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1.1 ! root 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 5.3 (Berkeley) 6/30/88"; ! 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: #if defined(vax)||defined(tahoe) /* VAX D format */ ! 102: #ifdef vax ! 103: #define _0x(A,B) 0x/**/A/**/B ! 104: #else /* vax */ ! 105: #define _0x(A,B) 0x/**/B/**/A ! 106: #endif /* vax */ ! 107: /*static double */ ! 108: /*athfhi = 4.6364760900080611433E-1 , /*Hex 2^ -1 * .ED63382B0DDA7B */ ! 109: /*athflo = 1.9338828231967579916E-19 , /*Hex 2^-62 * .E450059CFE92C0 */ ! 110: /*PIo4 = 7.8539816339744830676E-1 , /*Hex 2^ 0 * .C90FDAA22168C2 */ ! 111: /*at1fhi = 9.8279372324732906796E-1 , /*Hex 2^ 0 * .FB985E940FB4D9 */ ! 112: /*at1flo = -3.5540295636764633916E-18 , /*Hex 2^-57 * -.831EDC34D6EAEA */ ! 113: /*PIo2 = 1.5707963267948966135E0 , /*Hex 2^ 1 * .C90FDAA22168C2 */ ! 114: /*PI = 3.1415926535897932270E0 , /*Hex 2^ 2 * .C90FDAA22168C2 */ ! 115: /*a1 = 3.3333333333333473730E-1 , /*Hex 2^ -1 * .AAAAAAAAAAAB75 */ ! 116: /*a2 = -2.0000000000017730678E-1 , /*Hex 2^ -2 * -.CCCCCCCCCD946E */ ! 117: /*a3 = 1.4285714286694640301E-1 , /*Hex 2^ -2 * .92492492744262 */ ! 118: /*a4 = -1.1111111135032672795E-1 , /*Hex 2^ -3 * -.E38E38EBC66292 */ ! 119: /*a5 = 9.0909091380563043783E-2 , /*Hex 2^ -3 * .BA2E8BB31BD70C */ ! 120: /*a6 = -7.6922954286089459397E-2 , /*Hex 2^ -3 * -.9D89C827C37F18 */ ! 121: /*a7 = 6.6663180891693915586E-2 , /*Hex 2^ -3 * .8886B4AE379E58 */ ! 122: /*a8 = -5.8772703698290408927E-2 , /*Hex 2^ -4 * -.F0BBA58481A942 */ ! 123: /*a9 = 5.2170707402812969804E-2 , /*Hex 2^ -4 * .D5B0F3A1AB13AB */ ! 124: /*a10 = -4.4895863157820361210E-2 , /*Hex 2^ -4 * -.B7E4B97FD1048F */ ! 125: /*a11 = 3.3006147437343875094E-2 , /*Hex 2^ -4 * .8731743CF72D87 */ ! 126: /*a12 = -1.4614844866464185439E-2 ; /*Hex 2^ -6 * -.EF731A2F3476D9 */ ! 127: static long athfhix[] = { _0x(6338,3fed), _0x(da7b,2b0d)}; ! 128: #define athfhi (*(double *)athfhix) ! 129: static long athflox[] = { _0x(5005,2164), _0x(92c0,9cfe)}; ! 130: #define athflo (*(double *)athflox) ! 131: static long PIo4x[] = { _0x(0fda,4049), _0x(68c2,a221)}; ! 132: #define PIo4 (*(double *)PIo4x) ! 133: static long at1fhix[] = { _0x(985e,407b), _0x(b4d9,940f)}; ! 134: #define at1fhi (*(double *)at1fhix) ! 135: static long at1flox[] = { _0x(1edc,a383), _0x(eaea,34d6)}; ! 136: #define at1flo (*(double *)at1flox) ! 137: static long PIo2x[] = { _0x(0fda,40c9), _0x(68c2,a221)}; ! 138: #define PIo2 (*(double *)PIo2x) ! 139: static long PIx[] = { _0x(0fda,4149), _0x(68c2,a221)}; ! 140: #define PI (*(double *)PIx) ! 141: static long a1x[] = { _0x(aaaa,3faa), _0x(ab75,aaaa)}; ! 142: #define a1 (*(double *)a1x) ! 143: static long a2x[] = { _0x(cccc,bf4c), _0x(946e,cccd)}; ! 144: #define a2 (*(double *)a2x) ! 145: static long a3x[] = { _0x(4924,3f12), _0x(4262,9274)}; ! 146: #define a3 (*(double *)a3x) ! 147: static long a4x[] = { _0x(8e38,bee3), _0x(6292,ebc6)}; ! 148: #define a4 (*(double *)a4x) ! 149: static long a5x[] = { _0x(2e8b,3eba), _0x(d70c,b31b)}; ! 150: #define a5 (*(double *)a5x) ! 151: static long a6x[] = { _0x(89c8,be9d), _0x(7f18,27c3)}; ! 152: #define a6 (*(double *)a6x) ! 153: static long a7x[] = { _0x(86b4,3e88), _0x(9e58,ae37)}; ! 154: #define a7 (*(double *)a7x) ! 155: static long a8x[] = { _0x(bba5,be70), _0x(a942,8481)}; ! 156: #define a8 (*(double *)a8x) ! 157: static long a9x[] = { _0x(b0f3,3e55), _0x(13ab,a1ab)}; ! 158: #define a9 (*(double *)a9x) ! 159: static long a10x[] = { _0x(e4b9,be37), _0x(048f,7fd1)}; ! 160: #define a10 (*(double *)a10x) ! 161: static long a11x[] = { _0x(3174,3e07), _0x(2d87,3cf7)}; ! 162: #define a11 (*(double *)a11x) ! 163: static long a12x[] = { _0x(731a,bd6f), _0x(76d9,2f34)}; ! 164: #define a12 (*(double *)a12x) ! 165: #else /* defined(vax)||defined(tahoe) */ ! 166: static double ! 167: athfhi = 4.6364760900080609352E-1 , /*Hex 2^ -2 * 1.DAC670561BB4F */ ! 168: athflo = 4.6249969567426939759E-18 , /*Hex 2^-58 * 1.5543B8F253271 */ ! 169: PIo4 = 7.8539816339744827900E-1 , /*Hex 2^ -1 * 1.921FB54442D18 */ ! 170: at1fhi = 9.8279372324732905408E-1 , /*Hex 2^ -1 * 1.F730BD281F69B */ ! 171: at1flo = -2.4407677060164810007E-17 , /*Hex 2^-56 * -1.C23DFEFEAE6B5 */ ! 172: PIo2 = 1.5707963267948965580E0 , /*Hex 2^ 0 * 1.921FB54442D18 */ ! 173: PI = 3.1415926535897931160E0 , /*Hex 2^ 1 * 1.921FB54442D18 */ ! 174: a1 = 3.3333333333333942106E-1 , /*Hex 2^ -2 * 1.55555555555C3 */ ! 175: a2 = -1.9999999999979536924E-1 , /*Hex 2^ -3 * -1.9999999997CCD */ ! 176: a3 = 1.4285714278004377209E-1 , /*Hex 2^ -3 * 1.24924921EC1D7 */ ! 177: a4 = -1.1111110579344973814E-1 , /*Hex 2^ -4 * -1.C71C7059AF280 */ ! 178: a5 = 9.0908906105474668324E-2 , /*Hex 2^ -4 * 1.745CE5AA35DB2 */ ! 179: a6 = -7.6919217767468239799E-2 , /*Hex 2^ -4 * -1.3B0FA54BEC400 */ ! 180: a7 = 6.6614695906082474486E-2 , /*Hex 2^ -4 * 1.10DA924597FFF */ ! 181: a8 = -5.8358371008508623523E-2 , /*Hex 2^ -5 * -1.DE125FDDBD793 */ ! 182: a9 = 4.9850617156082015213E-2 , /*Hex 2^ -5 * 1.9860524BDD807 */ ! 183: a10 = -3.6700606902093604877E-2 , /*Hex 2^ -5 * -1.2CA6C04C6937A */ ! 184: a11 = 1.6438029044759730479E-2 ; /*Hex 2^ -6 * 1.0D52174A1BB54 */ ! 185: #endif /* defined(vax)||defined(tahoe) */ ! 186: ! 187: double atan2(y,x) ! 188: double y,x; ! 189: { ! 190: static double zero=0, one=1, small=1.0E-9, big=1.0E18; ! 191: double copysign(),logb(),scalb(),t,z,signy,signx,hi,lo; ! 192: int finite(), k,m; ! 193: ! 194: #if !defined(vax)&&!defined(tahoe) ! 195: /* if x or y is NAN */ ! 196: if(x!=x) return(x); if(y!=y) return(y); ! 197: #endif /* !defined(vax)&&!defined(tahoe) */ ! 198: ! 199: /* copy down the sign of y and x */ ! 200: signy = copysign(one,y) ; ! 201: signx = copysign(one,x) ; ! 202: ! 203: /* if x is 1.0, goto begin */ ! 204: if(x==1) { y=copysign(y,one); t=y; if(finite(t)) goto begin;} ! 205: ! 206: /* when y = 0 */ ! 207: if(y==zero) return((signx==one)?y:copysign(PI,signy)); ! 208: ! 209: /* when x = 0 */ ! 210: if(x==zero) return(copysign(PIo2,signy)); ! 211: ! 212: /* when x is INF */ ! 213: if(!finite(x)) ! 214: if(!finite(y)) ! 215: return(copysign((signx==one)?PIo4:3*PIo4,signy)); ! 216: else ! 217: return(copysign((signx==one)?zero:PI,signy)); ! 218: ! 219: /* when y is INF */ ! 220: if(!finite(y)) return(copysign(PIo2,signy)); ! 221: ! 222: /* compute y/x */ ! 223: x=copysign(x,one); ! 224: y=copysign(y,one); ! 225: if((m=(k=logb(y))-logb(x)) > 60) t=big+big; ! 226: else if(m < -80 ) t=y/x; ! 227: else { t = y/x ; y = scalb(y,-k); x=scalb(x,-k); } ! 228: ! 229: /* begin argument reduction */ ! 230: begin: ! 231: if (t < 2.4375) { ! 232: ! 233: /* truncate 4(t+1/16) to integer for branching */ ! 234: k = 4 * (t+0.0625); ! 235: switch (k) { ! 236: ! 237: /* t is in [0,7/16] */ ! 238: case 0: ! 239: case 1: ! 240: if (t < small) ! 241: { big + small ; /* raise inexact flag */ ! 242: return (copysign((signx>zero)?t:PI-t,signy)); } ! 243: ! 244: hi = zero; lo = zero; break; ! 245: ! 246: /* t is in [7/16,11/16] */ ! 247: case 2: ! 248: hi = athfhi; lo = athflo; ! 249: z = x+x; ! 250: t = ( (y+y) - x ) / ( z + y ); break; ! 251: ! 252: /* t is in [11/16,19/16] */ ! 253: case 3: ! 254: case 4: ! 255: hi = PIo4; lo = zero; ! 256: t = ( y - x ) / ( x + y ); break; ! 257: ! 258: /* t is in [19/16,39/16] */ ! 259: default: ! 260: hi = at1fhi; lo = at1flo; ! 261: z = y-x; y=y+y+y; t = x+x; ! 262: t = ( (z+z)-x ) / ( t + y ); break; ! 263: } ! 264: } ! 265: /* end of if (t < 2.4375) */ ! 266: ! 267: else ! 268: { ! 269: hi = PIo2; lo = zero; ! 270: ! 271: /* t is in [2.4375, big] */ ! 272: if (t <= big) t = - x / y; ! 273: ! 274: /* t is in [big, INF] */ ! 275: else ! 276: { big+small; /* raise inexact flag */ ! 277: t = zero; } ! 278: } ! 279: /* end of argument reduction */ ! 280: ! 281: /* compute atan(t) for t in [-.4375, .4375] */ ! 282: z = t*t; ! 283: #if defined(vax)||defined(tahoe) ! 284: z = t*(z*(a1+z*(a2+z*(a3+z*(a4+z*(a5+z*(a6+z*(a7+z*(a8+ ! 285: z*(a9+z*(a10+z*(a11+z*a12)))))))))))); ! 286: #else /* defined(vax)||defined(tahoe) */ ! 287: z = t*(z*(a1+z*(a2+z*(a3+z*(a4+z*(a5+z*(a6+z*(a7+z*(a8+ ! 288: z*(a9+z*(a10+z*a11))))))))))); ! 289: #endif /* defined(vax)||defined(tahoe) */ ! 290: z = lo - z; z += t; z += hi; ! 291: ! 292: return(copysign((signx>zero)?z:PI-z,signy)); ! 293: }
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