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/*
13  Long double expansions are
14  Copyright (C) 2001 Stephen L. Moshier <moshier@na-net.ornl.gov>
15  and are incorporated herein by permission of the author.  The author
16  reserves the right to distribute this material elsewhere under different
17  copying permissions.  These modifications are distributed here under
18  the following terms:
19
20    This library is free software; you can redistribute it and/or
21    modify it under the terms of the GNU Lesser General Public
22    License as published by the Free Software Foundation; either
23    version 2.1 of the License, or (at your option) any later version.
24
25    This library is distributed in the hope that it will be useful,
26    but WITHOUT ANY WARRANTY; without even the implied warranty of
27    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
28    Lesser General Public License for more details.
29
30    You should have received a copy of the GNU Lesser General Public
31    License along with this library; if not, see
32    <http://www.gnu.org/licenses/>.  */
33
34/* __quadmath_kernel_tanq( x, y, k )
35 * kernel tan function on [-pi/4, pi/4], pi/4 ~ 0.7854
36 * Input x is assumed to be bounded by ~pi/4 in magnitude.
37 * Input y is the tail of x.
38 * Input k indicates whether tan (if k=1) or
39 * -1/tan (if k= -1) is returned.
40 *
41 * Algorithm
42 *	1. Since tan(-x) = -tan(x), we need only to consider positive x.
43 *	2. if x < 2^-57, return x with inexact if x!=0.
44 *	3. tan(x) is approximated by a rational form x + x^3 / 3 + x^5 R(x^2)
45 *          on [0,0.67433].
46 *
47 *	   Note: tan(x+y) = tan(x) + tan'(x)*y
48 *		          ~ tan(x) + (1+x*x)*y
49 *	   Therefore, for better accuracy in computing tan(x+y), let
50 *		r = x^3 * R(x^2)
51 *	   then
52 *		tan(x+y) = x + (x^3 / 3 + (x^2 *(r+y)+y))
53 *
54 *      4. For x in [0.67433,pi/4],  let y = pi/4 - x, then
55 *		tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y))
56 *		       = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y)))
57 */
58
59#include "quadmath-imp.h"
60
61static const __float128
62  one = 1,
63  pio4hi = 7.8539816339744830961566084581987569936977E-1Q,
64  pio4lo = 2.1679525325309452561992610065108379921906E-35Q,
65
66  /* tan x = x + x^3 / 3 + x^5 T(x^2)/U(x^2)
67     0 <= x <= 0.6743316650390625
68     Peak relative error 8.0e-36  */
69 TH =  3.333333333333333333333333333333333333333E-1Q,
70 T0 = -1.813014711743583437742363284336855889393E7Q,
71 T1 =  1.320767960008972224312740075083259247618E6Q,
72 T2 = -2.626775478255838182468651821863299023956E4Q,
73 T3 =  1.764573356488504935415411383687150199315E2Q,
74 T4 = -3.333267763822178690794678978979803526092E-1Q,
75
76 U0 = -1.359761033807687578306772463253710042010E8Q,
77 U1 =  6.494370630656893175666729313065113194784E7Q,
78 U2 = -4.180787672237927475505536849168729386782E6Q,
79 U3 =  8.031643765106170040139966622980914621521E4Q,
80 U4 = -5.323131271912475695157127875560667378597E2Q;
81  /* 1.000000000000000000000000000000000000000E0 */
82
83
84__float128
85__quadmath_kernel_tanq (__float128 x, __float128 y, int iy)
86{
87  __float128 z, r, v, w, s;
88  int32_t ix, sign;
89  ieee854_float128 u, u1;
90
91  u.value = x;
92  ix = u.words32.w0 & 0x7fffffff;
93  if (ix < 0x3fc60000)		/* x < 2**-57 */
94    {
95      if ((int) x == 0)
96	{			/* generate inexact */
97	  if ((ix | u.words32.w1 | u.words32.w2 | u.words32.w3
98	       | (iy + 1)) == 0)
99	    return one / fabsq (x);
100	  else if (iy == 1)
101	    {
102	      math_check_force_underflow (x);
103	      return x;
104	    }
105	  else
106	    return -one / x;
107	}
108    }
109  if (ix >= 0x3ffe5942) /* |x| >= 0.6743316650390625 */
110    {
111      if ((u.words32.w0 & 0x80000000) != 0)
112	{
113	  x = -x;
114	  y = -y;
115	  sign = -1;
116	}
117      else
118	sign = 1;
119      z = pio4hi - x;
120      w = pio4lo - y;
121      x = z + w;
122      y = 0.0;
123    }
124  z = x * x;
125  r = T0 + z * (T1 + z * (T2 + z * (T3 + z * T4)));
126  v = U0 + z * (U1 + z * (U2 + z * (U3 + z * (U4 + z))));
127  r = r / v;
128
129  s = z * x;
130  r = y + z * (s * r + y);
131  r += TH * s;
132  w = x + r;
133  if (ix >= 0x3ffe5942)
134    {
135      v = (__float128) iy;
136      w = (v - 2.0 * (x - (w * w / (w + v) - r)));
137      /* SIGN is set for arguments that reach this code, but not
138	 otherwise, resulting in warnings that it may be used
139	 uninitialized although in the cases where it is used it has
140	 always been set.  */
141
142
143      if (sign < 0)
144	w = -w;
145
146      return w;
147    }
148  if (iy == 1)
149    return w;
150  else
151    {				/* if allow error up to 2 ulp,
152				   simply return -1.0/(x+r) here */
153      /*  compute -1.0/(x+r) accurately */
154      u1.value = w;
155      u1.words32.w2 = 0;
156      u1.words32.w3 = 0;
157      v = r - (u1.value - x);		/* u1+v = r+x */
158      z = -1.0 / w;
159      u.value = z;
160      u.words32.w2 = 0;
161      u.words32.w3 = 0;
162      s = 1.0 + u.value * u1.value;
163      return u.value + z * (s + u.value * v);
164    }
165}
166