1 
2 /* @(#)s_atan.c 5.1 93/09/24 */
3 /*
4  * ====================================================
5  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
6  *
7  * Developed at SunPro, a Sun Microsystems, Inc. business.
8  * Permission to use, copy, modify, and distribute this
9  * software is freely granted, provided that this notice
10  * is preserved.
11  * ====================================================
12  *
13  */
14 
15 /*
16 FUNCTION
17         <<atan>>, <<atanf>>---arc tangent
18 
19 INDEX
20    atan
21 INDEX
22    atanf
23 
24 SYNOPSIS
25         #include <math.h>
26         double atan(double <[x]>);
27         float atanf(float <[x]>);
28 
29 DESCRIPTION
30 
31 <<atan>> computes the inverse tangent (arc tangent) of the input value.
32 
33 <<atanf>> is identical to <<atan>>, save that it operates on <<floats>>.
34 
35 RETURNS
36 @ifnottex
37 <<atan>> returns a value in radians, in the range of -pi/2 to pi/2.
38 @end ifnottex
39 @tex
40 <<atan>> returns a value in radians, in the range of $-\pi/2$ to $\pi/2$.
41 @end tex
42 
43 PORTABILITY
44 <<atan>> is ANSI C.  <<atanf>> is an extension.
45 
46 */
47 
48 /* atan(x)
49  * Method
50  *   1. Reduce x to positive by atan(x) = -atan(-x).
51  *   2. According to the integer k=4t+0.25 chopped, t=x, the argument
52  *      is further reduced to one of the following intervals and the
53  *      arctangent of t is evaluated by the corresponding formula:
54  *
55  *      [0,7/16]      atan(x) = t-t^3*(a1+t^2*(a2+...(a10+t^2*a11)...)
56  *      [7/16,11/16]  atan(x) = atan(1/2) + atan( (t-0.5)/(1+t/2) )
57  *      [11/16.19/16] atan(x) = atan( 1 ) + atan( (t-1)/(1+t) )
58  *      [19/16,39/16] atan(x) = atan(3/2) + atan( (t-1.5)/(1+1.5t) )
59  *      [39/16,INF]   atan(x) = atan(INF) + atan( -1/t )
60  *
61  * Constants:
62  * The hexadecimal values are the intended ones for the following
63  * constants. The decimal values may be used, provided that the
64  * compiler will convert from decimal to binary accurately enough
65  * to produce the hexadecimal values shown.
66  */
67 
68 #include "fdlibm.h"
69 
70 #ifdef _NEED_FLOAT64
71 
72 static const __float64 atanhi[] = {
73     _F_64(4.63647609000806093515e-01), /* atan(0.5)hi 0x3FDDAC67, 0x0561BB4F */
74     _F_64(7.85398163397448278999e-01), /* atan(1.0)hi 0x3FE921FB, 0x54442D18 */
75     _F_64(9.82793723247329054082e-01), /* atan(1.5)hi 0x3FEF730B, 0xD281F69B */
76     _F_64(1.57079632679489655800e+00), /* atan(inf)hi 0x3FF921FB, 0x54442D18 */
77 };
78 
79 static const __float64 atanlo[] = {
80     _F_64(2.26987774529616870924e-17), /* atan(0.5)lo 0x3C7A2B7F, 0x222F65E2 */
81     _F_64(3.06161699786838301793e-17), /* atan(1.0)lo 0x3C81A626, 0x33145C07 */
82     _F_64(1.39033110312309984516e-17), /* atan(1.5)lo 0x3C700788, 0x7AF0CBBD */
83     _F_64(6.12323399573676603587e-17), /* atan(inf)lo 0x3C91A626, 0x33145C07 */
84 };
85 
86 static const __float64 aT[] = {
87     _F_64(3.33333333333329318027e-01), /* 0x3FD55555, 0x5555550D */
88     _F_64(-1.99999999998764832476e-01), /* 0xBFC99999, 0x9998EBC4 */
89     _F_64(1.42857142725034663711e-01), /* 0x3FC24924, 0x920083FF */
90     _F_64(-1.11111104054623557880e-01), /* 0xBFBC71C6, 0xFE231671 */
91     _F_64(9.09088713343650656196e-02), /* 0x3FB745CD, 0xC54C206E */
92     _F_64(-7.69187620504482999495e-02), /* 0xBFB3B0F2, 0xAF749A6D */
93     _F_64(6.66107313738753120669e-02), /* 0x3FB10D66, 0xA0D03D51 */
94     _F_64(-5.83357013379057348645e-02), /* 0xBFADDE2D, 0x52DEFD9A */
95     _F_64(4.97687799461593236017e-02), /* 0x3FA97B4B, 0x24760DEB */
96     _F_64(-3.65315727442169155270e-02), /* 0xBFA2B444, 0x2C6A6C2F */
97     _F_64(1.62858201153657823623e-02), /* 0x3F90AD3A, 0xE322DA11 */
98 };
99 
100 static const __float64 one = _F_64(1.0), huge = _F_64(1.0e300);
101 
102 __float64
atan64(__float64 x)103 atan64(__float64 x)
104 {
105     __float64 w, s1, s2, z;
106     __int32_t ix, hx, id;
107 
108     GET_HIGH_WORD(hx, x);
109     ix = hx & 0x7fffffff;
110     if (ix >= 0x44100000) { /* if |x| >= 2^66 */
111         __uint32_t low;
112         GET_LOW_WORD(low, x);
113         if (ix > 0x7ff00000 || (ix == 0x7ff00000 && (low != 0)))
114             return x + x; /* NaN */
115         if (hx > 0)
116             return atanhi[3] + atanlo[3];
117         else
118             return -atanhi[3] - atanlo[3];
119     }
120     if (ix < 0x3fdc0000) { /* |x| < 0.4375 */
121         if (ix < 0x3e200000) { /* |x| < 2^-29 */
122             if (huge + x > one)
123                 return x; /* raise inexact */
124         }
125         id = -1;
126     } else {
127         x = fabs64(x);
128         if (ix < 0x3ff30000) { /* |x| < 1.1875 */
129             if (ix < 0x3fe60000) { /* 7/16 <=|x|<11/16 */
130                 id = 0;
131                 x = (_F_64(2.0) * x - one) / (_F_64(2.0) + x);
132             } else { /* 11/16<=|x|< 19/16 */
133                 id = 1;
134                 x = (x - one) / (x + one);
135             }
136         } else {
137             if (ix < 0x40038000) { /* |x| < 2.4375 */
138                 id = 2;
139                 x = (x - _F_64(1.5)) / (one + 1.5 * x);
140             } else { /* 2.4375 <= |x| < 2^66 */
141                 id = 3;
142                 x = _F_64(-1.0) / x;
143             }
144         }
145     }
146     /* end of argument reduction */
147     z = x * x;
148     w = z * z;
149     /* break sum from i=0 to 10 aT[i]z**(i+1) into odd and even poly */
150     s1 = z *
151          (aT[0] +
152           w * (aT[2] + w * (aT[4] + w * (aT[6] + w * (aT[8] + w * aT[10])))));
153     s2 = w * (aT[1] + w * (aT[3] + w * (aT[5] + w * (aT[7] + w * aT[9]))));
154     if (id < 0)
155         return x - x * (s1 + s2);
156     else {
157         z = atanhi[id] - ((x * (s1 + s2) - atanlo[id]) - x);
158         return (hx < 0) ? -z : z;
159     }
160 }
161 
162 _MATH_ALIAS_d_d(atan)
163 
164 #endif /* _NEED_FLOAT64 */
165