1 /*
2 * ====================================================
3 * Copyright 2004 Sun Microsystems, Inc. All Rights Reserved.
4 *
5 * Permission to use, copy, modify, and distribute this
6 * software is freely granted, provided that this notice
7 * is preserved.
8 * ====================================================
9 */
10
11 /* __kernel_tan( x, y, k )
12 * kernel tan function on ~[-pi/4, pi/4] (except on -0), pi/4 ~ 0.7854
13 * Input x is assumed to be bounded by ~pi/4 in magnitude.
14 * Input y is the tail of x.
15 * Input k indicates whether tan (if k = 1) or -1/tan (if k = -1) is returned.
16 *
17 * Algorithm
18 * 1. Since tan(-x) = -tan(x), we need only to consider positive x.
19 * 2. Callers must return tan(-0) = -0 without calling here since our
20 * odd polynomial is not evaluated in a way that preserves -0.
21 * Callers may do the optimization tan(x) ~ x for tiny x.
22 * 3. tan(x) is approximated by a odd polynomial of degree 27 on
23 * [0,0.67434]
24 * 3 27
25 * tan(x) ~ x + T1*x + ... + T13*x
26 * where
27 *
28 * |tan(x) 2 4 26 | -59.2
29 * |----- - (1+T1*x +T2*x +.... +T13*x )| <= 2
30 * | x |
31 *
32 * Note: tan(x+y) = tan(x) + tan'(x)*y
33 * ~ tan(x) + (1+x*x)*y
34 * Therefore, for better accuracy in computing tan(x+y), let
35 * 3 2 2 2 2
36 * r = x *(T2+x *(T3+x *(...+x *(T12+x *T13))))
37 * then
38 * 3 2
39 * tan(x+y) = x + (T1*x + (x *(r+y)+y))
40 *
41 * 4. For x in [0.67434,pi/4], let y = pi/4 - x, then
42 * tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y))
43 * = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y)))
44 */
45
46 #include "math.h"
47 #include "math_private.h"
48 static const double xxx[] = {
49 3.33333333333334091986e-01, /* 3FD55555, 55555563 */
50 1.33333333333201242699e-01, /* 3FC11111, 1110FE7A */
51 5.39682539762260521377e-02, /* 3FABA1BA, 1BB341FE */
52 2.18694882948595424599e-02, /* 3F9664F4, 8406D637 */
53 8.86323982359930005737e-03, /* 3F8226E3, E96E8493 */
54 3.59207910759131235356e-03, /* 3F6D6D22, C9560328 */
55 1.45620945432529025516e-03, /* 3F57DBC8, FEE08315 */
56 5.88041240820264096874e-04, /* 3F4344D8, F2F26501 */
57 2.46463134818469906812e-04, /* 3F3026F7, 1A8D1068 */
58 7.81794442939557092300e-05, /* 3F147E88, A03792A6 */
59 7.14072491382608190305e-05, /* 3F12B80F, 32F0A7E9 */
60 -1.85586374855275456654e-05, /* BEF375CB, DB605373 */
61 2.59073051863633712884e-05, /* 3EFB2A70, 74BF7AD4 */
62 /* one */ 1.00000000000000000000e+00, /* 3FF00000, 00000000 */
63 /* pio4 */ 7.85398163397448278999e-01, /* 3FE921FB, 54442D18 */
64 /* pio4lo */ 3.06161699786838301793e-17 /* 3C81A626, 33145C07 */
65 };
66 #define one xxx[13]
67 #define pio4 xxx[14]
68 #define pio4lo xxx[15]
69 #define T xxx
70 /* INDENT ON */
71
72 double
__kernel_tan(double x,double y,int iy)73 __kernel_tan(double x, double y, int iy) {
74 double z, r, v, w, s;
75 int32_t ix, hx;
76
77 GET_HIGH_WORD(hx,x);
78 ix = hx & 0x7fffffff; /* high word of |x| */
79 if (ix >= 0x3FE59428) { /* |x| >= 0.6744 */
80 if (hx < 0) {
81 x = -x;
82 y = -y;
83 }
84 z = pio4 - x;
85 w = pio4lo - y;
86 x = z + w;
87 y = 0.0;
88 }
89 z = x * x;
90 w = z * z;
91 /*
92 * Break x^5*(T[1]+x^2*T[2]+...) into
93 * x^5(T[1]+x^4*T[3]+...+x^20*T[11]) +
94 * x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12]))
95 */
96 r = T[1] + w * (T[3] + w * (T[5] + w * (T[7] + w * (T[9] +
97 w * T[11]))));
98 v = z * (T[2] + w * (T[4] + w * (T[6] + w * (T[8] + w * (T[10] +
99 w * T[12])))));
100 s = z * x;
101 r = y + z * (s * (r + v) + y);
102 r += T[0] * s;
103 w = x + r;
104 if (ix >= 0x3FE59428) {
105 v = (double) iy;
106 return (double) (1 - ((hx >> 30) & 2)) *
107 (v - 2.0 * (x - (w * w / (w + v) - r)));
108 }
109 if (iy == 1)
110 return w;
111 else {
112 /*
113 * if allow error up to 2 ulp, simply return
114 * -1.0 / (x+r) here
115 */
116 /* compute -1.0 / (x+r) accurately */
117 double a, t;
118 z = w;
119 SET_LOW_WORD(z,0);
120 v = r - (z - x); /* z+v = r+x */
121 t = a = -1.0 / w; /* a = -1.0/w */
122 SET_LOW_WORD(t,0);
123 s = 1.0 + t * z;
124 return t + a * (s + t * v);
125 }
126 }
127