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