1 /*
2 * ====================================================
3 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
4 *
5 * Developed at SunPro, a Sun Microsystems, Inc. business.
6 * Permission to use, copy, modify, and distribute this
7 * software is freely granted, provided that this notice
8 * is preserved.
9 * ====================================================
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 #include <float.h>
33
34 #include "math.h"
35 #include "math_private.h"
36
37 static const double atanhi[] = {
38 4.63647609000806093515e-01, /* atan(0.5)hi 0x3FDDAC67, 0x0561BB4F */
39 7.85398163397448278999e-01, /* atan(1.0)hi 0x3FE921FB, 0x54442D18 */
40 9.82793723247329054082e-01, /* atan(1.5)hi 0x3FEF730B, 0xD281F69B */
41 1.57079632679489655800e+00, /* atan(inf)hi 0x3FF921FB, 0x54442D18 */
42 };
43
44 static const double atanlo[] = {
45 2.26987774529616870924e-17, /* atan(0.5)lo 0x3C7A2B7F, 0x222F65E2 */
46 3.06161699786838301793e-17, /* atan(1.0)lo 0x3C81A626, 0x33145C07 */
47 1.39033110312309984516e-17, /* atan(1.5)lo 0x3C700788, 0x7AF0CBBD */
48 6.12323399573676603587e-17, /* atan(inf)lo 0x3C91A626, 0x33145C07 */
49 };
50
51 static const double aT[] = {
52 3.33333333333329318027e-01, /* 0x3FD55555, 0x5555550D */
53 -1.99999999998764832476e-01, /* 0xBFC99999, 0x9998EBC4 */
54 1.42857142725034663711e-01, /* 0x3FC24924, 0x920083FF */
55 -1.11111104054623557880e-01, /* 0xBFBC71C6, 0xFE231671 */
56 9.09088713343650656196e-02, /* 0x3FB745CD, 0xC54C206E */
57 -7.69187620504482999495e-02, /* 0xBFB3B0F2, 0xAF749A6D */
58 6.66107313738753120669e-02, /* 0x3FB10D66, 0xA0D03D51 */
59 -5.83357013379057348645e-02, /* 0xBFADDE2D, 0x52DEFD9A */
60 4.97687799461593236017e-02, /* 0x3FA97B4B, 0x24760DEB */
61 -3.65315727442169155270e-02, /* 0xBFA2B444, 0x2C6A6C2F */
62 1.62858201153657823623e-02, /* 0x3F90AD3A, 0xE322DA11 */
63 };
64
65 static const double
66 one = 1.0,
67 huge = 1.0e300;
68
69 double
atan(double x)70 atan(double x)
71 {
72 double w,s1,s2,z;
73 int32_t ix,hx,id;
74
75 GET_HIGH_WORD(hx,x);
76 ix = hx&0x7fffffff;
77 if(ix>=0x44100000) { /* if |x| >= 2^66 */
78 u_int32_t low;
79 GET_LOW_WORD(low,x);
80 if(ix>0x7ff00000||
81 (ix==0x7ff00000&&(low!=0)))
82 return x+x; /* NaN */
83 if(hx>0) return atanhi[3]+*(volatile double *)&atanlo[3];
84 else return -atanhi[3]-*(volatile double *)&atanlo[3];
85 } if (ix < 0x3fdc0000) { /* |x| < 0.4375 */
86 if (ix < 0x3e400000) { /* |x| < 2^-27 */
87 if(huge+x>one) return x; /* raise inexact */
88 }
89 id = -1;
90 } else {
91 x = fabs(x);
92 if (ix < 0x3ff30000) { /* |x| < 1.1875 */
93 if (ix < 0x3fe60000) { /* 7/16 <=|x|<11/16 */
94 id = 0; x = (2.0*x-one)/(2.0+x);
95 } else { /* 11/16<=|x|< 19/16 */
96 id = 1; x = (x-one)/(x+one);
97 }
98 } else {
99 if (ix < 0x40038000) { /* |x| < 2.4375 */
100 id = 2; x = (x-1.5)/(one+1.5*x);
101 } else { /* 2.4375 <= |x| < 2^66 */
102 id = 3; x = -1.0/x;
103 }
104 }}
105 /* end of argument reduction */
106 z = x*x;
107 w = z*z;
108 /* break sum from i=0 to 10 aT[i]z**(i+1) into odd and even poly */
109 s1 = z*(aT[0]+w*(aT[2]+w*(aT[4]+w*(aT[6]+w*(aT[8]+w*aT[10])))));
110 s2 = w*(aT[1]+w*(aT[3]+w*(aT[5]+w*(aT[7]+w*aT[9]))));
111 if (id<0) return x - x*(s1+s2);
112 else {
113 z = atanhi[id] - ((x*(s1+s2) - atanlo[id]) - x);
114 return (hx<0)? -z:z;
115 }
116 }
117
118 #if LDBL_MANT_DIG == 53
119 __weak_reference(atan, atanl);
120 #endif
121