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Annotation of 43BSD/usr.lib/libm/IEEE/trig.c, revision 1.1

1.1     ! root        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: }

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