1 /* 2 * Copyright (c) 1985 Regents of the University of California. 3 * 4 * Use and reproduction of this software are granted in accordance with 5 * the terms and conditions specified in the Berkeley Software License 6 * Agreement (in particular, this entails acknowledgement of the programs' 7 * source, and inclusion of this notice) with the additional understanding 8 * that all recipients should regard themselves as participants in an 9 * ongoing research project and hence should feel obligated to report 10 * their experiences (good or bad) with these elementary function codes, 11 * using "sendbug 4bsd-bugs@BERKELEY", to the authors. 12 */ 13 14 #ifndef lint 15 static char sccsid[] = 16 "@(#)trig.c 1.2 (Berkeley) 8/22/85; 1.6 (ucb.elefunt) 07/13/87"; 17 #endif not lint 18 19 /* SIN(X), COS(X), TAN(X) 20 * RETURN THE SINE, COSINE, AND TANGENT OF X RESPECTIVELY 21 * DOUBLE PRECISION (VAX D format 56 bits, IEEE DOUBLE 53 BITS) 22 * CODED IN C BY K.C. NG, 1/8/85; 23 * REVISED BY W. Kahan and K.C. NG, 8/17/85. 24 * 25 * Required system supported functions: 26 * copysign(x,y) 27 * finite(x) 28 * drem(x,p) 29 * 30 * Static kernel functions: 31 * sin__S(z) ....sin__S(x*x) return (sin(x)-x)/x 32 * cos__C(z) ....cos__C(x*x) return cos(x)-1-x*x/2 33 * 34 * Method. 35 * Let S and C denote the polynomial approximations to sin and cos 36 * respectively on [-PI/4, +PI/4]. 37 * 38 * SIN and COS: 39 * 1. Reduce the argument into [-PI , +PI] by the remainder function. 40 * 2. For x in (-PI,+PI), there are three cases: 41 * case 1: |x| < PI/4 42 * case 2: PI/4 <= |x| < 3PI/4 43 * case 3: 3PI/4 <= |x|. 44 * SIN and COS of x are computed by: 45 * 46 * sin(x) cos(x) remark 47 * ---------------------------------------------------------- 48 * case 1 S(x) C(x) 49 * case 2 sign(x)*C(y) S(y) y=PI/2-|x| 50 * case 3 S(y) -C(y) y=sign(x)*(PI-|x|) 51 * ---------------------------------------------------------- 52 * 53 * TAN: 54 * 1. Reduce the argument into [-PI/2 , +PI/2] by the remainder function. 55 * 2. For x in (-PI/2,+PI/2), there are two cases: 56 * case 1: |x| < PI/4 57 * case 2: PI/4 <= |x| < PI/2 58 * TAN of x is computed by: 59 * 60 * tan (x) remark 61 * ---------------------------------------------------------- 62 * case 1 S(x)/C(x) 63 * case 2 C(y)/S(y) y=sign(x)*(PI/2-|x|) 64 * ---------------------------------------------------------- 65 * 66 * Notes: 67 * 1. S(y) and C(y) were computed by: 68 * S(y) = y+y*sin__S(y*y) 69 * C(y) = 1-(y*y/2-cos__C(x*x)) ... if y*y/2 < thresh, 70 * = 0.5-((y*y/2-0.5)-cos__C(x*x)) ... if y*y/2 >= thresh. 71 * where 72 * thresh = 0.5*(acos(3/4)**2) 73 * 74 * 2. For better accuracy, we use the following formula for S/C for tan 75 * (k=0): let ss=sin__S(y*y), and cc=cos__C(y*y), then 76 * 77 * y+y*ss (y*y/2-cc)+ss 78 * S(y)/C(y) = -------- = y + y * ---------------. 79 * C C 80 * 81 * 82 * Special cases: 83 * Let trig be any of sin, cos, or tan. 84 * trig(+-INF) is NaN, with signals; 85 * trig(NaN) is that NaN; 86 * trig(n*PI/2) is exact for any integer n, provided n*PI is 87 * representable; otherwise, trig(x) is inexact. 88 * 89 * Accuracy: 90 * trig(x) returns the exact trig(x*pi/PI) nearly rounded, where 91 * 92 * Decimal: 93 * pi = 3.141592653589793 23846264338327 ..... 94 * 53 bits PI = 3.141592653589793 115997963 ..... , 95 * 56 bits PI = 3.141592653589793 227020265 ..... , 96 * 97 * Hexadecimal: 98 * pi = 3.243F6A8885A308D313198A2E.... 99 * 53 bits PI = 3.243F6A8885A30 = 2 * 1.921FB54442D18 error=.276ulps 100 * 56 bits PI = 3.243F6A8885A308 = 4 * .C90FDAA22168C2 error=.206ulps 101 * 102 * In a test run with 1,024,000 random arguments on a VAX, the maximum 103 * observed errors (compared with the exact trig(x*pi/PI)) were 104 * tan(x) : 2.09 ulps (around 4.716340404662354) 105 * sin(x) : .861 ulps 106 * cos(x) : .857 ulps 107 * 108 * Constants: 109 * The hexadecimal values are the intended ones for the following constants. 110 * The decimal values may be used, provided that the compiler will convert 111 * from decimal to binary accurately enough to produce the hexadecimal values 112 * shown. 113 */ 114 115 #if (defined(VAX)||defined(TAHOE)) 116 #ifdef VAX 117 #define _0x(A,B) 0x/**/A/**/B 118 #else /* VAX */ 119 #define _0x(A,B) 0x/**/B/**/A 120 #endif /* VAX */ 121 /*thresh = 2.6117239648121182150E-1 , Hex 2^ -1 * .85B8636B026EA0 */ 122 /*PIo4 = 7.8539816339744830676E-1 , Hex 2^ 0 * .C90FDAA22168C2 */ 123 /*PIo2 = 1.5707963267948966135E0 , Hex 2^ 1 * .C90FDAA22168C2 */ 124 /*PI3o4 = 2.3561944901923449203E0 , Hex 2^ 2 * .96CBE3F9990E92 */ 125 /*PI = 3.1415926535897932270E0 , Hex 2^ 2 * .C90FDAA22168C2 */ 126 /*PI2 = 6.2831853071795864540E0 ; Hex 2^ 3 * .C90FDAA22168C2 */ 127 static long threshx[] = { _0x(b863,3f85), _0x(6ea0,6b02)}; 128 #define thresh (*(double*)threshx) 129 static long PIo4x[] = { _0x(0fda,4049), _0x(68c2,a221)}; 130 #define PIo4 (*(double*)PIo4x) 131 static long PIo2x[] = { _0x(0fda,40c9), _0x(68c2,a221)}; 132 #define PIo2 (*(double*)PIo2x) 133 static long PI3o4x[] = { _0x(cbe3,4116), _0x(0e92,f999)}; 134 #define PI3o4 (*(double*)PI3o4x) 135 static long PIx[] = { _0x(0fda,4149), _0x(68c2,a221)}; 136 #define PI (*(double*)PIx) 137 static long PI2x[] = { _0x(0fda,41c9), _0x(68c2,a221)}; 138 #define PI2 (*(double*)PI2x) 139 #else /* IEEE double */ 140 static double 141 thresh = 2.6117239648121182150E-1 , /*Hex 2^ -2 * 1.0B70C6D604DD4 */ 142 PIo4 = 7.8539816339744827900E-1 , /*Hex 2^ -1 * 1.921FB54442D18 */ 143 PIo2 = 1.5707963267948965580E0 , /*Hex 2^ 0 * 1.921FB54442D18 */ 144 PI3o4 = 2.3561944901923448370E0 , /*Hex 2^ 1 * 1.2D97C7F3321D2 */ 145 PI = 3.1415926535897931160E0 , /*Hex 2^ 1 * 1.921FB54442D18 */ 146 PI2 = 6.2831853071795862320E0 ; /*Hex 2^ 2 * 1.921FB54442D18 */ 147 #ifdef NATIONAL 148 static long fmaxx[] = { 0xffffffff, 0x7fefffff}; 149 #define fmax (*(double*)fmaxx) 150 #endif /* NATIONAL */ 151 #endif 152 static double zero=0, one=1, negone= -1, half=1.0/2.0, 153 small=1E-10, /* 1+small**2==1; better values for small: 154 small = 1.5E-9 for VAX D 155 = 1.2E-8 for IEEE Double 156 = 2.8E-10 for IEEE Extended */ 157 big=1E20; /* big = 1/(small**2) */ 158 159 double tan(x) 160 double x; 161 { 162 double copysign(),drem(),cos__C(),sin__S(),a,z,ss,cc,c; 163 int k; 164 #if (!defined(VAX)&&!defined(TAHOE)) 165 extern int finite(); 166 /* tan(NaN) and tan(INF) must be NaN */ 167 if(!finite(x)) return(x-x); 168 #endif 169 x=drem(x,PI); /* reduce x into [-PI/2, PI/2] */ 170 a=copysign(x,one); /* ... = abs(x) */ 171 if ( a >= PIo4 ) {k=1; x = copysign( PIo2 - a , x ); } 172 else { k=0; if(a < small ) { big + a; return(x); }} 173 174 z = x*x; 175 cc = cos__C(z); 176 ss = sin__S(z); 177 z = z*half ; /* Next get c = cos(x) accurately */ 178 c = (z >= thresh )? half-((z-half)-cc) : one-(z-cc); 179 if (k==0) return ( x + (x*(z-(cc-ss)))/c ); /* sin/cos */ 180 #ifdef NATIONAL 181 else if(x==0.0) return copysign(fmax,x); /* no inf on 32k */ 182 #endif /* NATIONAL */ 183 else return( c/(x+x*ss) ); /* ... cos/sin */ 184 185 186 } 187 double sin(x) 188 double x; 189 { 190 double copysign(),drem(),sin__S(),cos__C(),a,c,z; 191 #if (!defined(VAX)&&!defined(TAHOE)) 192 extern int finite(); 193 /* sin(NaN) and sin(INF) must be NaN */ 194 if(!finite(x)) return(x-x); 195 #endif 196 x=drem(x,PI2); /* reduce x into [-PI, PI] */ 197 a=copysign(x,one); 198 if( a >= PIo4 ) { 199 if( a >= PI3o4 ) /* .. in [3PI/4, PI ] */ 200 x=copysign((a=PI-a),x); 201 202 else { /* .. in [PI/4, 3PI/4] */ 203 a=PIo2-a; /* return sign(x)*C(PI/2-|x|) */ 204 z=a*a; 205 c=cos__C(z); 206 z=z*half; 207 a=(z>=thresh)?half-((z-half)-c):one-(z-c); 208 return(copysign(a,x)); 209 } 210 } 211 212 /* return S(x) */ 213 if( a < small) { big + a; return(x);} 214 return(x+x*sin__S(x*x)); 215 } 216 217 double cos(x) 218 double x; 219 { 220 double copysign(),drem(),sin__S(),cos__C(),a,c,z,s=1.0; 221 #if (!defined(VAX)&&!defined(TAHOE)) 222 extern int finite(); 223 /* cos(NaN) and cos(INF) must be NaN */ 224 if(!finite(x)) return(x-x); 225 #endif 226 x=drem(x,PI2); /* reduce x into [-PI, PI] */ 227 a=copysign(x,one); 228 if ( a >= PIo4 ) { 229 if ( a >= PI3o4 ) /* .. in [3PI/4, PI ] */ 230 { a=PI-a; s= negone; } 231 232 else /* .. in [PI/4, 3PI/4] */ 233 /* return S(PI/2-|x|) */ 234 { a=PIo2-a; return(a+a*sin__S(a*a));} 235 } 236 237 238 /* return s*C(a) */ 239 if( a < small) { big + a; return(s);} 240 z=a*a; 241 c=cos__C(z); 242 z=z*half; 243 a=(z>=thresh)?half-((z-half)-c):one-(z-c); 244 return(copysign(a,s)); 245 } 246 247 248 /* sin__S(x*x) 249 * DOUBLE PRECISION (VAX D format 56 bits, IEEE DOUBLE 53 BITS) 250 * STATIC KERNEL FUNCTION OF SIN(X), COS(X), AND TAN(X) 251 * CODED IN C BY K.C. NG, 1/21/85; 252 * REVISED BY K.C. NG on 8/13/85. 253 * 254 * sin(x*k) - x 255 * RETURN --------------- on [-PI/4,PI/4] , where k=pi/PI, PI is the rounded 256 * x 257 * value of pi in machine precision: 258 * 259 * Decimal: 260 * pi = 3.141592653589793 23846264338327 ..... 261 * 53 bits PI = 3.141592653589793 115997963 ..... , 262 * 56 bits PI = 3.141592653589793 227020265 ..... , 263 * 264 * Hexadecimal: 265 * pi = 3.243F6A8885A308D313198A2E.... 266 * 53 bits PI = 3.243F6A8885A30 = 2 * 1.921FB54442D18 267 * 56 bits PI = 3.243F6A8885A308 = 4 * .C90FDAA22168C2 268 * 269 * Method: 270 * 1. Let z=x*x. Create a polynomial approximation to 271 * (sin(k*x)-x)/x = z*(S0 + S1*z^1 + ... + S5*z^5). 272 * Then 273 * sin__S(x*x) = z*(S0 + S1*z^1 + ... + S5*z^5) 274 * 275 * The coefficient S's are obtained by a special Remez algorithm. 276 * 277 * Accuracy: 278 * In the absence of rounding error, the approximation has absolute error 279 * less than 2**(-61.11) for VAX D FORMAT, 2**(-57.45) for IEEE DOUBLE. 280 * 281 * Constants: 282 * The hexadecimal values are the intended ones for the following constants. 283 * The decimal values may be used, provided that the compiler will convert 284 * from decimal to binary accurately enough to produce the hexadecimal values 285 * shown. 286 * 287 */ 288 289 #if (defined(VAX)||defined(TAHOE)) 290 /*S0 = -1.6666666666666646660E-1 , Hex 2^ -2 * -.AAAAAAAAAAAA71 */ 291 /*S1 = 8.3333333333297230413E-3 , Hex 2^ -6 * .8888888888477F */ 292 /*S2 = -1.9841269838362403710E-4 , Hex 2^-12 * -.D00D00CF8A1057 */ 293 /*S3 = 2.7557318019967078930E-6 , Hex 2^-18 * .B8EF1CA326BEDC */ 294 /*S4 = -2.5051841873876551398E-8 , Hex 2^-25 * -.D73195374CE1D3 */ 295 /*S5 = 1.6028995389845827653E-10 , Hex 2^-32 * .B03D9C6D26CCCC */ 296 /*S6 = -6.2723499671769283121E-13 ; Hex 2^-40 * -.B08D0B7561EA82 */ 297 static long S0x[] = { _0x(aaaa,bf2a), _0x(aa71,aaaa)}; 298 #define S0 (*(double*)S0x) 299 static long S1x[] = { _0x(8888,3d08), _0x(477f,8888)}; 300 #define S1 (*(double*)S1x) 301 static long S2x[] = { _0x(0d00,ba50), _0x(1057,cf8a)}; 302 #define S2 (*(double*)S2x) 303 static long S3x[] = { _0x(ef1c,3738), _0x(bedc,a326)}; 304 #define S3 (*(double*)S3x) 305 static long S4x[] = { _0x(3195,b3d7), _0x(e1d3,374c)}; 306 #define S4 (*(double*)S4x) 307 static long S5x[] = { _0x(3d9c,3030), _0x(cccc,6d26)}; 308 #define S5 (*(double*)S5x) 309 static long S6x[] = { _0x(8d0b,ac30), _0x(ea82,7561)}; 310 #define S6 (*(double*)S6x) 311 #else /* IEEE double */ 312 static double 313 S0 = -1.6666666666666463126E-1 , /*Hex 2^ -3 * -1.555555555550C */ 314 S1 = 8.3333333332992771264E-3 , /*Hex 2^ -7 * 1.111111110C461 */ 315 S2 = -1.9841269816180999116E-4 , /*Hex 2^-13 * -1.A01A019746345 */ 316 S3 = 2.7557309793219876880E-6 , /*Hex 2^-19 * 1.71DE3209CDCD9 */ 317 S4 = -2.5050225177523807003E-8 , /*Hex 2^-26 * -1.AE5C0E319A4EF */ 318 S5 = 1.5868926979889205164E-10 ; /*Hex 2^-33 * 1.5CF61DF672B13 */ 319 #endif 320 321 static double sin__S(z) 322 double z; 323 { 324 #if (defined(VAX)||defined(TAHOE)) 325 return(z*(S0+z*(S1+z*(S2+z*(S3+z*(S4+z*(S5+z*S6))))))); 326 #else /* IEEE double */ 327 return(z*(S0+z*(S1+z*(S2+z*(S3+z*(S4+z*S5)))))); 328 #endif 329 } 330 331 332 /* cos__C(x*x) 333 * DOUBLE PRECISION (VAX D FORMAT 56 BITS, IEEE DOUBLE 53 BITS) 334 * STATIC KERNEL FUNCTION OF SIN(X), COS(X), AND TAN(X) 335 * CODED IN C BY K.C. NG, 1/21/85; 336 * REVISED BY K.C. NG on 8/13/85. 337 * 338 * x*x 339 * RETURN cos(k*x) - 1 + ----- on [-PI/4,PI/4], where k = pi/PI, 340 * 2 341 * PI is the rounded value of pi in machine precision : 342 * 343 * Decimal: 344 * pi = 3.141592653589793 23846264338327 ..... 345 * 53 bits PI = 3.141592653589793 115997963 ..... , 346 * 56 bits PI = 3.141592653589793 227020265 ..... , 347 * 348 * Hexadecimal: 349 * pi = 3.243F6A8885A308D313198A2E.... 350 * 53 bits PI = 3.243F6A8885A30 = 2 * 1.921FB54442D18 351 * 56 bits PI = 3.243F6A8885A308 = 4 * .C90FDAA22168C2 352 * 353 * 354 * Method: 355 * 1. Let z=x*x. Create a polynomial approximation to 356 * cos(k*x)-1+z/2 = z*z*(C0 + C1*z^1 + ... + C5*z^5) 357 * then 358 * cos__C(z) = z*z*(C0 + C1*z^1 + ... + C5*z^5) 359 * 360 * The coefficient C's are obtained by a special Remez algorithm. 361 * 362 * Accuracy: 363 * In the absence of rounding error, the approximation has absolute error 364 * less than 2**(-64) for VAX D FORMAT, 2**(-58.3) for IEEE DOUBLE. 365 * 366 * 367 * Constants: 368 * The hexadecimal values are the intended ones for the following constants. 369 * The decimal values may be used, provided that the compiler will convert 370 * from decimal to binary accurately enough to produce the hexadecimal values 371 * shown. 372 * 373 */ 374 375 #if (defined(VAX)||defined(TAHOE)) 376 /*C0 = 4.1666666666666504759E-2 , Hex 2^ -4 * .AAAAAAAAAAA9F0 */ 377 /*C1 = -1.3888888888865302059E-3 , Hex 2^ -9 * -.B60B60B60A0CCA */ 378 /*C2 = 2.4801587285601038265E-5 , Hex 2^-15 * .D00D00CDCD098F */ 379 /*C3 = -2.7557313470902390219E-7 , Hex 2^-21 * -.93F27BB593E805 */ 380 /*C4 = 2.0875623401082232009E-9 , Hex 2^-28 * .8F74C8FA1E3FF0 */ 381 /*C5 = -1.1355178117642986178E-11 ; Hex 2^-36 * -.C7C32D0A5C5A63 */ 382 static long C0x[] = { _0x(aaaa,3e2a), _0x(a9f0,aaaa)}; 383 #define C0 (*(double*)C0x) 384 static long C1x[] = { _0x(0b60,bbb6), _0x(0cca,b60a)}; 385 #define C1 (*(double*)C1x) 386 static long C2x[] = { _0x(0d00,38d0), _0x(098f,cdcd)}; 387 #define C2 (*(double*)C2x) 388 static long C3x[] = { _0x(f27b,b593), _0x(e805,b593)}; 389 #define C3 (*(double*)C3x) 390 static long C4x[] = { _0x(74c8,320f), _0x(3ff0,fa1e)}; 391 #define C4 (*(double*)C4x) 392 static long C5x[] = { _0x(c32d,ae47), _0x(5a63,0a5c)}; 393 #define C5 (*(double*)C5x) 394 #else /* IEEE double */ 395 static double 396 C0 = 4.1666666666666504759E-2 , /*Hex 2^ -5 * 1.555555555553E */ 397 C1 = -1.3888888888865301516E-3 , /*Hex 2^-10 * -1.6C16C16C14199 */ 398 C2 = 2.4801587269650015769E-5 , /*Hex 2^-16 * 1.A01A01971CAEB */ 399 C3 = -2.7557304623183959811E-7 , /*Hex 2^-22 * -1.27E4F1314AD1A */ 400 C4 = 2.0873958177697780076E-9 , /*Hex 2^-29 * 1.1EE3B60DDDC8C */ 401 C5 = -1.1250289076471311557E-11 ; /*Hex 2^-37 * -1.8BD5986B2A52E */ 402 #endif 403 404 static double cos__C(z) 405 double z; 406 { 407 return(z*z*(C0+z*(C1+z*(C2+z*(C3+z*(C4+z*C5)))))); 408 } 409