1
2 /* @(#)k_tan.c 5.1 93/09/24 */
3 /*
4 * ====================================================
5 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
6 *
7 * Developed at SunPro, a Sun Microsystems, Inc. business.
8 * Permission to use, copy, modify, and distribute this
9 * software is freely granted, provided that this notice
10 * is preserved.
11 * ====================================================
12 */
13
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 tan (if k=1) or
19 * -1/tan (if k= -1) is returned.
20 *
21 * Algorithm
22 * 1. Since tan(-x) = -tan(x), we need only to consider positive x.
23 * 2. if x < 2^-28 (hx<0x3e300000 0), return x with inexact if x!=0.
24 * 3. tan(x) is approximated by a odd polynomial of degree 27 on
25 * [0,0.67434]
26 * 3 27
27 * tan(x) ~ x + T1*x + ... + T13*x
28 * where
29 *
30 * |tan(x) 2 4 26 | -59.2
31 * |----- - (1+T1*x +T2*x +.... +T13*x )| <= 2
32 * | x |
33 *
34 * Note: tan(x+y) = tan(x) + tan'(x)*y
35 * ~ tan(x) + (1+x*x)*y
36 * Therefore, for better accuracy in computing tan(x+y), let
37 * 3 2 2 2 2
38 * r = x *(T2+x *(T3+x *(...+x *(T12+x *T13))))
39 * then
40 * 3 2
41 * tan(x+y) = x + (T1*x + (x *(r+y)+y))
42 *
43 * 4. For x in [0.67434,pi/4], let y = pi/4 - x, then
44 * tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y))
45 * = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y)))
46 */
47
48 #include "fdlibm.h"
49
50 #ifdef _NEED_FLOAT64
51
52 static const __float64
53 one = _F_64(1.00000000000000000000e+00), /* 0x3FF00000, 0x00000000 */
54 pio4 = _F_64(7.85398163397448278999e-01), /* 0x3FE921FB, 0x54442D18 */
55 pio4lo = _F_64(3.06161699786838301793e-17), /* 0x3C81A626, 0x33145C07 */
56 T[] = {
57 _F_64(3.33333333333334091986e-01), /* 0x3FD55555, 0x55555563 */
58 _F_64(1.33333333333201242699e-01), /* 0x3FC11111, 0x1110FE7A */
59 _F_64(5.39682539762260521377e-02), /* 0x3FABA1BA, 0x1BB341FE */
60 _F_64(2.18694882948595424599e-02), /* 0x3F9664F4, 0x8406D637 */
61 _F_64(8.86323982359930005737e-03), /* 0x3F8226E3, 0xE96E8493 */
62 _F_64(3.59207910759131235356e-03), /* 0x3F6D6D22, 0xC9560328 */
63 _F_64(1.45620945432529025516e-03), /* 0x3F57DBC8, 0xFEE08315 */
64 _F_64(5.88041240820264096874e-04), /* 0x3F4344D8, 0xF2F26501 */
65 _F_64(2.46463134818469906812e-04), /* 0x3F3026F7, 0x1A8D1068 */
66 _F_64(7.81794442939557092300e-05), /* 0x3F147E88, 0xA03792A6 */
67 _F_64(7.14072491382608190305e-05), /* 0x3F12B80F, 0x32F0A7E9 */
68 _F_64(-1.85586374855275456654e-05), /* 0xBEF375CB, 0xDB605373 */
69 _F_64(2.59073051863633712884e-05), /* 0x3EFB2A70, 0x74BF7AD4 */
70 };
71
72 __float64
__kernel_tan(__float64 x,__float64 y,int iy)73 __kernel_tan(__float64 x, __float64 y, int iy)
74 {
75 __float64 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 __uint32_t low;
82 GET_LOW_WORD(low, x);
83 if (((ix | low) | (iy + 1)) == 0)
84 return one / fabs64(x);
85 else {
86 if (iy == 1)
87 return x;
88 else {
89 __float64 a, t;
90 z = w = x + y;
91 SET_LOW_WORD(z, 0);
92 v = y - (z - x);
93 t = a = -one / w;
94 SET_LOW_WORD(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 = _F_64(0.0);
110 }
111 z = x * x;
112 w = z * z;
113 /* Break x^5*(T[1]+x^2*T[2]+...) into
114 * x^5(T[1]+x^4*T[3]+...+x^20*T[11]) +
115 * x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12]))
116 */
117 r = T[1] + w * (T[3] + w * (T[5] + w * (T[7] + w * (T[9] + w * T[11]))));
118 v = z *
119 (T[2] + w * (T[4] + w * (T[6] + w * (T[8] + w * (T[10] + w * T[12])))));
120 s = z * x;
121 r = y + z * (s * (r + v) + y);
122 r += T[0] * s;
123 w = x + r;
124 if (ix >= 0x3FE59428) {
125 v = (__float64)iy;
126 return (__float64)(1 - ((hx >> 30) & 2)) *
127 (v - _F_64(2.0) * (x - (w * w / (w + v) - r)));
128 }
129 if (iy == 1)
130 return w;
131 else { /* if allow error up to 2 ulp,
132 simply return -1.0/(x+r) here */
133 /* compute -1.0/(x+r) accurately */
134 __float64 a, t;
135 z = w;
136 SET_LOW_WORD(z, 0);
137 v = r - (z - x); /* z+v = r+x */
138 t = a = _F_64(-1.0) / w; /* a = _F_64(-1.0)/w */
139 SET_LOW_WORD(t, 0);
140 s = _F_64(1.0) + t * z;
141 return t + a * (s + t * v);
142 }
143 }
144
145 #endif /* _NEED_FLOAT64 */
146
