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BSD 4_3 development
[unix-history] / .ref-4f42a6a1ea1e02529aa6db238da5c5e1c488fad2 / usr / src / lib / libm / common / atan2.c
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5f1375d9
ZAL		 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
27c51c7b
GK		 15static char sccsid[] =
16"@(#)atan2.c 1.3 (Berkeley) 8/21/85; 1.2 (ucb.elefunt) %G%";
5f1375d9
ZAL		 17#endif not lint
18
19/* ATAN2(Y,X)
20 * RETURN ARG (X+iY)
21 * DOUBLE PRECISION (VAX D format 56 bits, IEEE DOUBLE 53 BITS)
22 * CODED IN C BY K.C. NG, 1/8/85;
23 * REVISED BY K.C. NG on 2/7/85, 2/13/85, 3/7/85, 3/30/85, 6/29/85.
24 *
25 * Required system supported functions :
26 * copysign(x,y)
27 * scalb(x,y)
28 * logb(x)
29 *
30 * Method :
31 * 1. Reduce y to positive by atan2(y,x)=-atan2(-y,x).
32 * 2. Reduce x to positive by (if x and y are unexceptional):
33 * ARG (x+iy) = arctan(y/x) ... if x > 0,
34 * ARG (x+iy) = pi - arctan[y/(-x)] ... if x < 0,
35 * 3. According to the integer k=4t+0.25 truncated , t=y/x, the argument
36 * is further reduced to one of the following intervals and the
37 * arctangent of y/x is evaluated by the corresponding formula:
38 *
39 * [0,7/16] atan(y/x) = t - t^3*(a1+t^2*(a2+...(a10+t^2*a11)...)
40 * [7/16,11/16] atan(y/x) = atan(1/2) + atan( (y-x/2)/(x+y/2) )
41 * [11/16.19/16] atan(y/x) = atan( 1 ) + atan( (y-x)/(x+y) )
42 * [19/16,39/16] atan(y/x) = atan(3/2) + atan( (y-1.5x)/(x+1.5y) )
43 * [39/16,INF] atan(y/x) = atan(INF) + atan( -x/y )
44 *
45 * Special cases:
46 * Notations: atan2(y,x) == ARG (x+iy) == ARG(x,y).
47 *
48 * ARG( NAN , (anything) ) is NaN;
49 * ARG( (anything), NaN ) is NaN;
50 * ARG(+(anything but NaN), +-0) is +-0 ;
51 * ARG(-(anything but NaN), +-0) is +-PI ;
52 * ARG( 0, +-(anything but 0 and NaN) ) is +-PI/2;
53 * ARG( +INF,+-(anything but INF and NaN) ) is +-0 ;
54 * ARG( -INF,+-(anything but INF and NaN) ) is +-PI;
55 * ARG( +INF,+-INF ) is +-PI/4 ;
56 * ARG( -INF,+-INF ) is +-3PI/4;
57 * ARG( (anything but,0,NaN, and INF),+-INF ) is +-PI/2;
58 *
59 * Accuracy:
60 * atan2(y,x) returns (PI/pi) * the exact ARG (x+iy) nearly rounded,
61 * where
62 *
63 * in decimal:
64 * pi = 3.141592653589793 23846264338327 .....
65 * 53 bits PI = 3.141592653589793 115997963 ..... ,
66 * 56 bits PI = 3.141592653589793 227020265 ..... ,
67 *
68 * in hexadecimal:
69 * pi = 3.243F6A8885A308D313198A2E....
70 * 53 bits PI = 3.243F6A8885A30 = 2 * 1.921FB54442D18 error=.276ulps
71 * 56 bits PI = 3.243F6A8885A308 = 4 * .C90FDAA22168C2 error=.206ulps
72 *
73 * In a test run with 356,000 random argument on [-1,1] * [-1,1] on a
74 * VAX, the maximum observed error was 1.41 ulps (units of the last place)
75 * compared with (PI/pi)*(the exact ARG(x+iy)).
76 *
77 * Note:
78 * We use machine PI (the true pi rounded) in place of the actual
79 * value of pi for all the trig and inverse trig functions. In general,
80 * if trig is one of sin, cos, tan, then computed trig(y) returns the
81 * exact trig(y*pi/PI) nearly rounded; correspondingly, computed arctrig
82 * returns the exact arctrig(y)*PI/pi nearly rounded. These guarantee the
83 * trig functions have period PI, and trig(arctrig(x)) returns x for
84 * all critical values x.
85 *
86 * Constants:
87 * The hexadecimal values are the intended ones for the following constants.
88 * The decimal values may be used, provided that the compiler will convert
89 * from decimal to binary accurately enough to produce the hexadecimal values
90 * shown.
91 */
92
93static double
94#ifdef VAX /* VAX D format */
95athfhi = 4.6364760900080611433E-1 , /*Hex 2^ -1 * .ED63382B0DDA7B */
96athflo = 1.9338828231967579916E-19 , /*Hex 2^-62 * .E450059CFE92C0 */
97PIo4 = 7.8539816339744830676E-1 , /*Hex 2^ 0 * .C90FDAA22168C2 */
98at1fhi = 9.8279372324732906796E-1 , /*Hex 2^ 0 * .FB985E940FB4D9 */
99at1flo = -3.5540295636764633916E-18 , /*Hex 2^-57 * -.831EDC34D6EAEA */
100PIo2 = 1.5707963267948966135E0 , /*Hex 2^ 1 * .C90FDAA22168C2 */
101PI = 3.1415926535897932270E0 , /*Hex 2^ 2 * .C90FDAA22168C2 */
102a1 = 3.3333333333333473730E-1 , /*Hex 2^ -1 * .AAAAAAAAAAAB75 */
103a2 = -2.0000000000017730678E-1 , /*Hex 2^ -2 * -.CCCCCCCCCD946E */
104a3 = 1.4285714286694640301E-1 , /*Hex 2^ -2 * .92492492744262 */
105a4 = -1.1111111135032672795E-1 , /*Hex 2^ -3 * -.E38E38EBC66292 */
106a5 = 9.0909091380563043783E-2 , /*Hex 2^ -3 * .BA2E8BB31BD70C */
107a6 = -7.6922954286089459397E-2 , /*Hex 2^ -3 * -.9D89C827C37F18 */
108a7 = 6.6663180891693915586E-2 , /*Hex 2^ -3 * .8886B4AE379E58 */
109a8 = -5.8772703698290408927E-2 , /*Hex 2^ -4 * -.F0BBA58481A942 */
110a9 = 5.2170707402812969804E-2 , /*Hex 2^ -4 * .D5B0F3A1AB13AB */
111a10 = -4.4895863157820361210E-2 , /*Hex 2^ -4 * -.B7E4B97FD1048F */
112a11 = 3.3006147437343875094E-2 , /*Hex 2^ -4 * .8731743CF72D87 */
113a12 = -1.4614844866464185439E-2 ; /*Hex 2^ -6 * -.EF731A2F3476D9 */
114#else /* IEEE double */
115athfhi = 4.6364760900080609352E-1 , /*Hex 2^ -2 * 1.DAC670561BB4F */
116athflo = 4.6249969567426939759E-18 , /*Hex 2^-58 * 1.5543B8F253271 */
117PIo4 = 7.8539816339744827900E-1 , /*Hex 2^ -1 * 1.921FB54442D18 */
118at1fhi = 9.8279372324732905408E-1 , /*Hex 2^ -1 * 1.F730BD281F69B */
119at1flo = -2.4407677060164810007E-17 , /*Hex 2^-56 * -1.C23DFEFEAE6B5 */
120PIo2 = 1.5707963267948965580E0 , /*Hex 2^ 0 * 1.921FB54442D18 */
121PI = 3.1415926535897931160E0 , /*Hex 2^ 1 * 1.921FB54442D18 */
122a1 = 3.3333333333333942106E-1 , /*Hex 2^ -2 * 1.55555555555C3 */
123a2 = -1.9999999999979536924E-1 , /*Hex 2^ -3 * -1.9999999997CCD */
124a3 = 1.4285714278004377209E-1 , /*Hex 2^ -3 * 1.24924921EC1D7 */
125a4 = -1.1111110579344973814E-1 , /*Hex 2^ -4 * -1.C71C7059AF280 */
126a5 = 9.0908906105474668324E-2 , /*Hex 2^ -4 * 1.745CE5AA35DB2 */
127a6 = -7.6919217767468239799E-2 , /*Hex 2^ -4 * -1.3B0FA54BEC400 */
128a7 = 6.6614695906082474486E-2 , /*Hex 2^ -4 * 1.10DA924597FFF */
129a8 = -5.8358371008508623523E-2 , /*Hex 2^ -5 * -1.DE125FDDBD793 */
130a9 = 4.9850617156082015213E-2 , /*Hex 2^ -5 * 1.9860524BDD807 */
131a10 = -3.6700606902093604877E-2 , /*Hex 2^ -5 * -1.2CA6C04C6937A */
132a11 = 1.6438029044759730479E-2 ; /*Hex 2^ -6 * 1.0D52174A1BB54 */
133#endif
134
135double atan2(y,x)
136double y,x;
137{
138 static double zero=0, one=1, small=1.0E-9, big=1.0E18;
139 double copysign(),logb(),scalb(),t,z,signy,signx,hi,lo;
140 int finite(), k,m;
141
142 /* if x or y is NAN */
143 if(x!=x) return(x); if(y!=y) return(y);
144
145 /* copy down the sign of y and x */
146 signy = copysign(one,y) ;
147 signx = copysign(one,x) ;
148
149 /* if x is 1.0, goto begin */
150 if(x==1) { y=copysign(y,one); t=y; if(finite(t)) goto begin;}
151
152 /* when y = 0 */
153 if(y==zero) return((signx==one)?y:copysign(PI,signy));
154
155 /* when x = 0 */
156 if(x==zero) return(copysign(PIo2,signy));
157
158 /* when x is INF */
159 if(!finite(x))
160 if(!finite(y))
161 return(copysign((signx==one)?PIo4:3*PIo4,signy));
162 else
163 return(copysign((signx==one)?zero:PI,signy));
164
165 /* when y is INF */
166 if(!finite(y)) return(copysign(PIo2,signy));
167
168
169 /* compute y/x */
170 x=copysign(x,one);
171 y=copysign(y,one);
172 if((m=(k=logb(y))-logb(x)) > 60) t=big+big;
173 else if(m < -80 ) t=y/x;
174 else { t = y/x ; y = scalb(y,-k); x=scalb(x,-k); }
175
176 /* begin argument reduction */
177begin:
178 if (t < 2.4375) {
179
180 /* truncate 4(t+1/16) to integer for branching */
181 k = 4 * (t+0.0625);
182 switch (k) {
183
184 /* t is in [0,7/16] */
185 case 0:
186 case 1:
187 if (t < small)
188 { big + small ; /* raise inexact flag */
189 return (copysign((signx>zero)?t:PI-t,signy)); }
190
191 hi = zero; lo = zero; break;
192
193 /* t is in [7/16,11/16] */
194 case 2:
195 hi = athfhi; lo = athflo;
196 z = x+x;
197 t = ( (y+y) - x ) / ( z + y ); break;
198
199 /* t is in [11/16,19/16] */
200 case 3:
201 case 4:
202 hi = PIo4; lo = zero;
203 t = ( y - x ) / ( x + y ); break;
204
205 /* t is in [19/16,39/16] */
206 default:
207 hi = at1fhi; lo = at1flo;
208 z = y-x; y=y+y+y; t = x+x;
209 t = ( (z+z)-x ) / ( t + y ); break;
210 }
211 }
212 /* end of if (t < 2.4375) */
213
214 else
215 {
216 hi = PIo2; lo = zero;
217
218 /* t is in [2.4375, big] */
219 if (t <= big) t = - x / y;
220
221 /* t is in [big, INF] */
222 else
223 { big+small; /* raise inexact flag */
224 t = zero; }
225 }
226 /* end of argument reduction */
227
228 /* compute atan(t) for t in [-.4375, .4375] */
229 z = t*t;
230#ifdef VAX
231 z = t*(z*(a1+z*(a2+z*(a3+z*(a4+z*(a5+z*(a6+z*(a7+z*(a8+
232 z*(a9+z*(a10+z*(a11+z*a12))))))))))));
233#else /* IEEE double */
234 z = t*(z*(a1+z*(a2+z*(a3+z*(a4+z*(a5+z*(a6+z*(a7+z*(a8+
235 z*(a9+z*(a10+z*a11)))))))))));
236#endif
237 z = lo - z; z += t; z += hi;
238
239 return(copysign((signx>zero)?z:PI-z,signy));
240}
Continuous source code history from original PDP-7 UNIX to FreeBSD 2, the first fully unencumbered FreeBSD release.