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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: (1) source distributions retain this entire copyright
7: * notice and comment, and (2) distributions including binaries display
8: * the following acknowledgement: ``This product includes software
9: * developed by the University of California, Berkeley and its contributors''
10: * in the documentation or other materials provided with the distribution
11: * and in all advertising materials mentioning features or use of this
12: * software. Neither the name of the University nor the names of its
13: * contributors may be used to endorse or promote products derived
14: * from this software without specific prior written permission.
15: * THIS SOFTWARE IS PROVIDED ``AS IS'' AND WITHOUT ANY EXPRESS OR
16: * IMPLIED WARRANTIES, INCLUDING, WITHOUT LIMITATION, THE IMPLIED
17: * WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE.
18: *
19: * All recipients should regard themselves as participants in an ongoing
20: * research project and hence should feel obligated to report their
21: * experiences (good or bad) with these elementary function codes, using
22: * the sendbug(8) program, to the authors.
23: */
24:
25: #ifndef lint
26: static char sccsid[] = "@(#)atan2.c 5.5 (Berkeley) 6/1/90";
27: #endif /* not lint */
28:
29: /* ATAN2(Y,X)
30: * RETURN ARG (X+iY)
31: * DOUBLE PRECISION (VAX D format 56 bits, IEEE DOUBLE 53 BITS)
32: * CODED IN C BY K.C. NG, 1/8/85;
33: * REVISED BY K.C. NG on 2/7/85, 2/13/85, 3/7/85, 3/30/85, 6/29/85.
34: *
35: * Required system supported functions :
36: * copysign(x,y)
37: * scalb(x,y)
38: * logb(x)
39: *
40: * Method :
41: * 1. Reduce y to positive by atan2(y,x)=-atan2(-y,x).
42: * 2. Reduce x to positive by (if x and y are unexceptional):
43: * ARG (x+iy) = arctan(y/x) ... if x > 0,
44: * ARG (x+iy) = pi - arctan[y/(-x)] ... if x < 0,
45: * 3. According to the integer k=4t+0.25 truncated , t=y/x, the argument
46: * is further reduced to one of the following intervals and the
47: * arctangent of y/x is evaluated by the corresponding formula:
48: *
49: * [0,7/16] atan(y/x) = t - t^3*(a1+t^2*(a2+...(a10+t^2*a11)...)
50: * [7/16,11/16] atan(y/x) = atan(1/2) + atan( (y-x/2)/(x+y/2) )
51: * [11/16.19/16] atan(y/x) = atan( 1 ) + atan( (y-x)/(x+y) )
52: * [19/16,39/16] atan(y/x) = atan(3/2) + atan( (y-1.5x)/(x+1.5y) )
53: * [39/16,INF] atan(y/x) = atan(INF) + atan( -x/y )
54: *
55: * Special cases:
56: * Notations: atan2(y,x) == ARG (x+iy) == ARG(x,y).
57: *
58: * ARG( NAN , (anything) ) is NaN;
59: * ARG( (anything), NaN ) is NaN;
60: * ARG(+(anything but NaN), +-0) is +-0 ;
61: * ARG(-(anything but NaN), +-0) is +-PI ;
62: * ARG( 0, +-(anything but 0 and NaN) ) is +-PI/2;
63: * ARG( +INF,+-(anything but INF and NaN) ) is +-0 ;
64: * ARG( -INF,+-(anything but INF and NaN) ) is +-PI;
65: * ARG( +INF,+-INF ) is +-PI/4 ;
66: * ARG( -INF,+-INF ) is +-3PI/4;
67: * ARG( (anything but,0,NaN, and INF),+-INF ) is +-PI/2;
68: *
69: * Accuracy:
70: * atan2(y,x) returns (PI/pi) * the exact ARG (x+iy) nearly rounded,
71: * where
72: *
73: * in decimal:
74: * pi = 3.141592653589793 23846264338327 .....
75: * 53 bits PI = 3.141592653589793 115997963 ..... ,
76: * 56 bits PI = 3.141592653589793 227020265 ..... ,
77: *
78: * in hexadecimal:
79: * pi = 3.243F6A8885A308D313198A2E....
80: * 53 bits PI = 3.243F6A8885A30 = 2 * 1.921FB54442D18 error=.276ulps
81: * 56 bits PI = 3.243F6A8885A308 = 4 * .C90FDAA22168C2 error=.206ulps
82: *
83: * In a test run with 356,000 random argument on [-1,1] * [-1,1] on a
84: * VAX, the maximum observed error was 1.41 ulps (units of the last place)
85: * compared with (PI/pi)*(the exact ARG(x+iy)).
86: *
87: * Note:
88: * We use machine PI (the true pi rounded) in place of the actual
89: * value of pi for all the trig and inverse trig functions. In general,
90: * if trig is one of sin, cos, tan, then computed trig(y) returns the
91: * exact trig(y*pi/PI) nearly rounded; correspondingly, computed arctrig
92: * returns the exact arctrig(y)*PI/pi nearly rounded. These guarantee the
93: * trig functions have period PI, and trig(arctrig(x)) returns x for
94: * all critical values x.
95: *
96: * Constants:
97: * The hexadecimal values are the intended ones for the following constants.
98: * The decimal values may be used, provided that the compiler will convert
99: * from decimal to binary accurately enough to produce the hexadecimal values
100: * shown.
101: */
102:
103: #include "mathimpl.h"
104:
105: vc(athfhi, 4.6364760900080611433E-1 ,6338,3fed,da7b,2b0d, -1, .ED63382B0DDA7B)
106: vc(athflo, 1.9338828231967579916E-19 ,5005,2164,92c0,9cfe, -62, .E450059CFE92C0)
107: vc(PIo4, 7.8539816339744830676E-1 ,0fda,4049,68c2,a221, 0, .C90FDAA22168C2)
108: vc(at1fhi, 9.8279372324732906796E-1 ,985e,407b,b4d9,940f, 0, .FB985E940FB4D9)
109: vc(at1flo,-3.5540295636764633916E-18 ,1edc,a383,eaea,34d6, -57,-.831EDC34D6EAEA)
110: vc(PIo2, 1.5707963267948966135E0 ,0fda,40c9,68c2,a221, 1, .C90FDAA22168C2)
111: vc(PI, 3.1415926535897932270E0 ,0fda,4149,68c2,a221, 2, .C90FDAA22168C2)
112: vc(a1, 3.3333333333333473730E-1 ,aaaa,3faa,ab75,aaaa, -1, .AAAAAAAAAAAB75)
113: vc(a2, -2.0000000000017730678E-1 ,cccc,bf4c,946e,cccd, -2,-.CCCCCCCCCD946E)
114: vc(a3, 1.4285714286694640301E-1 ,4924,3f12,4262,9274, -2, .92492492744262)
115: vc(a4, -1.1111111135032672795E-1 ,8e38,bee3,6292,ebc6, -3,-.E38E38EBC66292)
116: vc(a5, 9.0909091380563043783E-2 ,2e8b,3eba,d70c,b31b, -3, .BA2E8BB31BD70C)
117: vc(a6, -7.6922954286089459397E-2 ,89c8,be9d,7f18,27c3, -3,-.9D89C827C37F18)
118: vc(a7, 6.6663180891693915586E-2 ,86b4,3e88,9e58,ae37, -3, .8886B4AE379E58)
119: vc(a8, -5.8772703698290408927E-2 ,bba5,be70,a942,8481, -4,-.F0BBA58481A942)
120: vc(a9, 5.2170707402812969804E-2 ,b0f3,3e55,13ab,a1ab, -4, .D5B0F3A1AB13AB)
121: vc(a10, -4.4895863157820361210E-2 ,e4b9,be37,048f,7fd1, -4,-.B7E4B97FD1048F)
122: vc(a11, 3.3006147437343875094E-2 ,3174,3e07,2d87,3cf7, -4, .8731743CF72D87)
123: vc(a12, -1.4614844866464185439E-2 ,731a,bd6f,76d9,2f34, -6,-.EF731A2F3476D9)
124:
125: ic(athfhi, 4.6364760900080609352E-1 , -2, 1.DAC670561BB4F)
126: ic(athflo, 4.6249969567426939759E-18 , -58, 1.5543B8F253271)
127: ic(PIo4, 7.8539816339744827900E-1 , -1, 1.921FB54442D18)
128: ic(at1fhi, 9.8279372324732905408E-1 , -1, 1.F730BD281F69B)
129: ic(at1flo,-2.4407677060164810007E-17 , -56, -1.C23DFEFEAE6B5)
130: ic(PIo2, 1.5707963267948965580E0 , 0, 1.921FB54442D18)
131: ic(PI, 3.1415926535897931160E0 , 1, 1.921FB54442D18)
132: ic(a1, 3.3333333333333942106E-1 , -2, 1.55555555555C3)
133: ic(a2, -1.9999999999979536924E-1 , -3, -1.9999999997CCD)
134: ic(a3, 1.4285714278004377209E-1 , -3, 1.24924921EC1D7)
135: ic(a4, -1.1111110579344973814E-1 , -4, -1.C71C7059AF280)
136: ic(a5, 9.0908906105474668324E-2 , -4, 1.745CE5AA35DB2)
137: ic(a6, -7.6919217767468239799E-2 , -4, -1.3B0FA54BEC400)
138: ic(a7, 6.6614695906082474486E-2 , -4, 1.10DA924597FFF)
139: ic(a8, -5.8358371008508623523E-2 , -5, -1.DE125FDDBD793)
140: ic(a9, 4.9850617156082015213E-2 , -5, 1.9860524BDD807)
141: ic(a10, -3.6700606902093604877E-2 , -5, -1.2CA6C04C6937A)
142: ic(a11, 1.6438029044759730479E-2 , -6, 1.0D52174A1BB54)
143:
144: #ifdef vccast
145: #define athfhi vccast(athfhi)
146: #define athflo vccast(athflo)
147: #define PIo4 vccast(PIo4)
148: #define at1fhi vccast(at1fhi)
149: #define at1flo vccast(at1flo)
150: #define PIo2 vccast(PIo2)
151: #define PI vccast(PI)
152: #define a1 vccast(a1)
153: #define a2 vccast(a2)
154: #define a3 vccast(a3)
155: #define a4 vccast(a4)
156: #define a5 vccast(a5)
157: #define a6 vccast(a6)
158: #define a7 vccast(a7)
159: #define a8 vccast(a8)
160: #define a9 vccast(a9)
161: #define a10 vccast(a10)
162: #define a11 vccast(a11)
163: #define a12 vccast(a12)
164: #endif
165:
166: double atan2(y,x)
167: double y,x;
168: {
169: static const double zero=0, one=1, small=1.0E-9, big=1.0E18;
170: double t,z,signy,signx,hi,lo;
171: int k,m;
172:
173: #if !defined(vax)&&!defined(tahoe)
174: /* if x or y is NAN */
175: if(x!=x) return(x); if(y!=y) return(y);
176: #endif /* !defined(vax)&&!defined(tahoe) */
177:
178: /* copy down the sign of y and x */
179: signy = copysign(one,y) ;
180: signx = copysign(one,x) ;
181:
182: /* if x is 1.0, goto begin */
183: if(x==1) { y=copysign(y,one); t=y; if(finite(t)) goto begin;}
184:
185: /* when y = 0 */
186: if(y==zero) return((signx==one)?y:copysign(PI,signy));
187:
188: /* when x = 0 */
189: if(x==zero) return(copysign(PIo2,signy));
190:
191: /* when x is INF */
192: if(!finite(x))
193: if(!finite(y))
194: return(copysign((signx==one)?PIo4:3*PIo4,signy));
195: else
196: return(copysign((signx==one)?zero:PI,signy));
197:
198: /* when y is INF */
199: if(!finite(y)) return(copysign(PIo2,signy));
200:
201: /* compute y/x */
202: x=copysign(x,one);
203: y=copysign(y,one);
204: if((m=(k=logb(y))-logb(x)) > 60) t=big+big;
205: else if(m < -80 ) t=y/x;
206: else { t = y/x ; y = scalb(y,-k); x=scalb(x,-k); }
207:
208: /* begin argument reduction */
209: begin:
210: if (t < 2.4375) {
211:
212: /* truncate 4(t+1/16) to integer for branching */
213: k = 4 * (t+0.0625);
214: switch (k) {
215:
216: /* t is in [0,7/16] */
217: case 0:
218: case 1:
219: if (t < small)
220: { big + small ; /* raise inexact flag */
221: return (copysign((signx>zero)?t:PI-t,signy)); }
222:
223: hi = zero; lo = zero; break;
224:
225: /* t is in [7/16,11/16] */
226: case 2:
227: hi = athfhi; lo = athflo;
228: z = x+x;
229: t = ( (y+y) - x ) / ( z + y ); break;
230:
231: /* t is in [11/16,19/16] */
232: case 3:
233: case 4:
234: hi = PIo4; lo = zero;
235: t = ( y - x ) / ( x + y ); break;
236:
237: /* t is in [19/16,39/16] */
238: default:
239: hi = at1fhi; lo = at1flo;
240: z = y-x; y=y+y+y; t = x+x;
241: t = ( (z+z)-x ) / ( t + y ); break;
242: }
243: }
244: /* end of if (t < 2.4375) */
245:
246: else
247: {
248: hi = PIo2; lo = zero;
249:
250: /* t is in [2.4375, big] */
251: if (t <= big) t = - x / y;
252:
253: /* t is in [big, INF] */
254: else
255: { big+small; /* raise inexact flag */
256: t = zero; }
257: }
258: /* end of argument reduction */
259:
260: /* compute atan(t) for t in [-.4375, .4375] */
261: z = t*t;
262: #if defined(vax)||defined(tahoe)
263: z = t*(z*(a1+z*(a2+z*(a3+z*(a4+z*(a5+z*(a6+z*(a7+z*(a8+
264: z*(a9+z*(a10+z*(a11+z*a12))))))))))));
265: #else /* defined(vax)||defined(tahoe) */
266: z = t*(z*(a1+z*(a2+z*(a3+z*(a4+z*(a5+z*(a6+z*(a7+z*(a8+
267: z*(a9+z*(a10+z*a11)))))))))));
268: #endif /* defined(vax)||defined(tahoe) */
269: z = lo - z; z += t; z += hi;
270:
271: return(copysign((signx>zero)?z:PI-z,signy));
272: }
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