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Annotation of 43BSD/usr.lib/libm/IEEE/atan2.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[] = "@(#)atan2.c    1.3 (Berkeley) 8/21/85";
        !            16: #endif not lint
        !            17: 
        !            18: /* ATAN2(Y,X)
        !            19:  * RETURN ARG (X+iY)
        !            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 K.C. NG on 2/7/85, 2/13/85, 3/7/85, 3/30/85, 6/29/85.
        !            23:  *
        !            24:  * Required system supported functions :
        !            25:  *     copysign(x,y)
        !            26:  *     scalb(x,y)
        !            27:  *     logb(x)
        !            28:  *     
        !            29:  * Method :
        !            30:  *     1. Reduce y to positive by atan2(y,x)=-atan2(-y,x).
        !            31:  *     2. Reduce x to positive by (if x and y are unexceptional): 
        !            32:  *             ARG (x+iy) = arctan(y/x)           ... if x > 0,
        !            33:  *             ARG (x+iy) = pi - arctan[y/(-x)]   ... if x < 0,
        !            34:  *     3. According to the integer k=4t+0.25 truncated , t=y/x, the argument 
        !            35:  *        is further reduced to one of the following intervals and the 
        !            36:  *        arctangent of y/x is evaluated by the corresponding formula:
        !            37:  *
        !            38:  *         [0,7/16]       atan(y/x) = t - t^3*(a1+t^2*(a2+...(a10+t^2*a11)...)
        !            39:  *        [7/16,11/16]    atan(y/x) = atan(1/2) + atan( (y-x/2)/(x+y/2) )
        !            40:  *        [11/16.19/16]   atan(y/x) = atan( 1 ) + atan( (y-x)/(x+y) )
        !            41:  *        [19/16,39/16]   atan(y/x) = atan(3/2) + atan( (y-1.5x)/(x+1.5y) )
        !            42:  *        [39/16,INF]     atan(y/x) = atan(INF) + atan( -x/y )
        !            43:  *
        !            44:  * Special cases:
        !            45:  * Notations: atan2(y,x) == ARG (x+iy) == ARG(x,y).
        !            46:  *
        !            47:  *     ARG( NAN , (anything) ) is NaN;
        !            48:  *     ARG( (anything), NaN ) is NaN;
        !            49:  *     ARG(+(anything but NaN), +-0) is +-0  ;
        !            50:  *     ARG(-(anything but NaN), +-0) is +-PI ;
        !            51:  *     ARG( 0, +-(anything but 0 and NaN) ) is +-PI/2;
        !            52:  *     ARG( +INF,+-(anything but INF and NaN) ) is +-0 ;
        !            53:  *     ARG( -INF,+-(anything but INF and NaN) ) is +-PI;
        !            54:  *     ARG( +INF,+-INF ) is +-PI/4 ;
        !            55:  *     ARG( -INF,+-INF ) is +-3PI/4;
        !            56:  *     ARG( (anything but,0,NaN, and INF),+-INF ) is +-PI/2;
        !            57:  *
        !            58:  * Accuracy:
        !            59:  *     atan2(y,x) returns (PI/pi) * the exact ARG (x+iy) nearly rounded, 
        !            60:  *     where
        !            61:  *
        !            62:  *     in decimal:
        !            63:  *             pi = 3.141592653589793 23846264338327 ..... 
        !            64:  *    53 bits   PI = 3.141592653589793 115997963 ..... ,
        !            65:  *    56 bits   PI = 3.141592653589793 227020265 ..... ,  
        !            66:  *
        !            67:  *     in hexadecimal:
        !            68:  *             pi = 3.243F6A8885A308D313198A2E....
        !            69:  *    53 bits   PI = 3.243F6A8885A30  =  2 * 1.921FB54442D18   error=.276ulps
        !            70:  *    56 bits   PI = 3.243F6A8885A308 =  4 * .C90FDAA22168C2    error=.206ulps
        !            71:  *     
        !            72:  *     In a test run with 356,000 random argument on [-1,1] * [-1,1] on a
        !            73:  *     VAX, the maximum observed error was 1.41 ulps (units of the last place)
        !            74:  *     compared with (PI/pi)*(the exact ARG(x+iy)).
        !            75:  *
        !            76:  * Note:
        !            77:  *     We use machine PI (the true pi rounded) in place of the actual
        !            78:  *     value of pi for all the trig and inverse trig functions. In general, 
        !            79:  *     if trig is one of sin, cos, tan, then computed trig(y) returns the 
        !            80:  *     exact trig(y*pi/PI) nearly rounded; correspondingly, computed arctrig 
        !            81:  *     returns the exact arctrig(y)*PI/pi nearly rounded. These guarantee the 
        !            82:  *     trig functions have period PI, and trig(arctrig(x)) returns x for
        !            83:  *     all critical values x.
        !            84:  *     
        !            85:  * Constants:
        !            86:  * The hexadecimal values are the intended ones for the following constants.
        !            87:  * The decimal values may be used, provided that the compiler will convert
        !            88:  * from decimal to binary accurately enough to produce the hexadecimal values
        !            89:  * shown.
        !            90:  */
        !            91: 
        !            92: static double 
        !            93: #ifdef VAX     /* VAX D format */
        !            94: athfhi =  4.6364760900080611433E-1    , /*Hex  2^ -1   *  .ED63382B0DDA7B */
        !            95: athflo =  1.9338828231967579916E-19   , /*Hex  2^-62   *  .E450059CFE92C0 */
        !            96: PIo4   =  7.8539816339744830676E-1    , /*Hex  2^  0   *  .C90FDAA22168C2 */   
        !            97: at1fhi =  9.8279372324732906796E-1    , /*Hex  2^  0   *  .FB985E940FB4D9 */
        !            98: at1flo = -3.5540295636764633916E-18   , /*Hex  2^-57   * -.831EDC34D6EAEA */
        !            99: PIo2   =  1.5707963267948966135E0     , /*Hex  2^  1   *  .C90FDAA22168C2 */
        !           100: PI     =  3.1415926535897932270E0     , /*Hex  2^  2   *  .C90FDAA22168C2 */
        !           101: a1     =  3.3333333333333473730E-1    , /*Hex  2^ -1   *  .AAAAAAAAAAAB75 */
        !           102: a2     = -2.0000000000017730678E-1    , /*Hex  2^ -2   * -.CCCCCCCCCD946E */
        !           103: a3     =  1.4285714286694640301E-1    , /*Hex  2^ -2   *  .92492492744262 */
        !           104: a4     = -1.1111111135032672795E-1    , /*Hex  2^ -3   * -.E38E38EBC66292 */
        !           105: a5     =  9.0909091380563043783E-2    , /*Hex  2^ -3   *  .BA2E8BB31BD70C */
        !           106: a6     = -7.6922954286089459397E-2    , /*Hex  2^ -3   * -.9D89C827C37F18 */
        !           107: a7     =  6.6663180891693915586E-2    , /*Hex  2^ -3   *  .8886B4AE379E58 */
        !           108: a8     = -5.8772703698290408927E-2    , /*Hex  2^ -4   * -.F0BBA58481A942 */
        !           109: a9     =  5.2170707402812969804E-2    , /*Hex  2^ -4   *  .D5B0F3A1AB13AB */
        !           110: a10    = -4.4895863157820361210E-2    , /*Hex  2^ -4   * -.B7E4B97FD1048F */
        !           111: a11    =  3.3006147437343875094E-2    , /*Hex  2^ -4   *  .8731743CF72D87 */
        !           112: a12    = -1.4614844866464185439E-2    ; /*Hex  2^ -6   * -.EF731A2F3476D9 */
        !           113: #else  /* IEEE double */
        !           114: athfhi =  4.6364760900080609352E-1    , /*Hex  2^ -2   *  1.DAC670561BB4F */
        !           115: athflo =  4.6249969567426939759E-18   , /*Hex  2^-58   *  1.5543B8F253271 */
        !           116: PIo4   =  7.8539816339744827900E-1    , /*Hex  2^ -1   *  1.921FB54442D18 */
        !           117: at1fhi =  9.8279372324732905408E-1    , /*Hex  2^ -1   *  1.F730BD281F69B */
        !           118: at1flo = -2.4407677060164810007E-17   , /*Hex  2^-56   * -1.C23DFEFEAE6B5 */
        !           119: PIo2   =  1.5707963267948965580E0     , /*Hex  2^  0   *  1.921FB54442D18 */
        !           120: PI     =  3.1415926535897931160E0     , /*Hex  2^  1   *  1.921FB54442D18 */
        !           121: a1     =  3.3333333333333942106E-1    , /*Hex  2^ -2   *  1.55555555555C3 */
        !           122: a2     = -1.9999999999979536924E-1    , /*Hex  2^ -3   * -1.9999999997CCD */
        !           123: a3     =  1.4285714278004377209E-1    , /*Hex  2^ -3   *  1.24924921EC1D7 */
        !           124: a4     = -1.1111110579344973814E-1    , /*Hex  2^ -4   * -1.C71C7059AF280 */
        !           125: a5     =  9.0908906105474668324E-2    , /*Hex  2^ -4   *  1.745CE5AA35DB2 */
        !           126: a6     = -7.6919217767468239799E-2    , /*Hex  2^ -4   * -1.3B0FA54BEC400 */
        !           127: a7     =  6.6614695906082474486E-2    , /*Hex  2^ -4   *  1.10DA924597FFF */
        !           128: a8     = -5.8358371008508623523E-2    , /*Hex  2^ -5   * -1.DE125FDDBD793 */
        !           129: a9     =  4.9850617156082015213E-2    , /*Hex  2^ -5   *  1.9860524BDD807 */
        !           130: a10    = -3.6700606902093604877E-2    , /*Hex  2^ -5   * -1.2CA6C04C6937A */
        !           131: a11    =  1.6438029044759730479E-2    ; /*Hex  2^ -6   *  1.0D52174A1BB54 */
        !           132: #endif
        !           133: 
        !           134: double atan2(y,x)
        !           135: double  y,x;
        !           136: {  
        !           137:        static double zero=0, one=1, small=1.0E-9, big=1.0E18;
        !           138:        double copysign(),logb(),scalb(),t,z,signy,signx,hi,lo;
        !           139:        int finite(), k,m;
        !           140: 
        !           141:     /* if x or y is NAN */
        !           142:        if(x!=x) return(x); if(y!=y) return(y);
        !           143: 
        !           144:     /* copy down the sign of y and x */
        !           145:        signy = copysign(one,y) ;  
        !           146:        signx = copysign(one,x) ;  
        !           147: 
        !           148:     /* if x is 1.0, goto begin */
        !           149:        if(x==1) { y=copysign(y,one); t=y; if(finite(t)) goto begin;}
        !           150: 
        !           151:     /* when y = 0 */
        !           152:        if(y==zero) return((signx==one)?y:copysign(PI,signy));
        !           153: 
        !           154:     /* when x = 0 */
        !           155:        if(x==zero) return(copysign(PIo2,signy));
        !           156:            
        !           157:     /* when x is INF */
        !           158:        if(!finite(x))
        !           159:            if(!finite(y)) 
        !           160:                return(copysign((signx==one)?PIo4:3*PIo4,signy));
        !           161:            else
        !           162:                return(copysign((signx==one)?zero:PI,signy));
        !           163: 
        !           164:     /* when y is INF */
        !           165:        if(!finite(y)) return(copysign(PIo2,signy));
        !           166: 
        !           167: 
        !           168:     /* compute y/x */
        !           169:        x=copysign(x,one); 
        !           170:        y=copysign(y,one); 
        !           171:        if((m=(k=logb(y))-logb(x)) > 60) t=big+big; 
        !           172:            else if(m < -80 ) t=y/x;
        !           173:            else { t = y/x ; y = scalb(y,-k); x=scalb(x,-k); }
        !           174: 
        !           175:     /* begin argument reduction */
        !           176: begin:
        !           177:        if (t < 2.4375) {                
        !           178: 
        !           179:        /* truncate 4(t+1/16) to integer for branching */
        !           180:            k = 4 * (t+0.0625);
        !           181:            switch (k) {
        !           182: 
        !           183:            /* t is in [0,7/16] */
        !           184:            case 0:                    
        !           185:            case 1:
        !           186:                if (t < small) 
        !           187:                    { big + small ;  /* raise inexact flag */
        !           188:                      return (copysign((signx>zero)?t:PI-t,signy)); }
        !           189: 
        !           190:                hi = zero;  lo = zero;  break;
        !           191: 
        !           192:            /* t is in [7/16,11/16] */
        !           193:            case 2:                    
        !           194:                hi = athfhi; lo = athflo;
        !           195:                z = x+x;
        !           196:                t = ( (y+y) - x ) / ( z +  y ); break;
        !           197: 
        !           198:            /* t is in [11/16,19/16] */
        !           199:            case 3:                    
        !           200:            case 4:
        !           201:                hi = PIo4; lo = zero;
        !           202:                t = ( y - x ) / ( x + y ); break;
        !           203: 
        !           204:            /* t is in [19/16,39/16] */
        !           205:            default:                   
        !           206:                hi = at1fhi; lo = at1flo;
        !           207:                z = y-x; y=y+y+y; t = x+x;
        !           208:                t = ( (z+z)-x ) / ( t + y ); break;
        !           209:            }
        !           210:        }
        !           211:        /* end of if (t < 2.4375) */
        !           212: 
        !           213:        else                           
        !           214:        {
        !           215:            hi = PIo2; lo = zero;
        !           216: 
        !           217:            /* t is in [2.4375, big] */
        !           218:            if (t <= big)  t = - x / y;
        !           219: 
        !           220:            /* t is in [big, INF] */
        !           221:            else          
        !           222:              { big+small;      /* raise inexact flag */
        !           223:                t = zero; }
        !           224:        }
        !           225:     /* end of argument reduction */
        !           226: 
        !           227:     /* compute atan(t) for t in [-.4375, .4375] */
        !           228:        z = t*t;
        !           229: #ifdef VAX
        !           230:        z = t*(z*(a1+z*(a2+z*(a3+z*(a4+z*(a5+z*(a6+z*(a7+z*(a8+
        !           231:                        z*(a9+z*(a10+z*(a11+z*a12))))))))))));
        !           232: #else  /* IEEE double */
        !           233:        z = t*(z*(a1+z*(a2+z*(a3+z*(a4+z*(a5+z*(a6+z*(a7+z*(a8+
        !           234:                        z*(a9+z*(a10+z*a11)))))))))));
        !           235: #endif
        !           236:        z = lo - z; z += t; z += hi;
        !           237: 
        !           238:        return(copysign((signx>zero)?z:PI-z,signy));
        !           239: }

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