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 1.4 (Berkeley) 06/29/87";
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 #if defined(VAX) || defined(TAHOE) /* 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
atan2(y,x)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 #if defined(VAX) || defined(TAHOE)
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 }
240