1 /*
2 * CDDL HEADER START
3 *
4 * The contents of this file are subject to the terms of the
5 * Common Development and Distribution License (the "License").
6 * You may not use this file except in compliance with the License.
7 *
8 * You can obtain a copy of the license at usr/src/OPENSOLARIS.LICENSE
9 * or http://www.opensolaris.org/os/licensing.
10 * See the License for the specific language governing permissions
11 * and limitations under the License.
12 *
13 * When distributing Covered Code, include this CDDL HEADER in each
14 * file and include the License file at usr/src/OPENSOLARIS.LICENSE.
15 * If applicable, add the following below this CDDL HEADER, with the
16 * fields enclosed by brackets "[]" replaced with your own identifying
17 * information: Portions Copyright [yyyy] [name of copyright owner]
18 *
19 * CDDL HEADER END
20 */
21
22 /*
23 * Copyright 2011 Nexenta Systems, Inc. All rights reserved.
24 */
25 /*
26 * Copyright 2006 Sun Microsystems, Inc. All rights reserved.
27 * Use is subject to license terms.
28 */
29
30 #pragma weak atanl = __atanl
31
32 /*
33 * atanl(x)
34 * Table look-up algorithm
35 * By K.C. Ng, March 9, 1989
36 *
37 * Algorithm.
38 *
39 * The algorithm is based on atan(x)=atan(y)+atan((x-y)/(1+x*y)).
40 * We use poly1(x) to approximate atan(x) for x in [0,1/8] with
41 * error (relative)
42 * |(atan(x)-poly1(x))/x|<= 2^-115.94 long double
43 * |(atan(x)-poly1(x))/x|<= 2^-58.85 double
44 * |(atan(x)-poly1(x))/x|<= 2^-25.53 float
45 * and use poly2(x) to approximate atan(x) for x in [0,1/65] with
46 * error (absolute)
47 * |atan(x)-poly2(x)|<= 2^-122.15 long double
48 * |atan(x)-poly2(x)|<= 2^-64.79 double
49 * |atan(x)-poly2(x)|<= 2^-35.36 float
50 * Here poly1 and poly2 are odd polynomial with the following form:
51 * x + x^3*(a1+x^2*(a2+...))
52 *
53 * (0). Purge off Inf and NaN and 0
54 * (1). Reduce x to positive by atan(x) = -atan(-x).
55 * (2). For x <= 1/8, use
56 * (2.1) if x < 2^(-prec/2-2), atan(x) = x with inexact
57 * (2.2) Otherwise
58 * atan(x) = poly1(x)
59 * (3). For x >= 8 then
60 * (3.1) if x >= 2^(prec+2), atan(x) = atan(inf) - pio2lo
61 * (3.2) if x >= 2^(prec/3+2), atan(x) = atan(inf) - 1/x
62 * (3.3) if x > 65, atan(x) = atan(inf) - poly2(1/x)
63 * (3.4) Otherwise, atan(x) = atan(inf) - poly1(1/x)
64 *
65 * (4). Now x is in (0.125, 8)
66 * Find y that match x to 4.5 bit after binary (easy).
67 * If iy is the high word of y, then
68 * single : j = (iy - 0x3e000000) >> 19
69 * double : j = (iy - 0x3fc00000) >> 16
70 * quad : j = (iy - 0x3ffc0000) >> 12
71 *
72 * Let s = (x-y)/(1+x*y). Then
73 * atan(x) = atan(y) + poly1(s)
74 * = _TBL_atanl_hi[j] + (_TBL_atanl_lo[j] + poly2(s) )
75 *
76 * Note. |s| <= 1.5384615385e-02 = 1/65. Maxium occurs at x = 1.03125
77 *
78 */
79
80 #include "libm.h"
81
82 extern const long double _TBL_atanl_hi[], _TBL_atanl_lo[];
83 static const long double
84 one = 1.0L,
85 p1 = -3.333333333333333333333333333331344526118e-0001L,
86 p2 = 1.999999999999999999999999989931277668570e-0001L,
87 p3 = -1.428571428571428571428553606221309530901e-0001L,
88 p4 = 1.111111111111111111095219842737139747418e-0001L,
89 p5 = -9.090909090909090825503603835248061123323e-0002L,
90 p6 = 7.692307692307664052130743214708925258904e-0002L,
91 p7 = -6.666666666660213835187713228363717388266e-0002L,
92 p8 = 5.882352940152439399097283359608661949504e-0002L,
93 p9 = -5.263157780447533993046614040509529668487e-0002L,
94 p10 = 4.761895816878184933175855990886788439447e-0002L,
95 p11 = -4.347345005832274022681019724553538135922e-0002L,
96 p12 = 3.983031914579635037502589204647752042736e-0002L,
97 p13 = -3.348206704469830575196657749413894897554e-0002L,
98 q1 = -3.333333333333333333333333333195273650186e-0001L,
99 q2 = 1.999999999999999999999988146114392615808e-0001L,
100 q3 = -1.428571428571428571057630319435467111253e-0001L,
101 q4 = 1.111111111111105373263048208994541544098e-0001L,
102 q5 = -9.090909090421834209167373258681021816441e-0002L,
103 q6 = 7.692305377813692706850171767150701644539e-0002L,
104 q7 = -6.660896644393861499914731734305717901330e-0002L,
105 pio2hi = 1.570796326794896619231321691639751398740e+0000L,
106 pio2lo = 4.335905065061890512398522013021675984381e-0035L;
107
108 #define i0 0
109 #define i1 3
110
111 long double
112 atanl(long double x) {
113 long double y, z, r, p, s;
114 int *px = (int *) &x, *py = (int *) &y;
115 int ix, iy, sign, j;
116
117 ix = px[i0];
118 sign = ix & 0x80000000;
119 ix ^= sign;
120
121 /* for |x| < 1/8 */
122 if (ix < 0x3ffc0000) {
123 if (ix < 0x3feb0000) { /* when |x| < 2**(-prec/6-2) */
124 if (ix < 0x3fc50000) { /* if |x| < 2**(-prec/2-2) */
125 s = one;
126 *(3 - i0 + (int *) &s) = -1; /* s = 1-ulp */
127 *(1 + (int *) &s) = -1;
128 *(2 + (int *) &s) = -1;
129 *(i0 + (int *) &s) -= 1;
130 if ((int) (s * x) < 1)
131 return (x); /* raise inexact */
132 }
133 z = x * x;
134 if (ix < 0x3fe20000) { /* if |x| < 2**(-prec/4-1) */
135 return (x + (x * z) * p1);
136 } else { /* if |x| < 2**(-prec/6-2) */
137 return (x + (x * z) * (p1 + z * p2));
138 }
139 }
140 z = x * x;
141 return (x + (x * z) * (p1 + z * (p2 + z * (p3 + z * (p4 +
142 z * (p5 + z * (p6 + z * (p7 + z * (p8 + z * (p9 +
143 z * (p10 + z * (p11 + z * (p12 + z * p13)))))))))))));
144 }
145
146 /* for |x| >= 8.0 */
147 if (ix >= 0x40020000) {
148 px[i0] = ix;
149 if (ix < 0x40050400) { /* x < 65 */
150 r = one / x;
151 z = r * r;
152 /*
153 * poly1
154 */
155 y = r * (one + z * (p1 + z * (p2 + z * (p3 +
156 z * (p4 + z * (p5 + z * (p6 + z * (p7 +
157 z * (p8 + z * (p9 + z * (p10 + z * (p11 +
158 z * (p12 + z * p13)))))))))))));
159 y -= pio2lo;
160 } else if (ix < 0x40260000) { /* x < 2**(prec/3+2) */
161 r = one / x;
162 z = r * r;
163 /*
164 * poly2
165 */
166 y = r * (one + z * (q1 + z * (q2 + z * (q3 + z * (q4 +
167 z * (q5 + z * (q6 + z * q7)))))));
168 y -= pio2lo;
169 } else if (ix < 0x40720000) { /* x < 2**(prec+2) */
170 y = one / x - pio2lo;
171 } else if (ix < 0x7fff0000) { /* x < inf */
172 y = -pio2lo;
173 } else { /* x is inf or NaN */
174 if (((ix - 0x7fff0000) | px[1] | px[2] | px[i1]) != 0)
175 return (x - x);
176 y = -pio2lo;
177 }
178
179 if (sign == 0)
180 return (pio2hi - y);
181 else
182 return (y - pio2hi);
183 }
184
185 /* now x is between 1/8 and 8 */
186 px[i0] = ix;
187 iy = (ix + 0x00000800) & 0x7ffff000;
188 py[i0] = iy;
189 py[1] = py[2] = py[i1] = 0;
190 j = (iy - 0x3ffc0000) >> 12;
191
192 if (sign == 0)
193 s = (x - y) / (one + x * y);
194 else
195 s = (y - x) / (one + x * y);
196 z = s * s;
197 if (ix == iy)
198 p = s * (one + z * (q1 + z * (q2 + z * (q3 + z * q4))));
199 else
200 p = s * (one + z * (q1 + z * (q2 + z * (q3 + z * (q4 +
201 z * (q5 + z * (q6 + z * q7)))))));
202 if (sign == 0) {
203 r = p + _TBL_atanl_lo[j];
204 return (r + _TBL_atanl_hi[j]);
205 } else {
206 r = p - _TBL_atanl_lo[j];
207 return (r - _TBL_atanl_hi[j]);
208 }
209 }