__tan.c raw
1 /* origin: FreeBSD /usr/src/lib/msun/src/k_tan.c */
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 /* __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 odd indicates whether tan (if odd = 0) or -1/tan (if odd = 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 "libm.h"
47
48 static const double T[] = {
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 },
63 pio4 = 7.85398163397448278999e-01, /* 3FE921FB, 54442D18 */
64 pio4lo = 3.06161699786838301793e-17; /* 3C81A626, 33145C07 */
65
66 double __tan(double x, double y, int odd)
67 {
68 double_t z, r, v, w, s, a;
69 double w0, a0;
70 uint32_t hx;
71 int big, sign;
72
73 GET_HIGH_WORD(hx,x);
74 big = (hx&0x7fffffff) >= 0x3FE59428; /* |x| >= 0.6744 */
75 if (big) {
76 sign = hx>>31;
77 if (sign) {
78 x = -x;
79 y = -y;
80 }
81 x = (pio4 - x) + (pio4lo - y);
82 y = 0.0;
83 }
84 z = x * x;
85 w = z * z;
86 /*
87 * Break x^5*(T[1]+x^2*T[2]+...) into
88 * x^5(T[1]+x^4*T[3]+...+x^20*T[11]) +
89 * x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12]))
90 */
91 r = T[1] + w*(T[3] + w*(T[5] + w*(T[7] + w*(T[9] + w*T[11]))));
92 v = z*(T[2] + w*(T[4] + w*(T[6] + w*(T[8] + w*(T[10] + w*T[12])))));
93 s = z * x;
94 r = y + z*(s*(r + v) + y) + s*T[0];
95 w = x + r;
96 if (big) {
97 s = 1 - 2*odd;
98 v = s - 2.0 * (x + (r - w*w/(w + s)));
99 return sign ? -v : v;
100 }
101 if (!odd)
102 return w;
103 /* -1.0/(x+r) has up to 2ulp error, so compute it accurately */
104 w0 = w;
105 SET_LOW_WORD(w0, 0);
106 v = r - (w0 - x); /* w0+v = r+x */
107 a0 = a = -1.0 / w;
108 SET_LOW_WORD(a0, 0);
109 return a0 + a*(1.0 + a0*w0 + a0*v);
110 }
111