1 #pragma ident "@(#)k_tan.c 1.5 04/04/22 SMI"
2
3 /*
4 * ====================================================
5 * Copyright 2004 Sun Microsystems, Inc. All Rights Reserved.
6 *
7 * Permission to use, copy, modify, and distribute this
8 * software is freely granted, provided that this notice
9 * is preserved.
10 * ====================================================
11 */
12
13 /* INDENT OFF */
14 /* __kernel_tan( x, y, k )
15 * kernel tan function on [-pi/4, pi/4], pi/4 ~ 0.7854
16 * Input x is assumed to be bounded by ~pi/4 in magnitude.
17 * Input y is the tail of x.
18 * Input k indicates whether ieee_tan (if k = 1) or -1/tan (if k = -1) is returned.
19 *
20 * Algorithm
21 * 1. Since ieee_tan(-x) = -ieee_tan(x), we need only to consider positive x.
22 * 2. if x < 2^-28 (hx<0x3e300000 0), return x with inexact if x!=0.
23 * 3. ieee_tan(x) is approximated by a odd polynomial of degree 27 on
24 * [0,0.67434]
25 * 3 27
26 * tan(x) ~ x + T1*x + ... + T13*x
27 * where
28 *
29 * |ieee_tan(x) 2 4 26 | -59.2
30 * |----- - (1+T1*x +T2*x +.... +T13*x )| <= 2
31 * | x |
32 *
33 * Note: ieee_tan(x+y) = ieee_tan(x) + tan'(x)*y
34 * ~ ieee_tan(x) + (1+x*x)*y
35 * Therefore, for better accuracy in computing ieee_tan(x+y), let
36 * 3 2 2 2 2
37 * r = x *(T2+x *(T3+x *(...+x *(T12+x *T13))))
38 * then
39 * 3 2
40 * tan(x+y) = x + (T1*x + (x *(r+y)+y))
41 *
42 * 4. For x in [0.67434,pi/4], let y = pi/4 - x, then
43 * tan(x) = ieee_tan(pi/4-y) = (1-ieee_tan(y))/(1+ieee_tan(y))
44 * = 1 - 2*(ieee_tan(y) - (ieee_tan(y)^2)/(1+ieee_tan(y)))
45 */
46
47 #include "fdlibm.h"
48
49 static const double xxx[] = {
50 3.33333333333334091986e-01, /* 3FD55555, 55555563 */
51 1.33333333333201242699e-01, /* 3FC11111, 1110FE7A */
52 5.39682539762260521377e-02, /* 3FABA1BA, 1BB341FE */
53 2.18694882948595424599e-02, /* 3F9664F4, 8406D637 */
54 8.86323982359930005737e-03, /* 3F8226E3, E96E8493 */
55 3.59207910759131235356e-03, /* 3F6D6D22, C9560328 */
56 1.45620945432529025516e-03, /* 3F57DBC8, FEE08315 */
57 5.88041240820264096874e-04, /* 3F4344D8, F2F26501 */
58 2.46463134818469906812e-04, /* 3F3026F7, 1A8D1068 */
59 7.81794442939557092300e-05, /* 3F147E88, A03792A6 */
60 7.14072491382608190305e-05, /* 3F12B80F, 32F0A7E9 */
61 -1.85586374855275456654e-05, /* BEF375CB, DB605373 */
62 2.59073051863633712884e-05, /* 3EFB2A70, 74BF7AD4 */
63 /* one */ 1.00000000000000000000e+00, /* 3FF00000, 00000000 */
64 /* pio4 */ 7.85398163397448278999e-01, /* 3FE921FB, 54442D18 */
65 /* pio4lo */ 3.06161699786838301793e-17 /* 3C81A626, 33145C07 */
66 };
67 #define one xxx[13]
68 #define pio4 xxx[14]
69 #define pio4lo xxx[15]
70 #define T xxx
71 /* INDENT ON */
72
73 double
__kernel_tan(double x,double y,int iy)74 __kernel_tan(double x, double y, int iy) {
75 double z, r, v, w, s;
76 int ix, hx;
77
78 hx = __HI(x); /* high word of x */
79 ix = hx & 0x7fffffff; /* high word of |x| */
80 if (ix < 0x3e300000) { /* x < 2**-28 */
81 if ((int) x == 0) { /* generate inexact */
82 if (((ix | __LO(x)) | (iy + 1)) == 0)
83 return one / ieee_fabs(x);
84 else {
85 if (iy == 1)
86 return x;
87 else { /* compute -1 / (x+y) carefully */
88 double a, t;
89
90 z = w = x + y;
91 __LO(z) = 0;
92 v = y - (z - x);
93 t = a = -one / w;
94 __LO(t) = 0;
95 s = one + t * z;
96 return t + a * (s + t * v);
97 }
98 }
99 }
100 }
101 if (ix >= 0x3FE59428) { /* |x| >= 0.6744 */
102 if (hx < 0) {
103 x = -x;
104 y = -y;
105 }
106 z = pio4 - x;
107 w = pio4lo - y;
108 x = z + w;
109 y = 0.0;
110 }
111 z = x * x;
112 w = z * z;
113 /*
114 * Break x^5*(T[1]+x^2*T[2]+...) into
115 * x^5(T[1]+x^4*T[3]+...+x^20*T[11]) +
116 * x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12]))
117 */
118 r = T[1] + w * (T[3] + w * (T[5] + w * (T[7] + w * (T[9] +
119 w * T[11]))));
120 v = z * (T[2] + w * (T[4] + w * (T[6] + w * (T[8] + w * (T[10] +
121 w * T[12])))));
122 s = z * x;
123 r = y + z * (s * (r + v) + y);
124 r += T[0] * s;
125 w = x + r;
126 if (ix >= 0x3FE59428) {
127 v = (double) iy;
128 return (double) (1 - ((hx >> 30) & 2)) *
129 (v - 2.0 * (x - (w * w / (w + v) - r)));
130 }
131 if (iy == 1)
132 return w;
133 else {
134 /*
135 * if allow error up to 2 ulp, simply return
136 * -1.0 / (x+r) here
137 */
138 /* compute -1.0 / (x+r) accurately */
139 double a, t;
140 z = w;
141 __LO(z) = 0;
142 v = r - (z - x); /* z+v = r+x */
143 t = a = -1.0 / w; /* a = -1.0/w */
144 __LO(t) = 0;
145 s = 1.0 + t * z;
146 return t + a * (s + t * v);
147 }
148 }
149