1 /* $NetBSD: trig.h,v 1.4 1999/07/02 15:37:37 simonb Exp $ */ 2 /* 3 * Copyright (c) 1987, 1993 4 * The Regents of the University of California. All rights reserved. 5 * 6 * Redistribution and use in source and binary forms, with or without 7 * modification, are permitted provided that the following conditions 8 * are met: 9 * 1. Redistributions of source code must retain the above copyright 10 * notice, this list of conditions and the following disclaimer. 11 * 2. Redistributions in binary form must reproduce the above copyright 12 * notice, this list of conditions and the following disclaimer in the 13 * documentation and/or other materials provided with the distribution. 14 * 3. All advertising materials mentioning features or use of this software 15 * must display the following acknowledgement: 16 * This product includes software developed by the University of 17 * California, Berkeley and its contributors. 18 * 4. Neither the name of the University nor the names of its contributors 19 * may be used to endorse or promote products derived from this software 20 * without specific prior written permission. 21 * 22 * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND 23 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE 24 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE 25 * ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE 26 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL 27 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS 28 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) 29 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT 30 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY 31 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF 32 * SUCH DAMAGE. 33 * 34 * @(#)trig.h 8.1 (Berkeley) 6/4/93 35 */ 36 37 vc(thresh, 2.6117239648121182150E-1 ,b863,3f85,6ea0,6b02, -1, .85B8636B026EA0) 38 vc(PIo4, 7.8539816339744830676E-1 ,0fda,4049,68c2,a221, 0, .C90FDAA22168C2) 39 vc(PIo2, 1.5707963267948966135E0 ,0fda,40c9,68c2,a221, 1, .C90FDAA22168C2) 40 vc(PI3o4, 2.3561944901923449203E0 ,cbe3,4116,0e92,f999, 2, .96CBE3F9990E92) 41 vc(PI, 3.1415926535897932270E0 ,0fda,4149,68c2,a221, 2, .C90FDAA22168C2) 42 vc(PI2, 6.2831853071795864540E0 ,0fda,41c9,68c2,a221, 3, .C90FDAA22168C2) 43 44 ic(thresh, 2.6117239648121182150E-1 , -2, 1.0B70C6D604DD4) 45 ic(PIo4, 7.8539816339744827900E-1 , -1, 1.921FB54442D18) 46 ic(PIo2, 1.5707963267948965580E0 , 0, 1.921FB54442D18) 47 ic(PI3o4, 2.3561944901923448370E0 , 1, 1.2D97C7F3321D2) 48 ic(PI, 3.1415926535897931160E0 , 1, 1.921FB54442D18) 49 ic(PI2, 6.2831853071795862320E0 , 2, 1.921FB54442D18) 50 51 #ifdef vccast 52 #define thresh vccast(thresh) 53 #define PIo4 vccast(PIo4) 54 #define PIo2 vccast(PIo2) 55 #define PI3o4 vccast(PI3o4) 56 #define PI vccast(PI) 57 #define PI2 vccast(PI2) 58 #endif 59 60 #ifdef national 61 static long fmaxx[] = { 0xffffffff, 0x7fefffff}; 62 #define fmax (*(double*)fmaxx) 63 #endif /* national */ 64 65 static const double 66 zero = 0, 67 one = 1, 68 negone = -1, 69 half = 1.0/2.0, 70 small = 1E-10, /* 1+small**2 == 1; better values for small: 71 * small = 1.5E-9 for VAX D 72 * = 1.2E-8 for IEEE Double 73 * = 2.8E-10 for IEEE Extended 74 */ 75 big = 1E20; /* big := 1/(small**2) */ 76 77 /* sin__S(x*x) ... re-implemented as a macro 78 * DOUBLE PRECISION (VAX D format 56 bits, IEEE DOUBLE 53 BITS) 79 * STATIC KERNEL FUNCTION OF SIN(X), COS(X), AND TAN(X) 80 * CODED IN C BY K.C. NG, 1/21/85; 81 * REVISED BY K.C. NG on 8/13/85. 82 * 83 * sin(x*k) - x 84 * RETURN --------------- on [-PI/4,PI/4] , where k=pi/PI, PI is the rounded 85 * x 86 * value of pi in machine precision: 87 * 88 * Decimal: 89 * pi = 3.141592653589793 23846264338327 ..... 90 * 53 bits PI = 3.141592653589793 115997963 ..... , 91 * 56 bits PI = 3.141592653589793 227020265 ..... , 92 * 93 * Hexadecimal: 94 * pi = 3.243F6A8885A308D313198A2E.... 95 * 53 bits PI = 3.243F6A8885A30 = 2 * 1.921FB54442D18 96 * 56 bits PI = 3.243F6A8885A308 = 4 * .C90FDAA22168C2 97 * 98 * Method: 99 * 1. Let z=x*x. Create a polynomial approximation to 100 * (sin(k*x)-x)/x = z*(S0 + S1*z^1 + ... + S5*z^5). 101 * Then 102 * sin__S(x*x) = z*(S0 + S1*z^1 + ... + S5*z^5) 103 * 104 * The coefficient S's are obtained by a special Remez algorithm. 105 * 106 * Accuracy: 107 * In the absence of rounding error, the approximation has absolute error 108 * less than 2**(-61.11) for VAX D FORMAT, 2**(-57.45) for IEEE DOUBLE. 109 * 110 * Constants: 111 * The hexadecimal values are the intended ones for the following constants. 112 * The decimal values may be used, provided that the compiler will convert 113 * from decimal to binary accurately enough to produce the hexadecimal values 114 * shown. 115 * 116 */ 117 118 vc(S0, -1.6666666666666646660E-1 ,aaaa,bf2a,aa71,aaaa, -2, -.AAAAAAAAAAAA71) 119 vc(S1, 8.3333333333297230413E-3 ,8888,3d08,477f,8888, -6, .8888888888477F) 120 vc(S2, -1.9841269838362403710E-4 ,0d00,ba50,1057,cf8a, -12, -.D00D00CF8A1057) 121 vc(S3, 2.7557318019967078930E-6 ,ef1c,3738,bedc,a326, -18, .B8EF1CA326BEDC) 122 vc(S4, -2.5051841873876551398E-8 ,3195,b3d7,e1d3,374c, -25, -.D73195374CE1D3) 123 vc(S5, 1.6028995389845827653E-10 ,3d9c,3030,cccc,6d26, -32, .B03D9C6D26CCCC) 124 vc(S6, -6.2723499671769283121E-13 ,8d0b,ac30,ea82,7561, -40, -.B08D0B7561EA82) 125 126 ic(S0, -1.6666666666666463126E-1 , -3, -1.555555555550C) 127 ic(S1, 8.3333333332992771264E-3 , -7, 1.111111110C461) 128 ic(S2, -1.9841269816180999116E-4 , -13, -1.A01A019746345) 129 ic(S3, 2.7557309793219876880E-6 , -19, 1.71DE3209CDCD9) 130 ic(S4, -2.5050225177523807003E-8 , -26, -1.AE5C0E319A4EF) 131 ic(S5, 1.5868926979889205164E-10 , -33, 1.5CF61DF672B13) 132 133 #ifdef vccast 134 #define S0 vccast(S0) 135 #define S1 vccast(S1) 136 #define S2 vccast(S2) 137 #define S3 vccast(S3) 138 #define S4 vccast(S4) 139 #define S5 vccast(S5) 140 #define S6 vccast(S6) 141 #endif 142 143 #if defined(__vax__)||defined(tahoe) 144 # define sin__S(z) (z*(S0+z*(S1+z*(S2+z*(S3+z*(S4+z*(S5+z*S6))))))) 145 #else /* defined(__vax__)||defined(tahoe) */ 146 # define sin__S(z) (z*(S0+z*(S1+z*(S2+z*(S3+z*(S4+z*S5)))))) 147 #endif /* defined(__vax__)||defined(tahoe) */ 148 149 /* cos__C(x*x) ... re-implemented as a macro 150 * DOUBLE PRECISION (VAX D FORMAT 56 BITS, IEEE DOUBLE 53 BITS) 151 * STATIC KERNEL FUNCTION OF SIN(X), COS(X), AND TAN(X) 152 * CODED IN C BY K.C. NG, 1/21/85; 153 * REVISED BY K.C. NG on 8/13/85. 154 * 155 * x*x 156 * RETURN cos(k*x) - 1 + ----- on [-PI/4,PI/4], where k = pi/PI, 157 * 2 158 * PI is the rounded value of pi in machine precision : 159 * 160 * Decimal: 161 * pi = 3.141592653589793 23846264338327 ..... 162 * 53 bits PI = 3.141592653589793 115997963 ..... , 163 * 56 bits PI = 3.141592653589793 227020265 ..... , 164 * 165 * Hexadecimal: 166 * pi = 3.243F6A8885A308D313198A2E.... 167 * 53 bits PI = 3.243F6A8885A30 = 2 * 1.921FB54442D18 168 * 56 bits PI = 3.243F6A8885A308 = 4 * .C90FDAA22168C2 169 * 170 * 171 * Method: 172 * 1. Let z=x*x. Create a polynomial approximation to 173 * cos(k*x)-1+z/2 = z*z*(C0 + C1*z^1 + ... + C5*z^5) 174 * then 175 * cos__C(z) = z*z*(C0 + C1*z^1 + ... + C5*z^5) 176 * 177 * The coefficient C's are obtained by a special Remez algorithm. 178 * 179 * Accuracy: 180 * In the absence of rounding error, the approximation has absolute error 181 * less than 2**(-64) for VAX D FORMAT, 2**(-58.3) for IEEE DOUBLE. 182 * 183 * 184 * Constants: 185 * The hexadecimal values are the intended ones for the following constants. 186 * The decimal values may be used, provided that the compiler will convert 187 * from decimal to binary accurately enough to produce the hexadecimal values 188 * shown. 189 */ 190 191 vc(C0, 4.1666666666666504759E-2 ,aaaa,3e2a,a9f0,aaaa, -4, .AAAAAAAAAAA9F0) 192 vc(C1, -1.3888888888865302059E-3 ,0b60,bbb6,0cca,b60a, -9, -.B60B60B60A0CCA) 193 vc(C2, 2.4801587285601038265E-5 ,0d00,38d0,098f,cdcd, -15, .D00D00CDCD098F) 194 vc(C3, -2.7557313470902390219E-7 ,f27b,b593,e805,b593, -21, -.93F27BB593E805) 195 vc(C4, 2.0875623401082232009E-9 ,74c8,320f,3ff0,fa1e, -28, .8F74C8FA1E3FF0) 196 vc(C5, -1.1355178117642986178E-11 ,c32d,ae47,5a63,0a5c, -36, -.C7C32D0A5C5A63) 197 198 ic(C0, 4.1666666666666504759E-2 , -5, 1.555555555553E) 199 ic(C1, -1.3888888888865301516E-3 , -10, -1.6C16C16C14199) 200 ic(C2, 2.4801587269650015769E-5 , -16, 1.A01A01971CAEB) 201 ic(C3, -2.7557304623183959811E-7 , -22, -1.27E4F1314AD1A) 202 ic(C4, 2.0873958177697780076E-9 , -29, 1.1EE3B60DDDC8C) 203 ic(C5, -1.1250289076471311557E-11 , -37, -1.8BD5986B2A52E) 204 205 #ifdef vccast 206 #define C0 vccast(C0) 207 #define C1 vccast(C1) 208 #define C2 vccast(C2) 209 #define C3 vccast(C3) 210 #define C4 vccast(C4) 211 #define C5 vccast(C5) 212 #endif 213 214 #define cos__C(z) (z*z*(C0+z*(C1+z*(C2+z*(C3+z*(C4+z*C5)))))) 215