1 /* 2 * Copyright (c) 1987 Regents of the University of California. 3 * All rights reserved. 4 * 5 * %sccs.include.redist.c% 6 * 7 * All recipients should regard themselves as participants in an ongoing 8 * research project and hence should feel obligated to report their 9 * experiences (good or bad) with these elementary function codes, using 10 * the sendbug(8) program, to the authors. 11 * 12 * @(#)trig.h 5.5 (Berkeley) 06/01/90 13 */ 14 15 #include "mathimpl.h" 16 17 vc(thresh, 2.6117239648121182150E-1 ,b863,3f85,6ea0,6b02, -1, .85B8636B026EA0) 18 vc(PIo4, 7.8539816339744830676E-1 ,0fda,4049,68c2,a221, 0, .C90FDAA22168C2) 19 vc(PIo2, 1.5707963267948966135E0 ,0fda,40c9,68c2,a221, 1, .C90FDAA22168C2) 20 vc(PI3o4, 2.3561944901923449203E0 ,cbe3,4116,0e92,f999, 2, .96CBE3F9990E92) 21 vc(PI, 3.1415926535897932270E0 ,0fda,4149,68c2,a221, 2, .C90FDAA22168C2) 22 vc(PI2, 6.2831853071795864540E0 ,0fda,41c9,68c2,a221, 3, .C90FDAA22168C2) 23 24 ic(thresh, 2.6117239648121182150E-1 , -2, 1.0B70C6D604DD4) 25 ic(PIo4, 7.8539816339744827900E-1 , -1, 1.921FB54442D18) 26 ic(PIo2, 1.5707963267948965580E0 , 0, 1.921FB54442D18) 27 ic(PI3o4, 2.3561944901923448370E0 , 1, 1.2D97C7F3321D2) 28 ic(PI, 3.1415926535897931160E0 , 1, 1.921FB54442D18) 29 ic(PI2, 6.2831853071795862320E0 , 2, 1.921FB54442D18) 30 31 #ifdef vccast 32 #define thresh vccast(thresh) 33 #define PIo4 vccast(PIo4) 34 #define PIo2 vccast(PIo2) 35 #define PI3o4 vccast(PI3o4) 36 #define PI vccast(PI) 37 #define PI2 vccast(PI2) 38 #endif 39 40 #ifdef national 41 static long fmaxx[] = { 0xffffffff, 0x7fefffff}; 42 #define fmax (*(double*)fmaxx) 43 #endif /* national */ 44 45 static const double 46 zero = 0, 47 one = 1, 48 negone = -1, 49 half = 1.0/2.0, 50 small = 1E-10, /* 1+small**2 == 1; better values for small: 51 * small = 1.5E-9 for VAX D 52 * = 1.2E-8 for IEEE Double 53 * = 2.8E-10 for IEEE Extended 54 */ 55 big = 1E20; /* big := 1/(small**2) */ 56 57 /* sin__S(x*x) ... re-implemented as a macro 58 * DOUBLE PRECISION (VAX D format 56 bits, IEEE DOUBLE 53 BITS) 59 * STATIC KERNEL FUNCTION OF SIN(X), COS(X), AND TAN(X) 60 * CODED IN C BY K.C. NG, 1/21/85; 61 * REVISED BY K.C. NG on 8/13/85. 62 * 63 * sin(x*k) - x 64 * RETURN --------------- on [-PI/4,PI/4] , where k=pi/PI, PI is the rounded 65 * x 66 * value of pi in machine precision: 67 * 68 * Decimal: 69 * pi = 3.141592653589793 23846264338327 ..... 70 * 53 bits PI = 3.141592653589793 115997963 ..... , 71 * 56 bits PI = 3.141592653589793 227020265 ..... , 72 * 73 * Hexadecimal: 74 * pi = 3.243F6A8885A308D313198A2E.... 75 * 53 bits PI = 3.243F6A8885A30 = 2 * 1.921FB54442D18 76 * 56 bits PI = 3.243F6A8885A308 = 4 * .C90FDAA22168C2 77 * 78 * Method: 79 * 1. Let z=x*x. Create a polynomial approximation to 80 * (sin(k*x)-x)/x = z*(S0 + S1*z^1 + ... + S5*z^5). 81 * Then 82 * sin__S(x*x) = z*(S0 + S1*z^1 + ... + S5*z^5) 83 * 84 * The coefficient S's are obtained by a special Remez algorithm. 85 * 86 * Accuracy: 87 * In the absence of rounding error, the approximation has absolute error 88 * less than 2**(-61.11) for VAX D FORMAT, 2**(-57.45) for IEEE DOUBLE. 89 * 90 * Constants: 91 * The hexadecimal values are the intended ones for the following constants. 92 * The decimal values may be used, provided that the compiler will convert 93 * from decimal to binary accurately enough to produce the hexadecimal values 94 * shown. 95 * 96 */ 97 98 vc(S0, -1.6666666666666646660E-1 ,aaaa,bf2a,aa71,aaaa, -2, -.AAAAAAAAAAAA71) 99 vc(S1, 8.3333333333297230413E-3 ,8888,3d08,477f,8888, -6, .8888888888477F) 100 vc(S2, -1.9841269838362403710E-4 ,0d00,ba50,1057,cf8a, -12, -.D00D00CF8A1057) 101 vc(S3, 2.7557318019967078930E-6 ,ef1c,3738,bedc,a326, -18, .B8EF1CA326BEDC) 102 vc(S4, -2.5051841873876551398E-8 ,3195,b3d7,e1d3,374c, -25, -.D73195374CE1D3) 103 vc(S5, 1.6028995389845827653E-10 ,3d9c,3030,cccc,6d26, -32, .B03D9C6D26CCCC) 104 vc(S6, -6.2723499671769283121E-13 ,8d0b,ac30,ea82,7561, -40, -.B08D0B7561EA82) 105 106 ic(S0, -1.6666666666666463126E-1 , -3, -1.555555555550C) 107 ic(S1, 8.3333333332992771264E-3 , -7, 1.111111110C461) 108 ic(S2, -1.9841269816180999116E-4 , -13, -1.A01A019746345) 109 ic(S3, 2.7557309793219876880E-6 , -19, 1.71DE3209CDCD9) 110 ic(S4, -2.5050225177523807003E-8 , -26, -1.AE5C0E319A4EF) 111 ic(S5, 1.5868926979889205164E-10 , -33, 1.5CF61DF672B13) 112 113 #ifdef vccast 114 #define S0 vccast(S0) 115 #define S1 vccast(S1) 116 #define S2 vccast(S2) 117 #define S3 vccast(S3) 118 #define S4 vccast(S4) 119 #define S5 vccast(S5) 120 #define S6 vccast(S6) 121 #endif 122 123 #if defined(vax)||defined(tahoe) 124 # define sin__S(z) (z*(S0+z*(S1+z*(S2+z*(S3+z*(S4+z*(S5+z*S6))))))) 125 #else /* defined(vax)||defined(tahoe) */ 126 # define sin__S(z) (z*(S0+z*(S1+z*(S2+z*(S3+z*(S4+z*S5)))))) 127 #endif /* defined(vax)||defined(tahoe) */ 128 129 /* cos__C(x*x) ... re-implemented as a macro 130 * DOUBLE PRECISION (VAX D FORMAT 56 BITS, IEEE DOUBLE 53 BITS) 131 * STATIC KERNEL FUNCTION OF SIN(X), COS(X), AND TAN(X) 132 * CODED IN C BY K.C. NG, 1/21/85; 133 * REVISED BY K.C. NG on 8/13/85. 134 * 135 * x*x 136 * RETURN cos(k*x) - 1 + ----- on [-PI/4,PI/4], where k = pi/PI, 137 * 2 138 * PI is the rounded value of pi in machine precision : 139 * 140 * Decimal: 141 * pi = 3.141592653589793 23846264338327 ..... 142 * 53 bits PI = 3.141592653589793 115997963 ..... , 143 * 56 bits PI = 3.141592653589793 227020265 ..... , 144 * 145 * Hexadecimal: 146 * pi = 3.243F6A8885A308D313198A2E.... 147 * 53 bits PI = 3.243F6A8885A30 = 2 * 1.921FB54442D18 148 * 56 bits PI = 3.243F6A8885A308 = 4 * .C90FDAA22168C2 149 * 150 * 151 * Method: 152 * 1. Let z=x*x. Create a polynomial approximation to 153 * cos(k*x)-1+z/2 = z*z*(C0 + C1*z^1 + ... + C5*z^5) 154 * then 155 * cos__C(z) = z*z*(C0 + C1*z^1 + ... + C5*z^5) 156 * 157 * The coefficient C's are obtained by a special Remez algorithm. 158 * 159 * Accuracy: 160 * In the absence of rounding error, the approximation has absolute error 161 * less than 2**(-64) for VAX D FORMAT, 2**(-58.3) for IEEE DOUBLE. 162 * 163 * 164 * Constants: 165 * The hexadecimal values are the intended ones for the following constants. 166 * The decimal values may be used, provided that the compiler will convert 167 * from decimal to binary accurately enough to produce the hexadecimal values 168 * shown. 169 */ 170 171 vc(C0, 4.1666666666666504759E-2 ,aaaa,3e2a,a9f0,aaaa, -4, .AAAAAAAAAAA9F0) 172 vc(C1, -1.3888888888865302059E-3 ,0b60,bbb6,0cca,b60a, -9, -.B60B60B60A0CCA) 173 vc(C2, 2.4801587285601038265E-5 ,0d00,38d0,098f,cdcd, -15, .D00D00CDCD098F) 174 vc(C3, -2.7557313470902390219E-7 ,f27b,b593,e805,b593, -21, -.93F27BB593E805) 175 vc(C4, 2.0875623401082232009E-9 ,74c8,320f,3ff0,fa1e, -28, .8F74C8FA1E3FF0) 176 vc(C5, -1.1355178117642986178E-11 ,c32d,ae47,5a63,0a5c, -36, -.C7C32D0A5C5A63) 177 178 ic(C0, 4.1666666666666504759E-2 , -5, 1.555555555553E) 179 ic(C1, -1.3888888888865301516E-3 , -10, -1.6C16C16C14199) 180 ic(C2, 2.4801587269650015769E-5 , -16, 1.A01A01971CAEB) 181 ic(C3, -2.7557304623183959811E-7 , -22, -1.27E4F1314AD1A) 182 ic(C4, 2.0873958177697780076E-9 , -29, 1.1EE3B60DDDC8C) 183 ic(C5, -1.1250289076471311557E-11 , -37, -1.8BD5986B2A52E) 184 185 #ifdef vccast 186 #define C0 vccast(C0) 187 #define C1 vccast(C1) 188 #define C2 vccast(C2) 189 #define C3 vccast(C3) 190 #define C4 vccast(C4) 191 #define C5 vccast(C5) 192 #endif 193 194 #define cos__C(z) (z*z*(C0+z*(C1+z*(C2+z*(C3+z*(C4+z*C5)))))) 195