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s_atan.cpp (4119B)

      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 #include <math.h>
     38 
     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 }