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 __atan = atan
31
32/* INDENT OFF */
33/*
34 * atan(x)
35 * Accurate Table look-up algorithm with polynomial approximation in
36 * partially product form.
37 *
38 * -- K.C. Ng, October 17, 2004
39 *
40 * Algorithm
41 *
42 * (1). Purge off Inf and NaN and 0
43 * (2). Reduce x to positive by atan(x) = -atan(-x).
44 * (3). For x <= 1/8 and let z = x*x, return
45 *	(2.1) if x < 2^(-prec/2), atan(x) = x  with inexact flag raised
46 *	(2.2) if x < 2^(-prec/4-1), atan(x) = x+(x/3)(x*x)
47 *	(2.3) if x < 2^(-prec/6-2), atan(x) = x+(z-5/3)(z*x/5)
48 *	(2.4) Otherwise
49 *		atan(x) = poly1(x) = x + A * B,
50 *	where
51 *		A = (p1*x*z) * (p2+z(p3+z))
52 *		B = (p4+z)+z*z) * (p5+z(p6+z))
53 *	Note: (i) domain of poly1 is [0, 1/8], (ii) remez relative
54 *	approximation error of poly1 is bounded by
55 * 		|(atan(x)-poly1(x))/x| <= 2^-57.61
56 * (4). For x >= 8 then
57 *	(3.1) if x >= 2^prec,     atan(x) = atan(inf) - pio2lo
58 *	(3.2) if x >= 2^(prec/3), atan(x) = atan(inf) - 1/x
59 *	(3.3) if x <= 65,	  atan(x) = atan(inf) - poly1(1/x)
60 *	(3.4) otherwise           atan(x) = atan(inf) - poly2(1/x)
61 *	where
62 *		poly2(r) = (q1*r) * (q2+z(q3+z)) * (q4+z),
63 *	its domain is [0, 0.0154]; and its remez absolute
64 *	approximation error is bounded by
65 *		|atan(x)-poly2(x)|<= 2^-59.45
66 *
67 * (5). Now x is in (0.125, 8).
68 *	Recall identity
69 *		atan(x) = atan(y) + atan((x-y)/(1+x*y)).
70 *	Let j = (ix - 0x3fc00000) >> 16, 0 <= j < 96, where ix is the high
71 *	part of x in IEEE double format. Then
72 *		atan(x) = atan(y[j]) + poly2((x-y[j])/(1+x*y[j]))
73 *	where y[j] are carefully chosen so that it matches x to around 4.5
74 *	bits and at the same time atan(y[j]) is very close to an IEEE double
75 *	floating point number. Calculation indicates that
76 *		max|(x-y[j])/(1+x*y[j])| < 0.0154
77 *		j,x
78 *
79 * Accuracy: Maximum error observed is bounded by 0.6 ulp after testing
80 * more than 10 million random arguments
81 */
82/* INDENT ON */
83
84#include "libm.h"
85#include "libm_protos.h"
86
87extern const double _TBL_atan[];
88static const double g[] = {
89/* one	= */  1.0,
90/* p1	= */  8.02176624254765935351230154992663301527500152588e-0002,
91/* p2	= */  1.27223421700559402580665846471674740314483642578e+0000,
92/* p3	= */ -1.20606901800503640842521235754247754812240600586e+0000,
93/* p4	= */ -2.36088967922325565496066701598465442657470703125e+0000,
94/* p5	= */  1.38345799501389166152875986881554126739501953125e+0000,
95/* p6	= */  1.06742368078953453469637224770849570631980895996e+0000,
96/* q1   = */ -1.42796626333911796935538518482644576579332351685e-0001,
97/* q2   = */  3.51427110447873227059810477159863497078605962912e+0000,
98/* q3   = */  5.92129112708164262457444237952586263418197631836e-0001,
99/* q4   = */ -1.99272234785683144409063061175402253866195678711e+0000,
100/* pio2hi */  1.570796326794896558e+00,
101/* pio2lo */  6.123233995736765886e-17,
102/* t1   = */ -0.333333333333333333333333333333333,
103/* t2   = */  0.2,
104/* t3   = */ -1.666666666666666666666666666666666,
105};
106
107#define	one g[0]
108#define	p1 g[1]
109#define	p2 g[2]
110#define	p3 g[3]
111#define	p4 g[4]
112#define	p5 g[5]
113#define	p6 g[6]
114#define	q1 g[7]
115#define	q2 g[8]
116#define	q3 g[9]
117#define	q4 g[10]
118#define	pio2hi g[11]
119#define	pio2lo g[12]
120#define	t1 g[13]
121#define	t2 g[14]
122#define	t3 g[15]
123
124
125double
126atan(double x) {
127	double y, z, r, p, s;
128	int ix, lx, hx, j;
129
130	hx = ((int *) &x)[HIWORD];
131	lx = ((int *) &x)[LOWORD];
132	ix = hx & ~0x80000000;
133	j = ix >> 20;
134
135	/* for |x| < 1/8 */
136	if (j < 0x3fc) {
137		if (j < 0x3f5) {	/* when |x| < 2**(-prec/6-2) */
138			if (j < 0x3e3) {	/* if |x| < 2**(-prec/2-2) */
139				return ((int) x == 0 ? x : one);
140			}
141			if (j < 0x3f1) {	/* if |x| < 2**(-prec/4-1) */
142				return (x + (x * t1) * (x * x));
143			} else {	/* if |x| < 2**(-prec/6-2) */
144				z = x * x;
145				s = t2 * x;
146				return (x + (t3 + z) * (s * z));
147			}
148		}
149		z = x * x; s = p1 * x;
150		return (x + ((s * z) * (p2 + z * (p3 + z))) *
151				(((p4 + z) + z * z) * (p5 + z * (p6 + z))));
152	}
153
154	/* for |x| >= 8.0 */
155	if (j >= 0x402) {
156		if (j < 0x436) {
157			r = one / x;
158			if (hx >= 0) {
159				y =  pio2hi; p =  pio2lo;
160			} else {
161				y = -pio2hi; p = -pio2lo;
162			}
163			if (ix < 0x40504000) {	/* x <  65 */
164				z = r * r;
165				s = p1 * r;
166				return (y + ((p - r) - ((s * z) *
167					(p2 + z * (p3 + z))) *
168					(((p4 + z) + z * z) *
169					(p5 + z * (p6 + z)))));
170			} else if (j < 0x412) {
171				z = r * r;
172				return (y + (p - ((q1 * r) * (q4 + z)) *
173					(q2 + z * (q3 + z))));
174			} else
175				return (y + (p - r));
176		} else {
177			if (j >= 0x7ff) /* x is inf or NaN */
178				if (((ix - 0x7ff00000) | lx) != 0)
179#if defined(FPADD_TRAPS_INCOMPLETE_ON_NAN)
180					return (ix >= 0x7ff80000 ? x : x - x);
181					/* assumes sparc-like QNaN */
182#else
183					return (x - x);
184#endif
185			y = -pio2lo;
186			return (hx >= 0 ? pio2hi - y : y - pio2hi);
187		}
188	} else {	/* now x is between 1/8 and 8 */
189		double *w, w0, w1, s, z;
190		w = (double *) _TBL_atan + (((ix - 0x3fc00000) >> 16) << 1);
191		w0 = (hx >= 0)? w[0] : -w[0];
192		s = (x - w0) / (one + x * w0);
193		w1 = (hx >= 0)? w[1] : -w[1];
194		z = s * s;
195		return (((q1 * s) * (q4 + z)) * (q2 + z * (q3 + z)) + w1);
196	}
197}
198