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; 1.2 (ucb.elefunt) 09/12/85"; 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 #ifdef VAX 116 /*thresh = 2.6117239648121182150E-1 , Hex 2^ -1 * .85B8636B026EA0 */ 117 /*PIo4 = 7.8539816339744830676E-1 , Hex 2^ 0 * .C90FDAA22168C2 */ 118 /*PIo2 = 1.5707963267948966135E0 , Hex 2^ 1 * .C90FDAA22168C2 */ 119 /*PI3o4 = 2.3561944901923449203E0 , Hex 2^ 2 * .96CBE3F9990E92 */ 120 /*PI = 3.1415926535897932270E0 , Hex 2^ 2 * .C90FDAA22168C2 */ 121 /*PI2 = 6.2831853071795864540E0 ; Hex 2^ 3 * .C90FDAA22168C2 */ 122 static long threshx[] = { 0xb8633f85, 0x6ea06b02}; 123 #define thresh (*(double*)threshx) 124 static long PIo4x[] = { 0x0fda4049, 0x68c2a221}; 125 #define PIo4 (*(double*)PIo4x) 126 static long PIo2x[] = { 0x0fda40c9, 0x68c2a221}; 127 #define PIo2 (*(double*)PIo2x) 128 static long PI3o4x[] = { 0xcbe34116, 0x0e92f999}; 129 #define PI3o4 (*(double*)PI3o4x) 130 static long PIx[] = { 0x0fda4149, 0x68c2a221}; 131 #define PI (*(double*)PIx) 132 static long PI2x[] = { 0x0fda41c9, 0x68c2a221}; 133 #define PI2 (*(double*)PI2x) 134 #else /* IEEE double */ 135 static double 136 thresh = 2.6117239648121182150E-1 , /*Hex 2^ -2 * 1.0B70C6D604DD4 */ 137 PIo4 = 7.8539816339744827900E-1 , /*Hex 2^ -1 * 1.921FB54442D18 */ 138 PIo2 = 1.5707963267948965580E0 , /*Hex 2^ 0 * 1.921FB54442D18 */ 139 PI3o4 = 2.3561944901923448370E0 , /*Hex 2^ 1 * 1.2D97C7F3321D2 */ 140 PI = 3.1415926535897931160E0 , /*Hex 2^ 1 * 1.921FB54442D18 */ 141 PI2 = 6.2831853071795862320E0 ; /*Hex 2^ 2 * 1.921FB54442D18 */ 142 #endif 143 static double zero=0, one=1, negone= -1, half=1.0/2.0, 144 small=1E-10, /* 1+small**2==1; better values for small: 145 small = 1.5E-9 for VAX D 146 = 1.2E-8 for IEEE Double 147 = 2.8E-10 for IEEE Extended */ 148 big=1E20; /* big = 1/(small**2) */ 149 150 double tan(x) 151 double x; 152 { 153 double copysign(),drem(),cos__C(),sin__S(),a,z,ss,cc,c; 154 int finite(),k; 155 156 /* tan(NaN) and tan(INF) must be NaN */ 157 if(!finite(x)) return(x-x); 158 x=drem(x,PI); /* reduce x into [-PI/2, PI/2] */ 159 a=copysign(x,one); /* ... = abs(x) */ 160 if ( a >= PIo4 ) {k=1; x = copysign( PIo2 - a , x ); } 161 else { k=0; if(a < small ) { big + a; return(x); }} 162 163 z = x*x; 164 cc = cos__C(z); 165 ss = sin__S(z); 166 z = z*half ; /* Next get c = cos(x) accurately */ 167 c = (z >= thresh )? half-((z-half)-cc) : one-(z-cc); 168 if (k==0) return ( x + (x*(z-(cc-ss)))/c ); /* sin/cos */ 169 return( c/(x+x*ss) ); /* ... cos/sin */ 170 171 172 } 173 double sin(x) 174 double x; 175 { 176 double copysign(),drem(),sin__S(),cos__C(),a,c,z; 177 int finite(); 178 179 /* sin(NaN) and sin(INF) must be NaN */ 180 if(!finite(x)) return(x-x); 181 x=drem(x,PI2); /* reduce x into [-PI, PI] */ 182 a=copysign(x,one); 183 if( a >= PIo4 ) { 184 if( a >= PI3o4 ) /* .. in [3PI/4, PI ] */ 185 x=copysign((a=PI-a),x); 186 187 else { /* .. in [PI/4, 3PI/4] */ 188 a=PIo2-a; /* return sign(x)*C(PI/2-|x|) */ 189 z=a*a; 190 c=cos__C(z); 191 z=z*half; 192 a=(z>=thresh)?half-((z-half)-c):one-(z-c); 193 return(copysign(a,x)); 194 } 195 } 196 197 /* return S(x) */ 198 if( a < small) { big + a; return(x);} 199 return(x+x*sin__S(x*x)); 200 } 201 202 double cos(x) 203 double x; 204 { 205 double copysign(),drem(),sin__S(),cos__C(),a,c,z,s=1.0; 206 int finite(); 207 208 /* cos(NaN) and cos(INF) must be NaN */ 209 if(!finite(x)) return(x-x); 210 x=drem(x,PI2); /* reduce x into [-PI, PI] */ 211 a=copysign(x,one); 212 if ( a >= PIo4 ) { 213 if ( a >= PI3o4 ) /* .. in [3PI/4, PI ] */ 214 { a=PI-a; s= negone; } 215 216 else /* .. in [PI/4, 3PI/4] */ 217 /* return S(PI/2-|x|) */ 218 { a=PIo2-a; return(a+a*sin__S(a*a));} 219 } 220 221 222 /* return s*C(a) */ 223 if( a < small) { big + a; return(s);} 224 z=a*a; 225 c=cos__C(z); 226 z=z*half; 227 a=(z>=thresh)?half-((z-half)-c):one-(z-c); 228 return(copysign(a,s)); 229 } 230 231 232 /* sin__S(x*x) 233 * DOUBLE PRECISION (VAX D format 56 bits, IEEE DOUBLE 53 BITS) 234 * STATIC KERNEL FUNCTION OF SIN(X), COS(X), AND TAN(X) 235 * CODED IN C BY K.C. NG, 1/21/85; 236 * REVISED BY K.C. NG on 8/13/85. 237 * 238 * sin(x*k) - x 239 * RETURN --------------- on [-PI/4,PI/4] , where k=pi/PI, PI is the rounded 240 * x 241 * value of pi in machine precision: 242 * 243 * Decimal: 244 * pi = 3.141592653589793 23846264338327 ..... 245 * 53 bits PI = 3.141592653589793 115997963 ..... , 246 * 56 bits PI = 3.141592653589793 227020265 ..... , 247 * 248 * Hexadecimal: 249 * pi = 3.243F6A8885A308D313198A2E.... 250 * 53 bits PI = 3.243F6A8885A30 = 2 * 1.921FB54442D18 251 * 56 bits PI = 3.243F6A8885A308 = 4 * .C90FDAA22168C2 252 * 253 * Method: 254 * 1. Let z=x*x. Create a polynomial approximation to 255 * (sin(k*x)-x)/x = z*(S0 + S1*z^1 + ... + S5*z^5). 256 * Then 257 * sin__S(x*x) = z*(S0 + S1*z^1 + ... + S5*z^5) 258 * 259 * The coefficient S's are obtained by a special Remez algorithm. 260 * 261 * Accuracy: 262 * In the absence of rounding error, the approximation has absolute error 263 * less than 2**(-61.11) for VAX D FORMAT, 2**(-57.45) for IEEE DOUBLE. 264 * 265 * Constants: 266 * The hexadecimal values are the intended ones for the following constants. 267 * The decimal values may be used, provided that the compiler will convert 268 * from decimal to binary accurately enough to produce the hexadecimal values 269 * shown. 270 * 271 */ 272 273 #ifdef VAX 274 /*S0 = -1.6666666666666646660E-1 , Hex 2^ -2 * -.AAAAAAAAAAAA71 */ 275 /*S1 = 8.3333333333297230413E-3 , Hex 2^ -6 * .8888888888477F */ 276 /*S2 = -1.9841269838362403710E-4 , Hex 2^-12 * -.D00D00CF8A1057 */ 277 /*S3 = 2.7557318019967078930E-6 , Hex 2^-18 * .B8EF1CA326BEDC */ 278 /*S4 = -2.5051841873876551398E-8 , Hex 2^-25 * -.D73195374CE1D3 */ 279 /*S5 = 1.6028995389845827653E-10 , Hex 2^-32 * .B03D9C6D26CCCC */ 280 /*S6 = -6.2723499671769283121E-13 ; Hex 2^-40 * -.B08D0B7561EA82 */ 281 static long S0x[] = { 0xaaaabf2a, 0xaa71aaaa}; 282 #define S0 (*(double*)S0x) 283 static long S1x[] = { 0x88883d08, 0x477f8888}; 284 #define S1 (*(double*)S1x) 285 static long S2x[] = { 0x0d00ba50, 0x1057cf8a}; 286 #define S2 (*(double*)S2x) 287 static long S3x[] = { 0xef1c3738, 0xbedca326}; 288 #define S3 (*(double*)S3x) 289 static long S4x[] = { 0x3195b3d7, 0xe1d3374c}; 290 #define S4 (*(double*)S4x) 291 static long S5x[] = { 0x3d9c3030, 0xcccc6d26}; 292 #define S5 (*(double*)S5x) 293 static long S6x[] = { 0x8d0bac30, 0xea827561}; 294 #define S6 (*(double*)S6x) 295 #else /* IEEE double */ 296 static double 297 S0 = -1.6666666666666463126E-1 , /*Hex 2^ -3 * -1.555555555550C */ 298 S1 = 8.3333333332992771264E-3 , /*Hex 2^ -7 * 1.111111110C461 */ 299 S2 = -1.9841269816180999116E-4 , /*Hex 2^-13 * -1.A01A019746345 */ 300 S3 = 2.7557309793219876880E-6 , /*Hex 2^-19 * 1.71DE3209CDCD9 */ 301 S4 = -2.5050225177523807003E-8 , /*Hex 2^-26 * -1.AE5C0E319A4EF */ 302 S5 = 1.5868926979889205164E-10 ; /*Hex 2^-33 * 1.5CF61DF672B13 */ 303 #endif 304 305 static double sin__S(z) 306 double z; 307 { 308 #ifdef VAX 309 return(z*(S0+z*(S1+z*(S2+z*(S3+z*(S4+z*(S5+z*S6))))))); 310 #else /* IEEE double */ 311 return(z*(S0+z*(S1+z*(S2+z*(S3+z*(S4+z*S5)))))); 312 #endif 313 } 314 315 316 /* cos__C(x*x) 317 * DOUBLE PRECISION (VAX D FORMAT 56 BITS, IEEE DOUBLE 53 BITS) 318 * STATIC KERNEL FUNCTION OF SIN(X), COS(X), AND TAN(X) 319 * CODED IN C BY K.C. NG, 1/21/85; 320 * REVISED BY K.C. NG on 8/13/85. 321 * 322 * x*x 323 * RETURN cos(k*x) - 1 + ----- on [-PI/4,PI/4], where k = pi/PI, 324 * 2 325 * PI is the rounded value of pi in machine precision : 326 * 327 * Decimal: 328 * pi = 3.141592653589793 23846264338327 ..... 329 * 53 bits PI = 3.141592653589793 115997963 ..... , 330 * 56 bits PI = 3.141592653589793 227020265 ..... , 331 * 332 * Hexadecimal: 333 * pi = 3.243F6A8885A308D313198A2E.... 334 * 53 bits PI = 3.243F6A8885A30 = 2 * 1.921FB54442D18 335 * 56 bits PI = 3.243F6A8885A308 = 4 * .C90FDAA22168C2 336 * 337 * 338 * Method: 339 * 1. Let z=x*x. Create a polynomial approximation to 340 * cos(k*x)-1+z/2 = z*z*(C0 + C1*z^1 + ... + C5*z^5) 341 * then 342 * cos__C(z) = z*z*(C0 + C1*z^1 + ... + C5*z^5) 343 * 344 * The coefficient C's are obtained by a special Remez algorithm. 345 * 346 * Accuracy: 347 * In the absence of rounding error, the approximation has absolute error 348 * less than 2**(-64) for VAX D FORMAT, 2**(-58.3) for IEEE DOUBLE. 349 * 350 * 351 * Constants: 352 * The hexadecimal values are the intended ones for the following constants. 353 * The decimal values may be used, provided that the compiler will convert 354 * from decimal to binary accurately enough to produce the hexadecimal values 355 * shown. 356 * 357 */ 358 359 #ifdef VAX 360 /*C0 = 4.1666666666666504759E-2 , Hex 2^ -4 * .AAAAAAAAAAA9F0 */ 361 /*C1 = -1.3888888888865302059E-3 , Hex 2^ -9 * -.B60B60B60A0CCA */ 362 /*C2 = 2.4801587285601038265E-5 , Hex 2^-15 * .D00D00CDCD098F */ 363 /*C3 = -2.7557313470902390219E-7 , Hex 2^-21 * -.93F27BB593E805 */ 364 /*C4 = 2.0875623401082232009E-9 , Hex 2^-28 * .8F74C8FA1E3FF0 */ 365 /*C5 = -1.1355178117642986178E-11 ; Hex 2^-36 * -.C7C32D0A5C5A63 */ 366 static long C0x[] = { 0xaaaa3e2a, 0xa9f0aaaa}; 367 #define C0 (*(double*)C0x) 368 static long C1x[] = { 0x0b60bbb6, 0x0ccab60a}; 369 #define C1 (*(double*)C1x) 370 static long C2x[] = { 0x0d0038d0, 0x098fcdcd}; 371 #define C2 (*(double*)C2x) 372 static long C3x[] = { 0xf27bb593, 0xe805b593}; 373 #define C3 (*(double*)C3x) 374 static long C4x[] = { 0x74c8320f, 0x3ff0fa1e}; 375 #define C4 (*(double*)C4x) 376 static long C5x[] = { 0xc32dae47, 0x5a630a5c}; 377 #define C5 (*(double*)C5x) 378 #else /* IEEE double */ 379 static double 380 C0 = 4.1666666666666504759E-2 , /*Hex 2^ -5 * 1.555555555553E */ 381 C1 = -1.3888888888865301516E-3 , /*Hex 2^-10 * -1.6C16C16C14199 */ 382 C2 = 2.4801587269650015769E-5 , /*Hex 2^-16 * 1.A01A01971CAEB */ 383 C3 = -2.7557304623183959811E-7 , /*Hex 2^-22 * -1.27E4F1314AD1A */ 384 C4 = 2.0873958177697780076E-9 , /*Hex 2^-29 * 1.1EE3B60DDDC8C */ 385 C5 = -1.1250289076471311557E-11 ; /*Hex 2^-37 * -1.8BD5986B2A52E */ 386 #endif 387 388 static double cos__C(z) 389 double z; 390 { 391 return(z*z*(C0+z*(C1+z*(C2+z*(C3+z*(C4+z*C5)))))); 392 } 393