1 /* 2 * Copyright (c) 1985 Regents of the University of California. 3 * All rights reserved. 4 * 5 * Redistribution and use in source and binary forms are permitted 6 * provided that the above copyright notice and this paragraph are 7 * duplicated in all such forms and that any documentation, 8 * advertising materials, and other materials related to such 9 * distribution and use acknowledge that the software was developed 10 * by the University of California, Berkeley. The name of the 11 * University may not be used to endorse or promote products derived 12 * from this software without specific prior written permission. 13 * THIS SOFTWARE IS PROVIDED ``AS IS'' AND WITHOUT ANY EXPRESS OR 14 * IMPLIED WARRANTIES, INCLUDING, WITHOUT LIMITATION, THE IMPLIED 15 * WARRANTIES OF MERCHANTIBILITY AND FITNESS FOR A PARTICULAR PURPOSE. 16 * 17 * All recipients should regard themselves as participants in an ongoing 18 * research project and hence should feel obligated to report their 19 * experiences (good or bad) with these elementary function codes, using 20 * the sendbug(8) program, to the authors. 21 */ 22 23 #ifndef lint 24 static char sccsid[] = "@(#)atan2.c 5.4 (Berkeley) 09/22/88"; 25 #endif /* not lint */ 26 27 /* ATAN2(Y,X) 28 * RETURN ARG (X+iY) 29 * DOUBLE PRECISION (VAX D format 56 bits, IEEE DOUBLE 53 BITS) 30 * CODED IN C BY K.C. NG, 1/8/85; 31 * REVISED BY K.C. NG on 2/7/85, 2/13/85, 3/7/85, 3/30/85, 6/29/85. 32 * 33 * Required system supported functions : 34 * copysign(x,y) 35 * scalb(x,y) 36 * logb(x) 37 * 38 * Method : 39 * 1. Reduce y to positive by atan2(y,x)=-atan2(-y,x). 40 * 2. Reduce x to positive by (if x and y are unexceptional): 41 * ARG (x+iy) = arctan(y/x) ... if x > 0, 42 * ARG (x+iy) = pi - arctan[y/(-x)] ... if x < 0, 43 * 3. According to the integer k=4t+0.25 truncated , t=y/x, the argument 44 * is further reduced to one of the following intervals and the 45 * arctangent of y/x is evaluated by the corresponding formula: 46 * 47 * [0,7/16] atan(y/x) = t - t^3*(a1+t^2*(a2+...(a10+t^2*a11)...) 48 * [7/16,11/16] atan(y/x) = atan(1/2) + atan( (y-x/2)/(x+y/2) ) 49 * [11/16.19/16] atan(y/x) = atan( 1 ) + atan( (y-x)/(x+y) ) 50 * [19/16,39/16] atan(y/x) = atan(3/2) + atan( (y-1.5x)/(x+1.5y) ) 51 * [39/16,INF] atan(y/x) = atan(INF) + atan( -x/y ) 52 * 53 * Special cases: 54 * Notations: atan2(y,x) == ARG (x+iy) == ARG(x,y). 55 * 56 * ARG( NAN , (anything) ) is NaN; 57 * ARG( (anything), NaN ) is NaN; 58 * ARG(+(anything but NaN), +-0) is +-0 ; 59 * ARG(-(anything but NaN), +-0) is +-PI ; 60 * ARG( 0, +-(anything but 0 and NaN) ) is +-PI/2; 61 * ARG( +INF,+-(anything but INF and NaN) ) is +-0 ; 62 * ARG( -INF,+-(anything but INF and NaN) ) is +-PI; 63 * ARG( +INF,+-INF ) is +-PI/4 ; 64 * ARG( -INF,+-INF ) is +-3PI/4; 65 * ARG( (anything but,0,NaN, and INF),+-INF ) is +-PI/2; 66 * 67 * Accuracy: 68 * atan2(y,x) returns (PI/pi) * the exact ARG (x+iy) nearly rounded, 69 * where 70 * 71 * in decimal: 72 * pi = 3.141592653589793 23846264338327 ..... 73 * 53 bits PI = 3.141592653589793 115997963 ..... , 74 * 56 bits PI = 3.141592653589793 227020265 ..... , 75 * 76 * in hexadecimal: 77 * pi = 3.243F6A8885A308D313198A2E.... 78 * 53 bits PI = 3.243F6A8885A30 = 2 * 1.921FB54442D18 error=.276ulps 79 * 56 bits PI = 3.243F6A8885A308 = 4 * .C90FDAA22168C2 error=.206ulps 80 * 81 * In a test run with 356,000 random argument on [-1,1] * [-1,1] on a 82 * VAX, the maximum observed error was 1.41 ulps (units of the last place) 83 * compared with (PI/pi)*(the exact ARG(x+iy)). 84 * 85 * Note: 86 * We use machine PI (the true pi rounded) in place of the actual 87 * value of pi for all the trig and inverse trig functions. In general, 88 * if trig is one of sin, cos, tan, then computed trig(y) returns the 89 * exact trig(y*pi/PI) nearly rounded; correspondingly, computed arctrig 90 * returns the exact arctrig(y)*PI/pi nearly rounded. These guarantee the 91 * trig functions have period PI, and trig(arctrig(x)) returns x for 92 * all critical values x. 93 * 94 * Constants: 95 * The hexadecimal values are the intended ones for the following constants. 96 * The decimal values may be used, provided that the compiler will convert 97 * from decimal to binary accurately enough to produce the hexadecimal values 98 * shown. 99 */ 100 101 #include "mathimpl.h" 102 103 vc(athfhi, 4.6364760900080611433E-1 ,6338,3fed,da7b,2b0d, -1, .ED63382B0DDA7B) 104 vc(athflo, 1.9338828231967579916E-19 ,5005,2164,92c0,9cfe, -62, .E450059CFE92C0) 105 vc(PIo4, 7.8539816339744830676E-1 ,0fda,4049,68c2,a221, 0, .C90FDAA22168C2) 106 vc(at1fhi, 9.8279372324732906796E-1 ,985e,407b,b4d9,940f, 0, .FB985E940FB4D9) 107 vc(at1flo,-3.5540295636764633916E-18 ,1edc,a383,eaea,34d6, -57,-.831EDC34D6EAEA) 108 vc(PIo2, 1.5707963267948966135E0 ,0fda,40c9,68c2,a221, 1, .C90FDAA22168C2) 109 vc(PI, 3.1415926535897932270E0 ,0fda,4149,68c2,a221, 2, .C90FDAA22168C2) 110 vc(a1, 3.3333333333333473730E-1 ,aaaa,3faa,ab75,aaaa, -1, .AAAAAAAAAAAB75) 111 vc(a2, -2.0000000000017730678E-1 ,cccc,bf4c,946e,cccd, -2,-.CCCCCCCCCD946E) 112 vc(a3, 1.4285714286694640301E-1 ,4924,3f12,4262,9274, -2, .92492492744262) 113 vc(a4, -1.1111111135032672795E-1 ,8e38,bee3,6292,ebc6, -3,-.E38E38EBC66292) 114 vc(a5, 9.0909091380563043783E-2 ,2e8b,3eba,d70c,b31b, -3, .BA2E8BB31BD70C) 115 vc(a6, -7.6922954286089459397E-2 ,89c8,be9d,7f18,27c3, -3,-.9D89C827C37F18) 116 vc(a7, 6.6663180891693915586E-2 ,86b4,3e88,9e58,ae37, -3, .8886B4AE379E58) 117 vc(a8, -5.8772703698290408927E-2 ,bba5,be70,a942,8481, -4,-.F0BBA58481A942) 118 vc(a9, 5.2170707402812969804E-2 ,b0f3,3e55,13ab,a1ab, -4, .D5B0F3A1AB13AB) 119 vc(a10, -4.4895863157820361210E-2 ,e4b9,be37,048f,7fd1, -4,-.B7E4B97FD1048F) 120 vc(a11, 3.3006147437343875094E-2 ,3174,3e07,2d87,3cf7, -4, .8731743CF72D87) 121 vc(a12, -1.4614844866464185439E-2 ,731a,bd6f,76d9,2f34, -6,-.EF731A2F3476D9) 122 123 ic(athfhi, 4.6364760900080609352E-1 , -2, 1.DAC670561BB4F) 124 ic(athflo, 4.6249969567426939759E-18 , -58, 1.5543B8F253271) 125 ic(PIo4, 7.8539816339744827900E-1 , -1, 1.921FB54442D18) 126 ic(at1fhi, 9.8279372324732905408E-1 , -1, 1.F730BD281F69B) 127 ic(at1flo,-2.4407677060164810007E-17 , -56, -1.C23DFEFEAE6B5) 128 ic(PIo2, 1.5707963267948965580E0 , 0, 1.921FB54442D18) 129 ic(PI, 3.1415926535897931160E0 , 1, 1.921FB54442D18) 130 ic(a1, 3.3333333333333942106E-1 , -2, 1.55555555555C3) 131 ic(a2, -1.9999999999979536924E-1 , -3, -1.9999999997CCD) 132 ic(a3, 1.4285714278004377209E-1 , -3, 1.24924921EC1D7) 133 ic(a4, -1.1111110579344973814E-1 , -4, -1.C71C7059AF280) 134 ic(a5, 9.0908906105474668324E-2 , -4, 1.745CE5AA35DB2) 135 ic(a6, -7.6919217767468239799E-2 , -4, -1.3B0FA54BEC400) 136 ic(a7, 6.6614695906082474486E-2 , -4, 1.10DA924597FFF) 137 ic(a8, -5.8358371008508623523E-2 , -5, -1.DE125FDDBD793) 138 ic(a9, 4.9850617156082015213E-2 , -5, 1.9860524BDD807) 139 ic(a10, -3.6700606902093604877E-2 , -5, -1.2CA6C04C6937A) 140 ic(a11, 1.6438029044759730479E-2 , -6, 1.0D52174A1BB54) 141 142 #ifdef vccast 143 #define athfhi vccast(athfhi) 144 #define athflo vccast(athflo) 145 #define PIo4 vccast(PIo4) 146 #define at1fhi vccast(at1fhi) 147 #define at1flo vccast(at1flo) 148 #define PIo2 vccast(PIo2) 149 #define PI vccast(PI) 150 #define a1 vccast(a1) 151 #define a2 vccast(a2) 152 #define a3 vccast(a3) 153 #define a4 vccast(a4) 154 #define a5 vccast(a5) 155 #define a6 vccast(a6) 156 #define a7 vccast(a7) 157 #define a8 vccast(a8) 158 #define a9 vccast(a9) 159 #define a10 vccast(a10) 160 #define a11 vccast(a11) 161 #define a12 vccast(a12) 162 #endif 163 164 double atan2(y,x) 165 double y,x; 166 { 167 static const double zero=0, one=1, small=1.0E-9, big=1.0E18; 168 double t,z,signy,signx,hi,lo; 169 int k,m; 170 171 #if !defined(vax)&&!defined(tahoe) 172 /* if x or y is NAN */ 173 if(x!=x) return(x); if(y!=y) return(y); 174 #endif /* !defined(vax)&&!defined(tahoe) */ 175 176 /* copy down the sign of y and x */ 177 signy = copysign(one,y) ; 178 signx = copysign(one,x) ; 179 180 /* if x is 1.0, goto begin */ 181 if(x==1) { y=copysign(y,one); t=y; if(finite(t)) goto begin;} 182 183 /* when y = 0 */ 184 if(y==zero) return((signx==one)?y:copysign(PI,signy)); 185 186 /* when x = 0 */ 187 if(x==zero) return(copysign(PIo2,signy)); 188 189 /* when x is INF */ 190 if(!finite(x)) 191 if(!finite(y)) 192 return(copysign((signx==one)?PIo4:3*PIo4,signy)); 193 else 194 return(copysign((signx==one)?zero:PI,signy)); 195 196 /* when y is INF */ 197 if(!finite(y)) return(copysign(PIo2,signy)); 198 199 /* compute y/x */ 200 x=copysign(x,one); 201 y=copysign(y,one); 202 if((m=(k=logb(y))-logb(x)) > 60) t=big+big; 203 else if(m < -80 ) t=y/x; 204 else { t = y/x ; y = scalb(y,-k); x=scalb(x,-k); } 205 206 /* begin argument reduction */ 207 begin: 208 if (t < 2.4375) { 209 210 /* truncate 4(t+1/16) to integer for branching */ 211 k = 4 * (t+0.0625); 212 switch (k) { 213 214 /* t is in [0,7/16] */ 215 case 0: 216 case 1: 217 if (t < small) 218 { big + small ; /* raise inexact flag */ 219 return (copysign((signx>zero)?t:PI-t,signy)); } 220 221 hi = zero; lo = zero; break; 222 223 /* t is in [7/16,11/16] */ 224 case 2: 225 hi = athfhi; lo = athflo; 226 z = x+x; 227 t = ( (y+y) - x ) / ( z + y ); break; 228 229 /* t is in [11/16,19/16] */ 230 case 3: 231 case 4: 232 hi = PIo4; lo = zero; 233 t = ( y - x ) / ( x + y ); break; 234 235 /* t is in [19/16,39/16] */ 236 default: 237 hi = at1fhi; lo = at1flo; 238 z = y-x; y=y+y+y; t = x+x; 239 t = ( (z+z)-x ) / ( t + y ); break; 240 } 241 } 242 /* end of if (t < 2.4375) */ 243 244 else 245 { 246 hi = PIo2; lo = zero; 247 248 /* t is in [2.4375, big] */ 249 if (t <= big) t = - x / y; 250 251 /* t is in [big, INF] */ 252 else 253 { big+small; /* raise inexact flag */ 254 t = zero; } 255 } 256 /* end of argument reduction */ 257 258 /* compute atan(t) for t in [-.4375, .4375] */ 259 z = t*t; 260 #if defined(vax)||defined(tahoe) 261 z = t*(z*(a1+z*(a2+z*(a3+z*(a4+z*(a5+z*(a6+z*(a7+z*(a8+ 262 z*(a9+z*(a10+z*(a11+z*a12)))))))))))); 263 #else /* defined(vax)||defined(tahoe) */ 264 z = t*(z*(a1+z*(a2+z*(a3+z*(a4+z*(a5+z*(a6+z*(a7+z*(a8+ 265 z*(a9+z*(a10+z*a11))))))))))); 266 #endif /* defined(vax)||defined(tahoe) */ 267 z = lo - z; z += t; z += hi; 268 269 return(copysign((signx>zero)?z:PI-z,signy)); 270 } 271