libm/s_atan.c

   1 /*
   2  * ====================================================
   3  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
   4  *
   5  * Developed at SunPro, a Sun Microsystems, Inc. business.
   6  * Permission to use, copy, modify, and distribute this
   7  * software is freely granted, provided that this notice
   8  * is preserved.
   9  * ====================================================
  10  */
  11
  12 /* atan(x)
  13  * Method
  14  *   1. Reduce x to positive by atan(x) = -atan(-x).
  15  *   2. According to the integer k=4t+0.25 chopped, t=x, the argument
  16  *      is further reduced to one of the following intervals and the
  17  *      arctangent of t is evaluated by the corresponding formula:
  18  *
  19  *      [0,7/16]      atan(x) = t-t^3*(a1+t^2*(a2+...(a10+t^2*a11)...)
  20  *      [7/16,11/16]  atan(x) = atan(1/2) + atan( (t-0.5)/(1+t/2) )
  21  *      [11/16.19/16] atan(x) = atan( 1 ) + atan( (t-1)/(1+t) )
  22  *      [19/16,39/16] atan(x) = atan(3/2) + atan( (t-1.5)/(1+1.5t) )
  23  *      [39/16,INF]   atan(x) = atan(INF) + atan( -1/t )
  24  *
  25  * Constants:
  26  * The hexadecimal values are the intended ones for the following
  27  * constants. The decimal values may be used, provided that the
  28  * compiler will convert from decimal to binary accurately enough
  29  * to produce the hexadecimal values shown.
  30  */
  31
  32 #include "math.h"
  33 #include "math_private.h"
  34
  35 static const double atanhi[] = {
  36   4.63647609000806093515e-01, /* atan(0.5)hi 0x3FDDAC67, 0x0561BB4F */
  37   7.85398163397448278999e-01, /* atan(1.0)hi 0x3FE921FB, 0x54442D18 */
  38   9.82793723247329054082e-01, /* atan(1.5)hi 0x3FEF730B, 0xD281F69B */
  39   1.57079632679489655800e+00, /* atan(inf)hi 0x3FF921FB, 0x54442D18 */
  40 };
  41
  42 static const double atanlo[] = {
  43   2.26987774529616870924e-17, /* atan(0.5)lo 0x3C7A2B7F, 0x222F65E2 */
  44   3.06161699786838301793e-17, /* atan(1.0)lo 0x3C81A626, 0x33145C07 */
  45   1.39033110312309984516e-17, /* atan(1.5)lo 0x3C700788, 0x7AF0CBBD */
  46   6.12323399573676603587e-17, /* atan(inf)lo 0x3C91A626, 0x33145C07 */
  47 };
  48
  49 static const double aT[] = {
  50   3.33333333333329318027e-01, /* 0x3FD55555, 0x5555550D */
  51  -1.99999999998764832476e-01, /* 0xBFC99999, 0x9998EBC4 */
  52   1.42857142725034663711e-01, /* 0x3FC24924, 0x920083FF */
  53  -1.11111104054623557880e-01, /* 0xBFBC71C6, 0xFE231671 */
  54   9.09088713343650656196e-02, /* 0x3FB745CD, 0xC54C206E */
  55  -7.69187620504482999495e-02, /* 0xBFB3B0F2, 0xAF749A6D */
  56   6.66107313738753120669e-02, /* 0x3FB10D66, 0xA0D03D51 */
  57  -5.83357013379057348645e-02, /* 0xBFADDE2D, 0x52DEFD9A */
  58   4.97687799461593236017e-02, /* 0x3FA97B4B, 0x24760DEB */
  59  -3.65315727442169155270e-02, /* 0xBFA2B444, 0x2C6A6C2F */
  60   1.62858201153657823623e-02, /* 0x3F90AD3A, 0xE322DA11 */
  61 };
  62
  63 static const double
  64 one   = 1.0,
  65 huge   = 1.0e300;
  66
  67 double atan(double x)
  68 {
  69         double w,s1,s2,z;
  70         int32_t ix,hx,id;
  71
  72         GET_HIGH_WORD(hx,x);
  73         ix = hx&0x7fffffff;
  74         if(ix>=0x44100000) {    /* if |x| >= 2^66 */
  75             u_int32_t low;
  76             GET_LOW_WORD(low,x);
  77             if(ix>0x7ff00000||
  78                 (ix==0x7ff00000&&(low!=0)))
  79                 return x+x;             /* NaN */
  80             if(hx>0) return  atanhi[3]+atanlo[3];
  81             else     return -atanhi[3]-atanlo[3];
  82         } if (ix < 0x3fdc0000) {        /* |x| < 0.4375 */
  83             if (ix < 0x3e200000) {      /* |x| < 2^-29 */
  84                 if(huge+x>one) return x;        /* raise inexact */
  85             }
  86             id = -1;
  87         } else {
  88         x = fabs(x);
  89         if (ix < 0x3ff30000) {          /* |x| < 1.1875 */
  90             if (ix < 0x3fe60000) {      /* 7/16 <=|x|<11/16 */
  91                 id = 0; x = (2.0*x-one)/(2.0+x);
  92             } else {                    /* 11/16<=|x|< 19/16 */
  93                 id = 1; x  = (x-one)/(x+one);
  94             }
  95         } else {
  96             if (ix < 0x40038000) {      /* |x| < 2.4375 */
  97                 id = 2; x  = (x-1.5)/(one+1.5*x);
  98             } else {                    /* 2.4375 <= |x| < 2^66 */
  99                 id = 3; x  = -1.0/x;
 100             }
 101         }}
 102     /* end of argument reduction */
 103         z = x*x;
 104         w = z*z;
 105     /* break sum from i=0 to 10 aT[i]z**(i+1) into odd and even poly */
 106         s1 = z*(aT[0]+w*(aT[2]+w*(aT[4]+w*(aT[6]+w*(aT[8]+w*aT[10])))));
 107         s2 = w*(aT[1]+w*(aT[3]+w*(aT[5]+w*(aT[7]+w*aT[9]))));
 108         if (id<0) return x - x*(s1+s2);
 109         else {
 110             z = atanhi[id] - ((x*(s1+s2) - atanlo[id]) - x);
 111             return (hx<0)? -z:z;
 112         }
 113 }
 114 libm_hidden_def(atan)