4.3BSD - /usr/src/usr.lib/libm/IEEE/trig.c

   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) 8/22/85";
  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: }

Defined functions

Defined variables

Defined macros