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9b7ee67e UD |
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 | /* | |
cc7375ce RM |
13 | Long double expansions are |
14 | Copyright (C) 2001 Stephen L. Moshier <moshier@na-net.ornl.gov> | |
9cd2726c RM |
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: | |
cc7375ce RM |
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 | |
59ba27a6 PE |
31 | License along with this library; if not, see |
32 | <http://www.gnu.org/licenses/>. */ | |
9b7ee67e UD |
33 | |
34 | /* __kernel_tanl( 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 "math.h" | |
60 | #include "math_private.h" | |
9b7ee67e | 61 | static const long double |
9b7ee67e UD |
62 | one = 1.0L, |
63 | pio4hi = 7.8539816339744830961566084581987569936977E-1L, | |
64 | pio4lo = 2.1679525325309452561992610065108379921906E-35L, | |
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-1L, | |
70 | T0 = -1.813014711743583437742363284336855889393E7L, | |
71 | T1 = 1.320767960008972224312740075083259247618E6L, | |
72 | T2 = -2.626775478255838182468651821863299023956E4L, | |
73 | T3 = 1.764573356488504935415411383687150199315E2L, | |
74 | T4 = -3.333267763822178690794678978979803526092E-1L, | |
75 | ||
76 | U0 = -1.359761033807687578306772463253710042010E8L, | |
77 | U1 = 6.494370630656893175666729313065113194784E7L, | |
78 | U2 = -4.180787672237927475505536849168729386782E6L, | |
79 | U3 = 8.031643765106170040139966622980914621521E4L, | |
80 | U4 = -5.323131271912475695157127875560667378597E2L; | |
81 | /* 1.000000000000000000000000000000000000000E0 */ | |
82 | ||
83 | ||
9b7ee67e UD |
84 | long double |
85 | __kernel_tanl (long double x, long double y, int iy) | |
9b7ee67e UD |
86 | { |
87 | long double z, r, v, w, s; | |
88 | int32_t ix, sign; | |
89 | ieee854_long_double_shape_type u, u1; | |
90 | ||
91 | u.value = x; | |
92 | ix = u.parts32.w0 & 0x7fffffff; | |
93 | if (ix < 0x3fc60000) /* x < 2**-57 */ | |
94 | { | |
95 | if ((int) x == 0) | |
96 | { /* generate inexact */ | |
97 | if ((ix | u.parts32.w1 | u.parts32.w2 | u.parts32.w3 | |
98 | | (iy + 1)) == 0) | |
99 | return one / fabs (x); | |
100 | else | |
101 | return (iy == 1) ? x : -one / x; | |
102 | } | |
103 | } | |
104 | if (ix >= 0x3ffe5942) /* |x| >= 0.6743316650390625 */ | |
105 | { | |
106 | if ((u.parts32.w0 & 0x80000000) != 0) | |
107 | { | |
108 | x = -x; | |
109 | y = -y; | |
110 | sign = -1; | |
111 | } | |
112 | else | |
113 | sign = 1; | |
114 | z = pio4hi - x; | |
115 | w = pio4lo - y; | |
116 | x = z + w; | |
117 | y = 0.0; | |
118 | } | |
119 | z = x * x; | |
120 | r = T0 + z * (T1 + z * (T2 + z * (T3 + z * T4))); | |
121 | v = U0 + z * (U1 + z * (U2 + z * (U3 + z * (U4 + z)))); | |
122 | r = r / v; | |
123 | ||
124 | s = z * x; | |
125 | r = y + z * (s * r + y); | |
126 | r += TH * s; | |
127 | w = x + r; | |
128 | if (ix >= 0x3ffe5942) | |
129 | { | |
130 | v = (long double) iy; | |
131 | w = (v - 2.0 * (x - (w * w / (w + v) - r))); | |
132 | if (sign < 0) | |
133 | w = -w; | |
134 | return w; | |
135 | } | |
136 | if (iy == 1) | |
137 | return w; | |
138 | else | |
139 | { /* if allow error up to 2 ulp, | |
140 | simply return -1.0/(x+r) here */ | |
141 | /* compute -1.0/(x+r) accurately */ | |
142 | u1.value = w; | |
143 | u1.parts32.w2 = 0; | |
144 | u1.parts32.w3 = 0; | |
145 | v = r - (u1.value - x); /* u1+v = r+x */ | |
146 | z = -1.0 / w; | |
147 | u.value = z; | |
148 | u.parts32.w2 = 0; | |
149 | u.parts32.w3 = 0; | |
150 | s = 1.0 + u.value * u1.value; | |
151 | return u.value + z * (s + u.value * v); | |
152 | } | |
153 | } |