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: }
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