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#include <mathimpl.h> 37 38// Previously, PI and PI2 were defined as macros in math.h 39// however, the values there are inconsistent with those here. 40// Especially PI2 which in <math.h> is pi/2 and here is 2*pi. (ug!) 41#undef PI 42#undef PI2 43 44vc(thresh, 2.6117239648121182150E-1 ,b863,3f85,6ea0,6b02, -1, .85B8636B026EA0) 45vc(PIo4, 7.8539816339744830676E-1 ,0fda,4049,68c2,a221, 0, .C90FDAA22168C2) 46vc(PIo2, 1.5707963267948966135E0 ,0fda,40c9,68c2,a221, 1, .C90FDAA22168C2) 47vc(PI3o4, 2.3561944901923449203E0 ,cbe3,4116,0e92,f999, 2, .96CBE3F9990E92) 48vc(PI, 3.1415926535897932270E0 ,0fda,4149,68c2,a221, 2, .C90FDAA22168C2) 49vc(PI2, 6.2831853071795864540E0 ,0fda,41c9,68c2,a221, 3, .C90FDAA22168C2) 50 51ic(thresh, 2.6117239648121182150E-1 , -2, 1.0B70C6D604DD4) 52ic(PIo4, 7.8539816339744827900E-1 , -1, 1.921FB54442D18) 53ic(PIo2, 1.5707963267948965580E0 , 0, 1.921FB54442D18) 54ic(PI3o4, 2.3561944901923448370E0 , 1, 1.2D97C7F3321D2) 55ic(PI, 3.1415926535897931160E0 , 1, 1.921FB54442D18) 56ic(PI2, 6.2831853071795862320E0 , 2, 1.921FB54442D18) 57 58#ifdef vccast 59#define thresh vccast(thresh) 60#define PIo4 vccast(PIo4) 61#define PIo2 vccast(PIo2) 62#define PI3o4 vccast(PI3o4) 63#define PI vccast(PI) 64#define PI2 vccast(PI2) 65#endif 66 67#ifdef national 68static long fmaxx[] = { 0xffffffff, 0x7fefffff}; 69#define fmax (*(double*)fmaxx) 70#endif /* national */ 71 72static const double 73 zero = 0, 74 one = 1, 75 negone = -1, 76 half = 1.0/2.0, 77 small = 1E-10, /* 1+small**2 == 1; better values for small: 78 * small = 1.5E-9 for VAX D 79 * = 1.2E-8 for IEEE Double 80 * = 2.8E-10 for IEEE Extended 81 */ 82 big = 1E20; /* big := 1/(small**2) */ 83 84/* sin__S(x*x) ... re-implemented as a macro 85 * DOUBLE PRECISION (VAX D format 56 bits, IEEE DOUBLE 53 BITS) 86 * STATIC KERNEL FUNCTION OF SIN(X), COS(X), AND TAN(X) 87 * CODED IN C BY K.C. NG, 1/21/85; 88 * REVISED BY K.C. NG on 8/13/85. 89 * 90 * sin(x*k) - x 91 * RETURN --------------- on [-PI/4,PI/4] , where k=pi/PI, PI is the rounded 92 * x 93 * value of pi in machine precision: 94 * 95 * Decimal: 96 * pi = 3.141592653589793 23846264338327 ..... 97 * 53 bits PI = 3.141592653589793 115997963 ..... , 98 * 56 bits PI = 3.141592653589793 227020265 ..... , 99 * 100 * Hexadecimal: 101 * pi = 3.243F6A8885A308D313198A2E.... 102 * 53 bits PI = 3.243F6A8885A30 = 2 * 1.921FB54442D18 103 * 56 bits PI = 3.243F6A8885A308 = 4 * .C90FDAA22168C2 104 * 105 * Method: 106 * 1. Let z=x*x. Create a polynomial approximation to 107 * (sin(k*x)-x)/x = z*(S0 + S1*z^1 + ... + S5*z^5). 108 * Then 109 * sin__S(x*x) = z*(S0 + S1*z^1 + ... + S5*z^5) 110 * 111 * The coefficient S's are obtained by a special Remez algorithm. 112 * 113 * Accuracy: 114 * In the absence of rounding error, the approximation has absolute error 115 * less than 2**(-61.11) for VAX D FORMAT, 2**(-57.45) for IEEE DOUBLE. 116 * 117 * Constants: 118 * The hexadecimal values are the intended ones for the following constants. 119 * The decimal values may be used, provided that the compiler will convert 120 * from decimal to binary accurately enough to produce the hexadecimal values 121 * shown. 122 * 123 */ 124 125vc(S0, -1.6666666666666646660E-1 ,aaaa,bf2a,aa71,aaaa, -2, -.AAAAAAAAAAAA71) 126vc(S1, 8.3333333333297230413E-3 ,8888,3d08,477f,8888, -6, .8888888888477F) 127vc(S2, -1.9841269838362403710E-4 ,0d00,ba50,1057,cf8a, -12, -.D00D00CF8A1057) 128vc(S3, 2.7557318019967078930E-6 ,ef1c,3738,bedc,a326, -18, .B8EF1CA326BEDC) 129vc(S4, -2.5051841873876551398E-8 ,3195,b3d7,e1d3,374c, -25, -.D73195374CE1D3) 130vc(S5, 1.6028995389845827653E-10 ,3d9c,3030,cccc,6d26, -32, .B03D9C6D26CCCC) 131vc(S6, -6.2723499671769283121E-13 ,8d0b,ac30,ea82,7561, -40, -.B08D0B7561EA82) 132 133ic(S0, -1.6666666666666463126E-1 , -3, -1.555555555550C) 134ic(S1, 8.3333333332992771264E-3 , -7, 1.111111110C461) 135ic(S2, -1.9841269816180999116E-4 , -13, -1.A01A019746345) 136ic(S3, 2.7557309793219876880E-6 , -19, 1.71DE3209CDCD9) 137ic(S4, -2.5050225177523807003E-8 , -26, -1.AE5C0E319A4EF) 138ic(S5, 1.5868926979889205164E-10 , -33, 1.5CF61DF672B13) 139 140#ifdef vccast 141#define S0 vccast(S0) 142#define S1 vccast(S1) 143#define S2 vccast(S2) 144#define S3 vccast(S3) 145#define S4 vccast(S4) 146#define S5 vccast(S5) 147#define S6 vccast(S6) 148#endif 149 150#if defined(vax)||defined(tahoe) 151# define sin__S(z) (z*(S0+z*(S1+z*(S2+z*(S3+z*(S4+z*(S5+z*S6))))))) 152#else /* defined(vax)||defined(tahoe) */ 153# define sin__S(z) (z*(S0+z*(S1+z*(S2+z*(S3+z*(S4+z*S5)))))) 154#endif /* defined(vax)||defined(tahoe) */ 155 156/* cos__C(x*x) ... re-implemented as a macro 157 * DOUBLE PRECISION (VAX D FORMAT 56 BITS, IEEE DOUBLE 53 BITS) 158 * STATIC KERNEL FUNCTION OF SIN(X), COS(X), AND TAN(X) 159 * CODED IN C BY K.C. NG, 1/21/85; 160 * REVISED BY K.C. NG on 8/13/85. 161 * 162 * x*x 163 * RETURN cos(k*x) - 1 + ----- on [-PI/4,PI/4], where k = pi/PI, 164 * 2 165 * PI is the rounded value of pi in machine precision : 166 * 167 * Decimal: 168 * pi = 3.141592653589793 23846264338327 ..... 169 * 53 bits PI = 3.141592653589793 115997963 ..... , 170 * 56 bits PI = 3.141592653589793 227020265 ..... , 171 * 172 * Hexadecimal: 173 * pi = 3.243F6A8885A308D313198A2E.... 174 * 53 bits PI = 3.243F6A8885A30 = 2 * 1.921FB54442D18 175 * 56 bits PI = 3.243F6A8885A308 = 4 * .C90FDAA22168C2 176 * 177 * 178 * Method: 179 * 1. Let z=x*x. Create a polynomial approximation to 180 * cos(k*x)-1+z/2 = z*z*(C0 + C1*z^1 + ... + C5*z^5) 181 * then 182 * cos__C(z) = z*z*(C0 + C1*z^1 + ... + C5*z^5) 183 * 184 * The coefficient C's are obtained by a special Remez algorithm. 185 * 186 * Accuracy: 187 * In the absence of rounding error, the approximation has absolute error 188 * less than 2**(-64) for VAX D FORMAT, 2**(-58.3) for IEEE DOUBLE. 189 * 190 * 191 * Constants: 192 * The hexadecimal values are the intended ones for the following constants. 193 * The decimal values may be used, provided that the compiler will convert 194 * from decimal to binary accurately enough to produce the hexadecimal values 195 * shown. 196 */ 197 198vc(C0, 4.1666666666666504759E-2 ,aaaa,3e2a,a9f0,aaaa, -4, .AAAAAAAAAAA9F0) 199vc(C1, -1.3888888888865302059E-3 ,0b60,bbb6,0cca,b60a, -9, -.B60B60B60A0CCA) 200vc(C2, 2.4801587285601038265E-5 ,0d00,38d0,098f,cdcd, -15, .D00D00CDCD098F) 201vc(C3, -2.7557313470902390219E-7 ,f27b,b593,e805,b593, -21, -.93F27BB593E805) 202vc(C4, 2.0875623401082232009E-9 ,74c8,320f,3ff0,fa1e, -28, .8F74C8FA1E3FF0) 203vc(C5, -1.1355178117642986178E-11 ,c32d,ae47,5a63,0a5c, -36, -.C7C32D0A5C5A63) 204 205ic(C0, 4.1666666666666504759E-2 , -5, 1.555555555553E) 206ic(C1, -1.3888888888865301516E-3 , -10, -1.6C16C16C14199) 207ic(C2, 2.4801587269650015769E-5 , -16, 1.A01A01971CAEB) 208ic(C3, -2.7557304623183959811E-7 , -22, -1.27E4F1314AD1A) 209ic(C4, 2.0873958177697780076E-9 , -29, 1.1EE3B60DDDC8C) 210ic(C5, -1.1250289076471311557E-11 , -37, -1.8BD5986B2A52E) 211 212#ifdef vccast 213#define C0 vccast(C0) 214#define C1 vccast(C1) 215#define C2 vccast(C2) 216#define C3 vccast(C3) 217#define C4 vccast(C4) 218#define C5 vccast(C5) 219#endif 220 221#define cos__C(z) (z*z*(C0+z*(C1+z*(C2+z*(C3+z*(C4+z*C5)))))) 222