1 /* 2 * Copyright (c) 1985, 1993 3 * The Regents of the University of California. All rights reserved. 4 * 5 * %sccs.include.redist.c% 6 */ 7 8 #ifndef lint 9 static char sccsid[] = "@(#)atan2.c 8.1 (Berkeley) 06/04/93"; 10 #endif /* not lint */ 11 12 /* ATAN2(Y,X) 13 * RETURN ARG (X+iY) 14 * DOUBLE PRECISION (VAX D format 56 bits, IEEE DOUBLE 53 BITS) 15 * CODED IN C BY K.C. NG, 1/8/85; 16 * REVISED BY K.C. NG on 2/7/85, 2/13/85, 3/7/85, 3/30/85, 6/29/85. 17 * 18 * Required system supported functions : 19 * copysign(x,y) 20 * scalb(x,y) 21 * logb(x) 22 * 23 * Method : 24 * 1. Reduce y to positive by atan2(y,x)=-atan2(-y,x). 25 * 2. Reduce x to positive by (if x and y are unexceptional): 26 * ARG (x+iy) = arctan(y/x) ... if x > 0, 27 * ARG (x+iy) = pi - arctan[y/(-x)] ... if x < 0, 28 * 3. According to the integer k=4t+0.25 truncated , t=y/x, the argument 29 * is further reduced to one of the following intervals and the 30 * arctangent of y/x is evaluated by the corresponding formula: 31 * 32 * [0,7/16] atan(y/x) = t - t^3*(a1+t^2*(a2+...(a10+t^2*a11)...) 33 * [7/16,11/16] atan(y/x) = atan(1/2) + atan( (y-x/2)/(x+y/2) ) 34 * [11/16.19/16] atan(y/x) = atan( 1 ) + atan( (y-x)/(x+y) ) 35 * [19/16,39/16] atan(y/x) = atan(3/2) + atan( (y-1.5x)/(x+1.5y) ) 36 * [39/16,INF] atan(y/x) = atan(INF) + atan( -x/y ) 37 * 38 * Special cases: 39 * Notations: atan2(y,x) == ARG (x+iy) == ARG(x,y). 40 * 41 * ARG( NAN , (anything) ) is NaN; 42 * ARG( (anything), NaN ) is NaN; 43 * ARG(+(anything but NaN), +-0) is +-0 ; 44 * ARG(-(anything but NaN), +-0) is +-PI ; 45 * ARG( 0, +-(anything but 0 and NaN) ) is +-PI/2; 46 * ARG( +INF,+-(anything but INF and NaN) ) is +-0 ; 47 * ARG( -INF,+-(anything but INF and NaN) ) is +-PI; 48 * ARG( +INF,+-INF ) is +-PI/4 ; 49 * ARG( -INF,+-INF ) is +-3PI/4; 50 * ARG( (anything but,0,NaN, and INF),+-INF ) is +-PI/2; 51 * 52 * Accuracy: 53 * atan2(y,x) returns (PI/pi) * the exact ARG (x+iy) nearly rounded, 54 * where 55 * 56 * in decimal: 57 * pi = 3.141592653589793 23846264338327 ..... 58 * 53 bits PI = 3.141592653589793 115997963 ..... , 59 * 56 bits PI = 3.141592653589793 227020265 ..... , 60 * 61 * in hexadecimal: 62 * pi = 3.243F6A8885A308D313198A2E.... 63 * 53 bits PI = 3.243F6A8885A30 = 2 * 1.921FB54442D18 error=.276ulps 64 * 56 bits PI = 3.243F6A8885A308 = 4 * .C90FDAA22168C2 error=.206ulps 65 * 66 * In a test run with 356,000 random argument on [-1,1] * [-1,1] on a 67 * VAX, the maximum observed error was 1.41 ulps (units of the last place) 68 * compared with (PI/pi)*(the exact ARG(x+iy)). 69 * 70 * Note: 71 * We use machine PI (the true pi rounded) in place of the actual 72 * value of pi for all the trig and inverse trig functions. In general, 73 * if trig is one of sin, cos, tan, then computed trig(y) returns the 74 * exact trig(y*pi/PI) nearly rounded; correspondingly, computed arctrig 75 * returns the exact arctrig(y)*PI/pi nearly rounded. These guarantee the 76 * trig functions have period PI, and trig(arctrig(x)) returns x for 77 * all critical values x. 78 * 79 * Constants: 80 * The hexadecimal values are the intended ones for the following constants. 81 * The decimal values may be used, provided that the compiler will convert 82 * from decimal to binary accurately enough to produce the hexadecimal values 83 * shown. 84 */ 85 86 #include "mathimpl.h" 87 88 vc(athfhi, 4.6364760900080611433E-1 ,6338,3fed,da7b,2b0d, -1, .ED63382B0DDA7B) 89 vc(athflo, 1.9338828231967579916E-19 ,5005,2164,92c0,9cfe, -62, .E450059CFE92C0) 90 vc(PIo4, 7.8539816339744830676E-1 ,0fda,4049,68c2,a221, 0, .C90FDAA22168C2) 91 vc(at1fhi, 9.8279372324732906796E-1 ,985e,407b,b4d9,940f, 0, .FB985E940FB4D9) 92 vc(at1flo,-3.5540295636764633916E-18 ,1edc,a383,eaea,34d6, -57,-.831EDC34D6EAEA) 93 vc(PIo2, 1.5707963267948966135E0 ,0fda,40c9,68c2,a221, 1, .C90FDAA22168C2) 94 vc(PI, 3.1415926535897932270E0 ,0fda,4149,68c2,a221, 2, .C90FDAA22168C2) 95 vc(a1, 3.3333333333333473730E-1 ,aaaa,3faa,ab75,aaaa, -1, .AAAAAAAAAAAB75) 96 vc(a2, -2.0000000000017730678E-1 ,cccc,bf4c,946e,cccd, -2,-.CCCCCCCCCD946E) 97 vc(a3, 1.4285714286694640301E-1 ,4924,3f12,4262,9274, -2, .92492492744262) 98 vc(a4, -1.1111111135032672795E-1 ,8e38,bee3,6292,ebc6, -3,-.E38E38EBC66292) 99 vc(a5, 9.0909091380563043783E-2 ,2e8b,3eba,d70c,b31b, -3, .BA2E8BB31BD70C) 100 vc(a6, -7.6922954286089459397E-2 ,89c8,be9d,7f18,27c3, -3,-.9D89C827C37F18) 101 vc(a7, 6.6663180891693915586E-2 ,86b4,3e88,9e58,ae37, -3, .8886B4AE379E58) 102 vc(a8, -5.8772703698290408927E-2 ,bba5,be70,a942,8481, -4,-.F0BBA58481A942) 103 vc(a9, 5.2170707402812969804E-2 ,b0f3,3e55,13ab,a1ab, -4, .D5B0F3A1AB13AB) 104 vc(a10, -4.4895863157820361210E-2 ,e4b9,be37,048f,7fd1, -4,-.B7E4B97FD1048F) 105 vc(a11, 3.3006147437343875094E-2 ,3174,3e07,2d87,3cf7, -4, .8731743CF72D87) 106 vc(a12, -1.4614844866464185439E-2 ,731a,bd6f,76d9,2f34, -6,-.EF731A2F3476D9) 107 108 ic(athfhi, 4.6364760900080609352E-1 , -2, 1.DAC670561BB4F) 109 ic(athflo, 4.6249969567426939759E-18 , -58, 1.5543B8F253271) 110 ic(PIo4, 7.8539816339744827900E-1 , -1, 1.921FB54442D18) 111 ic(at1fhi, 9.8279372324732905408E-1 , -1, 1.F730BD281F69B) 112 ic(at1flo,-2.4407677060164810007E-17 , -56, -1.C23DFEFEAE6B5) 113 ic(PIo2, 1.5707963267948965580E0 , 0, 1.921FB54442D18) 114 ic(PI, 3.1415926535897931160E0 , 1, 1.921FB54442D18) 115 ic(a1, 3.3333333333333942106E-1 , -2, 1.55555555555C3) 116 ic(a2, -1.9999999999979536924E-1 , -3, -1.9999999997CCD) 117 ic(a3, 1.4285714278004377209E-1 , -3, 1.24924921EC1D7) 118 ic(a4, -1.1111110579344973814E-1 , -4, -1.C71C7059AF280) 119 ic(a5, 9.0908906105474668324E-2 , -4, 1.745CE5AA35DB2) 120 ic(a6, -7.6919217767468239799E-2 , -4, -1.3B0FA54BEC400) 121 ic(a7, 6.6614695906082474486E-2 , -4, 1.10DA924597FFF) 122 ic(a8, -5.8358371008508623523E-2 , -5, -1.DE125FDDBD793) 123 ic(a9, 4.9850617156082015213E-2 , -5, 1.9860524BDD807) 124 ic(a10, -3.6700606902093604877E-2 , -5, -1.2CA6C04C6937A) 125 ic(a11, 1.6438029044759730479E-2 , -6, 1.0D52174A1BB54) 126 127 #ifdef vccast 128 #define athfhi vccast(athfhi) 129 #define athflo vccast(athflo) 130 #define PIo4 vccast(PIo4) 131 #define at1fhi vccast(at1fhi) 132 #define at1flo vccast(at1flo) 133 #define PIo2 vccast(PIo2) 134 #define PI vccast(PI) 135 #define a1 vccast(a1) 136 #define a2 vccast(a2) 137 #define a3 vccast(a3) 138 #define a4 vccast(a4) 139 #define a5 vccast(a5) 140 #define a6 vccast(a6) 141 #define a7 vccast(a7) 142 #define a8 vccast(a8) 143 #define a9 vccast(a9) 144 #define a10 vccast(a10) 145 #define a11 vccast(a11) 146 #define a12 vccast(a12) 147 #endif 148 149 double atan2(y,x) 150 double y,x; 151 { 152 static const double zero=0, one=1, small=1.0E-9, big=1.0E18; 153 double t,z,signy,signx,hi,lo; 154 int k,m; 155 156 #if !defined(vax)&&!defined(tahoe) 157 /* if x or y is NAN */ 158 if(x!=x) return(x); if(y!=y) return(y); 159 #endif /* !defined(vax)&&!defined(tahoe) */ 160 161 /* copy down the sign of y and x */ 162 signy = copysign(one,y) ; 163 signx = copysign(one,x) ; 164 165 /* if x is 1.0, goto begin */ 166 if(x==1) { y=copysign(y,one); t=y; if(finite(t)) goto begin;} 167 168 /* when y = 0 */ 169 if(y==zero) return((signx==one)?y:copysign(PI,signy)); 170 171 /* when x = 0 */ 172 if(x==zero) return(copysign(PIo2,signy)); 173 174 /* when x is INF */ 175 if(!finite(x)) 176 if(!finite(y)) 177 return(copysign((signx==one)?PIo4:3*PIo4,signy)); 178 else 179 return(copysign((signx==one)?zero:PI,signy)); 180 181 /* when y is INF */ 182 if(!finite(y)) return(copysign(PIo2,signy)); 183 184 /* compute y/x */ 185 x=copysign(x,one); 186 y=copysign(y,one); 187 if((m=(k=logb(y))-logb(x)) > 60) t=big+big; 188 else if(m < -80 ) t=y/x; 189 else { t = y/x ; y = scalb(y,-k); x=scalb(x,-k); } 190 191 /* begin argument reduction */ 192 begin: 193 if (t < 2.4375) { 194 195 /* truncate 4(t+1/16) to integer for branching */ 196 k = 4 * (t+0.0625); 197 switch (k) { 198 199 /* t is in [0,7/16] */ 200 case 0: 201 case 1: 202 if (t < small) 203 { big + small ; /* raise inexact flag */ 204 return (copysign((signx>zero)?t:PI-t,signy)); } 205 206 hi = zero; lo = zero; break; 207 208 /* t is in [7/16,11/16] */ 209 case 2: 210 hi = athfhi; lo = athflo; 211 z = x+x; 212 t = ( (y+y) - x ) / ( z + y ); break; 213 214 /* t is in [11/16,19/16] */ 215 case 3: 216 case 4: 217 hi = PIo4; lo = zero; 218 t = ( y - x ) / ( x + y ); break; 219 220 /* t is in [19/16,39/16] */ 221 default: 222 hi = at1fhi; lo = at1flo; 223 z = y-x; y=y+y+y; t = x+x; 224 t = ( (z+z)-x ) / ( t + y ); break; 225 } 226 } 227 /* end of if (t < 2.4375) */ 228 229 else 230 { 231 hi = PIo2; lo = zero; 232 233 /* t is in [2.4375, big] */ 234 if (t <= big) t = - x / y; 235 236 /* t is in [big, INF] */ 237 else 238 { big+small; /* raise inexact flag */ 239 t = zero; } 240 } 241 /* end of argument reduction */ 242 243 /* compute atan(t) for t in [-.4375, .4375] */ 244 z = t*t; 245 #if defined(vax)||defined(tahoe) 246 z = t*(z*(a1+z*(a2+z*(a3+z*(a4+z*(a5+z*(a6+z*(a7+z*(a8+ 247 z*(a9+z*(a10+z*(a11+z*a12)))))))))))); 248 #else /* defined(vax)||defined(tahoe) */ 249 z = t*(z*(a1+z*(a2+z*(a3+z*(a4+z*(a5+z*(a6+z*(a7+z*(a8+ 250 z*(a9+z*(a10+z*a11))))))))))); 251 #endif /* defined(vax)||defined(tahoe) */ 252 z = lo - z; z += t; z += hi; 253 254 return(copysign((signx>zero)?z:PI-z,signy)); 255 } 256