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      1 /* @(#)s_atan.c 5.1 93/09/24 */
      2 /*
      3  * ====================================================
      4  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
      5  *
      6  * Developed at SunPro, a Sun Microsystems, Inc. business.
      7  * Permission to use, copy, modify, and distribute this
      8  * software is freely granted, provided that this notice
      9  * is preserved.
     10  * ====================================================
     11  */
     12 
     13 #include <sys/cdefs.h>
     14 __FBSDID("$FreeBSD$");
     15 
     16 /* atan(x)
     17  * Method
     18  *   1. Reduce x to positive by atan(x) = -atan(-x).
     19  *   2. According to the integer k=4t+0.25 chopped, t=x, the argument
     20  *      is further reduced to one of the following intervals and the
     21  *      arctangent of t is evaluated by the corresponding formula:
     22  *
     23  *      [0,7/16]      atan(x) = t-t^3*(a1+t^2*(a2+...(a10+t^2*a11)...)
     24  *      [7/16,11/16]  atan(x) = atan(1/2) + atan( (t-0.5)/(1+t/2) )
     25  *      [11/16.19/16] atan(x) = atan( 1 ) + atan( (t-1)/(1+t) )
     26  *      [19/16,39/16] atan(x) = atan(3/2) + atan( (t-1.5)/(1+1.5t) )
     27  *      [39/16,INF]   atan(x) = atan(INF) + atan( -1/t )
     28  *
     29  * Constants:
     30  * The hexadecimal values are the intended ones for the following
     31  * constants. The decimal values may be used, provided that the
     32  * compiler will convert from decimal to binary accurately enough
     33  * to produce the hexadecimal values shown.
     34  */
     35 
     36 #include <float.h>
     37 
     38 #include "math.h"
     39 #include "math_private.h"
     40 
     41 static const double atanhi[] = {
     42   4.63647609000806093515e-01, /* atan(0.5)hi 0x3FDDAC67, 0x0561BB4F */
     43   7.85398163397448278999e-01, /* atan(1.0)hi 0x3FE921FB, 0x54442D18 */
     44   9.82793723247329054082e-01, /* atan(1.5)hi 0x3FEF730B, 0xD281F69B */
     45   1.57079632679489655800e+00, /* atan(inf)hi 0x3FF921FB, 0x54442D18 */
     46 };
     47 
     48 static const double atanlo[] = {
     49   2.26987774529616870924e-17, /* atan(0.5)lo 0x3C7A2B7F, 0x222F65E2 */
     50   3.06161699786838301793e-17, /* atan(1.0)lo 0x3C81A626, 0x33145C07 */
     51   1.39033110312309984516e-17, /* atan(1.5)lo 0x3C700788, 0x7AF0CBBD */
     52   6.12323399573676603587e-17, /* atan(inf)lo 0x3C91A626, 0x33145C07 */
     53 };
     54 
     55 static const double aT[] = {
     56   3.33333333333329318027e-01, /* 0x3FD55555, 0x5555550D */
     57  -1.99999999998764832476e-01, /* 0xBFC99999, 0x9998EBC4 */
     58   1.42857142725034663711e-01, /* 0x3FC24924, 0x920083FF */
     59  -1.11111104054623557880e-01, /* 0xBFBC71C6, 0xFE231671 */
     60   9.09088713343650656196e-02, /* 0x3FB745CD, 0xC54C206E */
     61  -7.69187620504482999495e-02, /* 0xBFB3B0F2, 0xAF749A6D */
     62   6.66107313738753120669e-02, /* 0x3FB10D66, 0xA0D03D51 */
     63  -5.83357013379057348645e-02, /* 0xBFADDE2D, 0x52DEFD9A */
     64   4.97687799461593236017e-02, /* 0x3FA97B4B, 0x24760DEB */
     65  -3.65315727442169155270e-02, /* 0xBFA2B444, 0x2C6A6C2F */
     66   1.62858201153657823623e-02, /* 0x3F90AD3A, 0xE322DA11 */
     67 };
     68 
     69 	static const double
     70 one   = 1.0,
     71 huge   = 1.0e300;
     72 
     73 double
     74 atan(double x)
     75 {
     76 	double w,s1,s2,z;
     77 	int32_t ix,hx,id;
     78 
     79 	GET_HIGH_WORD(hx,x);
     80 	ix = hx&0x7fffffff;
     81 	if(ix>=0x44100000) {	/* if |x| >= 2^66 */
     82 	    u_int32_t low;
     83 	    GET_LOW_WORD(low,x);
     84 	    if(ix>0x7ff00000||
     85 		(ix==0x7ff00000&&(low!=0)))
     86 		return x+x;		/* NaN */
     87 	    if(hx>0) return  atanhi[3]+*(volatile double *)&atanlo[3];
     88 	    else     return -atanhi[3]-*(volatile double *)&atanlo[3];
     89 	} if (ix < 0x3fdc0000) {	/* |x| < 0.4375 */
     90 	    if (ix < 0x3e400000) {	/* |x| < 2^-27 */
     91 		if(huge+x>one) return x;	/* raise inexact */
     92 	    }
     93 	    id = -1;
     94 	} else {
     95 	x = fabs(x);
     96 	if (ix < 0x3ff30000) {		/* |x| < 1.1875 */
     97 	    if (ix < 0x3fe60000) {	/* 7/16 <=|x|<11/16 */
     98 		id = 0; x = (2.0*x-one)/(2.0+x);
     99 	    } else {			/* 11/16<=|x|< 19/16 */
    100 		id = 1; x  = (x-one)/(x+one);
    101 	    }
    102 	} else {
    103 	    if (ix < 0x40038000) {	/* |x| < 2.4375 */
    104 		id = 2; x  = (x-1.5)/(one+1.5*x);
    105 	    } else {			/* 2.4375 <= |x| < 2^66 */
    106 		id = 3; x  = -1.0/x;
    107 	    }
    108 	}}
    109     /* end of argument reduction */
    110 	z = x*x;
    111 	w = z*z;
    112     /* break sum from i=0 to 10 aT[i]z**(i+1) into odd and even poly */
    113 	s1 = z*(aT[0]+w*(aT[2]+w*(aT[4]+w*(aT[6]+w*(aT[8]+w*aT[10])))));
    114 	s2 = w*(aT[1]+w*(aT[3]+w*(aT[5]+w*(aT[7]+w*aT[9]))));
    115 	if (id<0) return x - x*(s1+s2);
    116 	else {
    117 	    z = atanhi[id] - ((x*(s1+s2) - atanlo[id]) - x);
    118 	    return (hx<0)? -z:z;
    119 	}
    120 }
    121 
    122 #if LDBL_MANT_DIG == 53
    123 __weak_reference(atan, atanl);
    124 #endif
    125