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[] = "@(#)trig.c 1.2 (Berkeley) 08/22/85"; 16 #endif not lint 17 18 /* SIN(X), COS(X), TAN(X) 19 * RETURN THE SINE, COSINE, AND TANGENT OF X RESPECTIVELY 20 * DOUBLE PRECISION (VAX D format 56 bits, IEEE DOUBLE 53 BITS) 21 * CODED IN C BY K.C. NG, 1/8/85; 22 * REVISED BY W. Kahan and K.C. NG, 8/17/85. 23 * 24 * Required system supported functions: 25 * copysign(x,y) 26 * finite(x) 27 * drem(x,p) 28 * 29 * Static kernel functions: 30 * sin__S(z) ....sin__S(x*x) return (sin(x)-x)/x 31 * cos__C(z) ....cos__C(x*x) return cos(x)-1-x*x/2 32 * 33 * Method. 34 * Let S and C denote the polynomial approximations to sin and cos 35 * respectively on [-PI/4, +PI/4]. 36 * 37 * SIN and COS: 38 * 1. Reduce the argument into [-PI , +PI] by the remainder function. 39 * 2. For x in (-PI,+PI), there are three cases: 40 * case 1: |x| < PI/4 41 * case 2: PI/4 <= |x| < 3PI/4 42 * case 3: 3PI/4 <= |x|. 43 * SIN and COS of x are computed by: 44 * 45 * sin(x) cos(x) remark 46 * ---------------------------------------------------------- 47 * case 1 S(x) C(x) 48 * case 2 sign(x)*C(y) S(y) y=PI/2-|x| 49 * case 3 S(y) -C(y) y=sign(x)*(PI-|x|) 50 * ---------------------------------------------------------- 51 * 52 * TAN: 53 * 1. Reduce the argument into [-PI/2 , +PI/2] by the remainder function. 54 * 2. For x in (-PI/2,+PI/2), there are two cases: 55 * case 1: |x| < PI/4 56 * case 2: PI/4 <= |x| < PI/2 57 * TAN of x is computed by: 58 * 59 * tan (x) remark 60 * ---------------------------------------------------------- 61 * case 1 S(x)/C(x) 62 * case 2 C(y)/S(y) y=sign(x)*(PI/2-|x|) 63 * ---------------------------------------------------------- 64 * 65 * Notes: 66 * 1. S(y) and C(y) were computed by: 67 * S(y) = y+y*sin__S(y*y) 68 * C(y) = 1-(y*y/2-cos__C(x*x)) ... if y*y/2 < thresh, 69 * = 0.5-((y*y/2-0.5)-cos__C(x*x)) ... if y*y/2 >= thresh. 70 * where 71 * thresh = 0.5*(acos(3/4)**2) 72 * 73 * 2. For better accuracy, we use the following formula for S/C for tan 74 * (k=0): let ss=sin__S(y*y), and cc=cos__C(y*y), then 75 * 76 * y+y*ss (y*y/2-cc)+ss 77 * S(y)/C(y) = -------- = y + y * ---------------. 78 * C C 79 * 80 * 81 * Special cases: 82 * Let trig be any of sin, cos, or tan. 83 * trig(+-INF) is NaN, with signals; 84 * trig(NaN) is that NaN; 85 * trig(n*PI/2) is exact for any integer n, provided n*PI is 86 * representable; otherwise, trig(x) is inexact. 87 * 88 * Accuracy: 89 * trig(x) returns the exact trig(x*pi/PI) nearly rounded, where 90 * 91 * Decimal: 92 * pi = 3.141592653589793 23846264338327 ..... 93 * 53 bits PI = 3.141592653589793 115997963 ..... , 94 * 56 bits PI = 3.141592653589793 227020265 ..... , 95 * 96 * Hexadecimal: 97 * pi = 3.243F6A8885A308D313198A2E.... 98 * 53 bits PI = 3.243F6A8885A30 = 2 * 1.921FB54442D18 error=.276ulps 99 * 56 bits PI = 3.243F6A8885A308 = 4 * .C90FDAA22168C2 error=.206ulps 100 * 101 * In a test run with 1,024,000 random arguments on a VAX, the maximum 102 * observed errors (compared with the exact trig(x*pi/PI)) were 103 * tan(x) : 2.09 ulps (around 4.716340404662354) 104 * sin(x) : .861 ulps 105 * cos(x) : .857 ulps 106 * 107 * Constants: 108 * The hexadecimal values are the intended ones for the following constants. 109 * The decimal values may be used, provided that the compiler will convert 110 * from decimal to binary accurately enough to produce the hexadecimal values 111 * shown. 112 */ 113 114 #ifdef VAX 115 /*thresh = 2.6117239648121182150E-1 , Hex 2^ -1 * .85B8636B026EA0 */ 116 /*PIo4 = 7.8539816339744830676E-1 , Hex 2^ 0 * .C90FDAA22168C2 */ 117 /*PIo2 = 1.5707963267948966135E0 , Hex 2^ 1 * .C90FDAA22168C2 */ 118 /*PI3o4 = 2.3561944901923449203E0 , Hex 2^ 2 * .96CBE3F9990E92 */ 119 /*PI = 3.1415926535897932270E0 , Hex 2^ 2 * .C90FDAA22168C2 */ 120 /*PI2 = 6.2831853071795864540E0 ; Hex 2^ 3 * .C90FDAA22168C2 */ 121 static long threshx[] = { 0xb8633f85, 0x6ea06b02}; 122 #define thresh (*(double*)threshx) 123 static long PIo4x[] = { 0x0fda4049, 0x68c2a221}; 124 #define PIo4 (*(double*)PIo4x) 125 static long PIo2x[] = { 0x0fda40c9, 0x68c2a221}; 126 #define PIo2 (*(double*)PIo2x) 127 static long PI3o4x[] = { 0xcbe34116, 0x0e92f999}; 128 #define PI3o4 (*(double*)PI3o4x) 129 static long PIx[] = { 0x0fda4149, 0x68c2a221}; 130 #define PI (*(double*)PIx) 131 static long PI2x[] = { 0x0fda41c9, 0x68c2a221}; 132 #define PI2 (*(double*)PI2x) 133 #else /* IEEE double */ 134 static double 135 thresh = 2.6117239648121182150E-1 , /*Hex 2^ -2 * 1.0B70C6D604DD4 */ 136 PIo4 = 7.8539816339744827900E-1 , /*Hex 2^ -1 * 1.921FB54442D18 */ 137 PIo2 = 1.5707963267948965580E0 , /*Hex 2^ 0 * 1.921FB54442D18 */ 138 PI3o4 = 2.3561944901923448370E0 , /*Hex 2^ 1 * 1.2D97C7F3321D2 */ 139 PI = 3.1415926535897931160E0 , /*Hex 2^ 1 * 1.921FB54442D18 */ 140 PI2 = 6.2831853071795862320E0 ; /*Hex 2^ 2 * 1.921FB54442D18 */ 141 #endif 142 static double zero=0, one=1, negone= -1, half=1.0/2.0, 143 small=1E-10, /* 1+small**2==1; better values for small: 144 small = 1.5E-9 for VAX D 145 = 1.2E-8 for IEEE Double 146 = 2.8E-10 for IEEE Extended */ 147 big=1E20; /* big = 1/(small**2) */ 148 149 double tan(x) 150 double x; 151 { 152 double copysign(),drem(),cos__C(),sin__S(),a,z,ss,cc,c; 153 int finite(),k; 154 155 /* tan(NaN) and tan(INF) must be NaN */ 156 if(!finite(x)) return(x-x); 157 x=drem(x,PI); /* reduce x into [-PI/2, PI/2] */ 158 a=copysign(x,one); /* ... = abs(x) */ 159 if ( a >= PIo4 ) {k=1; x = copysign( PIo2 - a , x ); } 160 else { k=0; if(a < small ) { big + a; return(x); }} 161 162 z = x*x; 163 cc = cos__C(z); 164 ss = sin__S(z); 165 z = z*half ; /* Next get c = cos(x) accurately */ 166 c = (z >= thresh )? half-((z-half)-cc) : one-(z-cc); 167 if (k==0) return ( x + (x*(z-(cc-ss)))/c ); /* sin/cos */ 168 return( c/(x+x*ss) ); /* ... cos/sin */ 169 170 171 } 172 double sin(x) 173 double x; 174 { 175 double copysign(),drem(),sin__S(),cos__C(),a,c,z; 176 int finite(); 177 178 /* sin(NaN) and sin(INF) must be NaN */ 179 if(!finite(x)) return(x-x); 180 x=drem(x,PI2); /* reduce x into [-PI, PI] */ 181 a=copysign(x,one); 182 if( a >= PIo4 ) { 183 if( a >= PI3o4 ) /* .. in [3PI/4, PI ] */ 184 x=copysign((a=PI-a),x); 185 186 else { /* .. in [PI/4, 3PI/4] */ 187 a=PIo2-a; /* return sign(x)*C(PI/2-|x|) */ 188 z=a*a; 189 c=cos__C(z); 190 z=z*half; 191 a=(z>=thresh)?half-((z-half)-c):one-(z-c); 192 return(copysign(a,x)); 193 } 194 } 195 196 /* return S(x) */ 197 if( a < small) { big + a; return(x);} 198 return(x+x*sin__S(x*x)); 199 } 200 201 double cos(x) 202 double x; 203 { 204 double copysign(),drem(),sin__S(),cos__C(),a,c,z,s=1.0; 205 int finite(); 206 207 /* cos(NaN) and cos(INF) must be NaN */ 208 if(!finite(x)) return(x-x); 209 x=drem(x,PI2); /* reduce x into [-PI, PI] */ 210 a=copysign(x,one); 211 if ( a >= PIo4 ) { 212 if ( a >= PI3o4 ) /* .. in [3PI/4, PI ] */ 213 { a=PI-a; s= negone; } 214 215 else /* .. in [PI/4, 3PI/4] */ 216 /* return S(PI/2-|x|) */ 217 { a=PIo2-a; return(a+a*sin__S(a*a));} 218 } 219 220 221 /* return s*C(a) */ 222 if( a < small) { big + a; return(s);} 223 z=a*a; 224 c=cos__C(z); 225 z=z*half; 226 a=(z>=thresh)?half-((z-half)-c):one-(z-c); 227 return(copysign(a,s)); 228 } 229 230 231 /* sin__S(x*x) 232 * DOUBLE PRECISION (VAX D format 56 bits, IEEE DOUBLE 53 BITS) 233 * STATIC KERNEL FUNCTION OF SIN(X), COS(X), AND TAN(X) 234 * CODED IN C BY K.C. NG, 1/21/85; 235 * REVISED BY K.C. NG on 8/13/85. 236 * 237 * sin(x*k) - x 238 * RETURN --------------- on [-PI/4,PI/4] , where k=pi/PI, PI is the rounded 239 * x 240 * value of pi in machine precision: 241 * 242 * Decimal: 243 * pi = 3.141592653589793 23846264338327 ..... 244 * 53 bits PI = 3.141592653589793 115997963 ..... , 245 * 56 bits PI = 3.141592653589793 227020265 ..... , 246 * 247 * Hexadecimal: 248 * pi = 3.243F6A8885A308D313198A2E.... 249 * 53 bits PI = 3.243F6A8885A30 = 2 * 1.921FB54442D18 250 * 56 bits PI = 3.243F6A8885A308 = 4 * .C90FDAA22168C2 251 * 252 * Method: 253 * 1. Let z=x*x. Create a polynomial approximation to 254 * (sin(k*x)-x)/x = z*(S0 + S1*z^1 + ... + S5*z^5). 255 * Then 256 * sin__S(x*x) = z*(S0 + S1*z^1 + ... + S5*z^5) 257 * 258 * The coefficient S's are obtained by a special Remez algorithm. 259 * 260 * Accuracy: 261 * In the absence of rounding error, the approximation has absolute error 262 * less than 2**(-61.11) for VAX D FORMAT, 2**(-57.45) for IEEE DOUBLE. 263 * 264 * Constants: 265 * The hexadecimal values are the intended ones for the following constants. 266 * The decimal values may be used, provided that the compiler will convert 267 * from decimal to binary accurately enough to produce the hexadecimal values 268 * shown. 269 * 270 */ 271 272 #ifdef VAX 273 /*S0 = -1.6666666666666646660E-1 , Hex 2^ -2 * -.AAAAAAAAAAAA71 */ 274 /*S1 = 8.3333333333297230413E-3 , Hex 2^ -6 * .8888888888477F */ 275 /*S2 = -1.9841269838362403710E-4 , Hex 2^-12 * -.D00D00CF8A1057 */ 276 /*S3 = 2.7557318019967078930E-6 , Hex 2^-18 * .B8EF1CA326BEDC */ 277 /*S4 = -2.5051841873876551398E-8 , Hex 2^-25 * -.D73195374CE1D3 */ 278 /*S5 = 1.6028995389845827653E-10 , Hex 2^-32 * .B03D9C6D26CCCC */ 279 /*S6 = -6.2723499671769283121E-13 ; Hex 2^-40 * -.B08D0B7561EA82 */ 280 static long S0x[] = { 0xaaaabf2a, 0xaa71aaaa}; 281 #define S0 (*(double*)S0x) 282 static long S1x[] = { 0x88883d08, 0x477f8888}; 283 #define S1 (*(double*)S1x) 284 static long S2x[] = { 0x0d00ba50, 0x1057cf8a}; 285 #define S2 (*(double*)S2x) 286 static long S3x[] = { 0xef1c3738, 0xbedca326}; 287 #define S3 (*(double*)S3x) 288 static long S4x[] = { 0x3195b3d7, 0xe1d3374c}; 289 #define S4 (*(double*)S4x) 290 static long S5x[] = { 0x3d9c3030, 0xcccc6d26}; 291 #define S5 (*(double*)S5x) 292 static long S6x[] = { 0x8d0bac30, 0xea827561}; 293 #define S6 (*(double*)S6x) 294 #else /* IEEE double */ 295 static double 296 S0 = -1.6666666666666463126E-1 , /*Hex 2^ -3 * -1.555555555550C */ 297 S1 = 8.3333333332992771264E-3 , /*Hex 2^ -7 * 1.111111110C461 */ 298 S2 = -1.9841269816180999116E-4 , /*Hex 2^-13 * -1.A01A019746345 */ 299 S3 = 2.7557309793219876880E-6 , /*Hex 2^-19 * 1.71DE3209CDCD9 */ 300 S4 = -2.5050225177523807003E-8 , /*Hex 2^-26 * -1.AE5C0E319A4EF */ 301 S5 = 1.5868926979889205164E-10 ; /*Hex 2^-33 * 1.5CF61DF672B13 */ 302 #endif 303 304 static double sin__S(z) 305 double z; 306 { 307 #ifdef VAX 308 return(z*(S0+z*(S1+z*(S2+z*(S3+z*(S4+z*(S5+z*S6))))))); 309 #else /* IEEE double */ 310 return(z*(S0+z*(S1+z*(S2+z*(S3+z*(S4+z*S5)))))); 311 #endif 312 } 313 314 315 /* cos__C(x*x) 316 * DOUBLE PRECISION (VAX D FORMAT 56 BITS, IEEE DOUBLE 53 BITS) 317 * STATIC KERNEL FUNCTION OF SIN(X), COS(X), AND TAN(X) 318 * CODED IN C BY K.C. NG, 1/21/85; 319 * REVISED BY K.C. NG on 8/13/85. 320 * 321 * x*x 322 * RETURN cos(k*x) - 1 + ----- on [-PI/4,PI/4], where k = pi/PI, 323 * 2 324 * PI is the rounded value of pi in machine precision : 325 * 326 * Decimal: 327 * pi = 3.141592653589793 23846264338327 ..... 328 * 53 bits PI = 3.141592653589793 115997963 ..... , 329 * 56 bits PI = 3.141592653589793 227020265 ..... , 330 * 331 * Hexadecimal: 332 * pi = 3.243F6A8885A308D313198A2E.... 333 * 53 bits PI = 3.243F6A8885A30 = 2 * 1.921FB54442D18 334 * 56 bits PI = 3.243F6A8885A308 = 4 * .C90FDAA22168C2 335 * 336 * 337 * Method: 338 * 1. Let z=x*x. Create a polynomial approximation to 339 * cos(k*x)-1+z/2 = z*z*(C0 + C1*z^1 + ... + C5*z^5) 340 * then 341 * cos__C(z) = z*z*(C0 + C1*z^1 + ... + C5*z^5) 342 * 343 * The coefficient C's are obtained by a special Remez algorithm. 344 * 345 * Accuracy: 346 * In the absence of rounding error, the approximation has absolute error 347 * less than 2**(-64) for VAX D FORMAT, 2**(-58.3) for IEEE DOUBLE. 348 * 349 * 350 * Constants: 351 * The hexadecimal values are the intended ones for the following constants. 352 * The decimal values may be used, provided that the compiler will convert 353 * from decimal to binary accurately enough to produce the hexadecimal values 354 * shown. 355 * 356 */ 357 358 #ifdef VAX 359 /*C0 = 4.1666666666666504759E-2 , Hex 2^ -4 * .AAAAAAAAAAA9F0 */ 360 /*C1 = -1.3888888888865302059E-3 , Hex 2^ -9 * -.B60B60B60A0CCA */ 361 /*C2 = 2.4801587285601038265E-5 , Hex 2^-15 * .D00D00CDCD098F */ 362 /*C3 = -2.7557313470902390219E-7 , Hex 2^-21 * -.93F27BB593E805 */ 363 /*C4 = 2.0875623401082232009E-9 , Hex 2^-28 * .8F74C8FA1E3FF0 */ 364 /*C5 = -1.1355178117642986178E-11 ; Hex 2^-36 * -.C7C32D0A5C5A63 */ 365 static long C0x[] = { 0xaaaa3e2a, 0xa9f0aaaa}; 366 #define C0 (*(double*)C0x) 367 static long C1x[] = { 0x0b60bbb6, 0x0ccab60a}; 368 #define C1 (*(double*)C1x) 369 static long C2x[] = { 0x0d0038d0, 0x098fcdcd}; 370 #define C2 (*(double*)C2x) 371 static long C3x[] = { 0xf27bb593, 0xe805b593}; 372 #define C3 (*(double*)C3x) 373 static long C4x[] = { 0x74c8320f, 0x3ff0fa1e}; 374 #define C4 (*(double*)C4x) 375 static long C5x[] = { 0xc32dae47, 0x5a630a5c}; 376 #define C5 (*(double*)C5x) 377 #else /* IEEE double */ 378 static double 379 C0 = 4.1666666666666504759E-2 , /*Hex 2^ -5 * 1.555555555553E */ 380 C1 = -1.3888888888865301516E-3 , /*Hex 2^-10 * -1.6C16C16C14199 */ 381 C2 = 2.4801587269650015769E-5 , /*Hex 2^-16 * 1.A01A01971CAEB */ 382 C3 = -2.7557304623183959811E-7 , /*Hex 2^-22 * -1.27E4F1314AD1A */ 383 C4 = 2.0873958177697780076E-9 , /*Hex 2^-29 * 1.1EE3B60DDDC8C */ 384 C5 = -1.1250289076471311557E-11 ; /*Hex 2^-37 * -1.8BD5986B2A52E */ 385 #endif 386 387 static double cos__C(z) 388 double z; 389 { 390 return(z*z*(C0+z*(C1+z*(C2+z*(C3+z*(C4+z*C5)))))); 391 } 392