1*34127Sbostic /* 231931Szliu * Copyright (c) 1987 Regents of the University of California. 3*34127Sbostic * All rights reserved. 4*34127Sbostic * 5*34127Sbostic * Redistribution and use in source and binary forms are permitted 6*34127Sbostic * provided that this notice is preserved and that due credit is given 7*34127Sbostic * to the University of California at Berkeley. The name of the University 8*34127Sbostic * may not be used to endorse or promote products derived from this 9*34127Sbostic * software without specific prior written permission. This software 10*34127Sbostic * is provided ``as is'' without express or implied warranty. 11*34127Sbostic * 12*34127Sbostic * All recipients should regard themselves as participants in an ongoing 13*34127Sbostic * research project and hence should feel obligated to report their 14*34127Sbostic * experiences (good or bad) with these elementary function codes, using 15*34127Sbostic * the sendbug(8) program, to the authors. 16*34127Sbostic * 17*34127Sbostic * @(#)trig.h 5.2 (Berkeley) 04/29/88 1831931Szliu */ 19*34127Sbostic 2031931Szliu #if defined(vax)||defined(tahoe) 2131931Szliu #ifdef vax 2231931Szliu #define _0x(A,B) 0x/**/A/**/B 2331931Szliu #else /* vax */ 2431931Szliu #define _0x(A,B) 0x/**/B/**/A 2531931Szliu #endif /* vax */ 2631931Szliu /*thresh = 2.6117239648121182150E-1 , Hex 2^ -1 * .85B8636B026EA0 */ 2731931Szliu /*PIo4 = 7.8539816339744830676E-1 , Hex 2^ 0 * .C90FDAA22168C2 */ 2831931Szliu /*PIo2 = 1.5707963267948966135E0 , Hex 2^ 1 * .C90FDAA22168C2 */ 2931931Szliu /*PI3o4 = 2.3561944901923449203E0 , Hex 2^ 2 * .96CBE3F9990E92 */ 3031931Szliu /*PI = 3.1415926535897932270E0 , Hex 2^ 2 * .C90FDAA22168C2 */ 3131931Szliu /*PI2 = 6.2831853071795864540E0 ; Hex 2^ 3 * .C90FDAA22168C2 */ 3231931Szliu static long threshx[] = { _0x(b863,3f85), _0x(6ea0,6b02)}; 3331931Szliu static long PIo4x[] = { _0x(0fda,4049), _0x(68c2,a221)}; 3431931Szliu static long PIo2x[] = { _0x(0fda,40c9), _0x(68c2,a221)}; 3531931Szliu static long PI3o4x[] = { _0x(cbe3,4116), _0x(0e92,f999)}; 3631931Szliu static long PIx[] = { _0x(0fda,4149), _0x(68c2,a221)}; 3731931Szliu static long PI2x[] = { _0x(0fda,41c9), _0x(68c2,a221)}; 3831931Szliu #define thresh (*(double*)threshx) 3931931Szliu #define PIo4 (*(double*)PIo4x) 4031931Szliu #define PIo2 (*(double*)PIo2x) 4131931Szliu #define PI3o4 (*(double*)PI3o4x) 4231931Szliu #define PI (*(double*)PIx) 4331931Szliu #define PI2 (*(double*)PI2x) 4431931Szliu #else /* defined(vax)||defined(tahoe) */ 4531931Szliu static double 4631931Szliu thresh = 2.6117239648121182150E-1 , /*Hex 2^ -2 * 1.0B70C6D604DD4 */ 4731931Szliu PIo4 = 7.8539816339744827900E-1 , /*Hex 2^ -1 * 1.921FB54442D18 */ 4831931Szliu PIo2 = 1.5707963267948965580E0 , /*Hex 2^ 0 * 1.921FB54442D18 */ 4931931Szliu PI3o4 = 2.3561944901923448370E0 , /*Hex 2^ 1 * 1.2D97C7F3321D2 */ 5031931Szliu PI = 3.1415926535897931160E0 , /*Hex 2^ 1 * 1.921FB54442D18 */ 5131931Szliu PI2 = 6.2831853071795862320E0 ; /*Hex 2^ 2 * 1.921FB54442D18 */ 5231931Szliu #ifdef national 5331931Szliu static long fmaxx[] = { 0xffffffff, 0x7fefffff}; 5431931Szliu #define fmax (*(double*)fmaxx) 5531931Szliu #endif /* national */ 5631931Szliu #endif /* defined(vax)||defined(tahoe) */ 5731931Szliu static double 5831931Szliu zero = 0, 5931931Szliu one = 1, 6031931Szliu negone = -1, 6131931Szliu half = 1.0/2.0, 6231931Szliu small = 1E-10, /* 1+small**2 == 1; better values for small: 6331931Szliu * small = 1.5E-9 for VAX D 6431931Szliu * = 1.2E-8 for IEEE Double 6531931Szliu * = 2.8E-10 for IEEE Extended 6631931Szliu */ 6731931Szliu big = 1E20; /* big := 1/(small**2) */ 6831931Szliu 6931931Szliu /* sin__S(x*x) ... re-implemented as a macro 7031931Szliu * DOUBLE PRECISION (VAX D format 56 bits, IEEE DOUBLE 53 BITS) 7131931Szliu * STATIC KERNEL FUNCTION OF SIN(X), COS(X), AND TAN(X) 7231931Szliu * CODED IN C BY K.C. NG, 1/21/85; 7331931Szliu * REVISED BY K.C. NG on 8/13/85. 7431931Szliu * 7531931Szliu * sin(x*k) - x 7631931Szliu * RETURN --------------- on [-PI/4,PI/4] , where k=pi/PI, PI is the rounded 7731931Szliu * x 7831931Szliu * value of pi in machine precision: 7931931Szliu * 8031931Szliu * Decimal: 8131931Szliu * pi = 3.141592653589793 23846264338327 ..... 8231931Szliu * 53 bits PI = 3.141592653589793 115997963 ..... , 8331931Szliu * 56 bits PI = 3.141592653589793 227020265 ..... , 8431931Szliu * 8531931Szliu * Hexadecimal: 8631931Szliu * pi = 3.243F6A8885A308D313198A2E.... 8731931Szliu * 53 bits PI = 3.243F6A8885A30 = 2 * 1.921FB54442D18 8831931Szliu * 56 bits PI = 3.243F6A8885A308 = 4 * .C90FDAA22168C2 8931931Szliu * 9031931Szliu * Method: 9131931Szliu * 1. Let z=x*x. Create a polynomial approximation to 9231931Szliu * (sin(k*x)-x)/x = z*(S0 + S1*z^1 + ... + S5*z^5). 9331931Szliu * Then 9431931Szliu * sin__S(x*x) = z*(S0 + S1*z^1 + ... + S5*z^5) 9531931Szliu * 9631931Szliu * The coefficient S's are obtained by a special Remez algorithm. 9731931Szliu * 9831931Szliu * Accuracy: 9931931Szliu * In the absence of rounding error, the approximation has absolute error 10031931Szliu * less than 2**(-61.11) for VAX D FORMAT, 2**(-57.45) for IEEE DOUBLE. 10131931Szliu * 10231931Szliu * Constants: 10331931Szliu * The hexadecimal values are the intended ones for the following constants. 10431931Szliu * The decimal values may be used, provided that the compiler will convert 10531931Szliu * from decimal to binary accurately enough to produce the hexadecimal values 10631931Szliu * shown. 10731931Szliu * 10831931Szliu */ 10931931Szliu 11031931Szliu #if defined(vax)||defined(tahoe) 11131931Szliu /*S0 = -1.6666666666666646660E-1 , Hex 2^ -2 * -.AAAAAAAAAAAA71 */ 11231931Szliu /*S1 = 8.3333333333297230413E-3 , Hex 2^ -6 * .8888888888477F */ 11331931Szliu /*S2 = -1.9841269838362403710E-4 , Hex 2^-12 * -.D00D00CF8A1057 */ 11431931Szliu /*S3 = 2.7557318019967078930E-6 , Hex 2^-18 * .B8EF1CA326BEDC */ 11531931Szliu /*S4 = -2.5051841873876551398E-8 , Hex 2^-25 * -.D73195374CE1D3 */ 11631931Szliu /*S5 = 1.6028995389845827653E-10 , Hex 2^-32 * .B03D9C6D26CCCC */ 11731931Szliu /*S6 = -6.2723499671769283121E-13 ; Hex 2^-40 * -.B08D0B7561EA82 */ 11831931Szliu static long S0x[] = { _0x(aaaa,bf2a), _0x(aa71,aaaa)}; 11931931Szliu static long S1x[] = { _0x(8888,3d08), _0x(477f,8888)}; 12031931Szliu static long S2x[] = { _0x(0d00,ba50), _0x(1057,cf8a)}; 12131931Szliu static long S3x[] = { _0x(ef1c,3738), _0x(bedc,a326)}; 12231931Szliu static long S4x[] = { _0x(3195,b3d7), _0x(e1d3,374c)}; 12331931Szliu static long S5x[] = { _0x(3d9c,3030), _0x(cccc,6d26)}; 12431931Szliu static long S6x[] = { _0x(8d0b,ac30), _0x(ea82,7561)}; 12531931Szliu #define S0 (*(double*)S0x) 12631931Szliu #define S1 (*(double*)S1x) 12731931Szliu #define S2 (*(double*)S2x) 12831931Szliu #define S3 (*(double*)S3x) 12931931Szliu #define S4 (*(double*)S4x) 13031931Szliu #define S5 (*(double*)S5x) 13131931Szliu #define S6 (*(double*)S6x) 13231931Szliu #else /* IEEE double */ 13331931Szliu static double 13431931Szliu S0 = -1.6666666666666463126E-1 , /*Hex 2^ -3 * -1.555555555550C */ 13531931Szliu S1 = 8.3333333332992771264E-3 , /*Hex 2^ -7 * 1.111111110C461 */ 13631931Szliu S2 = -1.9841269816180999116E-4 , /*Hex 2^-13 * -1.A01A019746345 */ 13731931Szliu S3 = 2.7557309793219876880E-6 , /*Hex 2^-19 * 1.71DE3209CDCD9 */ 13831931Szliu S4 = -2.5050225177523807003E-8 , /*Hex 2^-26 * -1.AE5C0E319A4EF */ 13931931Szliu S5 = 1.5868926979889205164E-10 ; /*Hex 2^-33 * 1.5CF61DF672B13 */ 14031931Szliu #endif 14131931Szliu 14231931Szliu #if defined(vax)||defined(tahoe) 14331931Szliu #define sin__S(z) (z*(S0+z*(S1+z*(S2+z*(S3+z*(S4+z*(S5+z*S6))))))) 14431931Szliu #else /* defined(vax)||defined(tahoe) */ 14531931Szliu #define sin__S(z) (z*(S0+z*(S1+z*(S2+z*(S3+z*(S4+z*S5)))))) 14631931Szliu #endif /* defined(vax)||defined(tahoe) */ 14731931Szliu 14831931Szliu /* cos__C(x*x) ... re-implemented as a macro 14931931Szliu * DOUBLE PRECISION (VAX D FORMAT 56 BITS, IEEE DOUBLE 53 BITS) 15031931Szliu * STATIC KERNEL FUNCTION OF SIN(X), COS(X), AND TAN(X) 15131931Szliu * CODED IN C BY K.C. NG, 1/21/85; 15231931Szliu * REVISED BY K.C. NG on 8/13/85. 15331931Szliu * 15431931Szliu * x*x 15531931Szliu * RETURN cos(k*x) - 1 + ----- on [-PI/4,PI/4], where k = pi/PI, 15631931Szliu * 2 15731931Szliu * PI is the rounded value of pi in machine precision : 15831931Szliu * 15931931Szliu * Decimal: 16031931Szliu * pi = 3.141592653589793 23846264338327 ..... 16131931Szliu * 53 bits PI = 3.141592653589793 115997963 ..... , 16231931Szliu * 56 bits PI = 3.141592653589793 227020265 ..... , 16331931Szliu * 16431931Szliu * Hexadecimal: 16531931Szliu * pi = 3.243F6A8885A308D313198A2E.... 16631931Szliu * 53 bits PI = 3.243F6A8885A30 = 2 * 1.921FB54442D18 16731931Szliu * 56 bits PI = 3.243F6A8885A308 = 4 * .C90FDAA22168C2 16831931Szliu * 16931931Szliu * 17031931Szliu * Method: 17131931Szliu * 1. Let z=x*x. Create a polynomial approximation to 17231931Szliu * cos(k*x)-1+z/2 = z*z*(C0 + C1*z^1 + ... + C5*z^5) 17331931Szliu * then 17431931Szliu * cos__C(z) = z*z*(C0 + C1*z^1 + ... + C5*z^5) 17531931Szliu * 17631931Szliu * The coefficient C's are obtained by a special Remez algorithm. 17731931Szliu * 17831931Szliu * Accuracy: 17931931Szliu * In the absence of rounding error, the approximation has absolute error 18031931Szliu * less than 2**(-64) for VAX D FORMAT, 2**(-58.3) for IEEE DOUBLE. 18131931Szliu * 18231931Szliu * 18331931Szliu * Constants: 18431931Szliu * The hexadecimal values are the intended ones for the following constants. 18531931Szliu * The decimal values may be used, provided that the compiler will convert 18631931Szliu * from decimal to binary accurately enough to produce the hexadecimal values 18731931Szliu * shown. 18831931Szliu * 18931931Szliu */ 19031931Szliu 19131931Szliu #if defined(vax)||defined(tahoe) 19231931Szliu /*C0 = 4.1666666666666504759E-2 , Hex 2^ -4 * .AAAAAAAAAAA9F0 */ 19331931Szliu /*C1 = -1.3888888888865302059E-3 , Hex 2^ -9 * -.B60B60B60A0CCA */ 19431931Szliu /*C2 = 2.4801587285601038265E-5 , Hex 2^-15 * .D00D00CDCD098F */ 19531931Szliu /*C3 = -2.7557313470902390219E-7 , Hex 2^-21 * -.93F27BB593E805 */ 19631931Szliu /*C4 = 2.0875623401082232009E-9 , Hex 2^-28 * .8F74C8FA1E3FF0 */ 19731931Szliu /*C5 = -1.1355178117642986178E-11 ; Hex 2^-36 * -.C7C32D0A5C5A63 */ 19831931Szliu static long C0x[] = { _0x(aaaa,3e2a), _0x(a9f0,aaaa)}; 19931931Szliu static long C1x[] = { _0x(0b60,bbb6), _0x(0cca,b60a)}; 20031931Szliu static long C2x[] = { _0x(0d00,38d0), _0x(098f,cdcd)}; 20131931Szliu static long C3x[] = { _0x(f27b,b593), _0x(e805,b593)}; 20231931Szliu static long C4x[] = { _0x(74c8,320f), _0x(3ff0,fa1e)}; 20331931Szliu static long C5x[] = { _0x(c32d,ae47), _0x(5a63,0a5c)}; 20431931Szliu #define C0 (*(double*)C0x) 20531931Szliu #define C1 (*(double*)C1x) 20631931Szliu #define C2 (*(double*)C2x) 20731931Szliu #define C3 (*(double*)C3x) 20831931Szliu #define C4 (*(double*)C4x) 20931931Szliu #define C5 (*(double*)C5x) 21031931Szliu #else /* defined(vax)||defined(tahoe) */ 21131931Szliu static double 21231931Szliu C0 = 4.1666666666666504759E-2 , /*Hex 2^ -5 * 1.555555555553E */ 21331931Szliu C1 = -1.3888888888865301516E-3 , /*Hex 2^-10 * -1.6C16C16C14199 */ 21431931Szliu C2 = 2.4801587269650015769E-5 , /*Hex 2^-16 * 1.A01A01971CAEB */ 21531931Szliu C3 = -2.7557304623183959811E-7 , /*Hex 2^-22 * -1.27E4F1314AD1A */ 21631931Szliu C4 = 2.0873958177697780076E-9 , /*Hex 2^-29 * 1.1EE3B60DDDC8C */ 21731931Szliu C5 = -1.1250289076471311557E-11 ; /*Hex 2^-37 * -1.8BD5986B2A52E */ 21831931Szliu #endif /* defined(vax)||defined(tahoe) */ 21931931Szliu 22031931Szliu #define cos__C(z) (z*z*(C0+z*(C1+z*(C2+z*(C3+z*(C4+z*C5)))))) 22131931Szliu 22231931Szliu extern int finite(); 22331931Szliu extern double copysign(),drem(); 224