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k_tan.c (4269B)
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1 /* @(#)k_tan.c 5.1 93/09/24 */
2 /*
3 * ====================================================
4 * Copyright 2004 Sun Microsystems, Inc. All Rights Reserved.
5 *
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 #ifndef lint
13 static char rcsid[] = "$FreeBSD: src/lib/msun/src/k_tan.c,v 1.9 2004/06/02 04:39:29 das Exp $";
14 #endif
15
16 /* __kernel_tan( x, y, k )
17 * kernel tan function on [-pi/4, pi/4], pi/4 ~ 0.7854
18 * Input x is assumed to be bounded by ~pi/4 in magnitude.
19 * Input y is the tail of x.
20 * Input k indicates whether tan (if k=1) or
21 * -1/tan (if k= -1) is returned.
22 *
23 * Algorithm
24 * 1. Since tan(-x) = -tan(x), we need only to consider positive x.
25 * 2. if x < 2^-28 (hx<0x3e300000 0), return x with inexact if x!=0.
26 * 3. tan(x) is approximated by an odd polynomial of degree 27 on
27 * [0,0.67434]
28 * 3 27
29 * tan(x) ~ x + T1*x + ... + T13*x
30 * where
31 *
32 * |tan(x) 2 4 26 | -59.2
33 * |----- - (1+T1*x +T2*x +.... +T13*x )| <= 2
34 * | x |
35 *
36 * Note: tan(x+y) = tan(x) + tan'(x)*y
37 * ~ tan(x) + (1+x*x)*y
38 * Therefore, for better accuracy in computing tan(x+y), let
39 * 3 2 2 2 2
40 * r = x *(T2+x *(T3+x *(...+x *(T12+x *T13))))
41 * then
42 * 3 2
43 * tan(x+y) = x + (T1*x + (x *(r+y)+y))
44 *
45 * 4. For x in [0.67434,pi/4], let y = pi/4 - x, then
46 * tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y))
47 * = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y)))
48 */
49
50 #include "math.h"
51 #include "math_private.h"
52 static const double
53 one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
54 pio4 = 7.85398163397448278999e-01, /* 0x3FE921FB, 0x54442D18 */
55 pio4lo= 3.06161699786838301793e-17, /* 0x3C81A626, 0x33145C07 */
56 T[] = {
57 3.33333333333334091986e-01, /* 0x3FD55555, 0x55555563 */
58 1.33333333333201242699e-01, /* 0x3FC11111, 0x1110FE7A */
59 5.39682539762260521377e-02, /* 0x3FABA1BA, 0x1BB341FE */
60 2.18694882948595424599e-02, /* 0x3F9664F4, 0x8406D637 */
61 8.86323982359930005737e-03, /* 0x3F8226E3, 0xE96E8493 */
62 3.59207910759131235356e-03, /* 0x3F6D6D22, 0xC9560328 */
63 1.45620945432529025516e-03, /* 0x3F57DBC8, 0xFEE08315 */
64 5.88041240820264096874e-04, /* 0x3F4344D8, 0xF2F26501 */
65 2.46463134818469906812e-04, /* 0x3F3026F7, 0x1A8D1068 */
66 7.81794442939557092300e-05, /* 0x3F147E88, 0xA03792A6 */
67 7.14072491382608190305e-05, /* 0x3F12B80F, 0x32F0A7E9 */
68 -1.85586374855275456654e-05, /* 0xBEF375CB, 0xDB605373 */
69 2.59073051863633712884e-05, /* 0x3EFB2A70, 0x74BF7AD4 */
70 };
71
72 double
73 __kernel_tan(double x, double y, int iy)
74 {
75 double z,r,v,w,s;
76 int32_t ix,hx;
77 GET_HIGH_WORD(hx,x);
78 ix = hx&0x7fffffff; /* high word of |x| */
79 if(ix<0x3e300000) { /* x < 2**-28 */
80 if ((int) x == 0) { /* generate inexact */
81 u_int32_t low;
82 GET_LOW_WORD(low,x);
83 if (((ix | low) | (iy + 1)) == 0)
84 return one / fabs(x);
85 else {
86 if (iy == 1)
87 return x;
88 else { /* compute -1 / (x+y) carefully */
89 double a, t;
90
91 z = w = x + y;
92 SET_LOW_WORD(z, 0);
93 v = y - (z - x);
94 t = a = -one / w;
95 SET_LOW_WORD(t, 0);
96 s = one + t * z;
97 return t + a * (s + t * v);
98 }
99 }
100 }
101 }
102 if(ix>=0x3FE59428) { /* |x|>=0.6744 */
103 if(hx<0) {x = -x; y = -y;}
104 z = pio4-x;
105 w = pio4lo-y;
106 x = z+w; y = 0.0;
107 }
108 z = x*x;
109 w = z*z;
110 /* Break x^5*(T[1]+x^2*T[2]+...) into
111 * x^5(T[1]+x^4*T[3]+...+x^20*T[11]) +
112 * x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12]))
113 */
114 r = T[1]+w*(T[3]+w*(T[5]+w*(T[7]+w*(T[9]+w*T[11]))));
115 v = z*(T[2]+w*(T[4]+w*(T[6]+w*(T[8]+w*(T[10]+w*T[12])))));
116 s = z*x;
117 r = y + z*(s*(r+v)+y);
118 r += T[0]*s;
119 w = x+r;
120 if(ix>=0x3FE59428) {
121 v = (double)iy;
122 return (double)(1-((hx>>30)&2))*(v-2.0*(x-(w*w/(w+v)-r)));
123 }
124 if(iy==1) return w;
125 else { /* if allow error up to 2 ulp,
126 simply return -1.0/(x+r) here */
127 /* compute -1.0/(x+r) accurately */
128 double a,t;
129 z = w;
130 SET_LOW_WORD(z,0);
131 v = r-(z - x); /* z+v = r+x */
132 t = a = -1.0/w; /* a = -1.0/w */
133 SET_LOW_WORD(t,0);
134 s = 1.0+t*z;
135 return t+a*(s+t*v);
136 }
137 }