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