1 /* 2 * Copyright (c) 1985 Regents of the University of California. 3 * All rights reserved. 4 * 5 * %sccs.include.redist.c% 6 * 7 * All recipients should regard themselves as participants in an ongoing 8 * research project and hence should feel obligated to report their 9 * experiences (good or bad) with these elementary function codes, using 10 * the sendbug(8) program, to the authors. 11 */ 12 13 #ifndef lint 14 static char sccsid[] = "@(#)atan2.c 5.5 (Berkeley) 06/01/90"; 15 #endif /* not lint */ 16 17 /* ATAN2(Y,X) 18 * RETURN ARG (X+iY) 19 * DOUBLE PRECISION (VAX D format 56 bits, IEEE DOUBLE 53 BITS) 20 * CODED IN C BY K.C. NG, 1/8/85; 21 * REVISED BY K.C. NG on 2/7/85, 2/13/85, 3/7/85, 3/30/85, 6/29/85. 22 * 23 * Required system supported functions : 24 * copysign(x,y) 25 * scalb(x,y) 26 * logb(x) 27 * 28 * Method : 29 * 1. Reduce y to positive by atan2(y,x)=-atan2(-y,x). 30 * 2. Reduce x to positive by (if x and y are unexceptional): 31 * ARG (x+iy) = arctan(y/x) ... if x > 0, 32 * ARG (x+iy) = pi - arctan[y/(-x)] ... if x < 0, 33 * 3. According to the integer k=4t+0.25 truncated , t=y/x, the argument 34 * is further reduced to one of the following intervals and the 35 * arctangent of y/x is evaluated by the corresponding formula: 36 * 37 * [0,7/16] atan(y/x) = t - t^3*(a1+t^2*(a2+...(a10+t^2*a11)...) 38 * [7/16,11/16] atan(y/x) = atan(1/2) + atan( (y-x/2)/(x+y/2) ) 39 * [11/16.19/16] atan(y/x) = atan( 1 ) + atan( (y-x)/(x+y) ) 40 * [19/16,39/16] atan(y/x) = atan(3/2) + atan( (y-1.5x)/(x+1.5y) ) 41 * [39/16,INF] atan(y/x) = atan(INF) + atan( -x/y ) 42 * 43 * Special cases: 44 * Notations: atan2(y,x) == ARG (x+iy) == ARG(x,y). 45 * 46 * ARG( NAN , (anything) ) is NaN; 47 * ARG( (anything), NaN ) is NaN; 48 * ARG(+(anything but NaN), +-0) is +-0 ; 49 * ARG(-(anything but NaN), +-0) is +-PI ; 50 * ARG( 0, +-(anything but 0 and NaN) ) is +-PI/2; 51 * ARG( +INF,+-(anything but INF and NaN) ) is +-0 ; 52 * ARG( -INF,+-(anything but INF and NaN) ) is +-PI; 53 * ARG( +INF,+-INF ) is +-PI/4 ; 54 * ARG( -INF,+-INF ) is +-3PI/4; 55 * ARG( (anything but,0,NaN, and INF),+-INF ) is +-PI/2; 56 * 57 * Accuracy: 58 * atan2(y,x) returns (PI/pi) * the exact ARG (x+iy) nearly rounded, 59 * where 60 * 61 * in decimal: 62 * pi = 3.141592653589793 23846264338327 ..... 63 * 53 bits PI = 3.141592653589793 115997963 ..... , 64 * 56 bits PI = 3.141592653589793 227020265 ..... , 65 * 66 * in hexadecimal: 67 * pi = 3.243F6A8885A308D313198A2E.... 68 * 53 bits PI = 3.243F6A8885A30 = 2 * 1.921FB54442D18 error=.276ulps 69 * 56 bits PI = 3.243F6A8885A308 = 4 * .C90FDAA22168C2 error=.206ulps 70 * 71 * In a test run with 356,000 random argument on [-1,1] * [-1,1] on a 72 * VAX, the maximum observed error was 1.41 ulps (units of the last place) 73 * compared with (PI/pi)*(the exact ARG(x+iy)). 74 * 75 * Note: 76 * We use machine PI (the true pi rounded) in place of the actual 77 * value of pi for all the trig and inverse trig functions. In general, 78 * if trig is one of sin, cos, tan, then computed trig(y) returns the 79 * exact trig(y*pi/PI) nearly rounded; correspondingly, computed arctrig 80 * returns the exact arctrig(y)*PI/pi nearly rounded. These guarantee the 81 * trig functions have period PI, and trig(arctrig(x)) returns x for 82 * all critical values x. 83 * 84 * Constants: 85 * The hexadecimal values are the intended ones for the following constants. 86 * The decimal values may be used, provided that the compiler will convert 87 * from decimal to binary accurately enough to produce the hexadecimal values 88 * shown. 89 */ 90 91 #include "mathimpl.h" 92 93 vc(athfhi, 4.6364760900080611433E-1 ,6338,3fed,da7b,2b0d, -1, .ED63382B0DDA7B) 94 vc(athflo, 1.9338828231967579916E-19 ,5005,2164,92c0,9cfe, -62, .E450059CFE92C0) 95 vc(PIo4, 7.8539816339744830676E-1 ,0fda,4049,68c2,a221, 0, .C90FDAA22168C2) 96 vc(at1fhi, 9.8279372324732906796E-1 ,985e,407b,b4d9,940f, 0, .FB985E940FB4D9) 97 vc(at1flo,-3.5540295636764633916E-18 ,1edc,a383,eaea,34d6, -57,-.831EDC34D6EAEA) 98 vc(PIo2, 1.5707963267948966135E0 ,0fda,40c9,68c2,a221, 1, .C90FDAA22168C2) 99 vc(PI, 3.1415926535897932270E0 ,0fda,4149,68c2,a221, 2, .C90FDAA22168C2) 100 vc(a1, 3.3333333333333473730E-1 ,aaaa,3faa,ab75,aaaa, -1, .AAAAAAAAAAAB75) 101 vc(a2, -2.0000000000017730678E-1 ,cccc,bf4c,946e,cccd, -2,-.CCCCCCCCCD946E) 102 vc(a3, 1.4285714286694640301E-1 ,4924,3f12,4262,9274, -2, .92492492744262) 103 vc(a4, -1.1111111135032672795E-1 ,8e38,bee3,6292,ebc6, -3,-.E38E38EBC66292) 104 vc(a5, 9.0909091380563043783E-2 ,2e8b,3eba,d70c,b31b, -3, .BA2E8BB31BD70C) 105 vc(a6, -7.6922954286089459397E-2 ,89c8,be9d,7f18,27c3, -3,-.9D89C827C37F18) 106 vc(a7, 6.6663180891693915586E-2 ,86b4,3e88,9e58,ae37, -3, .8886B4AE379E58) 107 vc(a8, -5.8772703698290408927E-2 ,bba5,be70,a942,8481, -4,-.F0BBA58481A942) 108 vc(a9, 5.2170707402812969804E-2 ,b0f3,3e55,13ab,a1ab, -4, .D5B0F3A1AB13AB) 109 vc(a10, -4.4895863157820361210E-2 ,e4b9,be37,048f,7fd1, -4,-.B7E4B97FD1048F) 110 vc(a11, 3.3006147437343875094E-2 ,3174,3e07,2d87,3cf7, -4, .8731743CF72D87) 111 vc(a12, -1.4614844866464185439E-2 ,731a,bd6f,76d9,2f34, -6,-.EF731A2F3476D9) 112 113 ic(athfhi, 4.6364760900080609352E-1 , -2, 1.DAC670561BB4F) 114 ic(athflo, 4.6249969567426939759E-18 , -58, 1.5543B8F253271) 115 ic(PIo4, 7.8539816339744827900E-1 , -1, 1.921FB54442D18) 116 ic(at1fhi, 9.8279372324732905408E-1 , -1, 1.F730BD281F69B) 117 ic(at1flo,-2.4407677060164810007E-17 , -56, -1.C23DFEFEAE6B5) 118 ic(PIo2, 1.5707963267948965580E0 , 0, 1.921FB54442D18) 119 ic(PI, 3.1415926535897931160E0 , 1, 1.921FB54442D18) 120 ic(a1, 3.3333333333333942106E-1 , -2, 1.55555555555C3) 121 ic(a2, -1.9999999999979536924E-1 , -3, -1.9999999997CCD) 122 ic(a3, 1.4285714278004377209E-1 , -3, 1.24924921EC1D7) 123 ic(a4, -1.1111110579344973814E-1 , -4, -1.C71C7059AF280) 124 ic(a5, 9.0908906105474668324E-2 , -4, 1.745CE5AA35DB2) 125 ic(a6, -7.6919217767468239799E-2 , -4, -1.3B0FA54BEC400) 126 ic(a7, 6.6614695906082474486E-2 , -4, 1.10DA924597FFF) 127 ic(a8, -5.8358371008508623523E-2 , -5, -1.DE125FDDBD793) 128 ic(a9, 4.9850617156082015213E-2 , -5, 1.9860524BDD807) 129 ic(a10, -3.6700606902093604877E-2 , -5, -1.2CA6C04C6937A) 130 ic(a11, 1.6438029044759730479E-2 , -6, 1.0D52174A1BB54) 131 132 #ifdef vccast 133 #define athfhi vccast(athfhi) 134 #define athflo vccast(athflo) 135 #define PIo4 vccast(PIo4) 136 #define at1fhi vccast(at1fhi) 137 #define at1flo vccast(at1flo) 138 #define PIo2 vccast(PIo2) 139 #define PI vccast(PI) 140 #define a1 vccast(a1) 141 #define a2 vccast(a2) 142 #define a3 vccast(a3) 143 #define a4 vccast(a4) 144 #define a5 vccast(a5) 145 #define a6 vccast(a6) 146 #define a7 vccast(a7) 147 #define a8 vccast(a8) 148 #define a9 vccast(a9) 149 #define a10 vccast(a10) 150 #define a11 vccast(a11) 151 #define a12 vccast(a12) 152 #endif 153 154 double atan2(y,x) 155 double y,x; 156 { 157 static const double zero=0, one=1, small=1.0E-9, big=1.0E18; 158 double t,z,signy,signx,hi,lo; 159 int k,m; 160 161 #if !defined(vax)&&!defined(tahoe) 162 /* if x or y is NAN */ 163 if(x!=x) return(x); if(y!=y) return(y); 164 #endif /* !defined(vax)&&!defined(tahoe) */ 165 166 /* copy down the sign of y and x */ 167 signy = copysign(one,y) ; 168 signx = copysign(one,x) ; 169 170 /* if x is 1.0, goto begin */ 171 if(x==1) { y=copysign(y,one); t=y; if(finite(t)) goto begin;} 172 173 /* when y = 0 */ 174 if(y==zero) return((signx==one)?y:copysign(PI,signy)); 175 176 /* when x = 0 */ 177 if(x==zero) return(copysign(PIo2,signy)); 178 179 /* when x is INF */ 180 if(!finite(x)) 181 if(!finite(y)) 182 return(copysign((signx==one)?PIo4:3*PIo4,signy)); 183 else 184 return(copysign((signx==one)?zero:PI,signy)); 185 186 /* when y is INF */ 187 if(!finite(y)) return(copysign(PIo2,signy)); 188 189 /* compute y/x */ 190 x=copysign(x,one); 191 y=copysign(y,one); 192 if((m=(k=logb(y))-logb(x)) > 60) t=big+big; 193 else if(m < -80 ) t=y/x; 194 else { t = y/x ; y = scalb(y,-k); x=scalb(x,-k); } 195 196 /* begin argument reduction */ 197 begin: 198 if (t < 2.4375) { 199 200 /* truncate 4(t+1/16) to integer for branching */ 201 k = 4 * (t+0.0625); 202 switch (k) { 203 204 /* t is in [0,7/16] */ 205 case 0: 206 case 1: 207 if (t < small) 208 { big + small ; /* raise inexact flag */ 209 return (copysign((signx>zero)?t:PI-t,signy)); } 210 211 hi = zero; lo = zero; break; 212 213 /* t is in [7/16,11/16] */ 214 case 2: 215 hi = athfhi; lo = athflo; 216 z = x+x; 217 t = ( (y+y) - x ) / ( z + y ); break; 218 219 /* t is in [11/16,19/16] */ 220 case 3: 221 case 4: 222 hi = PIo4; lo = zero; 223 t = ( y - x ) / ( x + y ); break; 224 225 /* t is in [19/16,39/16] */ 226 default: 227 hi = at1fhi; lo = at1flo; 228 z = y-x; y=y+y+y; t = x+x; 229 t = ( (z+z)-x ) / ( t + y ); break; 230 } 231 } 232 /* end of if (t < 2.4375) */ 233 234 else 235 { 236 hi = PIo2; lo = zero; 237 238 /* t is in [2.4375, big] */ 239 if (t <= big) t = - x / y; 240 241 /* t is in [big, INF] */ 242 else 243 { big+small; /* raise inexact flag */ 244 t = zero; } 245 } 246 /* end of argument reduction */ 247 248 /* compute atan(t) for t in [-.4375, .4375] */ 249 z = t*t; 250 #if defined(vax)||defined(tahoe) 251 z = t*(z*(a1+z*(a2+z*(a3+z*(a4+z*(a5+z*(a6+z*(a7+z*(a8+ 252 z*(a9+z*(a10+z*(a11+z*a12)))))))))))); 253 #else /* defined(vax)||defined(tahoe) */ 254 z = t*(z*(a1+z*(a2+z*(a3+z*(a4+z*(a5+z*(a6+z*(a7+z*(a8+ 255 z*(a9+z*(a10+z*a11))))))))))); 256 #endif /* defined(vax)||defined(tahoe) */ 257 z = lo - z; z += t; z += hi; 258 259 return(copysign((signx>zero)?z:PI-z,signy)); 260 } 261