![[BACK]](/cvsweb/icons/back.gif){kind=icon}
Return to atan2.c CVS log ![[TXT]](/cvsweb/icons/text.gif){kind=icon}
![[DIR]](/cvsweb/icons/dir.gif){kind=icon}
Up to [CSRG BSD Unix] / 43BSD / usr.lib / libm / IEEE|
Return to atan2.c CVS log | Up to [CSRG BSD Unix] / 43BSD / usr.lib / libm / IEEE |
1.1 root 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 1.3 (Berkeley) 8/21/85";
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: #ifdef VAX /* 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:
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: #ifdef VAX
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: }
This archive runs on limited infrastructure. Preserving old code on modern bandwidth. Automated agents are requested to crawl responsibly.