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[] = 16 "@(#)atan2.c 1.3 (Berkeley) 8/21/85; 1.5 (ucb.elefunt) 07/11/87"; 17 #endif not lint 18 19 /* ATAN2(Y,X) 20 * RETURN ARG (X+iY) 21 * DOUBLE PRECISION (VAX D format 56 bits, IEEE DOUBLE 53 BITS) 22 * CODED IN C BY K.C. NG, 1/8/85; 23 * REVISED BY K.C. NG on 2/7/85, 2/13/85, 3/7/85, 3/30/85, 6/29/85. 24 * 25 * Required system supported functions : 26 * copysign(x,y) 27 * scalb(x,y) 28 * logb(x) 29 * 30 * Method : 31 * 1. Reduce y to positive by atan2(y,x)=-atan2(-y,x). 32 * 2. Reduce x to positive by (if x and y are unexceptional): 33 * ARG (x+iy) = arctan(y/x) ... if x > 0, 34 * ARG (x+iy) = pi - arctan[y/(-x)] ... if x < 0, 35 * 3. According to the integer k=4t+0.25 truncated , t=y/x, the argument 36 * is further reduced to one of the following intervals and the 37 * arctangent of y/x is evaluated by the corresponding formula: 38 * 39 * [0,7/16] atan(y/x) = t - t^3*(a1+t^2*(a2+...(a10+t^2*a11)...) 40 * [7/16,11/16] atan(y/x) = atan(1/2) + atan( (y-x/2)/(x+y/2) ) 41 * [11/16.19/16] atan(y/x) = atan( 1 ) + atan( (y-x)/(x+y) ) 42 * [19/16,39/16] atan(y/x) = atan(3/2) + atan( (y-1.5x)/(x+1.5y) ) 43 * [39/16,INF] atan(y/x) = atan(INF) + atan( -x/y ) 44 * 45 * Special cases: 46 * Notations: atan2(y,x) == ARG (x+iy) == ARG(x,y). 47 * 48 * ARG( NAN , (anything) ) is NaN; 49 * ARG( (anything), NaN ) is NaN; 50 * ARG(+(anything but NaN), +-0) is +-0 ; 51 * ARG(-(anything but NaN), +-0) is +-PI ; 52 * ARG( 0, +-(anything but 0 and NaN) ) is +-PI/2; 53 * ARG( +INF,+-(anything but INF and NaN) ) is +-0 ; 54 * ARG( -INF,+-(anything but INF and NaN) ) is +-PI; 55 * ARG( +INF,+-INF ) is +-PI/4 ; 56 * ARG( -INF,+-INF ) is +-3PI/4; 57 * ARG( (anything but,0,NaN, and INF),+-INF ) is +-PI/2; 58 * 59 * Accuracy: 60 * atan2(y,x) returns (PI/pi) * the exact ARG (x+iy) nearly rounded, 61 * where 62 * 63 * in decimal: 64 * pi = 3.141592653589793 23846264338327 ..... 65 * 53 bits PI = 3.141592653589793 115997963 ..... , 66 * 56 bits PI = 3.141592653589793 227020265 ..... , 67 * 68 * in hexadecimal: 69 * pi = 3.243F6A8885A308D313198A2E.... 70 * 53 bits PI = 3.243F6A8885A30 = 2 * 1.921FB54442D18 error=.276ulps 71 * 56 bits PI = 3.243F6A8885A308 = 4 * .C90FDAA22168C2 error=.206ulps 72 * 73 * In a test run with 356,000 random argument on [-1,1] * [-1,1] on a 74 * VAX, the maximum observed error was 1.41 ulps (units of the last place) 75 * compared with (PI/pi)*(the exact ARG(x+iy)). 76 * 77 * Note: 78 * We use machine PI (the true pi rounded) in place of the actual 79 * value of pi for all the trig and inverse trig functions. In general, 80 * if trig is one of sin, cos, tan, then computed trig(y) returns the 81 * exact trig(y*pi/PI) nearly rounded; correspondingly, computed arctrig 82 * returns the exact arctrig(y)*PI/pi nearly rounded. These guarantee the 83 * trig functions have period PI, and trig(arctrig(x)) returns x for 84 * all critical values x. 85 * 86 * Constants: 87 * The hexadecimal values are the intended ones for the following constants. 88 * The decimal values may be used, provided that the compiler will convert 89 * from decimal to binary accurately enough to produce the hexadecimal values 90 * shown. 91 */ 92 93 #if (defined(VAX)||defined(TAHOE)) /* VAX D format */ 94 #ifdef VAX 95 #define _0x(A,B) 0x/**/A/**/B 96 #else /* VAX */ 97 #define _0x(A,B) 0x/**/B/**/A 98 #endif /* VAX */ 99 /*static double */ 100 /*athfhi = 4.6364760900080611433E-1 , /*Hex 2^ -1 * .ED63382B0DDA7B */ 101 /*athflo = 1.9338828231967579916E-19 , /*Hex 2^-62 * .E450059CFE92C0 */ 102 /*PIo4 = 7.8539816339744830676E-1 , /*Hex 2^ 0 * .C90FDAA22168C2 */ 103 /*at1fhi = 9.8279372324732906796E-1 , /*Hex 2^ 0 * .FB985E940FB4D9 */ 104 /*at1flo = -3.5540295636764633916E-18 , /*Hex 2^-57 * -.831EDC34D6EAEA */ 105 /*PIo2 = 1.5707963267948966135E0 , /*Hex 2^ 1 * .C90FDAA22168C2 */ 106 /*PI = 3.1415926535897932270E0 , /*Hex 2^ 2 * .C90FDAA22168C2 */ 107 /*a1 = 3.3333333333333473730E-1 , /*Hex 2^ -1 * .AAAAAAAAAAAB75 */ 108 /*a2 = -2.0000000000017730678E-1 , /*Hex 2^ -2 * -.CCCCCCCCCD946E */ 109 /*a3 = 1.4285714286694640301E-1 , /*Hex 2^ -2 * .92492492744262 */ 110 /*a4 = -1.1111111135032672795E-1 , /*Hex 2^ -3 * -.E38E38EBC66292 */ 111 /*a5 = 9.0909091380563043783E-2 , /*Hex 2^ -3 * .BA2E8BB31BD70C */ 112 /*a6 = -7.6922954286089459397E-2 , /*Hex 2^ -3 * -.9D89C827C37F18 */ 113 /*a7 = 6.6663180891693915586E-2 , /*Hex 2^ -3 * .8886B4AE379E58 */ 114 /*a8 = -5.8772703698290408927E-2 , /*Hex 2^ -4 * -.F0BBA58481A942 */ 115 /*a9 = 5.2170707402812969804E-2 , /*Hex 2^ -4 * .D5B0F3A1AB13AB */ 116 /*a10 = -4.4895863157820361210E-2 , /*Hex 2^ -4 * -.B7E4B97FD1048F */ 117 /*a11 = 3.3006147437343875094E-2 , /*Hex 2^ -4 * .8731743CF72D87 */ 118 /*a12 = -1.4614844866464185439E-2 ; /*Hex 2^ -6 * -.EF731A2F3476D9 */ 119 static long athfhix[] = { _0x(6338,3fed), _0x(da7b,2b0d)}; 120 #define athfhi (*(double *)athfhix) 121 static long athflox[] = { _0x(5005,2164), _0x(92c0,9cfe)}; 122 #define athflo (*(double *)athflox) 123 static long PIo4x[] = { _0x(0fda,4049), _0x(68c2,a221)}; 124 #define PIo4 (*(double *)PIo4x) 125 static long at1fhix[] = { _0x(985e,407b), _0x(b4d9,940f)}; 126 #define at1fhi (*(double *)at1fhix) 127 static long at1flox[] = { _0x(1edc,a383), _0x(eaea,34d6)}; 128 #define at1flo (*(double *)at1flox) 129 static long PIo2x[] = { _0x(0fda,40c9), _0x(68c2,a221)}; 130 #define PIo2 (*(double *)PIo2x) 131 static long PIx[] = { _0x(0fda,4149), _0x(68c2,a221)}; 132 #define PI (*(double *)PIx) 133 static long a1x[] = { _0x(aaaa,3faa), _0x(ab75,aaaa)}; 134 #define a1 (*(double *)a1x) 135 static long a2x[] = { _0x(cccc,bf4c), _0x(946e,cccd)}; 136 #define a2 (*(double *)a2x) 137 static long a3x[] = { _0x(4924,3f12), _0x(4262,9274)}; 138 #define a3 (*(double *)a3x) 139 static long a4x[] = { _0x(8e38,bee3), _0x(6292,ebc6)}; 140 #define a4 (*(double *)a4x) 141 static long a5x[] = { _0x(2e8b,3eba), _0x(d70c,b31b)}; 142 #define a5 (*(double *)a5x) 143 static long a6x[] = { _0x(89c8,be9d), _0x(7f18,27c3)}; 144 #define a6 (*(double *)a6x) 145 static long a7x[] = { _0x(86b4,3e88), _0x(9e58,ae37)}; 146 #define a7 (*(double *)a7x) 147 static long a8x[] = { _0x(bba5,be70), _0x(a942,8481)}; 148 #define a8 (*(double *)a8x) 149 static long a9x[] = { _0x(b0f3,3e55), _0x(13ab,a1ab)}; 150 #define a9 (*(double *)a9x) 151 static long a10x[] = { _0x(e4b9,be37), _0x(048f,7fd1)}; 152 #define a10 (*(double *)a10x) 153 static long a11x[] = { _0x(3174,3e07), _0x(2d87,3cf7)}; 154 #define a11 (*(double *)a11x) 155 static long a12x[] = { _0x(731a,bd6f), _0x(76d9,2f34)}; 156 #define a12 (*(double *)a12x) 157 #else /* IEEE double */ 158 static double 159 athfhi = 4.6364760900080609352E-1 , /*Hex 2^ -2 * 1.DAC670561BB4F */ 160 athflo = 4.6249969567426939759E-18 , /*Hex 2^-58 * 1.5543B8F253271 */ 161 PIo4 = 7.8539816339744827900E-1 , /*Hex 2^ -1 * 1.921FB54442D18 */ 162 at1fhi = 9.8279372324732905408E-1 , /*Hex 2^ -1 * 1.F730BD281F69B */ 163 at1flo = -2.4407677060164810007E-17 , /*Hex 2^-56 * -1.C23DFEFEAE6B5 */ 164 PIo2 = 1.5707963267948965580E0 , /*Hex 2^ 0 * 1.921FB54442D18 */ 165 PI = 3.1415926535897931160E0 , /*Hex 2^ 1 * 1.921FB54442D18 */ 166 a1 = 3.3333333333333942106E-1 , /*Hex 2^ -2 * 1.55555555555C3 */ 167 a2 = -1.9999999999979536924E-1 , /*Hex 2^ -3 * -1.9999999997CCD */ 168 a3 = 1.4285714278004377209E-1 , /*Hex 2^ -3 * 1.24924921EC1D7 */ 169 a4 = -1.1111110579344973814E-1 , /*Hex 2^ -4 * -1.C71C7059AF280 */ 170 a5 = 9.0908906105474668324E-2 , /*Hex 2^ -4 * 1.745CE5AA35DB2 */ 171 a6 = -7.6919217767468239799E-2 , /*Hex 2^ -4 * -1.3B0FA54BEC400 */ 172 a7 = 6.6614695906082474486E-2 , /*Hex 2^ -4 * 1.10DA924597FFF */ 173 a8 = -5.8358371008508623523E-2 , /*Hex 2^ -5 * -1.DE125FDDBD793 */ 174 a9 = 4.9850617156082015213E-2 , /*Hex 2^ -5 * 1.9860524BDD807 */ 175 a10 = -3.6700606902093604877E-2 , /*Hex 2^ -5 * -1.2CA6C04C6937A */ 176 a11 = 1.6438029044759730479E-2 ; /*Hex 2^ -6 * 1.0D52174A1BB54 */ 177 #endif 178 179 double atan2(y,x) 180 double y,x; 181 { 182 static double zero=0, one=1, small=1.0E-9, big=1.0E18; 183 double copysign(),logb(),scalb(),t,z,signy,signx,hi,lo; 184 int finite(), k,m; 185 186 #if (!defined(VAX)&&!defined(TAHOE)) 187 /* if x or y is NAN */ 188 if(x!=x) return(x); if(y!=y) return(y); 189 #endif 190 191 /* copy down the sign of y and x */ 192 signy = copysign(one,y) ; 193 signx = copysign(one,x) ; 194 195 /* if x is 1.0, goto begin */ 196 if(x==1) { y=copysign(y,one); t=y; if(finite(t)) goto begin;} 197 198 /* when y = 0 */ 199 if(y==zero) return((signx==one)?y:copysign(PI,signy)); 200 201 /* when x = 0 */ 202 if(x==zero) return(copysign(PIo2,signy)); 203 204 #if (!defined(VAX)&&!defined(TAHOE)) 205 /* when x is INF */ 206 if(!finite(x)) 207 if(!finite(y)) 208 return(copysign((signx==one)?PIo4:3*PIo4,signy)); 209 else 210 return(copysign((signx==one)?zero:PI,signy)); 211 212 /* when y is INF */ 213 if(!finite(y)) return(copysign(PIo2,signy)); 214 #endif 215 216 /* compute y/x */ 217 x=copysign(x,one); 218 y=copysign(y,one); 219 if((m=(k=logb(y))-logb(x)) > 60) t=big+big; 220 else if(m < -80 ) t=y/x; 221 else { t = y/x ; y = scalb(y,-k); x=scalb(x,-k); } 222 223 /* begin argument reduction */ 224 begin: 225 if (t < 2.4375) { 226 227 /* truncate 4(t+1/16) to integer for branching */ 228 k = 4 * (t+0.0625); 229 switch (k) { 230 231 /* t is in [0,7/16] */ 232 case 0: 233 case 1: 234 if (t < small) 235 { big + small ; /* raise inexact flag */ 236 return (copysign((signx>zero)?t:PI-t,signy)); } 237 238 hi = zero; lo = zero; break; 239 240 /* t is in [7/16,11/16] */ 241 case 2: 242 hi = athfhi; lo = athflo; 243 z = x+x; 244 t = ( (y+y) - x ) / ( z + y ); break; 245 246 /* t is in [11/16,19/16] */ 247 case 3: 248 case 4: 249 hi = PIo4; lo = zero; 250 t = ( y - x ) / ( x + y ); break; 251 252 /* t is in [19/16,39/16] */ 253 default: 254 hi = at1fhi; lo = at1flo; 255 z = y-x; y=y+y+y; t = x+x; 256 t = ( (z+z)-x ) / ( t + y ); break; 257 } 258 } 259 /* end of if (t < 2.4375) */ 260 261 else 262 { 263 hi = PIo2; lo = zero; 264 265 /* t is in [2.4375, big] */ 266 if (t <= big) t = - x / y; 267 268 /* t is in [big, INF] */ 269 else 270 { big+small; /* raise inexact flag */ 271 t = zero; } 272 } 273 /* end of argument reduction */ 274 275 /* compute atan(t) for t in [-.4375, .4375] */ 276 z = t*t; 277 #if (defined(VAX)||defined(TAHOE)) 278 z = t*(z*(a1+z*(a2+z*(a3+z*(a4+z*(a5+z*(a6+z*(a7+z*(a8+ 279 z*(a9+z*(a10+z*(a11+z*a12)))))))))))); 280 #else /* IEEE double */ 281 z = t*(z*(a1+z*(a2+z*(a3+z*(a4+z*(a5+z*(a6+z*(a7+z*(a8+ 282 z*(a9+z*(a10+z*a11))))))))))); 283 #endif 284 z = lo - z; z += t; z += hi; 285 286 return(copysign((signx>zero)?z:PI-z,signy)); 287 } 288