atan.c raw
1 /* origin: FreeBSD /usr/src/lib/msun/src/s_atan.c */
2 /*
3 * ====================================================
4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
5 *
6 * Developed at SunPro, a Sun Microsystems, Inc. business.
7 * Permission to use, copy, modify, and distribute this
8 * software is freely granted, provided that this notice
9 * is preserved.
10 * ====================================================
11 */
12 /* atan(x)
13 * Method
14 * 1. Reduce x to positive by atan(x) = -atan(-x).
15 * 2. According to the integer k=4t+0.25 chopped, t=x, the argument
16 * is further reduced to one of the following intervals and the
17 * arctangent of t is evaluated by the corresponding formula:
18 *
19 * [0,7/16] atan(x) = t-t^3*(a1+t^2*(a2+...(a10+t^2*a11)...)
20 * [7/16,11/16] atan(x) = atan(1/2) + atan( (t-0.5)/(1+t/2) )
21 * [11/16.19/16] atan(x) = atan( 1 ) + atan( (t-1)/(1+t) )
22 * [19/16,39/16] atan(x) = atan(3/2) + atan( (t-1.5)/(1+1.5t) )
23 * [39/16,INF] atan(x) = atan(INF) + atan( -1/t )
24 *
25 * Constants:
26 * The hexadecimal values are the intended ones for the following
27 * constants. The decimal values may be used, provided that the
28 * compiler will convert from decimal to binary accurately enough
29 * to produce the hexadecimal values shown.
30 */
31
32
33 #include "libm.h"
34
35 static const double atanhi[] = {
36 4.63647609000806093515e-01, /* atan(0.5)hi 0x3FDDAC67, 0x0561BB4F */
37 7.85398163397448278999e-01, /* atan(1.0)hi 0x3FE921FB, 0x54442D18 */
38 9.82793723247329054082e-01, /* atan(1.5)hi 0x3FEF730B, 0xD281F69B */
39 1.57079632679489655800e+00, /* atan(inf)hi 0x3FF921FB, 0x54442D18 */
40 };
41
42 static const double atanlo[] = {
43 2.26987774529616870924e-17, /* atan(0.5)lo 0x3C7A2B7F, 0x222F65E2 */
44 3.06161699786838301793e-17, /* atan(1.0)lo 0x3C81A626, 0x33145C07 */
45 1.39033110312309984516e-17, /* atan(1.5)lo 0x3C700788, 0x7AF0CBBD */
46 6.12323399573676603587e-17, /* atan(inf)lo 0x3C91A626, 0x33145C07 */
47 };
48
49 static const double aT[] = {
50 3.33333333333329318027e-01, /* 0x3FD55555, 0x5555550D */
51 -1.99999999998764832476e-01, /* 0xBFC99999, 0x9998EBC4 */
52 1.42857142725034663711e-01, /* 0x3FC24924, 0x920083FF */
53 -1.11111104054623557880e-01, /* 0xBFBC71C6, 0xFE231671 */
54 9.09088713343650656196e-02, /* 0x3FB745CD, 0xC54C206E */
55 -7.69187620504482999495e-02, /* 0xBFB3B0F2, 0xAF749A6D */
56 6.66107313738753120669e-02, /* 0x3FB10D66, 0xA0D03D51 */
57 -5.83357013379057348645e-02, /* 0xBFADDE2D, 0x52DEFD9A */
58 4.97687799461593236017e-02, /* 0x3FA97B4B, 0x24760DEB */
59 -3.65315727442169155270e-02, /* 0xBFA2B444, 0x2C6A6C2F */
60 1.62858201153657823623e-02, /* 0x3F90AD3A, 0xE322DA11 */
61 };
62
63 double atan(double x)
64 {
65 double_t w,s1,s2,z;
66 uint32_t ix,sign;
67 int id;
68
69 GET_HIGH_WORD(ix, x);
70 sign = ix >> 31;
71 ix &= 0x7fffffff;
72 if (ix >= 0x44100000) { /* if |x| >= 2^66 */
73 if (isnan(x))
74 return x;
75 z = atanhi[3] + 0x1p-120f;
76 return sign ? -z : z;
77 }
78 if (ix < 0x3fdc0000) { /* |x| < 0.4375 */
79 if (ix < 0x3e400000) { /* |x| < 2^-27 */
80 if (ix < 0x00100000)
81 /* raise underflow for subnormal x */
82 FORCE_EVAL((float)x);
83 return x;
84 }
85 id = -1;
86 } else {
87 x = fabs(x);
88 if (ix < 0x3ff30000) { /* |x| < 1.1875 */
89 if (ix < 0x3fe60000) { /* 7/16 <= |x| < 11/16 */
90 id = 0;
91 x = (2.0*x-1.0)/(2.0+x);
92 } else { /* 11/16 <= |x| < 19/16 */
93 id = 1;
94 x = (x-1.0)/(x+1.0);
95 }
96 } else {
97 if (ix < 0x40038000) { /* |x| < 2.4375 */
98 id = 2;
99 x = (x-1.5)/(1.0+1.5*x);
100 } else { /* 2.4375 <= |x| < 2^66 */
101 id = 3;
102 x = -1.0/x;
103 }
104 }
105 }
106 /* end of argument reduction */
107 z = x*x;
108 w = z*z;
109 /* break sum from i=0 to 10 aT[i]z**(i+1) into odd and even poly */
110 s1 = z*(aT[0]+w*(aT[2]+w*(aT[4]+w*(aT[6]+w*(aT[8]+w*aT[10])))));
111 s2 = w*(aT[1]+w*(aT[3]+w*(aT[5]+w*(aT[7]+w*aT[9]))));
112 if (id < 0)
113 return x - x*(s1+s2);
114 z = atanhi[id] - (x*(s1+s2) - atanlo[id] - x);
115 return sign ? -z : z;
116 }
117