![[BACK]](/cvsweb/icons/back.gif){kind=icon}
Return to atan2.c CVS log ![[TXT]](/cvsweb/icons/text.gif){kind=icon}
![[DIR]](/cvsweb/icons/dir.gif){kind=icon}
Up to [CSRG BSD Unix] / 43BSDTahoe / usr.lib / libm / common|
Return to atan2.c CVS log | Up to [CSRG BSD Unix] / 43BSDTahoe / usr.lib / libm / common |
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: }
This archive runs on limited infrastructure. Preserving old code on modern bandwidth. Automated agents are requested to crawl responsibly.