lib/libm/noieee_src/n_atan2.c

   1 /*      $NetBSD: n_atan2.c,v 1.5 2002/06/15 00:10:17 matt Exp $        */
   2 /*
   3  * Copyright (c) 1985, 1993
   4  *      The Regents of the University of California.  All rights reserved.
   5  *
   6  * Redistribution and use in source and binary forms, with or without
   7  * modification, are permitted provided that the following conditions
   8  * are met:
   9  * 1. Redistributions of source code must retain the above copyright
  10  *    notice, this list of conditions and the following disclaimer.
  11  * 2. Redistributions in binary form must reproduce the above copyright
  12  *    notice, this list of conditions and the following disclaimer in the
  13  *    documentation and/or other materials provided with the distribution.
  14  * 3. Neither the name of the University nor the names of its contributors
  15  *    may be used to endorse or promote products derived from this software
  16  *    without specific prior written permission.
  17  *
  18  * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
  19  * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
  20  * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
  21  * ARE DISCLAIMED.  IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
  22  * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
  23  * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
  24  * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
  25  * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
  26  * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
  27  * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
  28  * SUCH DAMAGE.
  29  */
  30
  31 #ifndef lint
  32 static char sccsid[] = "@(#)atan2.c     8.1 (Berkeley) 6/4/93";
  33 #endif /* not lint */
  34
  35 /* ATAN2(Y,X)
  36  * RETURN ARG (X+iY)
  37  * DOUBLE PRECISION (VAX D format 56 bits, IEEE DOUBLE 53 BITS)
  38  * CODED IN C BY K.C. NG, 1/8/85;
  39  * REVISED BY K.C. NG on 2/7/85, 2/13/85, 3/7/85, 3/30/85, 6/29/85.
  40  *
  41  * Required system supported functions :
  42  *      copysign(x,y)
  43  *      scalb(x,y)
  44  *      logb(x)
  45  *
  46  * Method :
  47  *      1. Reduce y to positive by atan2(y,x)=-atan2(-y,x).
  48  *      2. Reduce x to positive by (if x and y are unexceptional):
  49  *              ARG (x+iy) = arctan(y/x)           ... if x > 0,
  50  *              ARG (x+iy) = pi - arctan[y/(-x)]   ... if x < 0,
  51  *      3. According to the integer k=4t+0.25 truncated , t=y/x, the argument
  52  *         is further reduced to one of the following intervals and the
  53  *         arctangent of y/x is evaluated by the corresponding formula:
  54  *
  55  *         [0,7/16]        atan(y/x) = t - t^3*(a1+t^2*(a2+...(a10+t^2*a11)...)
  56  *         [7/16,11/16]    atan(y/x) = atan(1/2) + atan( (y-x/2)/(x+y/2) )
  57  *         [11/16.19/16]   atan(y/x) = atan( 1 ) + atan( (y-x)/(x+y) )
  58  *         [19/16,39/16]   atan(y/x) = atan(3/2) + atan( (y-1.5x)/(x+1.5y) )
  59  *         [39/16,INF]     atan(y/x) = atan(INF) + atan( -x/y )
  60  *
  61  * Special cases:
  62  * Notations: atan2(y,x) == ARG (x+iy) == ARG(x,y).
  63  *
  64  *      ARG( NAN , (anything) ) is NaN;
  65  *      ARG( (anything), NaN ) is NaN;
  66  *      ARG(+(anything but NaN), +-0) is +-0  ;
  67  *      ARG(-(anything but NaN), +-0) is +-PI ;
  68  *      ARG( 0, +-(anything but 0 and NaN) ) is +-PI/2;
  69  *      ARG( +INF,+-(anything but INF and NaN) ) is +-0 ;
  70  *      ARG( -INF,+-(anything but INF and NaN) ) is +-PI;
  71  *      ARG( +INF,+-INF ) is +-PI/4 ;
  72  *      ARG( -INF,+-INF ) is +-3PI/4;
  73  *      ARG( (anything but,0,NaN, and INF),+-INF ) is +-PI/2;
  74  *
  75  * Accuracy:
  76  *      atan2(y,x) returns (PI/pi) * the exact ARG (x+iy) nearly rounded,
  77  *      where
  78  *
  79  *      in decimal:
  80  *              pi = 3.141592653589793 23846264338327 .....
  81  *    53 bits   PI = 3.141592653589793 115997963 ..... ,
  82  *    56 bits   PI = 3.141592653589793 227020265 ..... ,
  83  *
  84  *      in hexadecimal:
  85  *              pi = 3.243F6A8885A308D313198A2E....
  86  *    53 bits   PI = 3.243F6A8885A30  =  2 * 1.921FB54442D18    error=.276ulps
  87  *    56 bits   PI = 3.243F6A8885A308 =  4 * .C90FDAA22168C2    error=.206ulps
  88  *
  89  *      In a test run with 356,000 random argument on [-1,1] * [-1,1] on a
  90  *      VAX, the maximum observed error was 1.41 ulps (units of the last place)
  91  *      compared with (PI/pi)*(the exact ARG(x+iy)).
  92  *
  93  * Note:
  94  *      We use machine PI (the true pi rounded) in place of the actual
  95  *      value of pi for all the trig and inverse trig functions. In general,
  96  *      if trig is one of sin, cos, tan, then computed trig(y) returns the
  97  *      exact trig(y*pi/PI) nearly rounded; correspondingly, computed arctrig
  98  *      returns the exact arctrig(y)*PI/pi nearly rounded. These guarantee the
  99  *      trig functions have period PI, and trig(arctrig(x)) returns x for
 100  *      all critical values x.
 101  *
 102  * Constants:
 103  * The hexadecimal values are the intended ones for the following constants.
 104  * The decimal values may be used, provided that the compiler will convert
 105  * from decimal to binary accurately enough to produce the hexadecimal values
 106  * shown.
 107  */
 108
 109 #define _LIBM_STATIC
 110 #include "mathimpl.h"
 111
 112 vc(athfhi, 4.6364760900080611433E-1  ,6338,3fed,da7b,2b0d,  -1, .ED63382B0DDA7B)
 113 vc(athflo, 1.9338828231967579916E-19 ,5005,2164,92c0,9cfe, -62, .E450059CFE92C0)
 114 vc(PIo4,   7.8539816339744830676E-1  ,0fda,4049,68c2,a221,   0, .C90FDAA22168C2)
 115 vc(at1fhi, 9.8279372324732906796E-1  ,985e,407b,b4d9,940f,   0, .FB985E940FB4D9)
 116 vc(at1flo,-3.5540295636764633916E-18 ,1edc,a383,eaea,34d6, -57,-.831EDC34D6EAEA)
 117 vc(PIo2,   1.5707963267948966135E0   ,0fda,40c9,68c2,a221,   1, .C90FDAA22168C2)
 118 vc(PI,     3.1415926535897932270E0   ,0fda,4149,68c2,a221,   2, .C90FDAA22168C2)
 119 vc(a1,     3.3333333333333473730E-1  ,aaaa,3faa,ab75,aaaa,  -1, .AAAAAAAAAAAB75)
 120 vc(a2,    -2.0000000000017730678E-1  ,cccc,bf4c,946e,cccd,  -2,-.CCCCCCCCCD946E)
 121 vc(a3,     1.4285714286694640301E-1  ,4924,3f12,4262,9274,  -2, .92492492744262)
 122 vc(a4,    -1.1111111135032672795E-1  ,8e38,bee3,6292,ebc6,  -3,-.E38E38EBC66292)
 123 vc(a5,     9.0909091380563043783E-2  ,2e8b,3eba,d70c,b31b,  -3, .BA2E8BB31BD70C)
 124 vc(a6,    -7.6922954286089459397E-2  ,89c8,be9d,7f18,27c3,  -3,-.9D89C827C37F18)
 125 vc(a7,     6.6663180891693915586E-2  ,86b4,3e88,9e58,ae37,  -3, .8886B4AE379E58)
 126 vc(a8,    -5.8772703698290408927E-2  ,bba5,be70,a942,8481,  -4,-.F0BBA58481A942)
 127 vc(a9,     5.2170707402812969804E-2  ,b0f3,3e55,13ab,a1ab,  -4, .D5B0F3A1AB13AB)
 128 vc(a10,   -4.4895863157820361210E-2  ,e4b9,be37,048f,7fd1,  -4,-.B7E4B97FD1048F)
 129 vc(a11,    3.3006147437343875094E-2  ,3174,3e07,2d87,3cf7,  -4, .8731743CF72D87)
 130 vc(a12,   -1.4614844866464185439E-2  ,731a,bd6f,76d9,2f34,  -6,-.EF731A2F3476D9)
 131
 132 ic(athfhi, 4.6364760900080609352E-1  ,  -2,  1.DAC670561BB4F)
 133 ic(athflo, 4.6249969567426939759E-18 , -58,  1.5543B8F253271)
 134 ic(PIo4,   7.8539816339744827900E-1  ,  -1,  1.921FB54442D18)
 135 ic(at1fhi, 9.8279372324732905408E-1  ,  -1,  1.F730BD281F69B)
 136 ic(at1flo,-2.4407677060164810007E-17 , -56, -1.C23DFEFEAE6B5)
 137 ic(PIo2,   1.5707963267948965580E0   ,   0,  1.921FB54442D18)
 138 ic(PI,     3.1415926535897931160E0   ,   1,  1.921FB54442D18)
 139 ic(a1,     3.3333333333333942106E-1  ,  -2,  1.55555555555C3)
 140 ic(a2,    -1.9999999999979536924E-1  ,  -3, -1.9999999997CCD)
 141 ic(a3,     1.4285714278004377209E-1  ,  -3,  1.24924921EC1D7)
 142 ic(a4,    -1.1111110579344973814E-1  ,  -4, -1.C71C7059AF280)
 143 ic(a5,     9.0908906105474668324E-2  ,  -4,  1.745CE5AA35DB2)
 144 ic(a6,    -7.6919217767468239799E-2  ,  -4, -1.3B0FA54BEC400)
 145 ic(a7,     6.6614695906082474486E-2  ,  -4,  1.10DA924597FFF)
 146 ic(a8,    -5.8358371008508623523E-2  ,  -5, -1.DE125FDDBD793)
 147 ic(a9,     4.9850617156082015213E-2  ,  -5,  1.9860524BDD807)
 148 ic(a10,   -3.6700606902093604877E-2  ,  -5, -1.2CA6C04C6937A)
 149 ic(a11,    1.6438029044759730479E-2  ,  -6,  1.0D52174A1BB54)
 150
 151 #ifdef vccast
 152 #define athfhi  vccast(athfhi)
 153 #define athflo  vccast(athflo)
 154 #define PIo4    vccast(PIo4)
 155 #define at1fhi  vccast(at1fhi)
 156 #define at1flo  vccast(at1flo)
 157 #define PIo2    vccast(PIo2)
 158 #define PI      vccast(PI)
 159 #define a1      vccast(a1)
 160 #define a2      vccast(a2)
 161 #define a3      vccast(a3)
 162 #define a4      vccast(a4)
 163 #define a5      vccast(a5)
 164 #define a6      vccast(a6)
 165 #define a7      vccast(a7)
 166 #define a8      vccast(a8)
 167 #define a9      vccast(a9)
 168 #define a10     vccast(a10)
 169 #define a11     vccast(a11)
 170 #define a12     vccast(a12)
 171 #endif
 172
 173 double
 174 atan2(double y, double x)
 175 {
 176         static const double zero=0, one=1, small=1.0E-9, big=1.0E18;
 177         double t,z,signy,signx,hi,lo;
 178         int k,m;
 179
 180 #if !defined(__vax__)&&!defined(tahoe)
 181     /* if x or y is NAN */
 182         if(x!=x) return(x); if(y!=y) return(y);
 183 #endif  /* !defined(__vax__)&&!defined(tahoe) */
 184
 185     /* copy down the sign of y and x */
 186         signy = copysign(one,y) ;
 187         signx = copysign(one,x) ;
 188
 189     /* if x is 1.0, goto begin */
 190         if(x==1) { y=copysign(y,one); t=y; if(finite(t)) goto begin;}
 191
 192     /* when y = 0 */
 193         if(y==zero) return((signx==one)?y:copysign(PI,signy));
 194
 195     /* when x = 0 */
 196         if(x==zero) return(copysign(PIo2,signy));
 197
 198     /* when x is INF */
 199         if(!finite(x))
 200             if(!finite(y))
 201                 return(copysign((signx==one)?PIo4:3*PIo4,signy));
 202             else
 203                 return(copysign((signx==one)?zero:PI,signy));
 204
 205     /* when y is INF */
 206         if(!finite(y)) return(copysign(PIo2,signy));
 207
 208     /* compute y/x */
 209         x=copysign(x,one);
 210         y=copysign(y,one);
 211         if((m=(k=logb(y))-logb(x)) > 60) t=big+big;
 212             else if(m < -80 ) t=y/x;
 213             else { t = y/x ; y = scalb(y,-k); x=scalb(x,-k); }
 214
 215     /* begin argument reduction */
 216 begin:
 217         if (t < 2.4375) {
 218
 219         /* truncate 4(t+1/16) to integer for branching */
 220             k = 4 * (t+0.0625);
 221             switch (k) {
 222
 223             /* t is in [0,7/16] */
 224             case 0:
 225             case 1:
 226                 if (t < small)
 227                     { big + small ;  /* raise inexact flag */
 228                       return (copysign((signx>zero)?t:PI-t,signy)); }
 229
 230                 hi = zero;  lo = zero;  break;
 231
 232             /* t is in [7/16,11/16] */
 233             case 2:
 234                 hi = athfhi; lo = athflo;
 235                 z = x+x;
 236                 t = ( (y+y) - x ) / ( z +  y ); break;
 237
 238             /* t is in [11/16,19/16] */
 239             case 3:
 240             case 4:
 241                 hi = PIo4; lo = zero;
 242                 t = ( y - x ) / ( x + y ); break;
 243
 244             /* t is in [19/16,39/16] */
 245             default:
 246                 hi = at1fhi; lo = at1flo;
 247                 z = y-x; y=y+y+y; t = x+x;
 248                 t = ( (z+z)-x ) / ( t + y ); break;
 249             }
 250         }
 251         /* end of if (t < 2.4375) */
 252
 253         else
 254         {
 255             hi = PIo2; lo = zero;
 256
 257             /* t is in [2.4375, big] */
 258             if (t <= big)  t = - x / y;
 259
 260             /* t is in [big, INF] */
 261             else
 262               { big+small;      /* raise inexact flag */
 263                 t = zero; }
 264         }
 265     /* end of argument reduction */
 266
 267     /* compute atan(t) for t in [-.4375, .4375] */
 268         z = t*t;
 269 #if defined(__vax__)||defined(tahoe)
 270         z = t*(z*(a1+z*(a2+z*(a3+z*(a4+z*(a5+z*(a6+z*(a7+z*(a8+
 271                         z*(a9+z*(a10+z*(a11+z*a12))))))))))));
 272 #else   /* defined(__vax__)||defined(tahoe) */
 273         z = t*(z*(a1+z*(a2+z*(a3+z*(a4+z*(a5+z*(a6+z*(a7+z*(a8+
 274                         z*(a9+z*(a10+z*a11)))))))))));
 275 #endif  /* defined(__vax__)||defined(tahoe) */
 276         z = lo - z; z += t; z += hi;
 277
 278         return(copysign((signx>zero)?z:PI-z,signy));
 279 }