trig.h revision 1.2
1 /* 2 * Copyright (c) 1987, 1993 3 * The Regents of the University of California. All rights reserved. 4 * 5 * Redistribution and use in source and binary forms, with or without 6 * modification, are permitted provided that the following conditions 7 * are met: 8 * 1. Redistributions of source code must retain the above copyright 9 * notice, this list of conditions and the following disclaimer. 10 * 2. Redistributions in binary form must reproduce the above copyright 11 * notice, this list of conditions and the following disclaimer in the 12 * documentation and/or other materials provided with the distribution. 13 * 3. All advertising materials mentioning features or use of this software 14 * must display the following acknowledgement: 15 * This product includes software developed by the University of 16 * California, Berkeley and its contributors. 17 * 4. Neither the name of the University nor the names of its contributors 18 * may be used to endorse or promote products derived from this software 19 * without specific prior written permission. 20 * 21 * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND 22 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE 23 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE 24 * ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE 25 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL 26 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS 27 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) 28 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT 29 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY 30 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF 31 * SUCH DAMAGE. 32 * 33 * @(#)trig.h 8.1 (Berkeley) 6/4/93 34 */ 35 36 vc(thresh, 2.6117239648121182150E-1 ,b863,3f85,6ea0,6b02, -1, .85B8636B026EA0) 37 vc(PIo4, 7.8539816339744830676E-1 ,0fda,4049,68c2,a221, 0, .C90FDAA22168C2) 38 vc(PIo2, 1.5707963267948966135E0 ,0fda,40c9,68c2,a221, 1, .C90FDAA22168C2) 39 vc(PI3o4, 2.3561944901923449203E0 ,cbe3,4116,0e92,f999, 2, .96CBE3F9990E92) 40 vc(PI, 3.1415926535897932270E0 ,0fda,4149,68c2,a221, 2, .C90FDAA22168C2) 41 vc(PI2, 6.2831853071795864540E0 ,0fda,41c9,68c2,a221, 3, .C90FDAA22168C2) 42 43 ic(thresh, 2.6117239648121182150E-1 , -2, 1.0B70C6D604DD4) 44 ic(PIo4, 7.8539816339744827900E-1 , -1, 1.921FB54442D18) 45 ic(PIo2, 1.5707963267948965580E0 , 0, 1.921FB54442D18) 46 ic(PI3o4, 2.3561944901923448370E0 , 1, 1.2D97C7F3321D2) 47 ic(PI, 3.1415926535897931160E0 , 1, 1.921FB54442D18) 48 ic(PI2, 6.2831853071795862320E0 , 2, 1.921FB54442D18) 49 50 #ifdef vccast 51 #define thresh vccast(thresh) 52 #define PIo4 vccast(PIo4) 53 #define PIo2 vccast(PIo2) 54 #define PI3o4 vccast(PI3o4) 55 #define PI vccast(PI) 56 #define PI2 vccast(PI2) 57 #endif 58 59 #ifdef national 60 static long fmaxx[] = { 0xffffffff, 0x7fefffff}; 61 #define fmax (*(double*)fmaxx) 62 #endif /* national */ 63 64 static const double 65 zero = 0, 66 one = 1, 67 negone = -1, 68 half = 1.0/2.0, 69 small = 1E-10, /* 1+small**2 == 1; better values for small: 70 * small = 1.5E-9 for VAX D 71 * = 1.2E-8 for IEEE Double 72 * = 2.8E-10 for IEEE Extended 73 */ 74 big = 1E20; /* big := 1/(small**2) */ 75 76 /* sin__S(x*x) ... re-implemented as a macro 77 * DOUBLE PRECISION (VAX D format 56 bits, IEEE DOUBLE 53 BITS) 78 * STATIC KERNEL FUNCTION OF SIN(X), COS(X), AND TAN(X) 79 * CODED IN C BY K.C. NG, 1/21/85; 80 * REVISED BY K.C. NG on 8/13/85. 81 * 82 * sin(x*k) - x 83 * RETURN --------------- on [-PI/4,PI/4] , where k=pi/PI, PI is the rounded 84 * x 85 * value of pi in machine precision: 86 * 87 * Decimal: 88 * pi = 3.141592653589793 23846264338327 ..... 89 * 53 bits PI = 3.141592653589793 115997963 ..... , 90 * 56 bits PI = 3.141592653589793 227020265 ..... , 91 * 92 * Hexadecimal: 93 * pi = 3.243F6A8885A308D313198A2E.... 94 * 53 bits PI = 3.243F6A8885A30 = 2 * 1.921FB54442D18 95 * 56 bits PI = 3.243F6A8885A308 = 4 * .C90FDAA22168C2 96 * 97 * Method: 98 * 1. Let z=x*x. Create a polynomial approximation to 99 * (sin(k*x)-x)/x = z*(S0 + S1*z^1 + ... + S5*z^5). 100 * Then 101 * sin__S(x*x) = z*(S0 + S1*z^1 + ... + S5*z^5) 102 * 103 * The coefficient S's are obtained by a special Remez algorithm. 104 * 105 * Accuracy: 106 * In the absence of rounding error, the approximation has absolute error 107 * less than 2**(-61.11) for VAX D FORMAT, 2**(-57.45) for IEEE DOUBLE. 108 * 109 * Constants: 110 * The hexadecimal values are the intended ones for the following constants. 111 * The decimal values may be used, provided that the compiler will convert 112 * from decimal to binary accurately enough to produce the hexadecimal values 113 * shown. 114 * 115 */ 116 117 vc(S0, -1.6666666666666646660E-1 ,aaaa,bf2a,aa71,aaaa, -2, -.AAAAAAAAAAAA71) 118 vc(S1, 8.3333333333297230413E-3 ,8888,3d08,477f,8888, -6, .8888888888477F) 119 vc(S2, -1.9841269838362403710E-4 ,0d00,ba50,1057,cf8a, -12, -.D00D00CF8A1057) 120 vc(S3, 2.7557318019967078930E-6 ,ef1c,3738,bedc,a326, -18, .B8EF1CA326BEDC) 121 vc(S4, -2.5051841873876551398E-8 ,3195,b3d7,e1d3,374c, -25, -.D73195374CE1D3) 122 vc(S5, 1.6028995389845827653E-10 ,3d9c,3030,cccc,6d26, -32, .B03D9C6D26CCCC) 123 vc(S6, -6.2723499671769283121E-13 ,8d0b,ac30,ea82,7561, -40, -.B08D0B7561EA82) 124 125 ic(S0, -1.6666666666666463126E-1 , -3, -1.555555555550C) 126 ic(S1, 8.3333333332992771264E-3 , -7, 1.111111110C461) 127 ic(S2, -1.9841269816180999116E-4 , -13, -1.A01A019746345) 128 ic(S3, 2.7557309793219876880E-6 , -19, 1.71DE3209CDCD9) 129 ic(S4, -2.5050225177523807003E-8 , -26, -1.AE5C0E319A4EF) 130 ic(S5, 1.5868926979889205164E-10 , -33, 1.5CF61DF672B13) 131 132 #ifdef vccast 133 #define S0 vccast(S0) 134 #define S1 vccast(S1) 135 #define S2 vccast(S2) 136 #define S3 vccast(S3) 137 #define S4 vccast(S4) 138 #define S5 vccast(S5) 139 #define S6 vccast(S6) 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 vc(C0, 4.1666666666666504759E-2 ,aaaa,3e2a,a9f0,aaaa, -4, .AAAAAAAAAAA9F0) 191 vc(C1, -1.3888888888865302059E-3 ,0b60,bbb6,0cca,b60a, -9, -.B60B60B60A0CCA) 192 vc(C2, 2.4801587285601038265E-5 ,0d00,38d0,098f,cdcd, -15, .D00D00CDCD098F) 193 vc(C3, -2.7557313470902390219E-7 ,f27b,b593,e805,b593, -21, -.93F27BB593E805) 194 vc(C4, 2.0875623401082232009E-9 ,74c8,320f,3ff0,fa1e, -28, .8F74C8FA1E3FF0) 195 vc(C5, -1.1355178117642986178E-11 ,c32d,ae47,5a63,0a5c, -36, -.C7C32D0A5C5A63) 196 197 ic(C0, 4.1666666666666504759E-2 , -5, 1.555555555553E) 198 ic(C1, -1.3888888888865301516E-3 , -10, -1.6C16C16C14199) 199 ic(C2, 2.4801587269650015769E-5 , -16, 1.A01A01971CAEB) 200 ic(C3, -2.7557304623183959811E-7 , -22, -1.27E4F1314AD1A) 201 ic(C4, 2.0873958177697780076E-9 , -29, 1.1EE3B60DDDC8C) 202 ic(C5, -1.1250289076471311557E-11 , -37, -1.8BD5986B2A52E) 203 204 #ifdef vccast 205 #define C0 vccast(C0) 206 #define C1 vccast(C1) 207 #define C2 vccast(C2) 208 #define C3 vccast(C3) 209 #define C4 vccast(C4) 210 #define C5 vccast(C5) 211 #endif 212 213 #define cos__C(z) (z*z*(C0+z*(C1+z*(C2+z*(C3+z*(C4+z*C5)))))) 214
Indexes created Mon Sep 22 13:09:51 GMT 2025