef national 47*31931Szliu static long fmaxx[] = { 0xffffffff, 0x7fefffff}; 48*31931Szliu #define fmax (*(double*)fmaxx) 49*31931Szliu #endif /* national */ 50*31931Szliu #endif /* defined(vax)||defined(tahoe) */ 51*31931Szliu static double 52*31931Szliu zero = 0, 53*31931Szliu one = 1, 54*31931Szliu negone = -1, 55*31931Szliu half = 1.0/2.0, 56*31931Szliu small = 1E-10, /* 1+small**2 == 1; better values for small: 57*31931Szliu * small = 1.5E-9 for VAX D 58*31931Szliu * = 1.2E-8 for IEEE Double 59*31931Szliu * = 2.8E-10 for IEEE Extended 60*31931Szliu */ 61*31931Szliu big = 1E20; /* big := 1/(small**2) */ 62*31931Szliu 63*31931Szliu /* sin__S(x*x) ... re-implemented as a macro 64*31931Szliu * DOUBLE PRECISION (VAX D format 56 bits, IEEE DOUBLE 53 BITS) 65*31931Szliu * STATIC KERNEL FUNCTION OF SIN(X), COS(X), AND TAN(X) 66*31931Szliu * CODED IN C BY K.C. NG, 1/21/85; 67*31931Szliu * REVISED BY K.C. NG on 8/13/85. 68*31931Szliu * 69*31931Szliu * sin(x*k) - x 70*31931Szliu * RETURN --------------- on [-PI/4,PI/4] , where k=pi/PI, PI is the rounded 71*31931Szliu * x 72*31931Szliu * value of pi in machine precision: 73*31931Szliu * 74*31931Szliu * Decimal: 75*31931Szliu * pi = 3.141592653589793 23846264338327 ..... 76*31931Szliu * 53 bits PI = 3.141592653589793 115997963 ..... , 77*31931Szliu * 56 bits PI = 3.141592653589793 227020265 ..... , 78*31931Szliu * 79*31931Szliu * Hexadecimal: 80*31931Szliu * pi = 3.243F6A8885A308D313198A2E.... 81*31931Szliu * 53 bits PI = 3.243F6A8885A30 = 2 * 1.921FB54442D18 82*31931Szliu * 56 bits PI = 3.243F6A8885A308 = 4 * .C90FDAA22168C2 83*31931Szliu * 84*31931Szliu * Method: 85*31931Szliu * 1. Let z=x*x. Create a polynomial approximation to 86*31931Szliu * (sin(k*x)-x)/x = z*(S0 + S1*z^1 + ... + S5*z^5). 87*31931Szliu * Then 88*31931Szliu * sin__S(x*x) = z*(S0 + S1*z^1 + ... + S5*z^5) 89*31931Szliu * 90*31931Szliu * The coefficient S's are obtained by a special Remez algorithm. 91*31931Szliu * 92*31931Szliu * Accuracy: 93*31931Szliu * In the absence of rounding error, the approximation has absolute error 94*31931Szliu * less than 2**(-61.11) for VAX D FORMAT, 2**(-57.45) for IEEE DOUBLE. 95*31931Szliu * 96*31931Szliu * Constants: 97*31931Szliu * The hexadecimal values are the intended ones for the following constants. 98*31931Szliu * The decimal values may be used, provided that the compiler will convert 99*31931Szliu * from decimal to binary accurately enough to produce the hexadecimal values 100*31931Szliu * shown. 101*31931Szliu * 102*31931Szliu */ 103*31931Szliu 104*31931Szliu #if defined(vax)||defined(tahoe) 105*31931Szliu /*S0 = -1.6666666666666646660E-1 , Hex 2^ -2 * -.AAAAAAAAAAAA71 */ 106*31931Szliu /*S1 = 8.3333333333297230413E-3 , Hex 2^ -6 * .8888888888477F */ 107*31931Szliu /*S2 = -1.9841269838362403710E-4 , Hex 2^-12 * -.D00D00CF8A1057 */ 108*31931Szliu /*S3 = 2.7557318019967078930E-6 , Hex 2^-18 * .B8EF1CA326BEDC */ 109*31931Szliu /*S4 = -2.5051841873876551398E-8 , Hex 2^-25 * -.D73195374CE1D3 */ 110*31931Szliu /*S5 = 1.6028995389845827653E-10 , Hex 2^-32 * .B03D9C6D26CCCC */ 111*31931Szliu /*S6 = -6.2723499671769283121E-13 ; Hex 2^-40 * -.B08D0B7561EA82 */ 112*31931Szliu static long S0x[] = { _0x(aaaa,bf2a), _0x(aa71,aaaa)}; 113*31931Szliu static long S1x[] = { _0x(8888,3d08), _0x(477f,8888)}; 114*31931Szliu static long S2x[] = { _0x(0d00,ba50), _0x(1057,cf8a)}; 115*31931Szliu static long S3x[] = { _0x(ef1c,3738), _0x(bedc,a326)}; 116*31931Szliu static long S4x[] = { _0x(3195,b3d7), _0x(e1d3,374c)}; 117*31931Szliu static long S5x[] = { _0x(3d9c,3030), _0x(cccc,6d26)}; 118*31931Szliu static long S6x[] = { _0x(8d0b,ac30), _0x(ea82,7561)}; 119*31931Szliu #define S0 (*(double*)S0x) 120*31931Szliu #define S1 (*(double*)S1x) 121*31931Szliu #define S2 (*(double*)S2x) 122*31931Szliu #define S3 (*(double*)S3x) 123*31931Szliu #define S4 (*(double*)S4x) 124*31931Szliu #define S5 (*(double*)S5x) 125*31931Szliu #define S6 (*(double*)S6x) 126*31931Szliu #else /* IEEE double */ 127*31931Szliu static double 128*31931Szliu S0 = -1.6666666666666463126E-1 , /*Hex 2^ -3 * -1.555555555550C */ 129*31931Szliu S1 = 8.3333333332992771264E-3 , /*Hex 2^ -7 * 1.111111110C461 */ 130*31931Szliu S2 = -1.9841269816180999116E-4 , /*Hex 2^-13 * -1.A01A019746345 */ 131*31931Szliu S3 = 2.7557309793219876880E-6 , /*Hex 2^-19 * 1.71DE3209CDCD9 */ 132*31931Szliu S4 = -2.5050225177523807003E-8 , /*Hex 2^-26 * -1.AE5C0E319A4EF */ 133*31931Szliu S5 = 1.5868926979889205164E-10 ; /*Hex 2^-33 * 1.5CF61DF672B13 */ 134*31931Szliu #endif 135*31931Szliu 136*31931Szliu #if defined(vax)||defined(tahoe) 137*31931Szliu #define sin__S(z) (z*(S0+z*(S1+z*(S2+z*(S3+z*(S4+z*(S5+z*S6))))))) 138*31931Szliu #else /* defined(vax)||defined(tahoe) */ 139*31931Szliu #define sin__S(z) (z*(S0+z*(S1+z*(S2+z*(S3+z*(S4+z*S5)))))) 140*31931Szliu #endif /* defined(vax)||defined(tahoe) */ 141*31931Szliu 142*31931Szliu /* cos__C(x*x) ... re-implemented as a macro 143*31931Szliu * DOUBLE PRECISION (VAX D FORMAT 56 BITS, IEEE DOUBLE 53 BITS) 144*31931Szliu * STATIC KERNEL FUNCTION OF SIN(X), COS(X), AND TAN(X) 145*31931Szliu * CODED IN C BY K.C. NG, 1/21/85; 146*31931Szliu * REVISED BY K.C. NG on 8/13/85. 147*31931Szliu * 148*31931Szliu * x*x 149*31931Szliu * RETURN cos(k*x) - 1 + ----- on [-PI/4,PI/4], where k = pi/PI, 150*31931Szliu * 2 151*31931Szliu * PI is the rounded value of pi in machine precision : 152*31931Szliu * 153*31931Szliu * Decimal: 154*31931Szliu * pi = 3.141592653589793 23846264338327 ..... 155*31931Szliu * 53 bits PI = 3.141592653589793 115997963 ..... , 156*31931Szliu * 56 bits PI = 3.141592653589793 227020265 ..... , 157*31931Szliu * 158*31931Szliu * Hexadecimal: 159*31931Szliu * pi = 3.243F6A8885A308D313198A2E.... 160*31931Szliu * 53 bits PI = 3.243F6A8885A30 = 2 * 1.921FB54442D18 161*31931Szliu * 56 bits PI = 3.243F6A8885A308 = 4 * .C90FDAA22168C2 162*31931Szliu * 163*31931Szliu * 164*31931Szliu * Method: 165*31931Szliu * 1. Let z=x*x. Create a polynomial approximation to 166*31931Szliu * cos(k*x)-1+z/2 = z*z*(C0 + C1*z^1 + ... + C5*z^5) 167*31931Szliu * then 168*31931Szliu * cos__C(z) = z*z*(C0 + C1*z^1 + ... + C5*z^5) 169*31931Szliu * 170*31931Szliu * The coefficient C's are obtained by a special Remez algorithm. 171*31931Szliu * 172*31931Szliu * Accuracy: 173*31931Szliu * In the absence of rounding error, the approximation has absolute error 174*31931Szliu * less than 2**(-64) for VAX D FORMAT, 2**(-58.3) for IEEE DOUBLE. 175*31931Szliu * 176*31931Szliu * 177*31931Szliu * Constants: 178*31931Szliu * The hexadecimal values are the intended ones for the following constants. 179*31931Szliu * The decimal values may be used, provided that the compiler will convert 180*31931Szliu * from decimal to binary accurately enough to produce the hexadecimal values 181*31931Szliu * shown. 182*31931Szliu * 183*31931Szliu */ 184*31931Szliu 185*31931Szliu #if defined(vax)||defined(tahoe) 186*31931Szliu /*C0 = 4.1666666666666504759E-2 , Hex 2^ -4 * .AAAAAAAAAAA9F0 */ 187*31931Szliu /*C1 = -1.3888888888865302059E-3 , Hex 2^ -9 * -.B60B60B60A0CCA */ 188*31931Szliu /*C2 = 2.4801587285601038265E-5 , Hex 2^-15 * .D00D00CDCD098F */ 189*31931Szliu /*C3 = -2.7557313470902390219E-7 , Hex 2^-21 * -.93F27BB593E805 */ 190*31931Szliu /*C4 = 2.0875623401082232009E-9 , Hex 2^-28 * .8F74C8FA1E3FF0 */ 191*31931Szliu /*C5 = -1.1355178117642986178E-11 ; Hex 2^-36 * -.C7C32D0A5C5A63 */ 192*31931Szliu static long C0x[] = { _0x(aaaa,3e2a), _0x(a9f0,aaaa)}; 193*31931Szliu static long C1x[] = { _0x(0b60,bbb6), _0x(0cca,b60a)}; 194*31931Szliu static long C2x[] = { _0x(0d00,38d0), _0x(098f,cdcd)}; 195*31931Szliu static long C3x[] = { _0x(f27b,b593), _0x(e805,b593)}; 196*31931Szliu static long C4x[] = { _0x(74c8,320f), _0x(3ff0,fa1e)}; 197*31931Szliu static long C5x[] = { _0x(c32d,ae47), _0x(5a63,0a5c)}; 198*31931Szliu #define C0 (*(double*)C0x) 199*31931Szliu #define C1 (*(double*)C1x) 200*31931Szliu #define C2 (*(double*)C2x) 201*31931Szliu #define C3 (*(double*)C3x) 202*31931Szliu #define C4 (*(double*)C4x) 203*31931Szliu #define C5 (*(double*)C5x) 204*31931Szliu #else /* defined(vax)||defined(tahoe) */ 205*31931Szliu static double 206*31931Szliu C0 = 4.1666666666666504759E-2 , /*Hex 2^ -5 * 1.555555555553E */ 207*31931Szliu C1 = -1.3888888888865301516E-3 , /*Hex 2^-10 * -1.6C16C16C14199 */ 208*31931Szliu C2 = 2.4801587269650015769E-5 , /*Hex 2^-16 * 1.A01A01971CAEB */ 209*31931Szliu C3 = -2.7557304623183959811E-7 , /*Hex 2^-22 * -1.27E4F1314AD1A */ 210*31931Szliu C4 = 2.0873958177697780076E-9 , /*Hex 2^-29 * 1.1EE3B60DDDC8C */ 211*31931Szliu C5 = -1.1250289076471311557E-11 ; /*Hex 2^-37 * -1.8BD5986B2A52E */ 212*31931Szliu #endif /* defined(vax)||defined(tahoe) */ 213*31931Szliu 214*31931Szliu #define cos__C(z) (z*z*(C0+z*(C1+z*(C2+z*(C3+z*(C4+z*C5)))))) 215*31931Szliu 216*31931Szliu extern int finite(); 217*31931Szliu extern double copysign(),drem(); 218