1 /* $NetBSD: n_atan2.c,v 1.7 2014/10/10 20:58:09 martin Exp $ */
2 /*
3 * Copyright (c) 1985, 1993
4 * The Regents of the University of California. All rights reserved.
5 *
6 * Redistribution and use in source and binary forms, with or without
7 * modification, are permitted provided that the following conditions
8 * are met:
9 * 1. Redistributions of source code must retain the above copyright
10 * notice, this list of conditions and the following disclaimer.
11 * 2. Redistributions in binary form must reproduce the above copyright
12 * notice, this list of conditions and the following disclaimer in the
13 * documentation and/or other materials provided with the distribution.
14 * 3. Neither the name of the University nor the names of its contributors
15 * may be used to endorse or promote products derived from this software
16 * without specific prior written permission.
17 *
18 * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
19 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
20 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
21 * ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
22 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
23 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
24 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
25 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
26 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
27 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
28 * SUCH DAMAGE.
29 */
30
31 #ifndef lint
32 static char sccsid[] = "@(#)atan2.c 8.1 (Berkeley) 6/4/93";
33 #endif /* not lint */
34
35 /* ATAN2(Y,X)
36 * RETURN ARG (X+iY)
37 * DOUBLE PRECISION (VAX D format 56 bits, IEEE DOUBLE 53 BITS)
38 * CODED IN C BY K.C. NG, 1/8/85;
39 * REVISED BY K.C. NG on 2/7/85, 2/13/85, 3/7/85, 3/30/85, 6/29/85.
40 *
41 * Required system supported functions :
42 * copysign(x,y)
43 * scalb(x,y)
44 * logb(x)
45 *
46 * Method :
47 * 1. Reduce y to positive by atan2(y,x)=-atan2(-y,x).
48 * 2. Reduce x to positive by (if x and y are unexceptional):
49 * ARG (x+iy) = arctan(y/x) ... if x > 0,
50 * ARG (x+iy) = pi - arctan[y/(-x)] ... if x < 0,
51 * 3. According to the integer k=4t+0.25 truncated , t=y/x, the argument
52 * is further reduced to one of the following intervals and the
53 * arctangent of y/x is evaluated by the corresponding formula:
54 *
55 * [0,7/16] atan(y/x) = t - t^3*(a1+t^2*(a2+...(a10+t^2*a11)...)
56 * [7/16,11/16] atan(y/x) = atan(1/2) + atan( (y-x/2)/(x+y/2) )
57 * [11/16.19/16] atan(y/x) = atan( 1 ) + atan( (y-x)/(x+y) )
58 * [19/16,39/16] atan(y/x) = atan(3/2) + atan( (y-1.5x)/(x+1.5y) )
59 * [39/16,INF] atan(y/x) = atan(INF) + atan( -x/y )
60 *
61 * Special cases:
62 * Notations: atan2(y,x) == ARG (x+iy) == ARG(x,y).
63 *
64 * ARG( NAN , (anything) ) is NaN;
65 * ARG( (anything), NaN ) is NaN;
66 * ARG(+(anything but NaN), +-0) is +-0 ;
67 * ARG(-(anything but NaN), +-0) is +-PI ;
68 * ARG( 0, +-(anything but 0 and NaN) ) is +-PI/2;
69 * ARG( +INF,+-(anything but INF and NaN) ) is +-0 ;
70 * ARG( -INF,+-(anything but INF and NaN) ) is +-PI;
71 * ARG( +INF,+-INF ) is +-PI/4 ;
72 * ARG( -INF,+-INF ) is +-3PI/4;
73 * ARG( (anything but,0,NaN, and INF),+-INF ) is +-PI/2;
74 *
75 * Accuracy:
76 * atan2(y,x) returns (PI/pi) * the exact ARG (x+iy) nearly rounded,
77 * where
78 *
79 * in decimal:
80 * pi = 3.141592653589793 23846264338327 .....
81 * 53 bits PI = 3.141592653589793 115997963 ..... ,
82 * 56 bits PI = 3.141592653589793 227020265 ..... ,
83 *
84 * in hexadecimal:
85 * pi = 3.243F6A8885A308D313198A2E....
86 * 53 bits PI = 3.243F6A8885A30 = 2 * 1.921FB54442D18 error=.276ulps
87 * 56 bits PI = 3.243F6A8885A308 = 4 * .C90FDAA22168C2 error=.206ulps
88 *
89 * In a test run with 356,000 random argument on [-1,1] * [-1,1] on a
90 * VAX, the maximum observed error was 1.41 ulps (units of the last place)
91 * compared with (PI/pi)*(the exact ARG(x+iy)).
92 *
93 * Note:
94 * We use machine PI (the true pi rounded) in place of the actual
95 * value of pi for all the trig and inverse trig functions. In general,
96 * if trig is one of sin, cos, tan, then computed trig(y) returns the
97 * exact trig(y*pi/PI) nearly rounded; correspondingly, computed arctrig
98 * returns the exact arctrig(y)*PI/pi nearly rounded. These guarantee the
99 * trig functions have period PI, and trig(arctrig(x)) returns x for
100 * all critical values x.
101 *
102 * Constants:
103 * The hexadecimal values are the intended ones for the following constants.
104 * The decimal values may be used, provided that the compiler will convert
105 * from decimal to binary accurately enough to produce the hexadecimal values
106 * shown.
107 */
108
109 #define _LIBM_STATIC
110 #include "mathimpl.h"
111
112 vc(athfhi, 4.6364760900080611433E-1 ,6338,3fed,da7b,2b0d, -1, .ED63382B0DDA7B)
113 vc(athflo, 1.9338828231967579916E-19 ,5005,2164,92c0,9cfe, -62, .E450059CFE92C0)
114 vc(PIo4, 7.8539816339744830676E-1 ,0fda,4049,68c2,a221, 0, .C90FDAA22168C2)
115 vc(at1fhi, 9.8279372324732906796E-1 ,985e,407b,b4d9,940f, 0, .FB985E940FB4D9)
116 vc(at1flo,-3.5540295636764633916E-18 ,1edc,a383,eaea,34d6, -57,-.831EDC34D6EAEA)
117 vc(PIo2, 1.5707963267948966135E0 ,0fda,40c9,68c2,a221, 1, .C90FDAA22168C2)
118 vc(PI, 3.1415926535897932270E0 ,0fda,4149,68c2,a221, 2, .C90FDAA22168C2)
119 vc(a1, 3.3333333333333473730E-1 ,aaaa,3faa,ab75,aaaa, -1, .AAAAAAAAAAAB75)
120 vc(a2, -2.0000000000017730678E-1 ,cccc,bf4c,946e,cccd, -2,-.CCCCCCCCCD946E)
121 vc(a3, 1.4285714286694640301E-1 ,4924,3f12,4262,9274, -2, .92492492744262)
122 vc(a4, -1.1111111135032672795E-1 ,8e38,bee3,6292,ebc6, -3,-.E38E38EBC66292)
123 vc(a5, 9.0909091380563043783E-2 ,2e8b,3eba,d70c,b31b, -3, .BA2E8BB31BD70C)
124 vc(a6, -7.6922954286089459397E-2 ,89c8,be9d,7f18,27c3, -3,-.9D89C827C37F18)
125 vc(a7, 6.6663180891693915586E-2 ,86b4,3e88,9e58,ae37, -3, .8886B4AE379E58)
126 vc(a8, -5.8772703698290408927E-2 ,bba5,be70,a942,8481, -4,-.F0BBA58481A942)
127 vc(a9, 5.2170707402812969804E-2 ,b0f3,3e55,13ab,a1ab, -4, .D5B0F3A1AB13AB)
128 vc(a10, -4.4895863157820361210E-2 ,e4b9,be37,048f,7fd1, -4,-.B7E4B97FD1048F)
129 vc(a11, 3.3006147437343875094E-2 ,3174,3e07,2d87,3cf7, -4, .8731743CF72D87)
130 vc(a12, -1.4614844866464185439E-2 ,731a,bd6f,76d9,2f34, -6,-.EF731A2F3476D9)
131
132 ic(athfhi, 4.6364760900080609352E-1 , -2, 1.DAC670561BB4F)
133 ic(athflo, 4.6249969567426939759E-18 , -58, 1.5543B8F253271)
134 ic(PIo4, 7.8539816339744827900E-1 , -1, 1.921FB54442D18)
135 ic(at1fhi, 9.8279372324732905408E-1 , -1, 1.F730BD281F69B)
136 ic(at1flo,-2.4407677060164810007E-17 , -56, -1.C23DFEFEAE6B5)
137 ic(PIo2, 1.5707963267948965580E0 , 0, 1.921FB54442D18)
138 ic(PI, 3.1415926535897931160E0 , 1, 1.921FB54442D18)
139 ic(a1, 3.3333333333333942106E-1 , -2, 1.55555555555C3)
140 ic(a2, -1.9999999999979536924E-1 , -3, -1.9999999997CCD)
141 ic(a3, 1.4285714278004377209E-1 , -3, 1.24924921EC1D7)
142 ic(a4, -1.1111110579344973814E-1 , -4, -1.C71C7059AF280)
143 ic(a5, 9.0908906105474668324E-2 , -4, 1.745CE5AA35DB2)
144 ic(a6, -7.6919217767468239799E-2 , -4, -1.3B0FA54BEC400)
145 ic(a7, 6.6614695906082474486E-2 , -4, 1.10DA924597FFF)
146 ic(a8, -5.8358371008508623523E-2 , -5, -1.DE125FDDBD793)
147 ic(a9, 4.9850617156082015213E-2 , -5, 1.9860524BDD807)
148 ic(a10, -3.6700606902093604877E-2 , -5, -1.2CA6C04C6937A)
149 ic(a11, 1.6438029044759730479E-2 , -6, 1.0D52174A1BB54)
150
151 #ifdef vccast
152 #define athfhi vccast(athfhi)
153 #define athflo vccast(athflo)
154 #define PIo4 vccast(PIo4)
155 #define at1fhi vccast(at1fhi)
156 #define at1flo vccast(at1flo)
157 #define PIo2 vccast(PIo2)
158 #define PI vccast(PI)
159 #define a1 vccast(a1)
160 #define a2 vccast(a2)
161 #define a3 vccast(a3)
162 #define a4 vccast(a4)
163 #define a5 vccast(a5)
164 #define a6 vccast(a6)
165 #define a7 vccast(a7)
166 #define a8 vccast(a8)
167 #define a9 vccast(a9)
168 #define a10 vccast(a10)
169 #define a11 vccast(a11)
170 #define a12 vccast(a12)
171 #endif
172
173 #ifdef __weak_alias
174 __weak_alias(_atan2l, atan2);
175 #endif
176
177 double
atan2(double y,double x)178 atan2(double y, double x)
179 {
180 static const double zero=0, one=1, small=1.0E-9, big=1.0E18;
181 double t,z,signy,signx,hi,lo;
182 int k,m;
183
184 #if !defined(__vax__)&&!defined(tahoe)
185 /* if x or y is NAN */
186 if(x!=x) return(x); if(y!=y) return(y);
187 #endif /* !defined(__vax__)&&!defined(tahoe) */
188
189 /* copy down the sign of y and x */
190 signy = copysign(one,y) ;
191 signx = copysign(one,x) ;
192
193 /* if x is 1.0, goto begin */
194 if(x==1) { y=copysign(y,one); t=y; if(finite(t)) goto begin;}
195
196 /* when y = 0 */
197 if(y==zero) return((signx==one)?y:copysign(PI,signy));
198
199 /* when x = 0 */
200 if(x==zero) return(copysign(PIo2,signy));
201
202 /* when x is INF */
203 if(!finite(x))
204 if(!finite(y))
205 return(copysign((signx==one)?PIo4:3*PIo4,signy));
206 else
207 return(copysign((signx==one)?zero:PI,signy));
208
209 /* when y is INF */
210 if(!finite(y)) return(copysign(PIo2,signy));
211
212 /* compute y/x */
213 x=copysign(x,one);
214 y=copysign(y,one);
215 if((m=(k=logb(y))-logb(x)) > 60) t=big+big;
216 else if(m < -80 ) t=y/x;
217 else { t = y/x ; y = scalb(y,-k); x=scalb(x,-k); }
218
219 /* begin argument reduction */
220 begin:
221 if (t < 2.4375) {
222
223 /* truncate 4(t+1/16) to integer for branching */
224 k = 4 * (t+0.0625);
225 switch (k) {
226
227 /* t is in [0,7/16] */
228 case 0:
229 case 1:
230 if (t < small)
231 { big + small ; /* raise inexact flag */
232 return (copysign((signx>zero)?t:PI-t,signy)); }
233
234 hi = zero; lo = zero; break;
235
236 /* t is in [7/16,11/16] */
237 case 2:
238 hi = athfhi; lo = athflo;
239 z = x+x;
240 t = ( (y+y) - x ) / ( z + y ); break;
241
242 /* t is in [11/16,19/16] */
243 case 3:
244 case 4:
245 hi = PIo4; lo = zero;
246 t = ( y - x ) / ( x + y ); break;
247
248 /* t is in [19/16,39/16] */
249 default:
250 hi = at1fhi; lo = at1flo;
251 z = y-x; y=y+y+y; t = x+x;
252 t = ( (z+z)-x ) / ( t + y ); break;
253 }
254 }
255 /* end of if (t < 2.4375) */
256
257 else
258 {
259 hi = PIo2; lo = zero;
260
261 /* t is in [2.4375, big] */
262 if (t <= big) t = - x / y;
263
264 /* t is in [big, INF] */
265 else
266 { big+small; /* raise inexact flag */
267 t = zero; }
268 }
269 /* end of argument reduction */
270
271 /* compute atan(t) for t in [-.4375, .4375] */
272 z = t*t;
273 #if defined(__vax__)||defined(tahoe)
274 z = t*(z*(a1+z*(a2+z*(a3+z*(a4+z*(a5+z*(a6+z*(a7+z*(a8+
275 z*(a9+z*(a10+z*(a11+z*a12))))))))))));
276 #else /* defined(__vax__)||defined(tahoe) */
277 z = t*(z*(a1+z*(a2+z*(a3+z*(a4+z*(a5+z*(a6+z*(a7+z*(a8+
278 z*(a9+z*(a10+z*a11)))))))))));
279 #endif /* defined(__vax__)||defined(tahoe) */
280 z = lo - z; z += t; z += hi;
281
282 return(copysign((signx>zero)?z:PI-z,signy));
283 }
284