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2aa62f2b 1/* @(#)k_tan.c 5.1 93/09/24 */\r
2/*\r
3 * ====================================================\r
4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.\r
5 *\r
6 * Developed at SunPro, a Sun Microsystems, Inc. business.\r
7 * Permission to use, copy, modify, and distribute this\r
8 * software is freely granted, provided that this notice\r
9 * is preserved.\r
10 * ====================================================\r
11 */\r
12#include <LibConfig.h>\r
13#include <sys/EfiCdefs.h>\r
14#if defined(LIBM_SCCS) && !defined(lint)\r
15__RCSID("$NetBSD: k_tan.c,v 1.12 2004/07/22 18:24:09 drochner Exp $");\r
16#endif\r
17\r
18/* __kernel_tan( x, y, k )\r
19 * kernel tan function on [-pi/4, pi/4], pi/4 ~ 0.7854\r
20 * Input x is assumed to be bounded by ~pi/4 in magnitude.\r
21 * Input y is the tail of x.\r
22 * Input k indicates whether tan (if k=1) or\r
23 * -1/tan (if k= -1) is returned.\r
24 *\r
25 * Algorithm\r
26 * 1. Since tan(-x) = -tan(x), we need only to consider positive x.\r
27 * 2. if x < 2^-28 (hx<0x3e300000 0), return x with inexact if x!=0.\r
28 * 3. tan(x) is approximated by a odd polynomial of degree 27 on\r
29 * [0,0.67434]\r
30 * 3 27\r
31 * tan(x) ~ x + T1*x + ... + T13*x\r
32 * where\r
33 *\r
34 * |tan(x) 2 4 26 | -59.2\r
35 * |----- - (1+T1*x +T2*x +.... +T13*x )| <= 2\r
36 * | x |\r
37 *\r
38 * Note: tan(x+y) = tan(x) + tan'(x)*y\r
39 * ~ tan(x) + (1+x*x)*y\r
40 * Therefore, for better accuracy in computing tan(x+y), let\r
41 * 3 2 2 2 2\r
42 * r = x *(T2+x *(T3+x *(...+x *(T12+x *T13))))\r
43 * then\r
44 * 3 2\r
45 * tan(x+y) = x + (T1*x + (x *(r+y)+y))\r
46 *\r
47 * 4. For x in [0.67434,pi/4], let y = pi/4 - x, then\r
48 * tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y))\r
49 * = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y)))\r
50 */\r
51\r
52#include "math.h"\r
53#include "math_private.h"\r
54\r
55static const double xxx[] = {\r
56 3.33333333333334091986e-01, /* 3FD55555, 55555563 */\r
57 1.33333333333201242699e-01, /* 3FC11111, 1110FE7A */\r
58 5.39682539762260521377e-02, /* 3FABA1BA, 1BB341FE */\r
59 2.18694882948595424599e-02, /* 3F9664F4, 8406D637 */\r
60 8.86323982359930005737e-03, /* 3F8226E3, E96E8493 */\r
61 3.59207910759131235356e-03, /* 3F6D6D22, C9560328 */\r
62 1.45620945432529025516e-03, /* 3F57DBC8, FEE08315 */\r
63 5.88041240820264096874e-04, /* 3F4344D8, F2F26501 */\r
64 2.46463134818469906812e-04, /* 3F3026F7, 1A8D1068 */\r
65 7.81794442939557092300e-05, /* 3F147E88, A03792A6 */\r
66 7.14072491382608190305e-05, /* 3F12B80F, 32F0A7E9 */\r
67 -1.85586374855275456654e-05, /* BEF375CB, DB605373 */\r
68 2.59073051863633712884e-05, /* 3EFB2A70, 74BF7AD4 */\r
69/* one */ 1.00000000000000000000e+00, /* 3FF00000, 00000000 */\r
70/* pio4 */ 7.85398163397448278999e-01, /* 3FE921FB, 54442D18 */\r
71/* pio4lo */ 3.06161699786838301793e-17 /* 3C81A626, 33145C07 */\r
72};\r
73#define one xxx[13]\r
74#define pio4 xxx[14]\r
75#define pio4lo xxx[15]\r
76#define T xxx\r
77\r
78double\r
79__kernel_tan(double x, double y, int iy)\r
80{\r
81 double z, r, v, w, s;\r
82 int32_t ix, hx;\r
83\r
84 GET_HIGH_WORD(hx, x); /* high word of x */\r
85 ix = hx & 0x7fffffff; /* high word of |x| */\r
86 if (ix < 0x3e300000) { /* x < 2**-28 */\r
87 if ((int) x == 0) { /* generate inexact */\r
88 u_int32_t low;\r
89 GET_LOW_WORD(low, x);\r
90 if(((ix | low) | (iy + 1)) == 0)\r
91 return one / fabs(x);\r
92 else {\r
93 if (iy == 1)\r
94 return x;\r
95 else { /* compute -1 / (x+y) carefully */\r
96 double a, t;\r
97\r
98 z = w = x + y;\r
99 SET_LOW_WORD(z, 0);\r
100 v = y - (z - x);\r
101 t = a = -one / w;\r
102 SET_LOW_WORD(t, 0);\r
103 s = one + t * z;\r
104 return t + a * (s + t * v);\r
105 }\r
106 }\r
107 }\r
108 }\r
109 if (ix >= 0x3FE59428) { /* |x| >= 0.6744 */\r
110 if (hx < 0) {\r
111 x = -x;\r
112 y = -y;\r
113 }\r
114 z = pio4 - x;\r
115 w = pio4lo - y;\r
116 x = z + w;\r
117 y = 0.0;\r
118 }\r
119 z = x * x;\r
120 w = z * z;\r
121 /*\r
122 * Break x^5*(T[1]+x^2*T[2]+...) into\r
123 * x^5(T[1]+x^4*T[3]+...+x^20*T[11]) +\r
124 * x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12]))\r
125 */\r
126 r = T[1] + w * (T[3] + w * (T[5] + w * (T[7] + w * (T[9] +\r
127 w * T[11]))));\r
128 v = z * (T[2] + w * (T[4] + w * (T[6] + w * (T[8] + w * (T[10] +\r
129 w * T[12])))));\r
130 s = z * x;\r
131 r = y + z * (s * (r + v) + y);\r
132 r += T[0] * s;\r
133 w = x + r;\r
134 if (ix >= 0x3FE59428) {\r
135 v = (double) iy;\r
136 return (double) (1 - ((hx >> 30) & 2)) *\r
137 (v - 2.0 * (x - (w * w / (w + v) - r)));\r
138 }\r
139 if (iy == 1)\r
140 return w;\r
141 else {\r
142 /*\r
143 * if allow error up to 2 ulp, simply return\r
144 * -1.0 / (x+r) here\r
145 */\r
146 /* compute -1.0 / (x+r) accurately */\r
147 double a, t;\r
148 z = w;\r
149 SET_LOW_WORD(z, 0);\r
150 v = r - (z - x); /* z+v = r+x */\r
151 t = a = -1.0 / w; /* a = -1.0/w */\r
152 SET_LOW_WORD(t, 0);\r
153 s = 1.0 + t * z;\r
154 return t + a * (s + t * v);\r
155 }\r
156}\r